Wau es an cy on ee tu LAs : a4 ue ' 3 erate ee vahatatea Th ee own bia’ I Pc ha ie eo : Whe eh ee etek a we si4e4 eae 4 -aew ee : ee ee aD An aoe o4 * cele, y ag, A AA ane ee ee ee ak, nu a Kaha vite « , he A ea eo yl 4 ; pwiae a wae ° 4 : adage ad he EW eee : A Ee ae ae '¢a&, Pewee s 8 Mek ly Pe” ek BED Coe bo Rae Vad) 64a Cew de ae ok r ur : i, ch Se AA CA See ee de Oe eee tee CUS SAR SEMA | SER He eG 8 awd & ern ‘ v THE LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY SIR ROBERT KANE, LL.D. F.R.S. M.R.LA. E.C.S. STR WILLIAM THOMSON, Kyr. LL.D. F.R.S. &e. AND WILLIAM FRANCIS, Pu.D. F.L.S. F.R.A.S. F.C.S. “ Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib. i. cap. 1. Not. VOL. XXII.—FIFTH SERIES. JULY—DECEMBER 1886. DEC 21 1886 LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, SOLD BY LONGMANS, GREEN, AND CO.; KENT AND CO.; SIMPKIN, MARSHALL, AND CQ.; AND WHITTAKER AND CO.;—-AND BY ADAM AND CHARLES BLACK, AND 7, AND T. CLARK, EDINBURGH; SMITH AND SON, GLASGOW 5-— HODGES, FOSTER, AND CO., DUBLIN ;—PUTNAM, NEW YORE ;— VEUVE J, BOYVEAU, PARIS ;—AND ASHER AND CO., BERLIN. “Meditationis est perscrutari occulta; contemplationis est admirari perspicua..... Admiratio generat quéestionem, quzestio investigationem, investigatio inventionem.”—Hugo de S. Victore. oe ths ae “Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, x Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ccelo, id Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Mazonium. CONTENTS OF VOL. XXII. (FIFTH SERIES). _. NUMBER OXXXIV.—JULY 1886. The Rey. O. Fisher on the Variations of Gravity at certain Stations of the Indian Arc of the Meridian in Relation to their Bearing upon the Constitution of the Earth’s Crust. . Mr. 8. Bidwell on a Modification of Wheatstone’s Rheostat . Drs. W. Ramsay and 8S. Young on some Thermodynamical erie —batiod Vs 0a. Veg ys opie ee he ee ee ee es Prof. E. A. Letts and Dr. N. Collie on a new Method for the eceartion,ob Mn letreth yl 2. ova le iets oe oe ee Mr. E. F. J. Love on M. Mascart’s Paper, “On Magnetiza- CULE. 4 ENRISE OE. ree ene € OD rane cee ee a Mr. W. Baily on a Theorem relating to Curved Diffraction- SaRPNM ERED I Fi 6 Fo octet, ioe a age sisi sy sie ie + we ss cige oe Prof. G. Widemann’s Magnetic Researches. (Plate I.) Prof. W. Ostwald on the Seat of the Hlectromotive Forces in PEP ice UN rea eo Pe ore tl ow oil spite ma edie oe Proceedings of the Geological Society:— , Mr. Rh. M. Deeley on the Pleistocene Succession in the MMe Sie Mee TSCM ee oP Laie sc. ayenss teh claee tie yas see a bose Mee Mr. R. N. Worth on the Existence of a Submarine Tri- assic Outlier in the English Channel off the Lizard .. Dr. H. Hicks on the Pre-Cambrian Age of Certain Grani- toid, Felsitic, and other Rocks in North-western Pem- REO MEE Fics sae,» yf ARO ee RO anc Cee ae 2 Prof. T. G. Bonney on some Rock-specimens collected by Dr. Hicks in North-western Pembrokeshire ........ Mr. Aubrey Strahan on the Glaciation of South Lan- cashire, Cheshire, and the Welsh Border .......... Mr. J. Durham on the Volcanic Rocks of North-eastern Pe eah soils 420. ade. «) Bey Penh ol oer eecv a « « ebasn ticks ere ote Mr. F. Rutley on some Eruptive Rocks from the Neigh- bourhood ot St. Vanver,, Cornwall... 0 Mr. H. W. Monckton and Mr. R.S8. Herries on the Bag- shot, Beds of the London Basin’... 2. oo... ges one Absolute Spherical Electrometer, by M. Lippmann ........ Simple Demonstration of the Electrical Residue, by Fr. ee aR EG side. pik es ea ha a 2s A Oe ee eine Page ail! 29 32 41 46 AT 50 70 72 73 73 74 75 77 78 78 79 lv CONTENTS OF VOL. XXII.—FIFTH SERIES. NUMBER CXXXV.—AUGUST. Page Mr. W. Sutherland on the Law of Attraction amongst the Moleculesiof a ‘Gas ...6 0...) t oe ha ee er 81 Prof. R. Shida on a new Instrument for continuously record- ing the Strength and Direction of a Varying Electric Current. (Plates Il. & WT)" 7. 2... 96 Prof. W. Ostwald’s Electrochemical Researches...........- 104 Mr. O. Heaviside on the Self-induction of Wires .......... 118 Mr. H. Cunynghame on a new Hyperbolagraph .......... 138 Mr. A. B. Basset on the Induction of Electric Currents, in an Infinite Plane Current Sheet, which is rotating in a Field of Maonetic Force... seu ss eee ga ee 140 Messrs. W. Emmott and W. Ackroyd on an Electric-light Mire-damp. Indicator ..’...°. 2-05 en- ae ee 145 Mr. F. J. Smith on certain Modifications of a Form of Spheri- cal Tnteoranor re icc ove ep ne sp Ce oh eee eee 147 Prof. 8. P. Langley on hitherto unrecognized Wave-lengths. (Plates TV—VL) on 5.0.0. ee ee eee tee ee 149 Mr. A. Buchheim on an Extension of a Theorem of Prof. Sylvester’s relating to Matrices “...2. 0.0. . 0 0. oe 173 Lord Rayleigh’s Notes on Magnetism.—I. On the Energy of Maenetized Tron |. oa. usw cee ces se bess se eee 175 Prof. E. A. Letts and Dr. N. Collie on the Salts of Tetrethy]- phosphonium and their Decomposition by Heat.......... 183 Me. John Aitken on Dew... .......-.-. 00 ++- e505! eo 206 Mr. A. P. Laurie on the Electromotive Force of Voltaic Cells having an Aluminium Plate as one Electrode ...... 213 Prof. S. P. Thompson on a Mode of maintaining Tuning Forks lay BIS CURCIVMIR ae neces «9 eke to he eas 2 aie sce 216 Proceedings of the Geological Society :— Prof. T. M*Kenny Hughes on some Perched Blocks and associated Phenomena, ss). ican cutee ee ee ee 220 Dr. C. Callaway on some derived Fragments in the Long- mynd and newer Archean Rocks of Shropshire...... 221 Mr. A. Strahan on the Relations of the Lincolnshire CANSHONG. 6 hss v0 (ne 4 Me eee ine ek eee 222 Mr. E. Gilpin, Jun.,on the Geology of Cape-Breton Island, INOVa SCO." e.g) a aus cate ogra 2 © ieee ee 223 Mr. W. Whitaker on some Well-sections in Middlesex.. 223 Mr. H. M. Becher on some Cupriferous Shales in the Province of Houpeh, China: 7) 7.0)... ee ee eee 224 Mr. W. H. Merritt on the Cascade Anthracitic Coal-field of the Rocky Mountains, Canada... 25.0. ee 224 Dr. A. B. Griffiths on certain Eocene Formations of Western Servia .. 2. O00 MSS OS See ee CONTENTS OF VOL. XXII.—FIFTH SERIES. Vv . Page On some Experiments relating to Hall’s Phenomenon, by Bateneebser WAWIATIN Varin Okie we dws ae ene. fo eae ew oe sR eeye os 226 On the Gold-leaf Electroscope, by Fr. Kolacek ............ 228 Application of Thermodynamics to Capillary Phenomena, by PSPS a hs eee halal eof no a as, Sicicy leo om hs ae wie aha 230 On Peltier’s Phenomenon in Liquids, by E. Naccari and A. Rasa NG Meet esa! cucieices gles wailed fo! a aifio cians id vai ote 6 otpid x 231 NUMBER CXXXVI.—SEPTEMBER. Prof. H. Hennessy on the Physical Structure of the Earth .. 233 Mr. S. Bidwell on the Magnetic Torsion of Iron and Nickel ED SS 7: SORE Se an area ip reece 251 Dr. P. E. Chase’s Tests of Herschel’s Athereal Physics .... 255 Mr. C. Chree on Bars and Wires of varying Elasticity ...... 259 Mr. C. Tomlinson’s Further Remarks on Mr. Aitken’s Theory OP LUGS Le a ieee eee er eT eee ree eee 270 Mr. O. Heaviside on the Self-induction of Wires.—Part II... 273 Prof. 8S. P. Thompson’s Further Notes on the Formule of the Electromagnet and the Equations of the Dynamo........ 288 Mr. R. H. M. Bosanquet on the Law of similar Electromag- META ARNEARIOM. GCs.) / a... uf leaky oi SRT (os is Seas awake paw cd le» 298 Measurement of Pitch by Manometric Flames, by M. Doumer. 309 On a New Method for Determining the Vertical Intensity of mMaenede, Preld, by bis, FErugeer 5. 42b.gy ie ceerels die desl eS 311 On the Constant of the Sun’s Heat, by M. Maurer ........ 312 On the Increase of Temperature produced by a Waterfall, by eA STN isp Ath PY AGA AE) cine tyler) «epee guerra) ds can east 312 NUMBER CXXXVII.—OCTOBER. Mr. J. Lester Woodbridge on Turbines ...............00. 313 Profs. W. E. Ayrton and J. Perry on the Expansion of Mer- cumyaver meen O° ©. amd 39° Coie cers dis wee ov See 325 Profs. W. E. Ayrton and J. Perry on the Expansion produced Iga ena A AERA TN oc l/S s o, e ge ky Oem eich a co nk wrk a eg 327 Prof. H. Hennessy on the Annual Precession calculated on the Hypothesis, of the! Harthis Solndipyy, 7.09.5) .4 36 UB. ws oo a 328 Mr. O. Heaviside on the Self-induction of Wires.—Part III. . 332 Sir W. Thomson on Stationary Waves in Flowing Water.— eat a ces eee bass ore Shel Bee oe EP aaa Gre 353 Mr. T. C. Mendenhall on the Electrical Resistance of Soft Carbonciwuder’ Pressure’. 0302 334 4.44 te wee es OPPS 358 iii John AuKea on: Dew 0 ee a et Bee Br 363 vl _ CONTENTS OF VOL, XXII.— FIFTH SERIES, Page Mr. T. Gray on a new Standard Sine-Galvanometer........ 368 Mr. I. Y. Edgeworth’s Problems in Probabilities .......... 371 On the Measurement of very High Pressures, and the Com- pressibility of Liquids, by M. F. Amagat .............. 384 On the Specific Induction Constants of Magnets in Magnetic Fields of Different Strengths, by Hilmar Sack :.:....... 386 On the Electrical Conductivity of Gases and Vapours, by M. Jean gw. Sh eS ee eee a sa One cee 387 An Electrical Experiment, by M. Busch................+- 388 NUMBER CXXXVIII._NOVEMBER. Mr. T.: Gray on the Electrolysis of Silver and of Copper, and the Application of Electrolysis to the Standardizing of Elec- tric Current- and Potential Meters. (Plate VII.)........ 389 Mr. H. Tomlinson on certain Sourees of Error in Connection with Experiments on Torsional Vibrations.............. 414 Mr. O. Heaviside on the Self-induction of Wires.—Part IV. 419 Mr. H. Tomlinson on the Effect of Stress and Strain on the Electrical Resistance of Carbon ©...... 2....0-5 04sec ene 442 Sir William Thomson on Stationary Waves in Flowing Water. Se Part Te bs EG PCE Pes Pll WE Le le ae 445 Mr. A. Lodge’s New Geometrical Representation of Moments and Produets of Inertia in a Plane Section ; and also of the Relations between Stresses and Strains in two Dimensions 453 Prof. M. A. Cornu on the Distinction between Spectral Lines of Solar and Terrestrial:Origin. (Plate VIII.).......... 458 Notices respecting New Books :— Mr, J. Milne’s Volcanoes‘of Japam .........-.. 79a 463 Dr. J. Croll’s Discussions on Climate and Cosmology .. 464 On the Magnetic Rotation of Mixtures of Water with some of the Acids of the Fatty Series, with Alcohol, and with Sulphuric Acid, and Observations on Water of Crystalliza- tion, ‘by. W ibs Perkin; PhiD., PuR.Sy (od. ae tvdk cee One 467 NUMBER CXXXIX.—DECEMBER. Lord Rayleigh’s Notes on Electricity and Magnetism.—II. The Self-induction and Resistance of Compound Conductors 469 Mr. R. H. M. Bosanquet on Permanent Magnets.—III. On 500 Magnetic Decay. .... +. eben ety sy epiemeed ue Seep ae he Dr. W. W. J. Nicol on the Vapour-pressures of Water from SA IEASOMMIOMST Seca ttein cele Ae meee aouuzincata mike GhucahemekaeelE CONTENTS OF VOL. XXII.—FIFTH SERIES. Sir William Thomson on Stationary Waves in Flowing Water. Se ataele Pree cA eR ONE eee as gE US OW e's ot Prof. R. Bunsen on the Decomposition of Glass by Carbon Dioxide held in Solution in Capillary Films of Water .... Messrs. A. Bartoli and E. Stracciati’s Reply to the Observa- tions of Messrs. Thorpe and Riicker upon their Essay entitled “ Intorno ad alcune formule date del Sig. Mende- lejeff e dai Siggn. T. H. Thorpe e A. W. Riicker per calco- lare la temperatura critica della dilatazione termica” . Mr, R. H. M. Bosanquet on EHlectromagnets.—VI. The Ten- ston of Lines of Force Cr ee ee | Silk v. Wire, or the “ Ghost” in the Galvanometer, by R. H. . M. Bosanquet 6 © P Cw. RD eH e Be eae eee ea Fee « 8 eo & eC 8s & Ce Re eeesvseeveeereeeee et eee ereeeeeereeseeaerereeeeeeeveeveeeeee eo @ ERRATA, Page 245, line 6, for intérieuse read intérieure —_ 249, Lingg , for Delauney read ay — 828, last line, for C—A ead A C PLATES. I. Illustrative of Prof. Wiedemann’s Magnetic Researches. II. & III. Illustrative of Prof. R. Shida’s Paper on a New Instrument for continuously recording the Strength and Direction of a Varying Electric Current. IV., V., VI. Illustrative of Prof. S. P. Langley’s Paper on hitherto unre- cognized Wave-lengths. VII. Illustrative of Mr. T. Gray’s Paper on the Electrolysis of Silver and of Copper. VIII. Illustrative of Prof. Cornu’s Paper on the Distinction between Spectral Lines of Solar and Terrestrial Origin. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SHRIES.] JULY 1886. I. On the Variations of Gravity at certain Stations of the Indian Are of the Meridian in Relation to their Bearing upon the Constitution of the Earth’s Crust. By Rev. O. FisHer, 1.A., #.G.S.* 1 & is well known that, during the geodetic operations for the measurement of the Indian are of the Meridian, it was found that the attraction of the Himalayas upon the plumb-line was less than it ought to have been, according to what Archdeacon Pratt calculated that their mass should have produced. This was attributed by Sir G. B. Airy to a downward protuberance of rock, having the same density as the mountains, into a substratum of greater density, so that their position would be approximately one of hydrostatic equilibrium ; and he showed that such an arrangement, by which their weight would be sustained, would have the effect upon the plumb-line of the kind, which had to be accounted for, greatly reducing the attraction of the mountains upon the plumb-line at a distant station, but much less so at a station near them f. Pratt himself, however, preferred to attribute the anomaly to a supposed deficient contraction of the crust of the Harth beneath the mountains during the secular cooling of the globe, causing their relative elevation; and to this he attributed the diminution of density. He retained the average density * Communicated by the Author. } Trans. Royal Soc. vol. cxlv. p. 101 (1855), quoted in ‘ Physics of the Earth’s Crust’ (Macmillan, 1881), p. 145. Phil. Mag. 8. 5. Vol. 22. No. 134. July 1886. B 2 Rev. O. Fisher on Variations of Gravity and their of the crust for the mountains themselves, and supposed an attenuation equal to their mass to affect the crust for a depth of 50 miles beneath them *. Geologists, however, would hardly be satisfied with this explanation. For, although there are some areas which appear to have been vertically lifted within comparatively recent times, such as notably the Colorado Plateau, neverthe- less the usual type of an elevated region is that of rocks which have been heaped together by lateral pressure, so that the material of which the mountains consist has been pushed horizontally towards the range over the nucleus ; and this is the case with the Himalayas. I have shown, in my ‘ Physics of the Earth’s Crust, that, if the substratum is plastic, this process would involve the production of the kind of downward protuberance that Airy postulates, and would explain the existence of a deficiency of the attraction towards the moun- tain-range. Accordingly, in the whole of my reasoning upon the subject, needing some working hypothesis, I have assumed that the crust is of the mean density of granitic rocks, viz. 2°68 ; and the substratum of the density of basic rocks, viz. 2°96. But I did not, in my book, make any quantitative estimate respecting the probable result of my assumptions upon the variation of gravity. My attention was again drawn to this question by readin the instructive lecture delivered by General Walker, the Superintendent of the Trigonometrical Survey of India, at the meeting of the British Association at Aberdeen in 18857. This led me to look into the description of the pendulum ex- periments referred to in that lecture, which are published in vol. v. of the ‘ Account of the Operations of the Great Trigo- nometrical Survey of India’{. If I understand the matter rightly, what was done was in principle this:—At certain stations of the Survey, of which the height and position had been already determined, the mean number of swings, called the “ vibration-number,” was observed, which were made by two pendulums in twenty-four hours which at the equator would have made about 86,000 § vibrations in the same in- terval. (It will be remembered that the number of seconds in twenty-four hours is 86,400.) In this manner the force of gravity at each station could be compared. ‘The effect of local attraction at the station was then estimated, as well as * ‘Figure of the Earth,’ 4th ed. pp. 201 and 208. + ‘Nature,’ vol. xxu. p. 481 (1885). ne Pees under the directions of Major-General J. T. Walker, C.B., R.E., F.R.S. Calcutta, 1879. § Ibid. p. [129]. relation to the Constitution of the Earth’s Crust. 3 that of the height ; and when these together had been allowed for, the corrected vibration-number so determined was re- garded as the vibration-number for that station reduced to the sea-level. Had these disturbing circumstances been cor- rectly estimated, and had no others of a hidden kind existed, this vibration-number ought to have tallied exactly with that belonging to the latitude of the station. It turned out, how- ever, that they were usually different, and for the most part in defect, often to a considerable extent. ‘ There appears to be no escape from the conclusion that there is a more or less marked negative variation of gravity over the whole of the Indian continent, and that the magnitude of this variation is somehow connected with the height. Let us group the stations as follows, omitting Ismailia and Kew as foreign.”* “‘T, Coast and Island Stations. Apparent varia- Station. Height tion in vibration- in feet. numbers. Panne) 3.5055 48 — 2°65 Kudankolam...| 168 —2°56 Minicoy ......... 6 +1:37 Milewpy 7-25. 2.05 6 -=—121 (SU. Fe) Se eee 5 (ak | Mangalore...... | 7 —3:24 Madras® 225. s.2. 27 — 3°39 Cocanada ...... | 9 —1°81 @olthar nak:...: 35 +0°75 Caleutta......... 18 =F Mean... a ge —1:43 “TI. Inland Stations less than 1000 feet high. Hi Apparent varia- Station. Height. tion in vibration- numbers. Mallapatti...... 288 —3°77 Pachapaliam...| 971 —4-41 Paiva ih) 2.6.22 810 —3°60 ball ee cnee 717 —4°30 Kaliana ......... 810 | —6'14 Mop |. ck scenes 879 —6°88 Meean Meer ...| 706 | —601 | tea Peagees — 5-02 * Loe. cit. p. [142]. B2 4 Rev. O. Fisher on Variations of Gravity and their “III. Inland Stations 1000 to 2000 feet high. Apparent varia- Station. Height. tion in vibration- numbers. Namthabad ...| 1173 —5'54 Kodangal ...... 1914 —4:59 Damargida ...| 1946 — 6°56 SOmianay. =. «75: 1714 —4:31 Badgaon ...... 1120. | —4:03 Ahmadpur ...| 1693 —4°38 Kalianpur ...... 1763 —3'61 Pahargarh ...... 1641 —5'60 Mean ...| 1620 —4'83 “TV. Inland Stations over 2000 feet high. Apparent varia- Station. Height. tion in vibration- numbers. Bangalore ...... 3064 — 5°68 1BYS) ohie eens ies 2242 —11:36 Mussoorie ......| 6920 — 8:22 ILGIG! kccortesces 15408 — 24:09 93 The method followed by Col. Herschel in estimating local attraction was to employ the formula for the attraction of a circular spherical cap, being “the part of a spherical shell bounded by a right cone, whose vertex is the centre of the sphere. Two such cones on a common axis intercept between them a zone, which is clearly the difference between two caps having a common axis.’’* The proper altitude of sections, every one of a set of such zones, taken around the station as their centre, each of appropriate width for its distance, was estimated, and also for the last zone the distance at which attraction would cease to be appreciable. ‘The density of the masses was taken at 2°75, being half the mean density of the Harth. It is obvious that such an estimate would stop at the sea-level, and take no account of variations of density below it; nor yet does it seem that the rise of the sea-level, which under the supposed conditions would be considerable, was allowed for. The for- mule used are essentially the same as those given by Pratt * Op, cit. p. [151]. relation to the Constitution of the Earth's Crust. 5 in the fourth edition of his ‘ Figure of the Earth,’ to which work Col. Herschel acknowledges himself indebted *. I propose, in what follows, to compare the vibration-num- bers at some of the meridian stations of the great Indian arc uncorrected for local attraction, but simply reduced to the sea-level +, with what they ought to be there did no elevated tract exist ; and the result will show how much local attrac- tion need be called upon to account for. Then, if a given hypothesis respecting the constitution of the Harth’s crust fairly meets the requirements in several instances, it will afford a fairly strong presumption that the hypothesis has a foundation in reality. It will be convenient to refer other stations to Punne as a base. It is situated near Cape Comorin, at the southern ex- tremity of the peninsula, in latitude 8° 9’ 28” N. But since Punne is 48 feet above the sea, it will be better to make the reduction, small though it be, to the sea-level there, once for all. We shall employ the following symbols in the calculations, the numerical values being taken from the ‘ Account of the Pendulum Operations’ already referred to. c=the radius of the earth=20,926,000 feet = 3963 miles. e=the elipticity, =0°0034483. m=ratio of centrifugal force to gravity at the equator, = 0°0034674. h=height of the station. /=latitude of the station. l’=latitude of Punne. N=number of vibrations at Punne in twenty-four hours when reduced to the sea-level. N +6N=the observed number at another station. g=the Harth’s attractive force at the equator. G =the force of gravity at the station. sG=the change in G corresponding to one vibration, where s=0:0000023148. k=the thickness of the cooled crust, taken as 25 miles. p=the density of the crust, mountain, and its root, taken as 2°68. ; o=the density of the substratum, taken as 2°96. h=the height of the station. u= the chord of the semiarc of the spherical cap. t=the depth of the root below the bottom of the crust. w= the chord of the semiarc of the root of the mountain. % Op. cit? p.| kali), Tt Op. cit. p, 1204 6 Rev. O. Fisher on Variations of Gravity and their The formula for the time of vibration of a pendulum, cg ah a enables us to connect N and G; anditis shown at p. [123] that, “as regards reduction for height of station, the formula )N=4N q is sufficiently exact;’’ N being in it taken as 86400. Hence it appears that, taking ON as one vibration, the change in G corresponding to one vibration is G =sG (suppose). The reduction of gravity for the height of the station may be taken as g a because the part of the reduction for height which would depend upon latitude is inappreciable. Hence the correction for the reduction of gravity to the sea-level at Punne for 48 feet is G x 0:0000045875. } And the correction to be added to the vibration-number, by the formula for 6N, will be 0719818. The observed vibration-number at Punne being 85982°75, we may say that at the sea-level at Punne, N = 85982°95. As an instance of the mode of making such a comparison as is proposed, let us take the case of Moré, the most northern station visited *, being also the most elevated (15,408 feet). Moré is about 150 miles, by the map f, distant from the sub- Himalayan plains to the south-west of it, and about the same distance N. by E. from Simla. Then we shall have: Gravity at: Moré—gravity at Punne : = difference for difference of latitude, — difference for height, + difference for local attraction. Now the difference for difference of latitude =9(1+1e—3m)(8m—e)(4$ cos 2l/—4 cos 2/)f. =9(4 cos 2l’—4 cos 21), suppose, * This was the last station visited by Captain Basevi, who there died, ‘a martyr to his work. See ‘Account &c.’ p. x. + Plate to “ Account &c.” Ser9) t Pratt’s ‘Figure of the Earth, 4th ed. art. 122, where his — corre- ) sponds to our g. a Relation to the Constitution of the Earth’s Crust. x where log B = 3°7159239. But Latitude of Punne= 8° 9 28”, and of Moré =33° 15/ 39”. The correction to gravity for difference of latitude then comes out =g x 0:0014589. Also correction for height 2h =—Jg9 a = —g X 00014726. Hence difference of gravity (6G) between Moré and Punne at sea-level ought to have been g x 0:0000137. Dividing this by g x 0:0000023148, which corresponds to one swing, it appears that there ought to have been 0°59 swing less at Moré, irrespective of local attraction, than at Punne. Now the observed vibration-number at Moré was . 85984°62 And at Punne, when reduced to sea-level ...... 85982°95 Actual difference, ON ...... = +1°67 So that there were in fact 1°67 swings more; whereas the difference, irrespective of local attraction, ought to have been —0°5. The result is that local attraction at Moré station needs to account for 2°26 swings in twenty-four hours. Reducing this to attraction measure, we may accept as a fact that Local attraction at Moré=g x 0:000052386, where g is the attraction of the sphere. WS eS | ne ee ee ae | Rey. O. Fisher on Variations of Gravity and their 8 “16— 4G. G+ 96-6 62-0 19-1 + 80FST Goptiecetionc. . .. + SOT es 006° Tet 90-6-++ 8G-F6+ $9-86 + 0¢69 elena eee 8 PTEOOSETL IAL 12-3-— 06-€ tes 09-6 — 1v-Spt 16-6¢+ GVGG Coors Sosa ae 66-7 — 98-T 0L9— 68-6— 9¢-LE+ LO-0+ 618 86-62-66 | ee Re ON, 6r-E — c6-1 86g — 86-6— 8¢-OF+ 08-FFt Ors ee EnOG | ee eS SUETEL VIS PILES = GL-T O09 — 69:0— Lv-FE+ SL-er+ LIL SAVERS ee ee c6-0 — VI-T cg + cT-O+ 1Z-8§+ 96-861 O18 Debt 0G S| ae 2 ayrete 6-6 — GGG 66 — Gr-0— LE-86 + EL-86+ IV9T L 9G ¥ Sarai eta ei thie 3s 96:0 — PL-G 90%+ GLeT+ 99-46+ 1P-L6+ S9LT Le eR EG 2g Pee tena dara ek eh-T — LGG est+ 6L-07 LeFEt 96-G6+ 691 LGnOS EC ie se. = PLT 7] ae GS-04 66-81+ 18-61+ O EPO eee 7 (suppose). And if ¢ is expressed in miles, we shall find that log y=5°8384723. Whence, in terms of the attraction of the sphere, oY 2cr The negative attraction of the root will be that of a cap- sector of thickness ¢ and of radius w, at a distance h+k, = that of a cap-sector of radius w and thickness h+k +t, — that of a cap-sector of radius wu and thickness h+k. The radius w therefore does not appear. The chord-radius w may be taken the same for the under as for the upper side of the plateau. Hence the negative attraction of the root Local attraction=<— p x &e. * “ Account” &c. p. [151]. + Ibid. p. [157]. relation to the Constitution of the Earth’s Crust. EZ =(o-p)a{uth+htt— Vu2+ (ht ht+tyP+ ae — GFE —(uth+k— vies (kp eet) _ +h) 6) C ) C =(c—p)a t— Vet tkhtiet+ Vw t(heky+ | ee C So that the whole vertical attraction of the cap-sector at the extremity of its axis is epee 2 te payurh Vue-bh? + - 2 —(o-p)a{ t— Vur+(htkh+tyP+ Vu? t+ ( ee +5. BEE C But the condition of hydrostatic equilibrium gives ph=(o—p)(1—7)e ; and it will appear that fractional quantities in — may be neglected. . ph=(o—p)t. : Hence, considering an Delmon sector of axial angle da, its attraction will be poa(u— Vu +h?)+(o—p)da( Vw? + (htk+ty — Vu +(htky), the terms in h, t, uh, and ué cancelling, by the condition of equilibrium. Thus the first part of Young’s correction dis- appears. The above expression affords a simple criterion of the amount of the vertical attractive force at the extremity of the axis, arising from the disturbance of gravity by the matter constituting a cap-sector, on the supposition of hydrostatic equilibrium. It is proportional to the difference between the extreme distances of the upper and lower edges of the root, multiplied by the relative density thereof, diminished by the difference of the distances of the upper and lower extreme edges of the visible mass, multiplied by its density. But when the distances are great, the thickness of the visible mass being small compared with that of the root, the attraction may be regarded as practically proportional to the difference of the distances of the upper and lower edges of the root. It follows that the attraction is less the longer the sector, and becomes insensible when that is very long. Phil. Mag. 8. 5. Vol. 22. No. 184. July 1886. C 18 ~~ Rey. O. Fisher on Variations of Gravity and their Suppose an elevated plateau of the form of an infinitely ~ long parallelepiped, and that the vertical attraction at a point on its surface, at a distance a from one edge and b from the other, is required. Let a be the angular distance of a sectorial radius from a. Then, dividing the plateau into two portions by a longitudinal plane through the station, and putting u successively =aseca and =bseca, and calling the above expression for the attraction /(a,«) and /(6, a), and intro- ducing the unit-factor ZY we shall have 20 Vertical attraction = = 2 ( (ft (a, a) +/(, a) )de. e (9) ~ A rather tedious integration gives V a2 sect at (h+k+t)?— Va? sec?at(h+k)?)da ) 2 2 tan?a+3(14(“t2**) )+a i 1+(“t2*) tan? «+ tan! « a a a = 9 le hak?) . J, (heh? ee tan? a+ 3(1+ = ))+ 1+(7 ) tan? @ + tant + (h+k+t)sin7! a —(h+h) sin7} — a a / 1+) V/1+(75) And this, when taken from «=0 to a= 53 gives 2 ; 1+( =) : ae ene (as +(h +k+t) "ne a 1+(3 1 . —(h +k) sinv) 4. eee is + 9 ) h+k From which ig (aseca— Va? sec? a+ h?) cde may be obtained 0 by putting h+k=0 and ¢=A, and changing the signs. I have calculated the vertical attraction, using these ex- pressions ; putting A=2°915 miles (the height of Moré), k= 25, t=27°920, p=p= 2°68, c=2'96,a=80,b=400. The result is that the attraction would produce 47155 swin gs of the pendulum per diem. relation to the Constitution of the Earth’s Crust. 19 To illustrate how much increasing the distance of the station from the edge of the plateau diminishes the attraction, it may be mentioned that the former of the above integrals gives, for a=80 The logarithmic term =—11:274, The two circular terms = + 24°651 ; whereas for a=b=400 they are The logarithmic term =—2°888, The two circular terms= + 5°807. In like manner the latter integral gives, for a= 80, The logarithmic term= +0-0519, The circular term =—0O°1061; whereas for a=b=400 these are respectively +0:00798 and —0°0185. As another instance of the effect of form and extension, it will be found that a circular plateau of the above height and of 80 miles radius (and therefore of infinitely smaller area than the parallelepiped) will, with its root, produce an attrac- tion corresponding to about 11 swings per diem. Archdeacon Pratt, in ‘a postscript on Himalayan attrac- tion,” in the Phil. Trans. 1859, p. 774, gives the rough dia- gram of the plateau here reproduced, referring to a work on the physical geography of the Himalayas hy Major R. Strachey, adding :—“ Ona careful consideration of all the data, Captain H. Strachey estimates the mean elevation of the tableland between the Himalayan and Turkish watersheds, and to the west of the ridge between the sources of the Indus and Brahmaputra, to be 15,000 feet (p. 56)”’*. This is the height of the station Moré. As far as I can judge by comparing the map in the ‘Account’ &e., the only one in which the locality is marked, with the relief-map by Major G. Strahan, and with Pratt’s diagram, it appears that Moré is distant about 80 miles from the nearly straight south-western escarpment of the range facing the plains, and 400 miles from the escarpment facing the north-west. If we imagine a series of cap-sectors drawn about the station, it will appear that the hypothesis of the parallelepiped makes the sectors on the north side too short near the cross section, and on the south side too long, while parallel to the range they are necessarily too long. This would make the attraction at the station too small. On the other hand, the actual plateau rises from the plains which are ® Toe; cit. POTTE: : C2 Rey..O. Fisher on Variations of Gravity and their 1) So . , 2 nia about 800 feet high, whereas the parallelepiped has been taken as rising from the sea-level. This would make its attraction too great. The difference could be calculated, if worth while. ee \ YK. 1) DP Wilts i 2 oe CO ~ ee ine LLL. aoe | Le ae i ed 7 eee Li y 7 \ NY . > Z <9 oie s. AW \\ \\ NS 5 ~ & 3 ES 20° | 150s Further, the slope of the range ought to have been taken into account. Thus a close agreement is not to be expected. However, referring to the fifth column of the Table, it appears that local attraction at Moré relative to Punne ou ght to produce 2:26 swings per diem ; whereas the attraction of our parallele- piped at a station situate as supposed would produce 4°15, or nearly two swings more. ‘The visible masses were estimated as. being capable of producing 23:57 swings. Hence the hypothesis of the parallelepiped, hydrostatically supported, accounts for 19°42 out of the 21°31 swings in defect which relation to the Constitution of the Earth's Crust. 21 have to be accounted for. So that the theory of hydrostatic equilibrium in this instance may be considered a not unsatis- factory explanation of the phenomena. To understand what has been done, we observe that the calculated effect of the local attraction of the parallelepiped is absolute, not relative, supposing that there would be no local attraction at the mean sea-level of a continental land-surface. It ought therefore to be compared with an absolute quantity. The 23°57 swings, given in column 7 of our Table, being the difference between the effect due to height and mass at Moré and that due to height alone, is also absolute. If, therefore, we take the parallelepiped to roughly represent the plateau, 23°57 —4:15=19-42 is the absolute difference between the effect of the plateau considered as supported by an excessively rigid crust and by hydrostatic equilibrium. Thus far observed vibration-numbers are not involved. When we consider these, we see that there were 1:67 more swings at Moré than at Punne. But without knowing what the effect of local attraction is at Punnee, we can have no exact knowledge of that which exists at Moré. If the number of swings of a given pendulum at the equator at the surface of the sea (but not subjected to the local attraction of an insular mass) was known, knowing the number at Punne, we could deduce the absolute attraction there, and consequently at Moré also. Here, however, we are at fault. Colonel Herschel has treated this question very fully in the second Appendix to the ‘ Account, &c.,’ explaining that he has adopted a novel method ; for, whereas local variations of gravity had previously been treated by the method of least squares, as if they were errors of observation, he has regarded them as having a real existence ; and has obtained his equa- tions for finding the vibration-number of a given pendulum at the equator from the numbers observed at known stations, on the two suppositions (1) that the mean of all observed local variations should be zero, and (2) that the mean of those in one narrow zone of north latitude should be equal to the mean of those in the corresponding zone of south latitude. But, seeing that the local variations are calculated subject to the attraction of the masses, and that the masses give very different attractions according as they are treated as if sup- ported by excessive rigidity of the crust, or by hydrostatic equilibrium, it appears that the resulting equatorial mean vibration-number obtained on the one supposition is not likely - to agree with that obtained upon the other. Colonel Herschel (Appendices, p. 45) makes local attraction at Punne=—4*2 swings. But if, by way of illustration, the 22 Rev. O. Fisher on Variations of Gravity and their hypothesis of hydrostatic equilibrium should turn out to so ~ alter the equatorial vibration-number as to make it instead = +1°89, then the absolute local attraction at Moré would be 1:89 + 2°26=4°'15, the same as that of our parallelepiped hy- drostatically supported. This is merely by way of illustration, because the parallelepiped only roughly represents the plateau; but it shows that the hypothesis may probably be competent to explain the phenomena. | If wu is larger than h+k-+¢, the expression, whose integral is given at p. 18, may be put under the approximate form if ak thtt, 2a sec a But since a is in the present case but little the larger, I have thought it best to calculate the attraction from the fuller formula. It seems, however, that the approximate formula would have given a result sufficiently exact ; for if we put a successively equal to 80 and 400 miles, using the above ap- proximate formula, we get the result 4:51 swings, whereas the fuller formula gives 4:15. These two numbers are suffi- ciently near for our purpose. Relying upon this, we can estimate the local vertical attraction at Kaliana. The attraction of a cap-sector upon a point beneath it, estimated downwards, is —pdo(u+h— VFR), . - 2e and of its root in the same direction, —(a—p)3a{ ¢— VF EEE TIP — VP 26)" The condition of equilibrium gives (c—p)t=ph. Hence the curvature terms cancel ; and, expanding in terms of iS fae u as and : we get Attraction = — phda( 2 - "et approximately. Now it appears from Major George Strahan’s Relief-map of India that the Himalayan plateau presents a nearly straight escarpment towards Kaliana, this straight face subtending an angle of about 120° at that station, so that radii drawn beyond - this will not enter the plateau on the western, and at a long distance only on the eastern side. relation to the Constitution of the Earth’s Crust. 23 Suppose, then, a series of cap-sectors, each of axial angle 6a, but of different radii, respectively long enough to reach the further edge of the plateau; and let a triangular block be cut off from these by a vertical plane through the crest of the . escarpment. ‘The vertical attraction of the mass beyond the vertical plane at the station will be that of the aggregate of __the sectors, minus that of the triangular block. There will remain the attraction of the slope of the plateau to be added. : H =P ae L 1) N FS vg Sue ae Beane CD ae a ao TC sl R Scale ;4, inch = 5 miles. P the station Kaliana. H the commencement of the plateau. H L=h=2°815 miles. F the foot of the slope. P L=a=140 miles. P F=6=60 miles. N R=depth of root of H L=t=9-57 x 2°815 miles. PK=k=26:621 miles. It will be observed that this value of & includes the height of the station, 0°153 mile, and of its root, 1:468 mile. The diagram shows a section through Kaliana at P, taken along the medial plane of the cap-sectors, and is intended to be on a scale of +, of an inch to five miles. The plateau commences at H. NRisthe depth of its root. PK is thick- ness of the crust, including the plain on which the station stands and its root. FH is the slope of the plateau. OR the slope of the corresponding root. | In calculating the attraction of the plateau and its root, we may reckon the height of the plateau from the level of Ka- liana, leaving out of consideration the attraction of the plain on which the latter stands, which may be regarded as an infinite plain extending under the plateau, and having its attraction balanced by that of its own root. The attraction of the plateau will then be 24 Rev. O. Fisher on Variations of Gravity and their —Spliba( 2 ee)- attraction of the triangular block. 2u And a being the distance of the edge of the plateau from Kaliana, Attraction of the triangular block 9 = — 2p (0B 2aseca re V3 Bh4+2h+1 = (A) But Sda= 20 5 =: ‘, the attraction of the plateau ey ee + Sphbe Bh+ Qh+t 2a 2u To this has to be ae the attraction of the slope of the plateau, which we must now calculate. An inspection of the diagram shows that the depth to the bottom of the root (h+k-+t) is always less than u, the distance from the station, which justifies the use of the approximate formula. Writing 7 for A and ny for t, we have from (A), Attraction of the triangular block of height Tv d+n =Kn)=—20( 3 ce (o+ yee And it is evident that the attraction of the elementar 'y layer of height 6 with its root will be afl). Ir V3 (8+n)n+hy Coe me The end of the slice 67, on account of the slope, will be dis- tant from the station, ” : b+7(a—b); so that, for its attraction at P, we must substitute this quantity for a, and subtract, and the ‘attraction of the slope and its root will be f(t 3 V3 ( a) V3 patel Tay, relation to the Constitution of the Earth’s Crust. 25 V3 (3+n)h bh k a =—2( ar an ge V3 (B+n)h? + Qkh he 2a ): Adding this to the attraction of the plateau, since the second term of the one cancels the first of the other, we have the attraction at the station et it 2 3+n)hb a =—pv3*} G+mi(C —k)log.5 f (8+n)h+k | Sphda Ou 5 If we substitute the values indicated above, and introduce the factor ee to express the effect in swings, the former of the above terms gives —2°73 swings in defect, due to the attraction of the plateau and other masses. The second term is not easily estimated, on account of the irregular out- line of the further boundary of the plateau ; but it will cer- tainly be quite smail, but being positive will make the negative effect less. Now there were observed relative to Punnee — 2°28 swings in defect due to local causes at Kaliana. The estimate —2°73, having been obtained on the suppo- sition that the slope of the plateau is a solid inclined plane, will be lessened by the circumstance that it is much inter- sected by valleys ; and this will be equivalent to assigning a lower mean value to the density p. Thus, on the whole, we may conclude that the calculated effect of local attraction at Kaliana brings the negative variation of gravity very close to the observed variation relative to Punne. The above number has (as was also done in the case of Moré) been estimated on the suppositions that the ellipticity is correctly assumed, and that there would be no local attrac- tion at the mean surface of the crust & considered as a land surface. Could we suppose Punne to be in such a situation, the above comparison would be exact. But since it is on the sea-coast, there is probably some local attraction there. In all but six instances the local attractions, given in the ‘ Account &c.’ as due to the visible masses, have been esti- mated as if they were caused by an infinite plain of the height of the station. The exceptions are Somtana, Ahmadpur, Usira, Dehra, Mussoorie, and Moré, at which places allowance has been made for inequality of surface. It might be thought that we need not have estimated the attractions afresh, as has * See pages 6 and 16 26 Rev. O. Fisher on Variations of Gravity and their been done roughly for Moré and Kaliana, because they have been already calculated and published. But it will be observed that the attraction of the “roots”’ of the masses cannot be discovered from that of the masses themselves, except in the case of plains. At the two former of the “ very irregularly surrounded stations,’ Dehra, Mussoorie, and Moré, we shall make no attempt to calculate the attractions, and conclude with some remarks upon certain stations in Peninsular India. There are no true mountain-ranges in Peninsular India, the so-called ‘‘ mountains” being only the escarpments of plateaus which have escaped denudation. “ Peninsular India is, in fact, a tableland, worn away by subaerial denudation, and perhaps to a minor extent on its margins by the sea.” * The Deccan traps are of Lower-Hocene age, covered in places by nummulitic rocks. Their total thickness may be 6000 feet f. The horizontality of the flows in these plateaus is remarkable, In considering the bearing of the gravitational phenomena at stations in this part of India, we ought to take its structure into account. No root has been formed by compression during the formation of its hills. As the country became gradually weighted by flow upon flow of the basalts, the crust must have sunk gradually intothe magma. The Geological Survey does not appear to have yet mastered the details ; but possibly it will be found that the country is faulted, and consequently deep roots will answer rather to low elevations than to high ones, and the equilibrium will be of the tract as a whole, instead of being established within every vertical boundary. In any case where the attraction of the mass above the sea- level.may be justly taken as due to an infinite plain, it appears by our formula that the negative attraction of the root would, on the hypothesis of hydrostatic equilibrium being established within every vertical boundary, exactly balance it; and the resulting local attraction at the station ought to be nil. Now there are seven stations of the great arc between latitudes 16° and 24° N. which are upon the basalt, viz. from Pahargarh to Kodangal inclusive ; and the local attractions, relative to the attraction at sea-level at Punne, in swings of the pendulum, range for these stations, as shown in column 5, from —0°79 to +1°75, being as below :—, * “Manual of the Geology of India,’ by Medlicott and Blanford (Cal- cutta, 1879), p. v. The contrast between the peninsular and Himalayan regions is strikingly shown by a large model now on view in the Indian Annexe of the Indo-Colonial Exhibition at South Kensington. + Ibid. p. 381. ¥ Ibid. p. 308, relation to the Constitution of the Earth’s Crust. 27 Local : : attraction Heights. Station. alee ohn Punne. ft. 1914 | Kodangal ......... +1:00 1946 | Damargida ...... —0-79 Erle WSomtuna 2720-6. c: +0-93 1130" | Badgson®) 5.2... +0°32 1693 | Abmadpur ...... +0°79 1763 | Kalianpur....... ae +1°75 1641 | Pabargarh......... —0-42 Wien PGS a). Sgdscass Seek +0°42 These numbers will be uniformly increased or diminished by any local attraction there may be at Punne, which is not situated on an infinite plain, but where proximity to the coast places it under conditions different from those at the stations which we are referring to it. To appreciate what these differences from zero attraction at the stations imply, we observe that one vibration per diem due to local attraction corresponds to about 645 feet of elevation of a plain ; and therefore, since the root (supposed of density p) displaces a layer of density o, one vibration in defect caused by it will correspond to aria x 645, or 6172 feet, 7. e. to 1:1707 mile of root. eee We see, then, that the root at Damargida, where the attrac- tion is — 0°79, would be about 0:92 mile too deep for local equi- librium ; and at Kalianpur, where it is +1:75, it would be 2-046 miles too shallow—these estimates being of course subject to the uncertainty belonging to local attraction at Punne. If we assume the crust at Punne to be 25 miles thick (which I have shown in my ‘ Physics &c.’ to be probable for places on the sea-coast), then, the depth for zero attraction of the root at Damargida being 3-527 miles, the actual depth there would be 5°171 and the whole thickness of the crust 30 miles. At Kalianpur, the depth for zero attraction being 3-195 miles, the actual depth would be 1:149 and the whole thickness 26 miles. These two instances are the greatest variations that occur throughout 8 degrees of latitude, and there are but two others of similar amount among the sixteen stations between Kaliana and Punne. Although these varying local attractions at the several stations make it clear that this region is not in hydrostatic equilibrium everywhere locally, nevertheless, the relative attractions being at some stations positive and at ‘others negative, it is quite possible that it may be as a whole sup- 28 Rey. O. Fisher on Variations of Gravity. ported in that manner, because we do not know how large the areas may be which would show positive or negative rela- tive attraction under column 5. Atany rate, the mean relative to Punne being only 0°42 swing, the result does not seem to discredit the hypothesis that there is a distribution of matter not far differing from what would accord with equilibrium for the region as a whole. ‘The attractions at these peninsular stations give no information about the mean thickness of the crust, assumed at 25 miles, because, in the case of an infinite plain, the terms involving it (k) are neglected. On the whole, it is apparent that the bottom of the crust is here irregular, and does not locally correspond for equilibrium with the surface-contour. In a basaltic region, as already remarked, this would seem natural ; for there will have been no compressing action tending to produce downward bulges corresponding to the elevated tracts, as would be the case in a mountain-chain ; so that any downward projection into the magma (1. é. root) will be due simply to the local depression of the crust, owing to its having become overweighted at the top, while rigidity of the crust, never crushed as in a mountain- chain, would extend the depression laterally and diminish it vertically. It is possible that the considerable local thickenings - and thinnings of the crust, which appear to occur at a few places in this region, may be due to faults of large throw, such as are not unknown in countries where there has been much out- pouring of basalts. The object of this paper has been (1) to summarize some of the results obtained during the Indian pendulum operations, as far as they bear upon the problem of the constitution of the _ Harth’s crust; and (2) to inquire whether, in a few of the instances where it seemed practicable to approach the subject by way of average, the theory of hydrostatic equilibrium gives a fair explanation of the phenomena. The cases that have been thus examined show that the theory of hydrostatic equilibrium makes, in swings of the pendulum, (1) Local attraction at Moré . . . (about) 4-15 Whereas relative to Punne itis . : 2°26 Difference . . el Bee) (2) It makes local attraction at Kaliana (about) —2°73 Whereas relative to Punne itis... — 2°28 Difference . . —0°45 (3) It makes mean of attraction at 7 Lada on the basalt iF
p/'v’. Hence, on the whole, the expanding gas does external work, and must accordingly draw on its supply of heat and get cooled. The thermal equivalent of pu—p’v’ at about 15° was calculated by Thomson and Joule from known data for air and CQ,, and was found to represent in the one case about a fourth, in the other about a third, of the actual cooling. The rest of the cooling effect is due to a gain of potential energy by the molecules at the expense of their heat ; in other words, the molecules of the expanding gas separate amongst the Molecules of a Gas. 83 against their mutual attractions. The two important general results obtained were :— | (1) The total cooling effect is directly proportional to the difference of the pressures on the two sides of the plug. (2) The total cooling effect is inversely proportional to the square of the absolute temperature of the gas. Although Thomson and Joule estimated the thermal equivalent of pu—p’v’ at 15°, they did not calculate its values at different temperatures and subtract them from the total cooling effect at the same temperatures, in order to get the parts of the cooling effect at these temperatures due to increase of the potential energy of the molecules. When this is done, the cooling effect due to increased potential energy, which we shall call 6, is, like the total cooling effe¢t in (1), directly proportional to the difference of the pressures on the two sides of the plug, because at a given temperature pu—p’v' is very nearly proportional top—p’. But (2) does not now hold for @. In its place we have this result, that the cooling effect due to increased potential energy is inversely proportional to the absolute temperature :— In obtaining the values of pu—p’v’ for air, Van der Waals’s formula was employed, . (p + 23") (v—-0026)=1-0011 (1+at): the unit of pressure being that of a metre of mercury, and the unit of volume that occupied by a kilogramme of gas at 0° C. and a pressure of one metre of mercury ; @ is the coefficient of expansion, and ¢ the temperature Centigrade. Thomson and Joule give the cooling effects at different temperatures corresponding to difference of pressure of 100 inches, or 2°54 metres of mercury; so that to get values of po—p'v’ corresponding to the same circumstances we must put p="76 m., p’=3'3 m. A kilogramme of gas is supposed to pass through the plug. Changing to ordinary units and dividing by the mechanical equivalent of heat J and the mR specific heat of air s, we get finally the cooling effects P° =P s due to departure from Boyle’s law, as tabulated below. The following table contains in the first column absolute temperatures, in the second the actual total cooling effects at the corresponding temperatures for a difference of pressure of 100 inches or 2°5 metres of mercury (these are taken from 84 Mr. W. Sutherland on the Law of Attraction Thomson and Joule’s paper in the Phil. Trans. 1862) ; in the third the calculated cooling effect ; in the fourth the value of @ or actual cooling effect minus ae ; and in the fifth the products Té. Air. Absolute | Cooling effect, | Difference of temperature, Jota 1 actual po—p'' the two ccoling Prod ae cooling effect. ee 1G: : s effects, 0. Le) ie} (e) Q 273'0 . 2 320 “600 164 280°1 88 ‘294 586 164 312°5 75 183 DOV cus uber 365°8 gayi 046 464 166 Thus within a range of nearly 93° the product T@ is practically constant, showing that the cooling effect due to increase of molecular potential energy is inversely proportional to the absolute temperature of the gas. The erratic number 177 is eliminated from the following table, in which, instead of the actual, experimental total cooling effects are used, those which Thomson and Joule calculated in the light of the fact that variation of the total cooling as the inverse square of the absolute temperature was the clear meaning of their experi- ments as a whole :— Absolute Calculated | Cooling effect, | Difference of temperature, actual pu—p'v' the two cooling Pr ee . cooling effect. Jo effects, 0. fo} (e} fe) Oo 2730 92 390 ‘600 164 280°1 ‘87 “994. ‘576 161 3125 ‘70 183 517 162 365°8 ‘D1 046 -464. 166 sil With the relation Oat thus established, it is not difficult to see that it means that the attraction between any two molecules of the gas is proportional to the product of their masses, and inversely proportional to the fourth power of the distance between them. For the mutual potential energy of the two molecules is inversely proportional to the cube of the distance between amongst the Molecules of a Gas. 85 them; or we may say that the potential of a particle of mass m at a point distant r from it is given by the equation V=A—. LP Let us assume that the constant A is of such magnitude that V becomes negligible for values of r greater than a certain length R, which corresponds to the radius of the sphere of action considered in most molecular theories, R being a large multiple of the distance between a molecule and its nearest neighbour and at the same time small compared to sensible distances. J is of the order of magnitude of, say, the thickness ofa capillary film. Under these circumstances we can consider each molecule as uniformly distributed through the small region of space round it, which may be said to belong to it. Then any one molecule P may be supposed to be gathered into a particle at its centre, leaving the space which belongs to it in the form ofa spherical vacuum, while all the other molecules have been spread out around it into a continuous matter of uniform density p. ‘To find the potential of a finite mass of gas at the centre of P, let us describe a cone of small solid angle w with its vertex at P, and terminating at the boundary of the gas AA’; it cuts off the small area aa! on the surface of the vacuous sphere. Then for the potential at P of any element xa’ distant r from P and of thickness dr we have Apwrdr r* frustrum aa’ A’/A,— Ri dp R Apu = = Apw log where Pa=7, and PA=R,. Let R,=nL, where n is a large number so that Lisa small but sensible length. ; and therefore for the whole potential of the 86 Mr. W. Sutherland on the Law of Attraction Then the above becomes Apw log a i.e. Apw log n+ Apw log 2; but 7, is so excessively small in comparison with the sen- sible length L that n is negligible in comparison with a ; Hence we may write Apw log = as equivalent to the above. ‘ Hence for the potential of the finite mass of gas round P, at the centre of P we can take that of the sphere of matter of radius L, which is, Am Ap log 2 Hence for the mutual potential energy of the particle P and the whole mass of gas, we have 477-Amp log ; 1 and for the total potential energy of n molecules, leaving out of count those so near the boundary thata sphere of radius L cannot be described about them so as to lie wholly in the matter under consideration, we have, 27 Anmp log 2 1 changing from the numerical coefficient 4 to 2, because we must not count the mutual potential energy of any two particles twice over. If now the mass of gas is allowed to expand (in Thomson and Joule’s experiments it expanded by various amounts up to six times the original volume), the value of log — remains | is practically constant, and the new value of the total potential energy is 2m7Anmp’ log 4 1 where p! is the density of the gas after expansion. Therefore the change of potential energy is proportional to M (p—p’); M being the mass of gas. Now the cooling effect corresponding to this will be obtained by dividing the above by JMs, where s is the specific heat of amongst the Molecules of a Gas. 87 the gas. Thus the cooling effect is proportional to pre Soe If T is the absolute temperature of the gas and p and p’ the pressures corresponding to the densities p and p’, p—p" P—P ; Js GS In the case of air s is practically constant, so that the theoretical cooling is directly proportional to the difference of pressures and inversely to the absolute temperature. Thus the hypothesis of a force attracting according to the law of the inverse fourth power and the product of the masses yields the two results deduced from the experimental data. It may be worth while mentioning that if the case is worked out for a very long cylinder of matter, attracting according to the Newtonian law, treated as a very prolate spheroid and expanded into another cylinder of the same section treated also as a spheroid, results in accordance with the above experimental results may be obtained, but with a third result, that the cooling effect would be proportional to the sectional area of the cylinder. Thus if the time ever comes when it will be practicable to look for the part of the cooling effect due to the mutual gravitation of the molecules, it will be found as a small fraction of the whole cooling effect, varying with the sectional area of the plug. is proportional to The only other gas on which Thomson and Joule conducted a sufficiently extended series of experiments to obtain definite results was CO,. They were able to enunciate the same two general results as for air, only the total cooling effects were not so accurately proportional to the inverse square of the temperature. To evaluate the thermal equivalent of pu—p’v’ at different temperatures for CO,, Clausius’ formula is used, ; a D033 a v—000426 T(v+:000494)* The unit of pressure is that of a kilogramme per square metre, and v is the volume in cubic metres of a kilogramme of CQ,. But, in the first place, the formula shows how at a given temperature the value of py—p’v’ is very nearly proportional to p—p', so that, as in the case of air, we can assert that the cooling effect due to increase of molecular potential energy is proportional to the difference of pressure on the two sides of the plug. 83 Mr. W. Sutherland on the Law of Attraction In passing from the value of pv—p'v' to the cooling effect Aas is) P —- , Which corresponds to it, account has been taken of the variation of s, the specific heat of CO,, with temperature. HK. Wiedemann’s determinations (*1952 at 0° C.,-2169 at 100°, "2837 at 200°) were adopted, the values at intermediate tem- peratures being obtained by interpolation. On reverting to the theoretical conclusion, it will be seen that the cooling is P a F and as s varies with the temperature, it will be necessary to test the theory by seeing whether the product T@s (for the constant value 100 inches of mercury for p—-p’) is constant. In the first of the following tables the total actual cooling effects are Thomson and Joule’s experimental numbers (Phil. Trans. 1862) ; in the second the calculated total cooling effects were obtained by them on the supposition that the cooling effects were inversely proportional to the square of the absolut temperature. to be proportional to CQ,. ‘ Absolute Mera tral Cooling effect, Difference of Predues emperature, line eff, puv—p'v the two cooling Os Tl. cooling eifect. Sr aed Pa effects, 6. Ss. ie) ie} 19) O° 2730 4-64 1:43 3-21 171 308'6 3°41 1:04 2°37 148 327-0 2°95 "89 2:0 141 370°5 2:14 63 151 120 Absolute Calculated | Cooling effect, | Difference of | p.cayot temperature, | total cooling pu—p'v! the two cooling} mg, ’ P effect. as effects, 0. " [o) to) oO ° 273°0 4°64 1-43 3:21 171 308°6 3°63 1:04 2°59 162 3270 3°23 "89 2°34 . 158 5705 2°52 | 63 1:89 - 152 In the first of the above tables for CO,, the products Tés show decided enough departure from constancy ; in fact T°@s would be nearly constant. The departure may be taken as showing the great value which similar experiments conducted on vapours compressed nearer and nearer to liquefaction and amongst the Molecules of a Gas. 89 then allowed to expand might possess. For in view of the facts that at 0° C. CO, liquefies under a pressure of about 40 atmospheres, while at 45° there is required 100 atmospheres to liquefy it, and that the pressure on the high-pressure side of the plug, in the experiments from which the above numbers were derived, reached 6 atmospheres, it becomes apparent that at the low temperatures the forces which are ultimately to produce cohesion in the liquid are hardly likely to be so closely Big hve: represented by the monomial expression wa as at the high temperatures. In fact, regarding the ultimate law of the action of one particle on another at any distance as a function of r of the form f S or f ( =) and cousidering the law of gravi- é tation as simply the first term of the expansion of the latter in ascending powers of 2 whick expresses the action accurately enough within the limits of astronomical distances, we may look upon Thomson and Joule’s experiments on air as showing how the second term, involving the inverse fourth power of 7, becomes appreciable at very small distances ; in the case of CO, we may regard the above table as showing I ; : how the term j= may begin to be appreciable, and how perhaps at still smaller distances still higher terms may appear and become predominant in producing cohesion and elasticity. There remains one application of our theory which throws an interesting light on a fact to which Thomson and Joule drew attention more than once as being very remarkable. When a mixture of the two gases, CO, and air, is expanded through a plug, it might be expected that each would contribute its proportion of cooling effect according to its own amount and its thermal capacity. But such is far from being the case. Indeed, experiment showed that the cooling effect for pure O is greater than for pure N, and yet in air and other mixtures of the two gases the cooling effect is less than in either of the constituents under the same circumstances. Let V, be the volume of a mixture of two gases before ex- pansion, V, the volume after. Let Va,, Vz,, be the volumes of the two constituent gases A and B before expansion; Va,, Vz,, the volumes after. Suppose that there are a molecules of A and } molecules of B in the mass under consideration. We must first make a hypothesis as to the action of a molecule of Aon a molecule of B. If the mutual potential of two 90 Mr. W. Sutherland on the Law of Attraction 2 molecules of A at distance r apart is — and of two molecules 12 of B is = , we will assume that the mutual potential of a V AB mm! molecule of A and one of B at distance r is C= Taree : where C is a constant. Then for the mutual potential energy of all the molecules of A before expansion we have an expression 27 A ampa, log, where p, means the density of the gas A when its a mole- cules are distributed through a volume V;. Similarly for the mutual potential of the molecules of B before expansion we have : fala 2arB bm'ps, log a the value of 7, being the same in each expression, because, according to Avogadro’s law, the molecules of different gases under the same circumstances own equal volumes of space. Leaving the quantity L of the same value in both expressions, amounts to asserting that the molecular forces in the two gases are quantities of the same order of magnitude. For the mutual potential energy of a molecule of A and one of B before expansion we have AdO /AB m'p » log Z : 1 and, therefore, for the mutual potential of the a molecules of A and the b molecules of B we have AoC / AB bm'p, log 2 1 But by proceeding in the other order, that is by writing down the mutual potential of the 6 particles of B and one of A and then summing for the a particles, we would obtain 4 / AB amp, log = 1 a Thus for the total energy of the mixed gases before expan- sion we have the expression (omitting common constants) Aamp ,. + Bb m'pp. 4+9C0VAB bm'py ; and after expansion, Aamp, + Bbm'pp +2C VW AB bm’p, , amongst the Molecules of a Gas. 91 p,, denoting the density of the gas A when its a molecules are distributed through a volume V>. Then for the increase of potential due to expansion we have Aam(p, —p,,) +B bm'(pg —pz,) +20 VAB bm'(p, —P,)- Now the last term in this has the same value as 29C VAB am(Pp— Pp); so that it may be replaced by the square root of the pro- duct of the two, namely 20 VAB am bm" (Pp. —Pa,) (P4,—Pa,)> Hence for the increase of potential energy we have the sum of the two expressions Aam(p, —p,) 3 Bbm'(pg, —p,) 3 and C times twice the product of their square roots. Now suppose that V, is the volume of the mixture at a pressure P,; V, ata pressure P,; the temperature T being the same in each case. Then py, is to the density of the gas A = and p, is to the density of 1 the gas A at pressure P, in the ratio - which is equal to 2 the previous ratio. Thus the term Aam(p,—p,) may be written Vie ae aa a—Pa), where p, p’, represent the densities of A at pressures P, and P,- But this is _“: times the gain of potential of a mass am of 1 the gas A escaping from under pressure P, to pressure P,; or, if we call @, the cooling effect for A corresponding to P2—P), we may write it Vv Jssp.Va vos where s, is the specific heat of A. For the other terms in the gain of potential by the mixed gases we can write corresponding expressions, and get for the result, V3 2 Jy (NGA cs ON. Ae JSsPavr 19 at Jsppasy" 16 5+ 2074 / yee v7 4 SoPavy 9p ee To obtain the cooling effect pane to this we must first divide by J, and then by the thermal capacity of the 92 _ Mr. W. Sutherland on the Law of Attraction mass operated on, which is SaP,V 4 + spppVp3.- Whieretore for the cooling effect @ of the mixture we have We 2 2 er 54Pa ya +r ‘sPay_? Bt ne v0 SpPayp Oe 1 eed sennre ees elms SsPaVa. + SpPpVp,- To show with this formula how the cooling for a mixture of O and N may be less than for either of the gases alone under the same circumstances, let us suppose that @ is less both than 0, and @, ; we will see whether the supposition leads to a pamela or impossible conclusion. Let us denote urs by D, and Ys, by D’, remembering that D+ D'=1 ; also Su, OY Gs ae OY [s Then the supposed inequalities become aD?0,+8D?0,+2CDD' /a80,0, < («D+ 6D")0@,, 6D’0,+«D?0,+2CDD!' V2680,0,< («D+ 6D')6, ; +, aD@,(D—1) +6D6,—8D'), + 2CDD' 480, 0, <0, or —«DD'd, + 6D”0,—BD/6,+2CDD’ VaB0,0,<9; *, —aDé,+6D’0, —B0,+2CD Va60,0, <0. Similarly —6D’'0,+e«D0,—«0,+2CD’ Va60,0,<0; -, adding —60,—«0,+2C(D+D’) Va80,6, <0. If C=1, this becomes, since D+ D/=1, —( VB, — Va8,)°<0, which is possible. If C is less than 1, the inequalities can also still exist together. Thus that the cooling effect for a mixture of two gases should prove less than that for either of the constituents has been shown to be a possible consequence of the theory of molecular attractions. By means of Thomson and Joule’ s experimental numbers for mixtures of air and CQ, in different proportions, we propose finally to calculate the values of C obtainable from the cooling amongst the Molecules of a Gas. 93 effects in different mixtures and see whether they agree; that is, whether our theory can be tested by its power to explain with any exactness a very peculiar experimental fact. There are no systematized experimental results by means of which the value of pv for different mixtures of air and CO, at different temperatures could be obtained directly ; but the following argument will show that we can make use of our previous numbers for the values of py—p’v’ at different tem- peratures in the case of air and CO,, to deduce the numbers for any mixture of air and CO,atthe same temperatures. Let V, be the volume of a mixture of the two gases at a certain tempera- ture and a pressure P); as before, let V,, Vg be the volumes which the constituent gases in V; would occupy if separated at pressure P,. Let W, be the potential energy of the mixed gases W,, Wz of the two separated gases ; then actually to separate the two gases will require work, Waa WN WV ot: Expand the separated gases to a condition represented by suffix 2, just as they were expanded in Thomson and Joule’s experiments, that is without doing external work other than that corresponding to the values of pv —p'v’ for each gas ; thus each of the separated gases would be cooled by the amount pope! ‘ ais previously calculated, and each would be cooled by its respective amount @, or @,, on account of the separation of molecules ; so that altogether the gain of potential energy during the expansion will be the sum of Js,6, and Js,0, and the two corrections pu—p’v'. Thirdly, allow the gases to diffuse into one another. In this case the work required will be pata Wig NN Hence the total gain of potential energy by the mixed gases on expanding from volume V;, to V, is Wiis Wow Wig Wak WN ya W,,+Js,0,+J5,0, Egass ky Vy) + (EV gic Paw aps but Wy —- Wy =Js,9, 5 W,.—W,, J 5,95. So that the total gain reduces to tele en ey By a) Emtec, and this corresponds to the total actual cooling effect observed by Thomson and Joule. The cooling effect denoted above by 6 is the equivalent of W,—W,; to obtain @ then from the 94 Mr. W. Sutherland on the Law of Attraction experimental numbers we have to subtract from them the cooling effects corresponding to the terms (P,V, —P,V, ). PLY 4 ee ee Given that a volume V a, of air is cooled by 1 Sa PLY. atta, PV, w J ? 8B and that a volume V, of CO, is cooled by 8 have for the cooling effect in the mixture :— ue ( (PiVy a5 PoVia)Va Pa aD (PiV, Gis =a) J Va Passat VuPpsp In this manner the pu—p’v’ part of the cooling effect of the mixed gases has been calculated from the previously found values for pure air and CQ,. , The table below contains in the first column the percentages of the two gases in the particular mixture ; the second contains the temperature (absolute) T at which the gas escaped; the third contains the total actual cooling effect for a difference of pressure of 100 in. or 2°54 metres of mercury, observed by Thomson and Joule (Phil. Trans. 1862) ; the fourth contains the calculated values of the above expression for the cooling effect to be subtracted; the fifth contains the cooling effect @ obtained by subtracting the numbers in the fourth column from those in the third ; while the sixth contains the values of C obtained by substituting the corresponding values of @ in the equation for 0. Mixtures of Air and CQ,. Percentage com- | Absolute tem- | moi.) actual | Cooling effect position of perature, cooling effect. to be 0. C. mixture. fe subtracted. Bs a0, \ Kittens 280 1-76 69 107 | -72 ico 280 1-17 “45 72 | 76 ee OO, \ She 280 1:86 76 11 | -57 . CO, \ a 6 323 1-29 49 8 | 8 oe CO, a a 323 88 33 55 | 6 Ee GO, } use 364 11 36 74 | -73 The agreement between these values of C calculated from different mixtures at different temperatures is very fair, espe- cially if the value ‘57 is left out of the count as being evidently affected by some error in the experiment from which it is de- duced ; because, if 1°:07 is correct for the first value of 0, amongst the Molecules of a Gas. 95 1-1 can hardly be right when the percentage of CO, has been increased from 32 to 388. Thus we can regard the experiments on mixed gases as furnishing confirmation of the truth of the law of the inverse fourth power for the attractions between molecules of gas. The value of C which we have obtained for air and CO, (mean 17) throws an interesting light on a certain aspect of the phenomenon of diffusion. Suppose a volume V, ofa gas A containing a molecules, and a volume V, of another gas B containing b molecules, separated by an infinitely thin partition, then the potential energy of the two masses of gas is (leaving out common factors) Aamp, + Bbm'p,. When the gases are mixed together without change of pressure we can see from what has gone before that the potential becomes A Bice sp pige 9 2cr/ AB bind TE ais Foo Peay Subtracting this from the previous expression, we get A Su BEE ee eae Mele 204/ AB folie ae amp . moey mM Pp ea ae ambm (V,+V,)" If C=1 this is a complete square, and therefore necessarily positive ; therefore when C=°7 the expression is also positive, and for all values of C less than 1 it must be positive; that is, the potential energy of the molecules diminishes by diffusion. Hence we may regard diffusion as partly due to the tendency of the molecules of the mixing gases to obey the dynamical principle that a position of stable equilibrium is a position of mininum potential energy. Diffusion is motion towards the position of stable equilibrium for the two gases. However, the kinetic factor in the diffusion of gases is so predominant that this aspectis not of much importance. But in the case of liquids it is otherwise, and the form of our last expression suggests how the tendency of some liquids to mix and of others to refuse to mix may depend on the magnitude ofa coefficient like C. Indeed, the study of the cooling effect of liquids and mixtures of liquids escaping from under pressure affords a splendid field for experimental inquiry. This paper will have possessed some value if it draws the attention of those who have facilities for such a research, to a field whose further exploration on the tracks of the pioneers must open valuable ground for Physical Science. Melbourne, April 1886. ei! XII. New Instrument for continuously recording the Strength and Direction of aVarying Electric Current. By R. SHIDA, M.E., Professor of Natural Philosophy in the Imperial College of Engineering, Tokio, Japan. [Plates II. & III.] To Sw William Thomson, F.R.S., LL.D., &. Duar Sir Wittam, ENCLOSE herewith a paper which I have just drawn up and which is a description of a new instrument I have devised and constructed for continuously recording the strength and direction of a varying electric current. My chief aim in designing such an instrument was to use it for making obser- vations of both regular and irregular variations of earth- currents which are present in the telegraph-wires of this country, just as they are present in the telegraph-wires of any other country. The importance of carrying on careful ob- servations of earth-currents has been felt more and more since you showed it before the Society of Telegraph Engineers and Hlectricians, about eleven years ago, in your presidential address. Indeed, so great an importance is now attached to such observations, that it was one of the main subjects discussed by the International Electric Congress which met at Paris last year. Now, since both regular and irregular earth-currents are so variable, that their strength and direction change, not only from day to day, but from hour to hour, or from minute to minute, or even from second to second, observations will be of very little value unless they are continuously made; hence the importance of a method of continuously registering the strength and direction of varying electric currents. The photographic method, such as is used in the Kew Observatory, is, of course, very satisfactory and accurate. But this method, besides requiring an elaborate arrangement of several pieces of appa- ratus, has a serious disadvantage, namely, that the observations must be made in a dark room. I have therefore felt for a long time the want of a method which, though not so accurate as the photographic method, is simple and convenient. It was thus that I was led to devise the apparatus described in the accompanying paper. As will be seen from the description given in the paper, the galvanometer-part of the apparatus is, in the main, the same as that of the more recent one of your Siphon Recorders ; that is to say, a coil containing a great number of turns of fine wire is suspended in a strong magnetic field produced by Strength and Direction of a Varying Electric Current. 97 permanent magnets. There is, however, one point in the apparatus which is quite new, at least quite new to my know- ledge: that is, that the advantage is taken of a singular pro- perty of matter, “ surface-tension of liquids.”” We know very well that a mercury-drop is often used for the purpose of making and breaking an electric circuit. But nobody has used a water-drop or an acidulated-water drop for the same purpose. The advantage of a water-drop over a mercury- drop, when employed for opening and closing an electric circuit, is that the former offers a far smaller resistance than the latter to the moving body which comes in contact with it. Now, in the instrument I am speaking of, a water-drop or, what is equivalent to it, a thin column of water drawn up between two narrow plates partly immersed in water, is used for the purpose of making and breaking the circuit, as will be seen from the description given in the paper. For the further details of the apparatus I ask you to be good enough to refer to the paper itself. The Instrument, I might mention, may of course be used as a “ coulombmeter,”’ because since in the paper ribbon, on which a record is obtained, the abscissas represent times and the ordinates represent currents, the area included by (the abscissa) the line of no current, and the ordinates correspond- ing to any two times and the curve of current, represents the quantity of electricity passed through the apparatus during the interval between the two times. * * * * . R. SHIDA. OnE of the principal subjects discussed by the International Electric Congress held in Paris in 1884, was that of Harth- currents ; and the result of the Congress as regards earth- currents was “ that the Conference expresses the wish that observations of earth-currents be pursued in all countries.” This resolution, together with the others, has been communi- cated to the various Governments; and our Government having conformed to the wish of the Conference, it was decided that the observations of earth-currents be made by the Tele- graph Department, in which I am a chief engineer. It thus devolved on me to take the subject up. A little consideration, suggested by the results of preliminary observations I have made of earth-currents, revealed to me that in order to carry out systematic observations of earth-currents, which, from time to time, vary in strength and direction, it is almost necessary, or at least extremely convenient, to have at. our disposal a simple instrument which will continuously record Phil. Mag. 8. 5. Vol. 22. No. 135. August 1886. H 98 Prof. R. Shida on a new Instrument for recording the the strength and direction of a varying electric current. In view of this, I have designed and constructed an instrument the description of which I have now the pleasure of communi- cating. In order that an instrument may continuously record a varying electric current, it is necessary that it should fulfil the two following conditions :— ? 1. That the motion of the needle of the galvanometer (which is a part of the instrument) be such that the same position of the needle always corresponds to the same strength of current, that is to say that the motion be non-oscillatory. 2. That the position of the needle of the galvanometer at any moment be recorded. I shall first explain generally how these two conditions are satisfied in the new instrument I am going to describe. As regards the first condition. This condition is satisfied by having a galvanometer whose needle consists of a coil of fine wire suspended in a powerful magnetic field, after the manner of the Siphon Recorder of Sir William Thomson. It is easy to show mathematically that in the case of an ordinary galvanometer, which consists of a magnetic needle suspended inside, or in the neighbourhood of a coil of wire, this condition cannot conveniently be fulfilled without diminishing its sensi- bility. On the other hand, in the case of a galvanometer consisting of a coil hung in a strong magnetic field as above described, it is easy to obtain a great sensibility and, at the same time, a non-oscillatory motion of the needle, as will be seen from the following investigations :— Let a be the angle of deflection of the coil at any time ¢, and let T be its period of oscillation when no current is circulating through it ; then we have for the equation of the motion da Am det TP | But when a current circulates through the coil, the equation of the motion will be altered owing to a retardation of the motion due to the current induced in the coil. Let us consider the magnitude of this retardation. If I be the intensity of the magnetic field which the coil occupies at time ¢, A the area included in all the turns of the coil, and if we neglect the self-induction of the coil on itself (which I think we can confidently do) ; then, plainly, N, the number of lines of force which pass through the coil at time ¢, is N=TIA sina ; hence aN da “a =IA COS a. Strength and Direction of a Varying Electric Current. 99 But = is the H.M.F. due to the inductive action ; hence, if R be the resistance of the circuit and «be small, the cur- rent induced in the coil at any time ¢, is approximately IA de rn ie Now the couple or torque due to the action of the field on the circuit is clA; and therefore the retardation of the angular velocity of the coil at any time ¢ is PA? da ae where yw is the moment of inertia of the coil. Hence we have, for equation of the motion of the coil, Ga VA da Am) 0 We an de The motion represented by this equation will be oscillatory or non-oscillatory according as 27/T is greater or less than T’A?/2uR, so that in order to make the motion of the coil non-oscillatory, all that is necessary is to have the magnetic field so strong that aie ple Now as regards the second condition. The method com- monly used of recording the motion of the needle of a galva- nometer is the photographic method, which is, undoubtedly, very satisfactory. But this method, besides requiring an elaborate arrangement of several pieces of apparatus, has a serious disadvantage, namely that the observations must be made in a darkened room. The method adopted in the new instrument is one which, though, perhaps, not so accurate as the photographic method, possesses the advantage of being very simple and convenient. In this method, which may be called the electrical method, there are several electrical circuits, each of which is closed when, and only when, the coil or the needle comes to a certain definite position corresponding to it, and each circuit, when closed, makes a mark on a moving paper ribbon chemically prepared, somewhat in the same way as in the Bain’s Telegraphs. If the coil turns round in one direction, it successively closes those circuits which make marks on one side of the centre of the ribbon, and if in the opposite direction, those circuits which make marks on the other side ; and further, the distance of the mark from the centre of the ribbon is greater or less according as the turning round of the coil is greater or less. H2 100 Prof. R. Shida on a new Instrument for recording the How these electrical circuits are exactly arranged will be seen later on with reference to the diagrams of the actual instrument. At present it suffices to explain how the coil or needle closes each electrical circuit separately, and without its motion being being checked or impeded. ‘This is effected by taking advantage of one of the well-known properties of matter, “ surface-tension of liquids.” When a capillary tube is partly immersed ina liquid which wets the tube, like water, the liquid ascends in the tube, and the smaller the diameter of the tube the greater the height to which the liquid ascends, and vice versa. In fact, it can be shown that if @ be the angle of capillarity, r the radius of the tube, w the weight of unit volume of the liquid, T the surface-tension per unit length of the liquid in contact with air, then h, the height to which the liquid rises, is an 1 2Tcosé Nl es wasp But the liquid is drawn up in the same way in the space between two parallel plates. In this case, if d be the distance between two plates, then 1 2Tcosé pee yh ee, d Ww which shows that the height to which a hquid rises between two parallel plates is equal to the height to which it rises in a tube whose radius is equal to the distance between the plates. Imagine now that there is a large number of capillary arrangements, each consisting of two very narrow plates standing in a vessel containing water at small distances from one another, and arranged in an arc of a circle, while the needle of the galvanometer is disposed in such a manner that, as it turns round, it successively comes in contact with the column of water drawn up between the plates of each of those capillary arrangements and thus closes several circuits in order; or else, that there is one such capillary arrangement, while the needle carries a large number of points so disposed that, when it turns round, these points successively come in contact with the column of water in the capillary arrange- ment, and thus close several circuits in order. Hither of these arrangements affords us the means of closing each circuit separately, and without the motion of the needle being checked. Jn the new instrument the latter plan is used, as will be more clearly seen with reference to the diagrams (Plates II. and III1.). Having now explained briefly the principles upon which Strength and Direction of a Varying Electric Current. 101 the action of the apparatus depends, I shall proceed to describe the construction and action of the apparatus. Fig. 1 (PI. IL.) shows the general view of the apparatus ; while figs. 2,3, and 4 (Pl. ILI.) shows the details of the arrangement of the coil, mag- net, &e. N and S are the poles of a powerful horseshoe magnet consisting ofa bundle of square bar-magnets made of very hard- tempered steel. Between the poles N and S there is sus- pended, by means of a fine silk thread, a coil(C), which contains a great many turns of a very fine insulated wire, and whose plane is at right angles to the line joining the two poles of the magnet; m is a piece of soft iron fixed inside the coil, nearly filling, but nowhere touching, it, and serves to intensify the magnetic field in which the coilhangs. When an electric current passes, the coil tends to turn round a vertical axis in one direction, or in the opposite direction, according as the current is positive or negative. The two weights, w, w, hanging from the coil can slide up and down the inclined plane (P). These weights resist the tendency of the coil to turn round caused by the passage of a current through it, and serve to bring the coil to its original position when the current ceases. The cords by which these weights are suspended pass through small holes in a piece of brass (#), whese dis- tance from the coil can be varied by moving it up and down along the vertical plane (P’), and thus the sensibility of the apparatus can be altered. The strength of the field is so great that the motion of the coil caused by the passage of a current is almost non-oscillatory. Attached to the coil (C) there is a thin circular disk of ebonite (D), whose axis coincides with the vertical axis about which the coil is free to turn, so that any angular motion of the coil causes exactly the same angular motion of the disk. This disk carries, on its underside and near to a portion of its circumference, a number of platinum teeth, ¢, ¢, ¢, Ke. Directly underneath these teeth, and rigidly fixed to the framework of the instrument, is a vessel (V), containing acidulated water, and in this vessel is provided a capillary arrangement which consists of two very narrow platinum plates p p (fig. 3) (which shall, hereafter, be called capillary plates), standing vertically up, side by side, from the central - part of the vessel, and drawing up the water of the vessel be- tween them. The position of these capillary plates, when everything is in its normal position, is such, that the platinum tooth midway between the ends of the series ¢...¢ 1s in con- tact with the column of water between the capillary plates, and that when the coil, and therefore the disk, is deflected to the right or left, the other platinum teeth on the left or right 102 Prof. R. Shida on a new Instrument for recording the successively come into contact with the column of water be- tween the capillary plates. Hvery time any one of the platinum teeth comes in contact with the water, it closes an electric circuit (to be described) corresponding to it, so that these platinum teeth may be called “ circuit-closers.”’ (L) isa cylinder of wood lacquered allover. It is covered with platinum sheet, and on this sheet is rolled a ribbon of white paper nearly as wide as the length of the cylinder. A portion of this cylinder is in the rectangular box (B), which contains a chemical solution, consisting of ferrocyanide of potassium, nitrate of ammonium, and water mixed in a certain proportion. Further, the cylinder (L) is made to revolve with uniform velocity by means of a clockwork arrangement placed inside the box (H). Thus the paper on the cylinder, as it rotates, comes out moistened with the chemical solution. Resting on the cylinder (L), and fitting tightly in a rod of ebonite (7), there are a number of platinum needles n,n, n, &c.; these needles may be called “mark- ing-needles,” for, if an electric current passes between any of these platinum needles and the revolving paper, a bluish mark is made on the paper directly underneath that needle. These marking-needles are electrically connected, each to each, with the circuit-closers in order, there being as many needles as there are circuit-closers ; that is to say, the first needle (on the right or left) is in connection with the first circuit-closer (on the right or left), the second needle with the second circuit-closer, the third with the third, and so on. The small terminal screws, a, a, a, &., on the ebonite plate (EH), which is fixed to the framework of the apparatus, and also the screws, 6, b, b, &e., are provided for facilita- ting these connections. The exceedingly fine wires (insulated) connect the screws a, a, a, &., with the circuit-closers, and they all hang down from the screws in the form of spiral springs, meeting together in the common axis of the disk (D), and the coil (C), and thence go to the circuit-closers ; so that it is to be understood that the resistance these wires offer to the motion of the disk and coil is so small as to be negligible. Now, the platinum sheet on the cylinder (L) is in connec- tion with one pole (Z) of the battery (CZ), by means of a pla- tinum spring (s) resting on it; while the other pole (C) of the battery is in connection with the capillary plates (see fig. 3). Consequently, when there is no current passing through the coil, the positive current flows from the copper pole of the battery through the capillary plates, and the circuit-closer in the centre, and thence through the corresponding marking needle (the centre one), rotating paper, and platinum sheet, Strength and Direction of a Varying Electric Current. 103 and back to the zinc pole of the battery, making a blue mark on the rotating paper just underneath that needle ; while if a current passes through the coil it is deflected to the right or left according to the direction of the current, the circuit- closers on the left or right of the centre successively coming into contact with the column of water between thercapillary plates in order, the result being that the corresponding needles make blue marks on the rotating paper. Butsince the paper revolves with uniform velocity, it is evident that the longer the time of contact between a circuit-closer and the water between the capillary plates, no matter which circuit-closer it is, the longer the length of the mark on the paper underneath the needle corresponding to that circuit-closer ; and the shorter the shorter. From the preceding description it will be clear that when an electric current, varying from time to time in strength and direction, passes through the coil (C), we shall get a curve made up of dots, or of dots and lines, on the moving paper ribbon, the nature of the curve determining the strength and direction of the current at any moment. Tig. 4 shows one of such curves experimentally obtained by allowing a varying _ current to pass through the coil. Now since the motion of the paper ribbon is uniform, it is easy to find out the point in the curve, or the position of the coil, corresponding to any moment; and since the motion of the coil is non-oscil- latory, each position of the coil corresponds to a certain definite strength of current, which can easily be determined by a simple experiment. So that by an examination of the curve thus obtained, it is easy to find out what was the strength of the current passing through the coil at any moment. To give a rough idea of the sensibility of the apparatus, it may be mentioned that when the record shown in fig. 4 was ‘obtained the apparatus was at its ordinary sensibility, which was such, that the superior and inferior limits of the current which it could record were respectively about 4 milliampéres and+ofamilliampére. But of course the sensitiveness of the apparatus can be varied within a considerable range in very much the same way as in Thomson’s Siphon Recorder. One defect of the instrument, it may be argued, is the fact that it does not record any current which produces such a deflection of the coil that no one of the circuit-closers is in contact with the water between the capillary plates. This defect, however, is not a very serious one, for since the in- strument is intended to be used for recording varying currents which give rise to a curve made up of dots, or of dots and lines, on the moving paper ribbon, it is easy, by examining 104 = Prof. W. Ostwald’s Electrochemical Researches. -the positions of dots and lines, to complete the curve to a certain degree of approximation. If, however, a greater accuracy be needed, all we have to do is to diminish the an- gular distance between the circuit-closers, and to increase their number. In the next instrument to be made, Iam going to introduce a few improvements, of which the most important is the mode of arranging the circuit-closers and capillary plates. Instead of having the circuit-closers movable with the coil, and capillary plates fixed, we may arrange so that the capillary plates move with the coil, while the circuit-closers are kept stationary ; and by this means, it is possible to di- minish the angular distances between the circuit-closers, and to increase their number without increasing the moment of inertia of the needle, and thus to obviate the above defect to a great extent, and, at the same time, to give to the instrument a greater sensibility. XIII. Electrochemical Researches. By W. OSTWALp, Pro- fessor of Chemistry in the Polytechnic School, Riga*. ss a reactions of acids, dependent on the characters, rather than on the nature of the constituents, of the acids, occur with an intensity which is different in each case, but is: always proportional to an affinity-constant which is itself dependent only on the character of the acid and not at all on the nature of the reaction. This fundamental fact, which throws new light on the old conception of affinity-constants, has been proved by the author for various reactions ; viz. the formation of salts in aqueous solutions, the actions of acids on insoluble salts, the change of acetamide into ammonium ace- tate, the catalytic decomposition of methylic acetate, and the inversion of cane-sugar{. These reactions, some of which are statical and others kinetical, led to the same numerical values for the affinity-constants of the acids examined. Adopting Clausius’s theory of electrolysis, and Williamson’s theory of chemical change, a distinct connection must exist between the reactions of acids and the electrical conductivity of these acids. ‘The theory of Williamson supposes that a continual exchange of atoms is occurring among the reacting molecules ; the velocity of a chemical action must therefore depend on the velocity of the atomic interchanges. The theory of Clausius says that electrolytic conduction is effected so that the free ions continually displace equivalent elements or * Abstract of Prof. Ostwald’s recent work, prepared by himself, and communicated by M. M. Pattison Muir, Cambridge. + See Pattison Muir’s ‘ Principles of Chemistry,’ p. 418 e¢ seg, Prof. W. Ostwald’s Electrochemical Researches. 105 radicles from the molecules, and that the parts of molecules set free again change places with others. The more fre- quently the ions can interchange the more rapidly will elec- tricity be conducted, because the electricity can only pass along with the ions. Now as, according to Faraday’s law, the ions always transmit the same quantity of electricity independently of their chemical character, it must be con- cluded from the theories of Clausius and Williamson that the velocities of reactions taking place under the influence of acids must be proportional to the velocities with which the acids transmit equal quantities of electricity, 7. e. to the elec- trical conductivities of the acids*. This inference from the theories of Williamson and Clausius has been verified by the author, by a series of measurements of the electrical conductivities of acids the velocities of reac- tions of which he had already determined ; the chief reactions in question were the catalyticdecomposition of methylic acetate and the inversion of cane-sugar. The following Table (p. 106) presents the results. Columns IJ. and II. give the velocities of the reactions referred to that of hydrochloric acid as 100 ; the numbers hold good for solutions containing respectively 2 and 4 equivalents, in grammes, in 1000 cubic centimetres. Columns III., [V., and V. give the electrical conductivities, determined by the method of Kohlrausch, referred to that of hydrochloric acid as 100; the numbers in III. apply to normal solutions containing one gram-equivalent in 1000 cubic. centim. ; in 1V., to 75 normal; and in V., to 745 normal solutions. These numbers show that the electrical conductivities of the acids in the table are proportional to the velocities of the change of methylic acetate into methylic alcohol and acetic acid, and the inversion of cane-sugar, brought about by these acids. The differences between the actual numbers in columns I, IL, I11., and columns [V. and V., may be explained by the occurrence of secondary actions among the first products (methy] alcohol, acetic acid, inverted sugar) of the two changes, measurements of the velocities of which were made. But determinations of the electrical conductivities of the acids are entirely free from the influences of secondary changes. By means of these determinations accurate values may be found for the affinity-consiants of the acids. These values are as important in the theory of chemical affinity as the values of the equivalent weights of the elements are in Stoichiometry. * This conclusion has been already stated by Arrhenius (Bigh. till. V. Svensk, Ak, Hand. 8, Nos. 13 & 14, 1884); but it was based on a com- paratively small number of experiments. 106 Prof. W. Ostwald’s Electrochemical Researches. 98 eeveee eeccoe eeeves Acid. Formula. Hydrochloric...... FICK Geese ae Hydrobromic...... TB psceus Snuat teenenc DNTEING 05. ocedaonien NO ces es ecakine sen Morne’ 2. 0....0425 ELC OSEP OS a ratneetes ACCC. a1.4.. eke Re CH, / COS ya. 018 Chloracetic....... CH,Cl.CO,H ...... Dichloracetic...... CHC CORE ks. Mrichloravetic’ .. 2) CCl. (COME ott... Glycollic ......... CH,OH2CO,Hy iL... Methoxyacetic .... CH,OCH,.CO,H... Ethoxyacetic ...... CH,OC,H, . CO,H Diglycollic......... CCHICO LED), a... Propionic’ .....%... Cort SCONE Pie. IDEY OR Re eee One C,H,0OH .CO,H Oxypropionicf ...| C,H,OH .CO,H Glyeeriew, vl C,H.,(OH), . CO,H Pyvuigie aac. seheaes CH,.CO.CO,H IBULY MIC ac cgogertols C.F, COS cgmareat Tsobutyric ......... CAH, - COLE ier... -s Oxyisobutyric ...|C,H,OH .CO,H Oxcaite 53 Seis andes « COLE) shes lb saat.e Mal@RiC) 0. s3ever0e CE (COL) oe cca SUCCIMICE cso shoe COBY (COLE), ov aase. Wales ieee. C,H,OH(CO,H), Dartanie: eaiacs.e C,H,(OH),(CO.H), RACEMIC reaper. ane C,H,(OH), (CO.H), Pyrotartarie’ ...... COB (CODED) aa. an Cirle Ake. Me. hake ve C,H,OH(CO,H), ... Phosphoric ..,... HPO oaed JANETC! ectiey.r > vac EE ASO ds lomadsieeten « II. eorces eccves * TE 100 101 99 1-72 0-436 5:06 seoeee 7:16 5:32 1h 531 1:56 153 ences 15:4 12:4 eee ee 118 120 117 pon S1 (0°) In looking over the table we notice many relations between the affinity-constants and the chemical composition and con- stitution of the acids. Before, however, these relations can be inquired into it will be necessary to ask, What is the influ- ence of dilution on the electrical conductivities of the acids? The numbers in columns III., IV., and V., which apply to solutions diluted in the ratio 1:10: 100, show that the quantity of water present exerts a great and varying influence on the conductivities. It has been found by the author (Journ. fir prakt. Chemie, [2] xxxi. p. 807, 1885) that the amount of water also influences the actions of acids in the inversion of cane-sugar, and that these actions vary in the same way as the electrical conductivites of the acids, for the various degrees of dilution. The numbers in columns III., IV., and V. do not express the specific conductivities of the various acids, as this term is understood in Physics, but rather the products of the specific conductivities into the equivalent volumes of the solutions. The meaning here given to the term equivalent-conductivity Prof. W. Ostwald’s Electrochemical Researches. 107 is as follows :—Let a vessel contain two parallel electrodes at a distance of 1 centim. apart, and let there be used one equivalent in grams of a specified acid; then the conduc- tivity of the acid, when diluted with a definite quantity of water, is the equivalent-conductivity for that degree of di- lution. It will afterwards be found more convenient to refer the conductivities of the acids to molecular rather than to equivalent weights. The molecular conductivity of an acid is found in a similar way to that whereby the equivalent- conductivity is determined. The degree of dilution is ex- pressed by the number of litres of solution containing one molecular weight, in grams, of any specified acid. It would be more accurate to express the ratio of acid to water by a molecular ratio, e.g. HCl: 100H,0; but, as only very dilute solutions are considered here, no essential inaccuracy will be introduced by adopting the more convenient method. If the dilution =O (that is, if no water is present), the con- ductivity of an acid is usually equal to, or is very little greater than, zero. As water is added the molecular conductivity increases, and approaches a maximum which is reached when the dilution is infinite. There is no exception to this general law. The conductivities of the stronger acids, HCl, HBr, HI, HNO;, HC1O;, HC1O,, HBrO3, HIO;, &e., are nearly at their maxima in moderately dilute solutions ; the molecular con- ductivities of these acids vary but little with the dilution. The following table presents the conductivities of some of the stronger acids when the dilution increases in the ratio of the powers of 2. The unit, in terms of which the conduc- tivities are expressed, is 4°248 times greater than the mercury unit. » | HO. | HBr. | HL | HNO,. | HClO,. | HCIO,. 2 | 79 | 804 | 804 | 779 | 779 | 79:1 4 | 809 | 884 | 832 | 804 | 802 | 822 8 | 836 | 851 | 849 | 824 | 823 | 84-6 16 | 854 | 866 | 864 | 849 | 840 | 86-2 32 | 870 | 879 | 876 | 863 | 854 | 881. 64 | 881 | 889 | 887 | 874 | 864 | 892 128 | 887 | 894 | 894 | 882 | 879 | 89-7 256 | 892 | 896 | 897 | 884 | 887 | 899 512 | 896 | 897 | 897 | 888 | 887 | 89:8 1024 | 895 | 895 | 893 | 889 | 886 | 89:8 The molecular conductivities of these acids are nearly the same (Kohlrausch had already observed this for some of these acids) ; they slowly increase and reach a maximum, equal to 108 = Prof. W. Ostwald’s Electrochemical Researches. about 90 for all these acids, at a dilution of 512 litres; in more dilute solutions the conductivities are slightly dimi- nished, owing to the impurities in the distilled water (Journ. fir prakt. Chemie, [2] xxxi. p. 440). Whether the molecular conductivities of all acids reach a maximum equal to about 90 in very dilute solutions cannot be determined by direct, expe- riments, because even with dilutions to 542 and 1024 litres the impurities present in the purest distilled water begin to exert an influence which cannot be accurately measured. An answer may, however, be given to this question if it is found possible to draw conclusions as to the behaviour of acids in more dilute solutions from their observed behaviour in less dilute solutions. : The observed values of the molecular conductivities of several acids are given in the following table. B = butyric, A= acetic, F = formic, M = monochloracetic, D = dichlor- acetic, H = hypophosphorous, I = iodic, acid; v = dilution, in litres. : 4096 ESOS) 19359 a teens 68°69 79°74 80:48 818 The molecular conductivity increases in every case, but in a very varying degree, with increasing dilution. The increase is greater, for a given increase of dilution, the smaller the conductivity of the acid ; it is also greater for weak than for strong acids, and greater for small than for large dilutions. The value of the increase in molecular conductivity for each dilution follows a special course : in the case of weak acids it increases as dilution increases ; the increase attains a maxi- mum value, equal to 8°9, in the case of monochloracetic acid for a dilution of 512 litres (mol. conduc. at 256 litres = 37°8, and at 512 litres = 46°7, diff. = 8:9) ; the same value is attained for dichloracetic acid for the dilution from 8 to 16 litres (mol. c. = 43 and 52:1) ; approximately the same value is attained for hypophosphorous acid for the dilution from 8 Prof. W. Ostwald’s Electrochemical Researches. 109 to 16 litres (mol. ec. = 45°8 and 54:1), and for iodic acid for the dilution from 4 to 8 litres (mol. c. = 50°5 and 59). As dilution increases in these cases, the increase in molecular conductivity for each dilution begins to decrease; for the strong acids the increase of conductivity is very small. It appears as if the maximum increase in molecular conductivity occurs where the conductivity is equal to 45—that is, is equal to half the maximum limit, 90. The relations between dilu- tion and molecular conductivity are more definite in the various series of acids. The mol. c. of formic acid is 1°76 for the dilution 2 litres; almost the same value (1°81) is reached by butyric acid for the dilution 32 litres. The following table exhibits the numbers for these two acids :— Formic Acid.) Dilution, in litres. Butyric Acid. EL Dicmjck 2 32 1:81 2°47 + 64 2:56 3°43 8 128 3°59 4:80 16 256 5:04 6°33 32 512 7:02 9:18 64 1024 9°74 12°6 128 2048 13°4 17:0 256 4096 18:0 22°4 512 8192 23°8 | 29:0 1024 16384 31°5 The numbers run parallel to each other ; those for butyric acid are about 5 per cent. greater than those for formic acid. Aqueous solutions of formic and butyric acids exhibit there- fore about equal molecular conductivities when the latter is sixteen times more dilute than the former. If the data for the other acids are examined, it is apparent that the dilutions at which the molecular conductivities of the monobasic acids exhibit equal values always bear a constant relation to each other. This fundamental fact is exhibited by the following table, wherein equal molecular conductivities are placed in the same horizontal lines. The different series exhibit a closely corre- sponding course. The dilution is expressed by means of the exponent p, which is defined by the relation, dilution = 2?. Ifthe values of p and the molecular conductivities are regarded as coordinates of a curve, then the lines for the individual acids all form parts of one and the same curve which is common to all the acids. The point of origin for each acid must be specially chosen onthe axis of the dilution-exponents p; in other words, a special constant must be given for the value of » for each acid. Prof. W. Ostwald’s Electrochemical Researches. 110 1-88 8 6-18 L 79g «| «9 | G.G8 q Gre 1 el oe | 6 . ecg Pe] 818 : Z0e. ol-6-[e ies ee Te gen cot 28 S| 1 cone alg LPL L eee | Oe he a | gan | J Oe Sota tee el | ee | 94g | OI 9.0G Z Zeq | «FP B.Gp sale 3.97 | 6 9.6 I Ose | 6.08 See ‘ 9.66 | 2 0-66 | OL 62e | 9 ree. |G esq. |G OLE 18 6ZL | > 9-31 |b ece | @ Fame & eoe[ = 2) al Gri: eae Ol 96-9 Z| ee9 |¢ ac alee ZOL 16 66-F 11 o3b |F ee oe G0-G | 8 re je | BE (8 | voc |o Oe ele cle | 4 Ist |¢ 80 18 | oo |g ieponnd Eee Be see ormog |pa] “oney [nal ong ji mor | TA] POL |) soudoddsy | UE] -sopgorq | UE]-xopqoouoyg] FE Powe see a SO ee oe eee SS Se Prof. W. Ostwald’s Electrochemical Researches. 111 This result is identical with that already stated ; for, if equal conductivities of the various acids are exhibited at equally proportional dilutions, then the dilution-exponents, which can be represented as logarithms of the dilutions (with the base 2), must show constant differences. The form of the curve is indicated by the results obtained ; it runs asymptotic ' between the axis of p and a parallel placed at a distance equal to 90 units, and appears to be symmetric to right and left and also above and below. The maximum of increase, as already observed, lies at 45, where the curve shows an inflexion-point. \ As we have here undoubtedly to do with a natural law, no exception to which is shown by any of the results obtained for 90 to 100 monobasic acids, it is reasonable to suppose that the curve must be capable of representation by a fairly simple analytical expression. At first sight a tangent-function is suggested. The results were therefore reduced so that the maximum fell at the value 90; they were then regarded as angles, and the corresponding tangents were found. The tangents, however, formed not an arithmetical but a geome- trical series ; the logarithms of the tangents gave approxi- mately constant differences, they increased proportionately with the dilution-exponents. A few examples are given. _ Acetic Acid. Pp. mol. c. log tan mol.c.| Difference. 1 0:5196 7-9576 : 9 0°7550 8-1199 ar A carn 8:4990 ‘147 Bon Vii tos 85690 01470 6 2-943 87110 fe 7 4-084 88537 vee 8 5642 89947 ae 9 7°753 9-1340 aEee 10 10-47 9.2667 | 1827 se EO OE Se ne a | 112 Prof. W. Ostwald’s Electrochemical Researches. Formic Acid. pp mol. c. logtan mol. c.| Difference. 1 1-758 8-4870 D) 2-465 8-6340 eels 4 4-796 8-9240 0-1416 | 5 6°634 9:0656 0-1429 | r¢ 12°59 9°3490 0°1358 | 8 16:98 9:4848 0-1309 | 9 22°43 96157 0-1284 10 29-02 9°7441 | Hypophosphorous Acid. p. mol.c. |logtanmol.c.| Difference. 1 30°89 9°7'767 ; 9 37-91 9:8914 iaeae 4 54°13 0°1408 0:1354 5 62°10 0:2762 5 0°1410 6 69:06 0:4172 0-1267 Fi 74:05 0°5439 0-1997 8 17°84 0:6666 The differences do not exhibit a constant value ; they are somewhat greater for the strong than for the weaker acids. The details are not given for the other acids, but only the mean values of the differences; these mean values are as follows :— Butyric... 0°155 Formic...... 0:140 Dichloracetic...... 0:133 TIodic... 0°126 Acetic ...0°145 Chloracetic.. 0°1386 Hypophosphorous 0:°127 Chloric 0°140 The mean of these is 0°136; but the individual means deviate from this much more than can be accounted for by experimental errors. ‘These deviations may be considered as follows :—If po represent the dilution-exponent at which the conductivity is equal to half the maximum, 45 in the present units, then tan 45°=unity, and log tan ae ail Then for every other dilution-exponent p, log tan m=°136 (p—po) ; where m is the molecular conductivity, referred to 90 as the maximum. Putting the quantity of water, v, in place of the di- lution-exponent,p,we have v= 2?, and logu=plog2=" 30103p, or p=3'032 log v, and the equation given above becomes log tan m=°1386 x 3°082 (log v— log vp) = "4124 log = Si ee Syne ae enum g Prof. W. Ostwald’s Electrochemical Researches. alte whence Vv °4124 tan m={— é Vo This final equation seems to be sufficiently simple. lt includes one constant, vp, dependent on the nature of the acid ; the other constant, -4124, depends on the units chosen. But there seem to be two objections to the expression. ‘There is a general agreement between the observed and calculated results, but individual acids show greater divergences than can be accounted for by experimental errors. If one attempts to explain these discrepancies, one is forced to admit that the formula has no rational foundation. An angle-function is never used in mathematical physics, so far as the author is aware, for a quantity which has no evident connection with angles. : The author has attempted to develop other, rationally grounded, expressions for the nature of the action of the water on the acid molecules ; but none has yet been found to agree so closely with the observed results as that given above. The mathematical treatment of the problem must be reserved for future consideration. The author then proceeds to discuss the results obtained with the polybasic acids. The behaviour of polybasic acids as regards dilution differs from that of monobasic acids. In one of his earlier papers the author suggested that when a solu- tion of a polybasic acid is electrolyzed, only one of the hy- drogen atoms of the acid goes to the cathode ; the electrolysis takes place according to the scheme H| HR” and H| H.R”. Sulphuric acid appeared as an exception. Further investi- gation has, however, shown that when the maximum conduc- tivity is nearly reached by dilution, the second, and eventually the third, hydrogen atom takes part in the transfer of elec- tricity, and electrolysis proceeds in accordance with the scheme H,|R” and H;|R’”’. The participation of the second and third atom of hydrogen depends upon the nature of the acids ; those which have but feeble acid properties, e. g. selenious, phosphorous, or phosphoric acid (shown by the impossibility of titrating these acids by the use of litmus) exhibit molecular conductivities which follow much the same course as the con- ductivities of the monobasic acids. The results for some of the polybasic acids are given in the following tables; the molecular conductivities are referred to the same units as before. The values of log tan m and the differences are also given, so that comparisons may be made between the poly- basic and monobasic acids. Phil. Mag. 8. 5. Vol. 22. No. 185. August 1886. I Prof. W. Ostwald’s Electrochemical Researches. 114 Selenious Acid, H,SeQs. ps V. My. Ms. m. ~ jlogtanm.| Diff. 2 4 9°752 9°720 | 9786 | 9:2345 0-1184 3 8 | .12°73 12-66 12°70 9°3529 0-1215 4 16 | 16°57 16°62 16°60 9°4744 0-126 L 5 5 ypu Paar (CS 2ifo” | 2NGS 9-6005 01296 6 64 | 28:25 » | 28:23 | 28:24 97301 0:1336 gig de 128. | 36°23 | 36:07 86°15 9°8637 0:1380 8 256 | 45:21 45:00 | 45-11 00017 0-1414 9 512 | 5444 | 54:10 | 54:27 01431 0-1458 10 1024. | 63:00 | 62°58 | 62°79 02889 1] 2048 | 69°80 | 69:00 | 69-40 12 4096 | 73°98 | 73:18 | 73°58 The behaviour of selenious acid is very similar to that of a monobasic acid, especially that of monochloracetic acid. If the numbers are divided by 2, the results represent the con- ‘ductivities referred to an equivalent weight of selenious acid ; the numbers thus obtained are not comparable with those tabulated in former series. Phosphorous Acid, H3;PO3. 1p Vv. My. Me -m. jlogtanm.| Diff. ~ cd; 4 2 28°63 28°62 28:63 | 9°7371 0-0966 2 4 34°30 34:28 34:29 | 99337 01076 4° 16 49-17 49-00 49:09 | 0:0622 0-1246 5 32 56°96 56:96 56:96 | 0°1868 0-1351 6 64 64°65 64:39 64°52 | 03219 01220 7 128 70°28 70°14 70:21 0:-4439 O-1L 43 9 512 77°84 77°30 77:57 | 06568 0:0598 10 1024 | 7930 | 7892.|° 79-11 | 0-7158 " ait 2048 80:00 79°50 79°75 + 12 4096 79°60 78°54 79:07 The differences are very small when the dilutions are large ; this is probably due to formation of some phosphoric acid the molecular conductivity of which is smaller than that of phos- phorous acid. In other respects the course of the change of conductivity of phosphorous acid is similar to that of the monobasic acids. Those dibasic acids whose normal salts are not alkaline but neutral behave very differently. ven the weaker acids of this class show an increase of conductivity over the monobasic acids when the solutions become dilute ; this advance of con- ductivity is exhibited sooner and to a greater extent the Prof. W. Ostwald’s Llectrochemical Researches. 115 stronger the acid. As examples, malonic and oxalic acids are given, Malonic Acid, CH, (COOH), | p- v. mM. My. m. \logtanm.| Diff. 1 2 4-48 4-48 4:48 | 88940 ; 2 4 | 632 | 635 | 634 | 9:0457 ear 3 8 8-90 8:83 887 | 91933 | 64397 4 16 | d216 | 1214 | 1215 | 93330 | oiiog 5 32 16-60 | 1654 | 1657 | 94736 | 9.3907 6 64 | 2239 | 22:15 | 22:27 | 96123 | 9.7308 7 128 29-45 | 29-43 | 29-44 | 9-7516 | 9.1327 8 256 8795 | 3753 | 87-74 | 98887 | 0.7493 9 512 | 47-40 | 4682 | 47-11 | 0°03820| 5454 10 1024 5660 | 56-10 | 56:39 | O1774 | Oj 4r8 ee 2048 6518 | 6420 | 64-69 | 0:3252 ‘pom e: 4096 71-66 | 70-48 | 71-07 | 13 8192 | 76°70 | 75:26 | 75-98 Ai a dilution of 2 litres the conductivity of malonic acid is about 10 per cent. less than that of monochloracetic acid ; as dilution increases the difference between the conductivities of ‘these two acids decreases, until the conductivities are equal at 512 litres dilution; from this point onwards malonic acid surpasses acetic acid; when the half maximum is past, the second hydrogen atom of malonic acid begins to take part in the electrolysis. Oxalic acid, which is a stronger acid than malonic, shows this greater conductivity than the monobasic Oxalic Acid, H,C,0,. p v. My Mg. m. flogtanm.| Diff. | a Finnie’ 316! | esr7 | %oye88 | ae 2 4 | 35°85 | 35°79 | 85°82 | 98584] 6.1598 3 8 | 4414 | 4410 | 4412 | 99867 | Qi 599 4 16 | 527 | 528 | 528 | 01197] 6.7450 5 32 | 613 | 61-4 | 614 | 02634 | 6554 6 64 | 690 | 690 | 690 | 0-4158| 61599 Z 128 | 751 | 750 | 751- | 05750) 9.4609 8 256) 798. | 198 | 798. | 07449) noes 9 512 | 836 | 835 | 836 | 09501) args 10 1024} 872 | 873 | 873 3264 : 11 | 2648 | 921 | 919 | 920 | 12 4096 | 989 | 984 | 98-7 13 8192 | 1185 | 1185 | 1185 14 | 16384 acids in a very marked way ; the maximum of the monobasic acids is passed by oxalic acid. The falling off in the conduc- tivity, which is eustomary with the monobasic acids when the dilution is greater than 1024 litres, is hidden by the partici- i 116 Prof. W. Ostwald’s Electrochemical Researches. pation in the electrolysis of the second hydrogen atom of oxalic acid. The marked increase in the value of log tan m is very characteristic. Sulphuric acid is stronger than oxalic acid; the second hydrogen atom of sulphuric acid will therefore probably sooner take part in the electrolysis, the maximum of the mo- nobasic acids will be overstepped, and another maximum will be approached which we may suppose will be double as large as that of the monobasic acids. The following numbers were obtained :— Sulphuric Acid, H,SQ,. p v. My My m 1 2 92:7 92°7 92:7 2 4 96°4 96-4 96°4 3 8 100°7 100°5 100°6 4 16 107°5 107-2 107-4 9) 32 116°3 116-2 116°3 6 64 127-0 127-4 127-2 7 128 139°5 138-9 139°2 8 256 150°6 150°6 150°6 9 512 161-0 160°3 160°9 10 1024 169°3 168°9 169-1 yt 2048 1745 1743 V4 12 4096 17 A 177-0 177-1 13 8192 177-1 1766 176°9 14 16384 1743 1741 174-2 Neither these numbers, nor those obtained by referring the conductivities to the equivalent weight of sulphuric acid, can be made to agree with the normal curve. The separate actions of the two hydrogen atoms of sulphuric acid are shown very markedly in the results recently obtained by F. Kohlrausch (Gétt. Nachr. 1885, p. 80). The conducti- vities of the stronger monobasic acids appear as straight lines in the system of coordinates chosen by Kohlrausch ; the curve of sulphuric acid forms two straight lines at different inclina- tions, joined by a short curve which falls at the dilution 2-8 litres. The table given by the author contains therefore the second part of the complete curve. As Kohlrausch did not examine any other dibasic acid of nearly the same strength as sulphuric, the behaviour of this acid remains unexplained by his results. _ The point at which the second hydrogen atom of a very strong dibasic acid begins to take part in the electrolysis is situated in the concentrated solution. Dilute solutions of such an acid behave similarly with the monobasic acids, pro- vided that molecule is not compared with molecule, but Prof. W. Ostwald’s Electrochemical Researches. 117 equivalent with equivalent. Methylene-disulphonic acid, CH, (SOQ,OH)., is a very strong dibasic acid; solutions of this acid behave similarly to those of nitric acid. Methylene-disulphonic Acid, CH, (SO,0H),. D. VU. My. Me Me Equiv. I 2 133-7 | 1385°1 134-4 67:2 2 4 1460 | 1467 | 146-4 73:2 3 8 1530 | 1536 | 153:3 76:7 4 16 1589 | 1585 | 158°7 79-4 4) 32 163-5 | 1640 | 1633 81-9 6 64 Hees be LGT Se | 16-5 83°38 7 128 BEES oll) Lite2 85°6 8 256 MO elias || vias 8771 9 512 LT (se RS an nC) 88°5 10 1024 1788 177-2 178-0 89:0 1a 2048 179°3 178°5 1789 89:5 12 4096 179-2 177-2 178-2 89-1 13 8192 177-1 175-7 176-4 88:2 The last column, headed Equiv., shows that the conductivity referred to the equivalent weight (half the molecular weight) of this acid varies in the same way as the conductivity of a strong monobasic acid. The limiting value of the molecular conductivity of dibasic acids, when the second phase of elec- trolysis is traversed, is shown by the numbers in the above table to be twice as large as that for monobasic acids. The relations between dilution and conductivity of tribasic and polybasic acids are analogous to those already discussed. Phosphoric acid reacts with two equivalents of alkali to form nearly neutral salts; the third hydrogen atom is that of a very weak acid. In conformity with these facts, we find the conductivity only slightly increased by great dilution. Phosphoric Acid, H3PQx,. | a ew. My. My. m. fjlogtanm.| Diff. | 1 2 | 14a | 1493 | 1422 | 94038 | coors | ene: 4} 1699 | 17-01 | 17-00 | 94853 | Oyeae pig 8 | 2122 | 21:30 | 21:26 | 95900] 53499 4 16 | 27-05 | 27-12 | 27-09 | 9-7089 | idee 5 32 34°41 3441 84°41 9:8357 0:1347 6 64 | 4985 | 4324 | 4305 | 9:9704| Osa, 8 6 1 °6818 | 61-8 | 618 | 02707)|-¢ 1ac9 9 | 512 | 699 | 696 | 699 | 04366 | Oiine | 10 1024 75°4 75°4 75-4 05842 11 2048 79:0 78:9 79:0 12 | 4098 | 798 | 798 | 798 13 8192 | 78:9 78:7 788 118 . Mr. O. Heaviside on the Phosphoric acid closely resembles dichloracetie acid ; only in very dilute solutions does it surpass that acid. Citric acid corresponds with malonic acid in behaviour on dilution ; it forms no salts with alkaline reaction, but is much weaker than phosphoric acid. Strong tribasic acids have not yet been examined ; but their behaviour may be deduced from the results already obtained. The results for dibasic and tribasic acids cannot be brought into mathematical form until a rational expression is found for the law of dilution of the monobasic acids. It is evident that the dibasic acids must be regarded as to some extent the sums of two monobasic acids having different conductivities. The prospect presents itself of finding numerical expressions for the function of the replaceable hydrogen of the polybasic acids. XIV. On the Self-induction of Wires. By OLIVER HEAVISIDE*. SERIES of experiments made some years ago, in which I used the Wheatstone-bridge and the differential tele- phone as balances of induction as well as of resistance, led me to undertake a theoretical investigation of the phenomena occur- ring when conducting-cores are placed in long solenoidal coils, in which impressed electromotive force is made to act, in order to explain the. disturbances of balance which are pro- duced by the dissipation of energy in the cores. The simpler portions of this investigation, leaving out those of greater mathematical difficulty. and less practical interest, relating to hollow cores and the effect of allowing dielectric displacement, were published in the ‘ Electrician’ from May 38, 1884, to January 3, 1885. This investigation led me to the mathematically similar investigation of the transmission of current into wires. I say into wires, instead of through wires, because the current is really transmitted by diffusion from the boundary into a wire from the external dielectric, under all ordinarily. occurring circumstances. In the case of a core placed in a coil the magnetic force is longitudinal and the current circular ; in the case of a straight round wire the current is longitudinal and the magnetic force circular. . The transmission of the lon- gitudinal current into the wire takes place, however, exactly -in the same manner as the transmission of the longitudinal magnetic force into the core within the coil, when the boun- * Communicated by the Author. Self-induction of Wires. 119 dary conditions are made similar, which is easily to be realized. Similarly we may compare the circular electric current in the core to the circular magnetic current in the wire. I also found the transfer of energy to be similar in both cases, viz. radially inward or outward to or from the axis of the core or the wire. It was therefore necessary to consider the dielectric, in order to complete the course of the transfer of energy from its source, say a voltaic cell, to its sink, the wire or the core where it is finally dissipated in the"form of heat, and its temporary storage as electric and magnetic energy in the field generally, including the conductorsi’ Terminating the paper above referred to, having ‘so ‘much other matter, | started a fresh one under the title of “ Electro- magnetic Induction and its Propagation,’ commencing in the ‘ Electrician,’ January 3, 1885. Having, according to my sketched plan, to get’ rid of general matter first, before proceeding to special solutions, I took occasion near the com- mencement of the paper to give a general account of some of my results regarding the propagation of current, in which the following occurs, describing the way the current rises in a wire, and the consequent approximation, under certain cir- cumstances, to mere surface-conduction. It was meant to illustrate the previously-mentioned stoppage of current-con- duction by high conductivity. After an account of the transfer of energy through the dielectric (concerning which I shall say a few words later) I continue (‘ Electrician,’ Janvary 10, 1885) :— “ Since, gn starting a current, the energy reaches the wire from the medium without, it may be expected that the electric current is first set up in the outer part, and takes time to penetrate tothe middle. This Ihave verified by investigating some special cases. “Increase the conductivity enormously, still keeping it finite, however. Let it, for instance, take minutes to set up a current % the axis. Then ordinary rapid signalling ‘through the wire’ would be accompanied by a surface-current only, penetrating to buta small depth. The disturbance is thus propagated parallel to the wire in the manner of waves, with reflection at the end, and hardly any tailing off. With infi- nite conductivity thefe can be no current set up in the wire atall. There is no dissipativity ; wave-propagation is perfect. The wire-current is wholly superficial, an abstraction, yet it is “nearly the same with very high conductivity. This illustrates the impenetrability of a perfect conductor to magnetic induc- tion (and similarly to electric currents) applied by Maxwell to the molecular theory of magnetism.” ..... 120 . Mr. O. Heaviside on the Attention has recently been forcibly directed towards the phenomenon above described of the inward transmission of current into wires by Professor Hughes’s Inaugural Address to the Society of Telegraph HEngineers and Electricians, January 1886. This paper was, for many reasons, perhaps the most remarkable one ever written. It was remarkable for the extraordinary ignoration of well-known facts, thoroughly worked out already ; it was remarkable for a quite phenomenal mixing up of the effects due to induction and to resistance, and the author’s apparent inability to separate them, or to see the real meaning of his results; one might indeed imagine that an entirely new science of induction was in its earliest stages. It was remarkable that the great experimental skill of the author should have led him to employ a method which was in itself highly objectionable, being capable of giving, in general, neither a true resistance nor a true induction-balance (as may be very easily seen by simple experiments with coils, without any mathematical examination of the theory)—a method which does not therefore admit of any exact interpre- tation of results without the fullest particulars being given and subjected to laborious calculations ; and finally, it was remarkable as containing, so far as could be safely guessed at, many verifications of the approximation towards mere surface- conduction in wires. ‘This is, after all, the really important matter, against which all the rest is insignificant. As regards the method employed, I have shown its inaccu- racy in a paper ‘‘ On the Use of the Bridge as an Induction- balance,” in ‘ The Electrician,’ April 80, 1886, wherein I also described correct methods, including the simple bridge with- out mutual induction, and also methods in which mutual induction is employed to get balance, giving the requisite formule, which are of the simplest character. As regards the interpretation of Prof. Hughes’s thick-wire results, showing departure from the linear theory, by which I mean the theory that ignores differences in the current-density in wires, I made the following remarks in the ‘ Hlectrician,? April 28, 1886. After commenting upon the difficulty of exact interpretation, I proceed :— ; “The most interesting of the experiments are those relating to the effect of increased diameter on what Prof. Hughes terms the inductive capacity of wires. My own interpretation is roughly this. That the time-constant of a wire first in- creases with the diameter ”’ [this is of course what the linear theory shows], “and then, later, decreases rapidly ; and that the decrease sets in the sooner the higher the conductivity and the higher the inductivity (or magnetic permeability) of the Self-induction of Wires. 125 wires. If this be correct, it is exactly what I should have expected and predicted. In fact, I have already described the phenomenon in this Journal ; or, rather, the phenomenon I described contains.in itself the above interpretation. In the ‘Hlectrician’ for January 12, 1884” [corrected to January 10, 1885, in a subsequent letter, May 7, 1886], “I described how the current starts in a wire. It begins on its boundary, and is propagated inward. Thus during the rise of the current it is less strong at the centre than at the boun- dary. As regards the manner of inward propagation, it takes place according to the same laws as the propagation of mag- netic force and current into cores from an enveloping coil, which I have described in considerable detail in this Journal. The retardation depends upon the conductivity, upon the inductivity, and upon the section, under similar boundary- conditions. Ifthe conductivity be high enough, or the induc- tivity, or the section be large enough, to make the central current appreciably less than the boundary-current during the greater part of the time of rise of the current, there will be an apparent reduction in the time-constant. Go to an ex- treme case—yery rapid short currents, and large retardation to inward transmission. Here we have the current in layers, strong on the boundary, weak in the middle. Clearly then, if we wish to regard the wire as a mere linear circuit, which it is not, and as we can only do to a first approximation, we should remove the central part of the wire—that is, increase its resistance, regarded as a line, or reduce its time-constant. This will happen the sooner the greater the inductivity and the conductivity, as the section is continuously increased. It is only thin wires that can be treated as mere lines, and even they, if the speed be only great enough, must be treated as solid conductors. I ought also to mention that the influence of external conductors, as of the return conductor, is of im- portance, sometimes of very great importance, in modifying the distribution of current in the transient state. I have had for years in manuscript some solutions valltin g to round wires, and hope some day to arrive at them in the course of the paper I am at present publishing, or, rather, not publishing, as the editor has been able to afford so little space for it lately. “‘ As a general assistance to those who go by old methods, a rising current inducing an opposite current in itself and in parallel conductors, this may be useful. Parallel currents are said to attract or repel, according as the currents are together or opposed, ‘This is, however, mechanical force on the con- ductors. The distribution of current is not affected by it. But when currents are increasing or decreasing, there is an 129 Mr. O. Heaviside on the apparent attraction or repulsion between them. Oppositely- going currents repel when they are decreasing and attract when they are increasing. Thus, send a current into a loop, one wire. the return to the other, both being close together. During the rise of the current it will be denser on the sides ot the wires nearest ore another than on the remote sides. .... 2 _ An iron wire, through which rapid reversals are sent, should afterwards be found, by reason of its magnetic retentiveness, magnetized in concentric cylindrical shells, of alternately positive and negative magnetization. This would only occur superficially. he thickness of the layers would give infor- mation regarding the amount of retardation, from which the inductivity could be deduced. The case is similar to that of the superficial layers of magnetization produced in a core in a coil through which reversals are sent, the magnetization being then, however, longitudinal instead of circular. The linear theory is departed from in the most extreme manner, when the return-current closely envelops the wire. The theory of the rise of the current in this case I have given in the ‘ Hlectrician’ for May 14, 1886, and the case of the return-current at any distance is considered June 11, and will be followed by others. The investigation following in this paper is more comprehensive, taking into account both electrostatic and electromagnetic induction, working down to the electromagnetic theory on the one hand, and approxima- ting towards the electrostatic theory (long submarine cable) on the other; with this difference, that inertia is not so wholly ignorable in the long-line case as is elastic yielding in the case of a’ short wire. Nor is the variation of current- density wholly ignorable. But first as regerds the transfer of energy in the electro- magnetic field. ‘This is 'a‘very important matter theoretically. It is a necessity of a ratignally inte ligible scheme (even if it be only on paper) that: rae transfer of energy should be expli- citly defipable. It the absence of this definiteness that makes the Germar* methods so repulsive to a plain man who likes to #. whers he is going and what he is doing, and hates metaphy sics In science. ~' -I found- that I had been anticipated by Prof. Poynting“ in the deduction of thé transfer of energy formula appropriate to Maxwell’s electromagnetic scheme, in the main. It is, therefore, only as having given the equation of activity i in a more general form, the most general that Maxwell’s scheme admits of, and having deduced it in a simple manner, that I can attach myself to the matter. In connection with it, how- ever, there is another matter of some importance, viz. the use Self-induction of Wires. 123 of a certain fundamental equation. That I should have been able to arrive at the most general form, taking into account intrinsic magnetization, as well as not confining myself to media homogeneous and isotropic as regards the three quan- tities conductivity, inductivity, and dielectric capacity, in a simple and direct manner, without any volume-integrations or complications, arose from my method of treating the general equations. I here sketch out the scheme, in the form I give it. Let H, be the magnetic force and F the current. (Thick letters here for vectors. The later investigation is wholly scalar.) Then, “curl” denoting the well-known rotatory operator, Maxwell’s fundamental current equation is Carre eee Ae One ety and is his definition of electric current in terms of magnetic force. It necessitates closure of the electric current, and, at a surface, tangential continuity of H, and normal continuity of T. The electric current may be conductive, or the variation of the elastic ‘‘ displacement,” say T=C+D. C being the conduction-current, D the displacement, linear functions of the electric force E, thus C=kk,, D=cE,/47 ; k being the conductivity, and c¢ the dielectric capacity (or ¢/4a the condenser-capacity per unit volume). Equation (1) thus connects the electric and the magnetic forces one way. But this is not enough to make a complete system. A second relation between E, and H, is wanted. Maxwell’s second relation is his equation of electric force in terms of two highly artificial quantities, a vector and a scalar potential, say A and P, thus E,= —A—VB, . (2) ignoring impressed force for the present. From A we get down to H, again, thus, curl A=B, B= pH, ; B being the magnetic induction, and yw the inductivity. (Here we ignore intrinsic magnetization.) The equation (2) is arrived at through a rather complex investigation. From these equations are deduced the general equations of electromagnetic disturbances in vol. ii. art. 783. They contain both A and P. One or other of them must go before we can practically work them, which are, independently 124 Mr. O. Heaviside on the of this, rather unmanageable, although they are not really general, for impressed forces are omitted, and the intrinsic magnetization must be zero, and the medium isotropic. Again, and this is an objection of some magnitude, the two potentials A and P, if given everywhere, are not sufficient to specify the state of the electromagnetic field. Try it; and fail. — Hven without using these complex general equations referred to, but those on which they are based, (1) and (2), the very artificial nature of A and P greatly obscures and complicates many investigations. Not being able to work practically in terms of A and P in a general manner, and yet knowing there was nothing absolutely wrong, I went to the root of the evil, and cured it, thus:— As a companion to equation (1) use this, —curl] E;=47G5 2. eee where G is the magnetic current, or B/47. That this may be derived at once from (2) is obvious. But what is of greater importance in view of the difficult establishment of (2), is that (3) can be got immediately independently, and that (2) is its consequence. Equation (3) is in fact the mathematical ex- pression of the Faraday law of induction, that the electromotive force of induction in any closed circuit is to be measured by the rate of decrease of the induction through it. _ Now make (1) and (3) the fundamental equations of motion, and ignore (2) altogether, except for special purposes. There are several great advantages in the use of (3). First, the abolition of the two potentials. Next, we are brought into immediate contact with KE, and H,, which have physical sig- nificance in really defining the state of the medium anywhere (k, w, and ¢ of course to be known), which A and P do not, and cannot, even if given over all space. Thirdly, by reason of the close parallelism between (1) and (3), electric force related to magnetic current, as magnetic force to electric cur- rent, we are enabled to easily perceive many important relations which are not at all obvious when the potentials A and P are used, and (3) ignored. Fourthly, we are enabled with con- siderable ease, if we have obtained solutions relating to variable states in which the lines of E, and H, are related in one way, to at once get the solutions of problems of quite different hysical meaning, in which KE; and Hy, or quantities directly related to them, change places. For example, the variation of magnetic force in a core in a coil, and of electric current in a round wire; and many others. That the advantages attending the use of (3) as a funda- mental equation are not imaginary, I have repeatedly verified. Self-induction of Wires. 125 The establishment of the general equation of activity, however, which I now give (‘ Hlectrician, February 21, 1885), shows that (3) is really the proper and natural fundamental equation to use. But we must first introduce impressed forces, allow- ing energy to be taken in by the electric and magnetic currents. In (1) and (38) E, and 4H, are not the effective electric and magnetic forces concerned in producing the fluxes conduction-current, displacement, and induction, but require impressed forces, say e and h, to be added. Let E=E,+e, &e. ; then we shall have B=pH, C=£E, D=cH/47, . . . (4) as the three linear relations between forces and fluxes ; two equations, F=C+D, G=B/4r, . ... . (5) showing the structure of the currents ; and two equations of cross-connection, eur (Hh) = 4a P si wl ody ait ae (6) eur (Me) =4rGe ds ail a Next, let Q be the dissipativity, U the electric energy, and T the magnetic energy per unit volume, defined thus : Ones) USP EDs) = SEB Aor iene 18) (according to the notation of scalar products used in my paper in the Philosophical Magazine, June 1880 ; ¢, &, and uw are in general the operators appropriate to linear connection between forces and fluxes). Then we get the full equation of activity at once, by multiplying (6) by E and (7) by H and adding the results. Itis . el +hG=EI+HG+ div. VE—e)(H—h)/47, | a 9 =Q+U+T + diy. V(E—e)(H—h)/4zr, J where div. stands for divergence, the negative of Maxwell’s convergence. The left side showing the energy taken in per second per unit volume by reason of impressed forces, and Q+U+T being expended on the spot in heating, and in- creasing the electric and magnetic energies, we see that V(E—e)(H—h)/7 is the vector transfer of energy per unit area per second, or the energy-current density. The appro- priateness of (7) as a companion to (6) is very clearly shown. The scheme expressed by (4), (5), (6), (7) is, however, in ‘one respect too general. The magnetic current is closed, by 126 Mr. O. Heaviside on the (7); but that does not necessitate the closure of the magnetic induction,which is necessary to avoid having unipolar magnets. Hence div. B=0. .j6e +, +. er is required to meet facts, in addition to (4), (5), (6), (7). There is no magnetic conduction-current with dissipation o energy, analogous to the electric conduction-current. As regards the meanings of e and h, in the light of dynamics they define themselves in the equation of activity ; that is, so far as the mere measure of impressed forces is concerned, apart from physical causation. Thus e is the amount of energy taken in by the electromagnetic field per second per unit volume per unit electric current, and h is similarly related to magnetic current. Under e has to be included the recognized voltaic and thermoelectric forces. But besides them, it has to include the impressed electric force due to motion in a magnetic field, or VvB if v is the vector velocity, necessitating a mechanical force VFB. It-has also to in- clude intrinsic electrization, the state which is set up in solid dielectrics under the continued application of electric force. — Thus J=ce/4ar connects the intensity of intrinsic electrization J with the corresponding e. I can only find two kinds of h. First, due to motion in an electric field, viz. 42 VDv, necessitating a mechanical force 4a VDG; and, secondly, much more importantly, intrinsic magnetization I, connected with the corresponding h thus, T= wh / Az. As regards potentials, there are, to match the two electric potentials A and P, two magnetic potentials, say Z and 0; © being the single-valued scalar magnetic potential, and Z the vector potential of the magnetic current, some of whose properties in relation to dielectric and conductive displacement I have worked out in the paper referred to before. As regards the general equations of disturbances, like Max- well’s (7), chapter xx. vol. il., they are far more a hindrance than an assistance in general investigations. But when we come to a special investigation, and need to know the forms of the functions involved, then we may eliminate either E or H between (6) and (7), and use the suitable coordinates. We may make use of the above equations at the start, in passing to the question of the propagation of disturbances along a wire, after which the investigation will be wholly scalar. Put e=0 in (7) ; then we see that we cannot alter Self-induction of Wires. 127 the magnetic force at a point without giving rotation to the electric force. Now as in a steady state the electric force has no rotation (away from the seat of impressed force), it follows that under no circumstances (except by artificial arrangements of impressed force) can we set up the steady state in a con- ductor strictly according to the linear theory. We may approximate to it very closely throughout the greater part of the variable period, but it will be widely departed from in the very early stages. Let there be a straight round wire of radius a, conduc- tivity k,, inductivity ~,, and dielectric capacity ¢, ; surrounded up to radius a, by a dielectric of conductivity £2, inductivity 1, and dielectric capacity ¢,: in its turn surrounded to radius a; by a conductor of k3, 43, and cz. This might be carried on to any extent; but we stop at r=a3, r being distance from the axis of the wire, as the outer conductor is to be the return to the inner wire. Let the magnetic lines be such as would be produced by longitudinal impressed electric force, viz. circles in planes perpendicular to the axis of the wire and centered thereon. Let H be the intensity of magnetic force at distance r from the axis, and distance z along it from a fixed point. Use (6), with h=0, to find the electric current. It has two compo- nents, say I‘ longitudinal or parallel to z, and y radial, or parallel to 7, given by 1 a dH 40D = ~ rH, = ae re sitet a LA) We have also E=pfF, if p is a generalized resistivity, or oe gt a p OEE Se green tA Sut be laetipags (12) Now use equation (7), with e=0. The curl of the longitu- dinal and of the radial electric force are both circular, like H giving y] = Pees: Gay pli=p (7-2). ht) oe tony In this use (11), and we get the H equation, which is did ?H dre det ae = The suffixes 1,5, and 3 to be used, according as the wire, dielec- tric or sheath, is in question. In a normal state of free subsidence, d/dé = p a constant. Let also d?/dz* = —im?”, where m’ isa constant, depending upon dopkH+poH.. . . (14) 128. Mr. O. Heaviside on the ' the terminal conditions. Also, let —s'=4rpkpt+pep?+m. . . . se Then (14) becomes dl us dr r which is the equation " the J,(sr) and its complementary function, which call pes Thus, for reference, Grats H= 0; » a Jo(sr)=1— Ae a | Ten)=— 76, pares Big K,(sr) =Jo(sr) . log os ae EM yay ce ra —jor{1-agp +. }. J * We have therefore the following sets of solutions, in the wire, dielectric, and sheath respectively, the A’s and B’s being constants :— Hy = AqJ (sy) cos (mz + 0)", 7 Amy, = A,J ,(syr)m sin (mz + @)e?*, | AT, = AyJo(s1")s, cos (mz+ 0)eP", | Hy = {AgJ (sr) + ByKy(syr)} cos (me +0)«?*, | Amys = {AgJ, (sor) + BoK,(syr) }msin (met Oe", Am, = § AgJ o( ser) + By Ko(sgr)} 82 cos (mz+ 9)e?*, | H;= {AsJ1(s3r) + B3Ky(s37”)! cos (mz + )e? , Amys; = {A3J1(s3r) + B3K,(sgr)}m sin (me + @)e?*, 4713 = {AsJ (ss) + Bs Ko(s37)}s3 cos (mz+0)er?. | To harmonize these, we have the boundary conditions of continuity of tangential electric and magnetic forces, and of normal electric and magnetic currents (or of magnetic induc- tion). Thus y=, and pyly=p.Ts, at r=a,, give us A,/Ai(J1Ko Tip JoKy )(s2a) = J (81a) Ko(sgay) — (P11/P282) Jo(s,41) Ky (soa), . (19) B,/Ay(S1 Ko —J o Ky) (8201) = (0181/P282)I o( $141) J 1(s201) —J (814, )J o( 82a )- Self-induction of Wires. 129 As there is to be no current beyond the sheath, y3;=0, or H;=0, at r=a3. This gives J B= —A; K, (s3a13). ° ° e «ats (20) This, and the conditions y,=y2, and p3l'3=p.l, at r=ag, give us (ATs + BuK) (2902) =As | Jal sots)— (sets) Kas) b (21) (p282/0353)(AgJ ot B,Ko) (spd2) = Ag { J o( S32) — Fees) Kast) \ 3 whence, eliminating A; by division, and putting for A, and B, their values in terms of A, through (19), we obtain the determinantal equation of the p’s for a particular value of m7”. It is 2383 J o(83@o) Ky (sss) —Jx(s3c@3) Ko(sgae) P282 J1( S342) Ky (s3a3) —J (8343) Ky (83a) J 1( 81) Ko( soa) er (p15;/ P2S2)J o( $141) Ky (soar) = (ors) Sea) Ixy) —Fi(ovts Toleyay) CH) + Pate) bates! tiieii, « : Sou te STF PP ich acter Let cere J1(S9aq) -- Ky (soa) J where the dots indicate repetition of the fraction immediately over them. Before proceeding to practical simplifications, we may in outline continue the process of finding the complete solution to correspond to any given initial state. The m’s must be found from the terminal conditions. Suppose, for example, that the wire, of length J, forms a closed circuit, and that the sheath and the dielectric are similarly closed on themselves. Then clearly we shall have Fourier periodic series, with m=0, 27/l, Ar/l, G67/l, &e. If, again, we desire to make the sheath the return to the wire, without external resistance, join them at the end z=0 by a conducting-plate of no resistance, placed perpendicular to the axis ; and do the same at the other end, where z=1. This will make y=0 at e=0 and at z=1; will make the 6s vanish, and m=0, m/l, 2/l, 3cr/l, &e. Each of these m’s has its infinite series of p’s, by the equa- tion (22). Now, as regards the initial state, the electric field and the magnetic field must be both given. For, although the quan- tity H, fully expressed, alone settles the complete state of the Phil. Mag. 8. 5. Vol. 22. No. 135. August 1886. K 130. Mr. O. Heaviside on the system after the first moment, yet at the first moment (when the previously acting impressed forces finally cease) the elec- tric field and the magnetic field are independent. The energy which is dissipated according to Joule’s law has two sources, the electric and the magnetic energies. Now we may, by longitudinal impressed force, set up a certain distribution of magnetic energy, but no electric energy. Or, having set up a certain magnetic and a certain electric field by a particular distribution of impressed force, we may alter it in various ways, so as to keep the magnetic field the same whilst we vary the electric field. So both fields require to be known, or. equivalent information given. We may then decompose them into the proper normal systems by means of the universal conjugate property derived from the equation of activity, that of the equality of the mutual electric energy of two complete normal systems to their mutual magnetic energy (‘ Electrician,’ November 27, 1885). Thus, if Uy, and T,, are the doubles of the complete electric and magnetic energies of any normal system, and Uo, is the mutual electric energy of the initial electric field and the normal electric field in question, and Ty, is the mutual magnetic energy of the initial magnetic field and the normal magnetic EUG relebee eme na 2 Gig 1 as the expression for the value of the coefficient A,, which settles the actual size of the normal system in question. Equal roots require further investigation. This would complete the theoretical treatment. It is best to use the electric and mag- netic forces .as initial data in the general case. As regards potentials, we cannot express the electric energy in terms of merely the electric potential and the electrification, but require to use also the vector potential Z and the magnetic current. Now there are several important practical simplifications. Suppose, first, that the thickness of the sheath is only a small fraction of its distance from the axis. Then it may be treated as if it were infinitely thin, making the sheath a linear conductor; of course its resistance may remain the same as if of finite thickness. Let a, be the very small thickness of the sheath, then the big fraction on the left side of (22) will become . (Jot 834431) Ky — (Ko + 83044) J, (Jy + 83439) Ky — (Ky + 53a, Ko) J, 1 Ji Ko—JdoKy 1 (sss) ——)—_— | 8303 )= — —— 35 S3%4 Tikal ieee ) Saha” Self-induction of Wires. 131 wherein J, and K, are derived from J; and K, as the latter are derived from Jp and Ky. So the left side of (21) will become Pat ah ENCE Se Go P2592 83% =. P2%So The inductivity of the sheath is now of noimportance. Being on the outer edge of the magnetic field, the thinness of the sheath makes its contribution to the magnetic energy be diminished indefinitely. Again, in important practical cases, the resistance of the return is next to nothing in comparison with that of the wire. Then put p3=0 in (21). This makes the left side vanish, and then we sweep away the denominator on the right side, and get the determinantal or differential equation _ J1( 8141) Ko( 821) — (P181/p282) 3 o( 8141) Ky (5904) ; (0,81/P252)Jo(s141)I1( S201) —F1(s1%1) So S241) fd are hee! Although we may have the return of nearly no resistance and yet of low conductivity (as in the case of the Earth), yet it cannot be quite zero without infinite conductivity, which is what is here assumed. The result is that we shut out the return conductor from participation, except superficially, in the phenomena. (24) will result from the condition p.I’,=0, or [,=0, at r=a,; that is, no tangential current, or electric force, in the dielectric close to the sheath. If there could be any, it would involve infinite current-density in the sheath. As it is, there is none, and the return-current has become a mere abstraction, to be measured by the tangential magnetic force divided by 47, and turned round through a right angle on the inner boundary of the sheath. In a similar manner, if we make the wire infinitely conducting (or of infinitely great inductivity either) the wire will be shut out. Then the mag- netic and electric fields are confined to the dielectric only, and we shall have purely wave-propagation, unless it be a conductor as well.. Now, with the return of no resistance, let the dielectric be nonconducting and the wire non-dielectric, or c,=0, k,=0. The most important simplification arises from the smallness of Sod. +For we have —st=p.cp? +m. If the length / of the line is a large multiple of the greatest transverse length a, we are concerned with, m” is made a small quantity—very small when the line is miles in length, except in case of the insignificant terms involving large mul- tiples of #7 in m=n7/l. eam (mc)? is the speed of light 2 132 Mr. O. Heaviside on the through the dielectric, so that unless p be extravagantly large icp” is exceedingly small also. Thus, with moderate distance of return-current, sa) is in general exceedingly small. Therefore in the expressions (17) take first terms only, making Jo (sor) =1, Ji (sor) = 3 S97 ; } (26) Ko (sr) = log ser; Ky(sgr) = — (ser) 7" These, used in (25), bring it down to cps loge ° Jy (sya) = eae Jo(5)41) 5 te ete (27) concerning which, so far as numerical accuracy is concerned, the only assumption made is that the return has no resistance. We have now the following complete normal system :— H,=AJ,(s17) cos (mz+O)er*, Amey, = AJi(syr)m sin (mz + 0)e?, ArT’; = AJ o(s17)s, cos (mz +0 )e?, H,= B(sir)—1 cos (mz + 0)e?*, Amy, = B(s3r)—1m sin (mz + 0)e?*, 471,=B log (ag/”) cos (mz + 0)e?*, i where B= A((¢15;/p2)J 0(814,) + log (a,/a,). The longitudinal current and electric force in the dielectric vary as the logarithm of the ratio a,/7, vanishing at r=dp. The radial components vary inversely as the distance. Nume- rically considered, the longitudinal electric force is negligible against the radial, which is important as causing the elec- trostatic retardation on long lines. But, theoretically, the longitudinal component of the electric force is very important when we look to the physical actions that take place, as it determines the passage of energy from the dielectric, its seat of transmission along the wire, into the conductor, where it is dissipated. ) Regarding (28), however, it is to be remarked that, on account of the approximations, the dielectric solutions do not satisfy the fundamental equation (6). Applying it, we get Y'=0. But the other fundamental (7) is satisfied. To satisfy (6), take (28) Ky (sor) = — (ser) —1 + 3 syn (log sor —1); leading to the determinantal equation 1 log”. Tr(sra,) = Sols) | Sp +4a;(log? +1) i Self-induction of Wires. 133 and requiring us to substitute — (337) +37 log (a/r) +47 for (s?r)-1 in the H, and y2 formule in (28). Then (6) is nearly satisfied, and is quite satisfied if we change the last term in the last expression to $7. But the other fundamental is violated. Now in (27) take m=0, making —s2=y,cp’, and bringing (27) down to 4.8,a49 9(81) = RPI (se): Stare ts (29) where Lip=2p1z log (42/01), the coefficient of self-induction of the surface-current, and Ro= (what); the resistance of the wire, both per unit length of wire ; so that L/Ro is the time-constant of the linear theory, on the supposition that the resistance of the wire fully operates, although the current is confined to the surface. This case of m=O is appropriate when the line is so short that the electro- static induction is really negligible in its effects on the wire- current. In fact we shall arrive at (29) from purely electro- magnetic considerations, with c=0 everywhere. But it is also the proper equation in the m=O case when the electro- static retardation is not negligible. It must be taken into account, for instance, in the subsidence of an initially steady current, independently of the electrostatic charge. Equation (22) in powers of p, by means of +s?a?= —pp/Ro We get Be) ee ee ) ee = $= Bet E+...) (0) Taking first powers only, we get —p~*= (py + Lo)/Ro which is greater than the linear theory time-constant of the wire by the amount $y,/Ro, since 4, is the coefficient of self- induction per unit length of wire when the return-current is upon its surface. But taking second powers as well, we get, if L=$y,+ Lo, —p'=L/Rp and 344/Ro, of which the first is exactly the linear-theory value. The real time-constant of the first normal system of current, therefore, exceeds the linear-theory value by an amount which is less 134 Mr. O. Heaviside on the than $4;/Ro, when the return is so distant, or the retardation (44,07) of the wire is so small that a steady current subsides with very nearly uniform current-density, being very slightly less at the boundary than at the axis. It is not, however, to be inferred that the subsidence of the ‘‘ current in the wire’ is delayed. It is accelerated, at least at first. (29) may be written (4/Lio) J o(sia1) = —. L $44) 1(811), ore of eer (31) the appropriate form when a full investigation is desired. Draw the curves y; = right member, and y,= left member, the abscissa being s,a,. Their intersections will give the values of s,a, satisfying (31). The first root has been already considered, when p,;/Ly is very small. The rest, under the same circumstances, will be nearly those of J Hee = 0. But if the wire is of iron p;/Ly may be very large, and there will be no approach to the linear theory. Many normal systems must be taken into account to get numerical solutions. Simi- larly if the sheath be close to the wire, whether it be magnetic or not. Hlectrostatic charge being ignored, join the wire and sheath to make a closed circuit, in which insert a steady impressed force ¢ at time t=0. Let I be the current at distance r from the axis at time ¢. (There isno y now). The rise of I to the final steady value, say a is given by J (sy) €?! bia rip 1 au -3 “oo . (82) where g= Loa,/2m4. The values of s,a, are to be got by (81). The total current C, or the current in the wire, in ordinary language, rises thus to its final value Co :— ) ept 0=0,4 1- -3(=) fee Pay The boundary condition of I’ is that, at r=ay, av J, Dota y at =0, “7 (sim) =a9. ~ oe eee Considering the first term only in the summation in (33), as may be done except in the first stage of Me rise, when the linear theory is nearly followed, put —p-!'=(L+L,)/R,, where L, must be very small compared with a then a alte Lai )7e8? O=04 I aetadas When the current is started, by a steady impressed force in Self-induction of Wires. 135 the coil circuit, in a long solenoidal coil of small thickness, containing a solid conducting-core, the magnetic force in the core rises in the saiae manner as the current in the wire, according to (32); as the boundary condition of the magnetic force is of the same form as (34), g being then a function of the number of windings, &c. There is also the water-pipe analogy, which is always turn- ing up. This I made use of in the ‘ Hlectrician,’ July 12,1884. Water in a round pipe is started from rest and set into a state of steady motion by the sudden and continued application of a steady longitudinal dragging or shearing-force applied to its boundary, according to the equation (32). This analogue is useful because every one is familiar with the setting of water in motion by friction on its boundary, transmitted inward by viscosity. Graphically representing (32), abscisse the time, and ordi- nates I’, at the centre, intermediate points, and the boundary, by what we may call the arrival-curves of the current, and comparing them with . T=1)(1—e7 Bo), the linear-theory arrival-curve at all parts of the wire, we may notice these characteristics. The current rises much more rapidly at the boundary than according to the linear theory, at first, but much more slowly in the later stages. Going inward from the boundary we find that an inflection is pro- duced in the arrival-curve near its commencement ; the rapid rise being delayed for an appreciable interval of time. This dead period is very marked of course at the axis of the wire, there being practically no current at all there until a certain time has elapsed. That the central part of the wire is nearly inoperative when rapid reversals are sent is easily understood from this, or perhaps more easily by the use of the water-pipe analogue. Some curves of (32), for two special values of g, I gave in the ‘ Electrician,’ September 6, 1884. Let there be a simple harmonic impressed force esin nt in the circuit of wire and sheath, with no external resistance, making a total circuit-resistance R. (I translate the core solution into the wire solution.) The boundary condition is esin nt dt Rae ='t+97, ok. EAS aad peers ehe) ) and the solution is t= Raa (P2+ Q2)-?{(PyM + Qi) sin nt +(PoN—Q oM) cos nt}; . (36) 136 | Mr. O. Heaviside on the where M and N are the following functions, . =1J((ar Vi) +3I5o(ar v—i), \, ae =4iJ (ar Vi )—41) (ar / —1); i standing for /—1, and 2 for V4rp,k,n. Also P=M+i9@W’, ‘Q=N+ aN’, '°: > 2 eas the ‘ denoting differentiation to 7. In (36) M and N have the values at distance 7, and Po, Q, the values at r=a,, the boundary. We have P? + Q?= M?+ N?+ 2¢(MM’'+ NN’) +¢?(M?+N”). (89) If y=(ar)*= (4a, kyr’?n)*, we have the following series :— M?+N?— J (ittea(ittz" ‘rete OMS a carillsae a: g(1+475 rAlCurcreer (8. M?+N?= a (1+ Be 102 (14 + pra (1+ + ete ig (dte MM’+NN'= 37 (+e GIR? aie (1+ on (1+ pape 120 (ite These are suitable for calculating the amplitude of I’ or of C when y is not a very large quantity. The wire current C is given by M?+N” 2 t N’+ PM’ C= ae preg) sin(nt— tan~! Byron) (41) where P, Q, M, N, M’, N’ have the boundary values. As for M and N themselves, their expansions are ce PD Sy) illest M=1— gogo + gagzgige +> (42) sali ie sie F PEG RIO But these series are quite unsuitable when y is very large. Then use the approximate formule Jo(sr)= (= -)" cos (sr— i) Ji(sr) = ey cos (s ST — ), (48) (4 Self-induction of Wires. 137 which make, if f=y? M24 N=J)(fV/ iJ o( P/ —2) = ef"2/2arf, M?2+N?= pev 2 Qarr?, . (44) MM’ + NN =cf¥2/2arr/2. In the extreme, very high speed, or large retardation, or both combined, making y very great, the amplitude of the wire current C tends to be represented by e/Lioln ; . (45) showing that the current is stronger than according to the linear theory, and far stronger in the case of an iron wire, or very close return. The amplitude of the current-density at the axis, under the same circumstances, e 2 Qaray 1° imag” ay ; n a, (9) 0 The forces which act on the pole are given by the fol- lowing equations :— 1dQ__ 2mRy ("_reo™ —=, eo dp oe Y, BPFH n%a? Jn(e)ar, dO. eine oe prs ae gps 2 2 2 oO ee Ae te RNG dz rare \ BPA? + 2c? Ae) aA, > (10) dQ, : n Ne 2An = dp’ —2ma"Sn? | Pe awe” (Ac)In(Ac)da. J Now we have supposed the magnetic pole to be rotating in the same direction as that in which d is measured, therefore the first equation indicates a force tending to oppose the motion of the magnet; hence the magnet exerts a force on the sheet tending to pull it round in the direction of rotation. If therefore the magnet is fixed and the sheet be made to rotate, the magnet will experience a dragging force parallel to the direction of motion of the sheet. 144 On the Induction of Electric Currents. The second equation represents a repulsive force acting from the disk ; and the third equation represents a force acting towards the axis of the disk. These results are equivalent to those obtained by Maxwell and observed by Arago. (see art. 669). If the magnet were not sufficiently long to enable us to leave the action of the negative pole out of account, the mag- netic potential of the currents excited by the latter will be found by changing m into —m, h into h+a, where a is the length of the magnet; and the combined effect of the two poles will be obtained by adding the results. It is, however, simpler to place a short magnet, of length 2c, at right angles to the axis of revolution, and consider the combined effect due to both poles. The magnetic potential of the currents due to the action of the negative pole will be obtained in this case by changing m into —m, and ¢ into a+; whence the expressions for the magnetic forces acting on the positive pole will be found from equations (10) by changing m into 2m, and x into 2n+1, and effecting the summation with respect to n from to 0. The calculation can then be effected as follows :—Hxpand the Bessel’s functions in powers of Ac; the expressions for the forces will then assume the form of series every term of which i is of the form ie Are Mid, Rv? + Qn +1)?o” where p and v are positive integers. Now , ae AX 1 (, RA? + (Qn+1)2o? ae ee AF me 1) + singQig } where g=2(2n+1)hw/R. Hence, by differentiating with respect to h, the first-mentioned integral can be expressed in terms of the sine and cosine integrals, the values of which have been calculated by Mr. Glaisher, whence numerical- values of the magnetic forces might be found. If we neglect powers of c higher than the fifth, the mag- netic forces in the @, z, and p directions will be found to be respectively equal to 0 3 .—OAh 273 5r4¢ 5 o NU —2dh me Ne ci any) at eel e dNE RA? + 4 192 64 RA? + 9?’ Xe —2\h _ Met dr: mo" {” BA? + ~T)a Ft creas a Pe? a ee) aia 3mw2e? 0° A8e-?*dn, Cy eo | 2a ee 64 J, Ra? + 9a%" [ 145 J XVII. On an Electric-light Fire-damp Indicator. By Water Emmort and WILLIAM ACKROYD. * 5 Mage Royal Commission on Accidents in Mines point out, in their recently issued Report, a serious objection to the use of the electric light in mines, notwithstanding its many other great advantages, in that the light of an incan- descent lamp being produced within a vacuum cannot admit of any device for the indication of fire-damp such as is employed in the Davy for example. This difficulty was Bios 1; experienced by one of us in the course of an installation of the electric light in the Lofthouse pit, Wyke, Yorks, in the summer of 1885; and we have since made a series of experi- ments with the object of devising a method of making the electric light an indicator of fire-damp. The apparatus placed before the Physical Society is the outcome of our work. It consists of two incandescent lamps, one with white glass and the other with red, and other necessary adjuncts, such that in an ordinary atmosphere the white incandescent lamp alone shines, but in fire-damp the white lamp goes out and the red one begins to emit its light. This is effected as follows :—A porous pot of unglazed hard- baked porcelain is joined by air-tight connections to a tube a portion of which is represented by TT!, fig. 2. This tube is of such an internal diameter that it will readily admit of being sealed with a small quantity of mercury, Hg. A platinum wire runs the whole length of the tube and is * Communicated by the Physical Society: read June 12, 1886. Phil. Mag. 8. 5. Vol. 22. No. 185. August 1886. L 146 On an Electric-light Fire-damp Indicator. connected with one of the poles of the battery B or other source of electricity. ‘Two other platinum wires in the tube run parallel with this for part of the way, as in fig. 2, and each is connected with a lamp. The lamps W and R are joined, and a branch wire connects them to the other pole of the battery. In fig. 2 the current is represented as flowing through W ; when from diffusion in an atmosphere of fire- damp, the conducting plug Hg is driven up to T 1, the current will flow througn R, and the red light may then be taken to indicate the presence of fire-damp. Fig. 2. p. Porous pot. a and 6, desiccating tubes, The wires being within the tube, one or other of the lamps must always be shining so long as there is a current, whether the apparatus be in an atmosphere of fire-damp, choke-damp, or air ; and to prevent the mercury being driven out of the tube by too much pressure, bulbs are arranged on either side as in fig. 3, which presents a diagrammatic view of the apparatus. We have found an internal diameter of tubing of about 3 millim, best adapted for ensuring easy mobility of the mercury, The presence of the wires within the tube has interfered with the perfection of the seal ; this we have over- come bythe introduction of a little concentrated sulphuric acid, which also serves the purpose of preventing sparking and of lubricating the interior, The use of sulphuric acid necessi- tates the addition of desiccators a and 8, fig. 3, to each end of the tube ; but in cases where it has been found advisable not to use sulphuric acid, both the acid and the desiccators have been dispensed with by slightly modifying the arrange- ment of the wires at the lower part of the tube. With this form of apparatus we are readily able to detect the presence of 5 per cent. of coal-gas in a mixture of this gas with air; and witha mercury seal of less weight and closer proximity of the wires at T and T° (fig. 2), it appears possible to get any required degree of sensitiveness. It is proposed to have the apparatus fixed in the main roads and hauling roads in pit installations. Caer) XVIII. On certain Modifications of a Form of Spherical Integrator. By FREDERICK JouN SmitH, B.A. Oxon.* Ay he working at the subject of Dynamometric Measure- ments, a great number of different forms of mechanical integrators were attached to a transmission-dynamometer (Phil. Mag. vol. xv. p. 87). In one of these the small disk of a Morin integrator (Phil. Mag. vol. xvi. p. 59) was replaced by a sphere carried between four little cylinders ; the sphere was made of phosphorbronze (Ashmol. Soc. 1884). As long as the dynamometer was driven with but little variation, the results were satisfactory; but as soon as one had to deal with quick variation, either of tension of belt or of velocity, it was found that the moment of inertia of the sphere was a serious obstacle to accurate results. It appeared probable that no instrument dependent upon the action of gravity could be relied upon. This being the case, it was evident that if a sphere was to be used it ought, in the first place, to be as light Fig. 1. as practicable and, secondly, as firmly held as possible from any slip. These two ends have been arrived at as follows :—The sphere Q is made hollow in order to be light, and it moves. between six wheels ; of these, four wheels, D EF G (fig. 1), are * aa aanene by the Author, 2 148 Modifications of a Form of Spherical Integrator. fixed upon a frame, while two others K L, held in a frame SMN'V (fig. 2) rocking about points at S V, keep the sphere Fig. 2. in compression ; the frame is so moved, by means of a slot R attached to the spring rod of the dynamometer, that the line PM, and therefore the circle on the sphere rolling upon the eylinder D, is proportional to the tension of the driving-belt ; the cylinder K is driven at a rate proportional to the velocity of the belt-speed, and the cylinder D is attached to a revo- lution-counter. When the tension of the belt is constant, then the revolution of D is proportional to the velocity. When the velocity is constant, the revoluticn is proportional to the tension of the belt. When both velocity and tension vary together, then the revolution of D is proportional to the pro- duct of velocity and tension which is the work at any instant, and the number of revolutions of D during any time is pro- portional to the work done during that time, and shows, therefore, how much energy has been transmitted by the dynamometer. In the figures, for the sake of clearness, the frame which carries the cylinders is omitted. The instrument which the author of this communication has arrived at, is quite unaffected by any rapid change either of velocity or tension of belt that may take place while it isrunning. The beautiful three-wheel spherical integrator of Prof. Hele Shaw, which was applied to the dynamometer, as long as it had to deal with slowly varying velocity, acted very well indeed ; the idea of changing the direction of the angle at which the poles of the sphere lie in is taken from it. Algiers, March 1886, pe k4g. XIX. On hitherto unrecognized Wave-lengths. By Prof. 8. P. Laneury, Allegheny, Pa.* [Plates IV.—-VI. ] Bj have already} presented a description of the method by which we are able to fix the wave-lengths in the solar spectrum: by direct measurement as far as about 23,000 of Angstrém’s scale. At this point the heat in the solar normal spectrum had become so feeble that it taxed the utmost limits of our capacity in 1881 to measure it; for it will be remem- bered that we are able to study the prismatic spectrum of the infra-red with comparative ease, because the prism condenses the heat; but the grating greatly diffuses and weakens it, so that, were it due to this cause alone, we should find measure- menis in this part of the grating spectrum enormously more difficult than those in the prismatic. But, independently of this, of the heat which belongs to any ray, our grating in general employs not over the tenth part. These causes com- bine to make the heat in certain portions, where we have been compelled to measure, almost infinitesimal. Weare led to take this labour; not primarily to settle the theoretical questions involved in determining the relations between dispersion and wave-length (though these are most interesting), but with the object of providing a way which will hereafter enable any observer to determine the visible or invisible wave-lengths of any heat, whether from a celestial or terrestrial source, observed in any prism; and thus to gain that knowledge of the intimate constitution of radiant bodies which an acquaintance with the vibratory period of their molecules can usually alone afford us. It is this considerable end—the attempt to open up more fruitful means of research in the whole unexplored region of infra-red energy, not only from celestial but from terrestrial sourees—which will, we hope, justify the labour devoted to the following determina- tions. it may be hoped that wider interest will attach to our task of demonstrating the character of a certain curve ; when it is seen that a knowledge of its true form has ceased to be a matter of abstract speculation only, but will, in connection with what has already appeared, now introduce to us such large regions of research as we have just indicated. Over and above this, however, we shall find our results also affecting opinion on the theoretical considerations regarding the relation of wave-length and dispersion just alluded to. * Communicated by the Author. + This Journal, March 1884, p. 194. "150.2 Prof. 8. P. Langley on In previous communications I have given a representation of the solar-heat spectrum terminating near 2"7 or 2""8, and I have stated that while there were feeble indications of solar heat below this point, yet that the solar radiation beyond seemed sensibly cut off, as though below this were a nearly unlimited cold band. I do not mean then, in saying that solar heat sensibly ceases below this point, to say that abso- lutely none can exist, but that none, at any rate, does exist sensible to the delicate apparatus with which these first deter- minations were made, and that none in any case exists of an order in the least comparable with the smaller portions of that already described. The reader will gather a more clear conception of the diffi- culty of decision and of the almost infinitesimal amount of this solar heat, if itexist, by looking at Plate IV. fig. 1,in connection with the statement that if there be any solar heat at 4 the highest ordinate representing it, on the same scale as that shown on the left of the Plate, would at any rate not occupy the thickness of the horizontal line which represents the axis of abscisse. However, since we are rather inclined to admit, from our final experiments with our latest and most sensitive apparatus, that heat of some kind reappears here (near 4"), whether from the atmosphere of the sun, or elsewhere, insen- sible to the most delicate thermopile, and in any case, if it be real, almost infinitesimal in degree, or of the same order of intensity with that in the lunar spectrum, our statement that no sensible solar heat exists in this part of the spectrum must be taken under this qualification. New APPARATUS. The apparatus for the determination of wave-lengths in connection with the flint-glass prism has been already described*. The following is the same in principle, with certain changes to adapt it either to the solar or electric heat. Let 8, be the first slit (see Plate V). For solar heat it has doubly-moving jaws, controlled by a micro-. meter-screw, while in the case of the electric are we use a special form of slit surrounded by water, to be directly de- scribed. G is the large concave grating. The massive beam A carries the large rock-salt train 8,, L,, P, L,, B. G is fixed at the extremity of the beam, so that its collimating axis coincides with that of L,; and by means of an automatic ap- paratus, not shown here, the slits 8; 8, are caused to lie always in the same straight line at right angles to GS,. Under * This Journal, March 1884, p, 194. unrecognized Wave-lengths. 151 these circumstances, it has been demonstrated by Prof. Row- land that the wave-length of light, passing through the slit 8, to fall upon the grating and there be diffracted to 8,, is directly proportional to the distance 8, 8. Accordingly, owing to this extremely simple relation, we are able to state at once what invisible ray or rays are at any moment passing through the slit into our rock-salt train. Our engraving represents the arrangement as fitted up for the heat of the electric arc, which is placed immediately in front of the special nozzle n carrying the slit 8; We wish to employ the arc chiefly in the extreme infra-red beyond the solar heat, and where any heat is exclusively minute. The hottest part of the electric arc is found in the pit or crater of the positive carbon, concealed from direct vision, and occupying a space of only 3 or 4 millim. square, even in large arcs. The car- bons, then, must be inclined in order that a horizontal beam may escape from this almost hidden crater, which, owing to its small size, should be brought nearly in contact with the slit, in order to utilize the whole of its very minute area, while in this case the inclination of the carbons will prevent such approach. Hxperiments with various forms of incandescent strips and carbons, directed by clockwork in the ordinary position, have proved the necessity of adopting the special device by which we have finally overcome these difficulties. Figure 3, Plate V., shows in section and in front view a special slit, conical in form, around which a current of water is forced to circulate. Figure 2 shows the carbons on a smaller scale and the apparatus which permits them to be set at any height, inclined at any angle to the vertical, drawn back or approached to any distance. They are usually placed almost in contact with the special slit 8,;; and the need of the water circulation is obvious, were it only to prevent the sides of the jaws of the slit from melting, as they would otherwise soon do. ‘There is, however, another necessity for this water- circulation. ‘The need of a slit which may be artificially cooled for measurements in the extreme infra-red of the spec- trum from the electric arc, is rendered evident when we state, that for these extreme wave-lengths the are radiation is com- paratively so small, that the heat from the hottest part of the dazzling bright carbon does not very greatly exceed that from a piece of melting ice. If, then, we are to distinguish here between the radiation which passes through the open slit from the incandescent carbon, and that which comes from the adjacent edges of the slit, which inevitably mingles more or less with the former, the difference between the two tempera- tures must be made as great as possible. © 152 Prof. 8. P. Langley on Following now on Plate V. the course of the ray from this electric arc, we observe that it falls on the grating G (to be presently described), which spreads it out into not one but many superposed spectra, distributed along the circumference of a circle whose centre is at O equidistant from G, S;, and S.. For clearer illustration; let us suppose ourselves about to measure the heat in some ray of the visible spectrum such as that near D, (whose wave-length is nearly 06), and that the line 8, S, is a scale of equal parts. In this case, the beam A will be moved so that, while the grating remains at the inter- section of NS and GA and normal to the latter, the slit 8, will be brought close to 8, in the position 0#°6 on our scale of equal parts, whose zero is at 8). Here (if we suppose sun- light to be employed) we shall see a brilliant spectrum filled with Fraunhofer lines crossing the front of the plate of the slit 8. Beyond this, the second, third, and other higher orders of spectra are distributed on the circumference of the circle in which 8, always lies. Were it our only object to discriminate the heat in this particular visible ray, we should not in this case need the slit 8, or the prismatic train, but could place the bolometer directly at 8. Since we make the ordinary use of the slit 8, however, we suppose ourselves to be determining the prismatic dispersion for a given wave- length; that is, it forms with the prismatic train an approxi- mately homogeneous spectral image at B, which can be viewed or measured with the bolometer, giving the value of n for a known value of X. For the mere purpose of measuring the heat in the ray, or determining its wave-length here in the visible part, where there is but a single sensible heat-spectrum, we do not need slit S, at all; while the refractive index for a glass prism could be as easily determined as that for rock-salt. Besides this, we should find here a relatively abundant heat, and could use so narrow a bolometer as to fix the position by the heat alone quite accurately. Very different, however, are al] the conditions if we wish to measure, for example, a wave- length corresponding to 3" or 4“ in the invisible spectrum and in the new region which we are for the first time exploring. Glass is impermeable to this kind of heat, but with our rock- salt train and with the delicate apparatus previously described, there is little difficulty in discriminating it by the bolometer, where the prism alone is employed, and in mapping the deviation of each spectral ray, as shown in plate vi. (Phil. Mag. May 1886). But now that we wish to determine wave-lengths, the conditions are altogether different; for now not only does the grating enormously expand this part of the spectrum and diminish the heat correspondingly, just where that heat is itself unrecognized Wave-lengths. : 153 feeblest (so that the heat in parts of this region is something like z5o part of that in the corresponding prismatic spectrum), but instead of one visible, we have now to deal with numerous invisible spectra, overlapping each other. Here, then, the slit S, has an additional function to fulfil, namely, with the aid of the prism, to discriminate these invisible spectra from each other. Thus in the position actually shown in the drawing, which corresponds to a wave- length of 35341, ze. of 35,341 of Angstrom, we should see the slit covered by a bright spectrum due to several of the higher orders, while we know that energy of the wave-length we are seeking is wholly invisible. If we place a pellet of sodium in our electric arc, we shall see the two sodium-lines on the slit-plate, of which D, will fall exactly on the slit, if it . be in adjustment, but this sodium-line evidently does not belong to the wave-length we are seeking. There are, in fact, passing through the same slit and lying superposed on one another by an unavoidable property of the grating, an infinite number of spectra in theory, of which in this case nearly 20 are actually recognizable by photography, by the eye, or by the bolometer; and of which, to consider only those where the wave-length is equal to or greater than that of the sodium-line D,*, we have six spectra as follows :-— a (visible) 6th spectrum. . D,- 2=0°5890 5 th Fly sta tebe 0°7068 ¢ (invisible) 4th es erie aes 0°3835 d = ord Bod Ait 1:1780 é ee Pads) siti pate ee 1-7670 tf . 1st “ taki Ds 3°5341 It is in this invisible underlying 1st spectrum, buried, so to speak, beneath five others, of which three are themselves in- visible also, that lies the wave-length we are seeking ; conse- quently there are (to consider no others) at least six qualities of heat, of six distinct refrangibilities, whose wave-lengths are equal to or greater than that of D,, which pass simultaneously through the slit $.. They pass through the prism,and on looking through a telescope occupying the position of the bolometer- tube, we shall, by suitably directing the arm of the spectroscope, see the light from the sixth one at a. Its wave-length will be 05890, corresponding to a measured deviation (in the case of the rock-salt prism of an angle of 60° 00’ 00” and a tem- * We have heretofore adopted Angstrém’s notation in calling the more- refrangible sodium-line “ D,”. We shall hereafter, however, in conformity with the now more general usage, call this line, whose wave-length in Anestrém is 5889, “D,’”’.. The corrections to Angstrom are due to the researches of Messrs, Peirce and Rowland. 154 Prof. 8. P. Langley on perature of 20°C.) of 41°05'40”. Now, on replacing the telescope by the bolometer, the bolometer-wire will feel this same ray which the eye has just recognized by its light; and, if the galvanometer be in a sensitive condition, the image will be thrown by the heat off the scale, while a little on either side of this position no indication will be given. The beam and the slit S, remaining in the same position, let us next suppose that the bolometer-arm is carried towards 6, in the direction of B. There will be no sensible deflection until it reaches the position b in the red, corresponding to a wave- length of 0*-7068, and in the prism to an angle of 40° 33/ nearly; for there is no sensible heat except in the successive images of slit 8, formed by the prism P in the line PB. Passing further toward B we come into the heat in ¢, and next to the heat in d, which is less than +}5 that in the direct prismatic image, when no grating is employed. This was the utmost limit of our power of measurement in 1883, beyond this point radiations from the grating being then absolutely insensible, and the radiation at the point d itself being excessively minute, even in the solar spectrum, where the heat, so far as any is found, is asa rule far greater than that in the spectrum of the arc. Accordingly I have else- where observed that these measures could be carried on as well by a large electric arc as by the sun; but in fact, owing to the difficulties attendant on bringing the arc, which must be of immense heat, close to slit 8, and to other causes, the sun- light would be preferable wherever it could be used. Our observation of June 7, 1882, gave the value of the index of refraction corresponding to X=2'356, which was the lowest possibly attainable by our then apparatus. Inces- sant practise and study, resulting in improvements already referred to, have enabled us finally to measure down to a wave-length of 9xAD,, corresponding to a position much below 7. We may add that in doing so, it is sometimes con- venient to employ a bolometer wide enough to overlap the images in the other adjacent spectra of the higher orders, which we may usually do without confusing them, owing to their feebleness compared with that of the first spectrum in which we are searching. We usually, however, employ a bolometer of not more than 1 millim. aperture, and this demands excessive delicacy in the heat-measuring apparatus, since the heat here is, approxi- mately speaking, about 5006 of that in the region between the sodium-lines in the direct spectrum of a rock-salt prism. Errata. Page 158, line 4, after amperes add through the coils of 20 ohms resistance. — 163, — 20, for 89° 15'7" read 38° 15"°7, and so to end of column. — 168, — 8, for +2 read +Q. unrecognized Wave-lengths. 155 This is near the limit of our present measuring-powers with the grating, even when every possible device is used to increase the extremely feeble heat in this part of the spectrum. We commenced by using an electric arc with carbons 12 millim. in diameter in the position indicated. These were supplied by an engine of three-horse power ; but even in this case the pit of the crater did not nearly cover the very short slit (its length is 8 millim.). For these last and most difficult measurements we have been obliged to procure the use of an engine of twelve-horse power and carbons 25 millim. in diameter. With this enormous current the hottest part is not easily maintained in place. ‘To keep it directly in front of the slit we have tried various plans, such as boring out the carbons lengthwise so as to form hollow cylinders of them, and filling the core with a very pure carbon tempered to the requisite solidity. Ordinarily it will be sufficient, however, to first form the central crater by a drill. This gives us a persistent crater, whose light, in the position shown in the engraving, filled a slit whose vertical height is 8 millim. It is probably the intensest artificial heat ever subjected to analysis. BOLOMETER. The changes in the bolometer since it was first described (‘ Proceedings American Academy,’ 1881) are superficial rather than radical, and refer chiefly to the form of the case, and facilities for its accurate pointing. ‘The linear bolometer is now made to expose to the radiant heat a vertical tape or wire of platinum, iron, or carbon. This is usually about 10 millim. long and only from z¢455 to ydq millim. thick; but according to its special purpose it is made from 1 millim. to 0-04 millim. wide. In the latter case it appears like the vertical strand of an ordinary reticule in the focus of a posi- tive eyepiece attached to the case, and is movable by a micrometer-screw. It is in fact in appearance a micrometer- thread, controlled in the usual way, but which is connected with the galvanometer and endowed with the power of feeling the radiations, visible or invisible, from any object to which it is directed. or very feeble sources of heat, such as those with which we are here concerned, the strip is made as much as a millimeter in width, and is not provided with a micro- meter-screw, but moves with the arm carrying it, and its positions are read by the divided circle of the spectrometer - to 10” (ten seconds of arc). It is this simple form of the instrument which has been used in the present investigation, and whichjis shown in Plate IV. fig. 2. The bolometer is shown in position in the middle of the case, where its central strip 156 Prof. 8. P. Langley on is accurately self-centred in the cylinder. For. protection from air-currents, since the obscure heat studied will be stopped by a glass cover, we must make use of the special device I have described in the memoir just referred to, of successive chambers or drums separated by diaphragms with a common central aperture. With these precautions, and with the special adjuncts before described, a bolometer with a strip 5 millim. wide can be set by the invisible heat alone to within 10” of are, while in the ordinary use of the linear thermopile we are liable to errors of a considerable fraction of a degree. Hven with a bolometer 1 millim. wide, it will be subsequently seen, we can set to one minute of arc. This refers only to the precision of pointing attainable ; we will consider the sensi- tiveness of the instrument later in connection with the galvanometer. 7 GALVANOMETER. It must be remembered that while the nominal sensitive- ness of a galvanometer can be increased to any extent by increasing the astaticism of the needle (quite as nominal power can be multiplied to any extent on a telescope by altering lenses at the eyepiece), that the real or working capacity depends upon the ability to always obtain a like result under given conditions. Accordingly we have con- tinued to devote great pains to extend our original concep- tion, so as to make the galvanometer, as well as the bolometer, not merely an indicator of heat, but a real ‘ meter,” which shall distinctly answer the question “how much ?”’ as to almost infinitely minute amounts of energy. For the benefit of any physicists who may desire to repeat these experiments, we may observe that we have found the bolometer capable of almost unlimited delicacy of perception of heat, but that our chief trouble has arisen from the diffi- culty of constructing a galvanometer suitable to develop its full capacity for exact measurement. We have been unable to find among galvanometers ordinarily constructed one capable of indicating with precision changes in the amount of current of much less than RK of an ampere. It was in the construction of a galvanometer designed to measure the heat in the spectrum of the moon, that we acquired the ex- perience which we have utilized in the present researches. A reflecting-galvanometer of the form devised by Sir William Thomson has been used for the basis of our construc- tion and altered as follows. (For several of the changes described I am indebted for suggestions due to the great unrecognized Wave-lengths. 157 kindness of Sir William Thomson and Professor Rowland.) First, the short suspending fibre supplied by the makers has been teplaced by one 33 centim. in length, stretched and pre- pared with particular care. Next, since the effect of a given minute change of current is proportionable (other things being equal) to the magnetic moment, and to the minuteness with which the angular deflection of the needle can be read, we have reconstructed the mirror and needles as follows :— For the magnets* soft sheet steel 35 of a millim. thick is rolled up into minute hollow ee, each about 8 millim. long and about 1 millim. diameter. These are hardened and made to take a permanent charge of nearly 900 Gaussian units. Ten of these are placed behind the back of the mirror and ten below, making twenty in all. The reflecting mirror is accurately concave, being specially worked for the pur- pose, 9°5 millim. in diameter, 1 metre radius of curvature, weighing 63 milligrammes, and platinized on the front face by the discharge in vacuo of platinum electrodes, by the process of Prof. Wright}, of Yale College. The stem which unites the upper and lower system of the magnets is a hair- like and hollow tube of glass, while it occurred to me to re- place the aluminum vane of the ordinary instruments by the wings of a dragon-fly (Libellula), in which nature offers a model of lightness and rigidity quite inimitable by art. The glass plate which encloses the front of the galvanometer has optically plane and parallel sides, and the screen, placed at 1 metre distance from the mirror, is a portion of a cylinder 1 metre in radius, divided into 500 divisions of 1 millim. each. The optical arrangements for illuminating and forming an image of the wire form one of such precision that a motion of =, of one of these divisions can be distinctly noted. There is an independent provision, by means of which the image of a second opaque and inverted scale can be viewed by the observer through a telescope, not, asin the ordinary construc- tion, directed onto a flat attached to the needles, but in which the concave mirror, already described, becomes the mirror of a Herschelian telescope itself. Ordinarily the condition of astaticism of the needle is such that, without any damping- magnet, it will execute a single vibration in not less than 15 nor more than 30 seconds. Much greater sensitiveness can be given to it, of course, but without, as we have found, cor- responding advantage. * The design and construction of the hollow magnets is due to Mr. F. W. Very of this Observatory. + Prof. Wright has had the goodness to platinize these delicate mirrors for us himself. 158 Prof. 8. P. Langley on For the purpose of forming an estimate of the sensitiveness of the instrument it may be stated that, when making a single vibration in 20 seconds, a deflection of 1 millim. division of the scale is given by a current of 0:000,000,000,5 amperes ; and, as we have just remarked, a tenth of one of these divi- sions can be discriminated. That this degree of sensitiveness is associated with a real, and not nominal corresponding degree of accuracy, is shown by the fact that many series of accord- ant measurements have been made when the maximum de- flection did not exceed three such divisions ; and that similar measures have been made in the invisible spectrum given by ice melting in a dark room, when the maximum deflection observed was 1°6 millim., and most deflections less than one millimetre. On the other hand the exposure of the same bolometer to ordinary direct sunlight with only y955 of the current passing, 7.e. with the galvanometer shunted 1000 times, would drive the needle immediately, and with violence, off the scale. Our experience has shown us that this galvanometer, in conjunction with such a bolometer as we have described, is capable of recording a disturbance of rather less than STMT part. To attain corresponding accuracy in gravity determi- nations we should need to have a balance capable of weighing a kilogramme, which would give atthe same time an unequivocal deflection for a difference of one one-thousandth of a milli- gramme ia either pan. A deflection of 1 millim. corresponds, in the case of such a bolometer as we have used in the lunar spectrum, or in that of melting ice, to a change of temperature in the bolometer-strips considerably less than 100,506 of a degree Centigrade, and we have just seen that about 75 of this can be shown. In other words, about one one-millionth of a degree can be indicated by it, and a quantity less than one one- hundred-thousandth of a degree, not only indicated, but measured. It will be obvious tothe practised observer that this degree of accuracy will not be in reality reached unless the bolometer-strips are perfectly protected from all extra- neous radiations and air-currents, and especially unless the image is fixed upon the scale when the bolometer is not ex- posed to heat. This degree of precision we believe ourselves to have actually obtained. GRATINGS. Of the concave gratings we have three, of the very largest size. These magnificent instruments we owe, not only to the skill, but to the special kindness of Prof. Rowland, who has unrecognized Wave-lengths. 159 been good enough to execute them for us of the very short focus and open ruling necessary for our particular work. Let us designate them as grating No. 1, No. 2, and No. 3. The dimensions of grating No. 1 have been given in a prior memoir, but we repeat them here with some other data, for the reader’s convenience. The limit of precision imposed by the use of the bolometer makes it superfluous to introduce any temperature-correction, or to give the figures with more exactness than we here do. . Grating | Grating Grating No. 2. No. 1. No. 3. Radius of curvature............0cec..00+ 1626 mm. | 1753 mm.| 1627 mm. We. ot, lines to. millim......<.<.<--+3.5- 142°1 142°1 Hi Height of ruled portion ..............- 102 mm. 80 mm. 75 mm. Width <; -; Ect ee Pact Ope. bak 146 132 133 Distance corresponding to 10,000 of fe) Angstrém’s units on the line of wave-lengths S, S, ......-seeeeee eee 231-0 249°1 1850 The ruled portion of each of these truly superb instruments occupies from 100 to 150 centim. On their exquisite defini- tion we need not enlarge, since sufficient of them are now in the hands of physicists to make our commendation super- fluous. We have already described the action of the grating. The essential feature, for our purpose is that, under the stated conditions, we can in theory be absolutely sure of the wave- length of the invisible ray under examination by choosing it a multiple of the wave-length of some visible line in the superposed spectrum which is coincident with slit 8,. Thus in the case of our illustration, we have supposed the sodium-line to be used, since this is conspicuous in that of the sun and easily reproduced in that of the arc. The wave- length we are in search of is always a times the wave-length of D, (a being some aliquot number). In practice we thus for greater certainty always form the image of some line in the visible spectrum on slit S,, although, as already explained, its mere position on the line 8, 8, is, if the apparatus be in ad- justment, a guarantee that none but the exact ray and its multiples come under examination. LENSES AND PRISMS. The rock-salt lenses L, L, are of different focal lengths for different occasions. For the extremely feeble heat we are considering, we are using very clear and perfectly figured & 160 Prof. 8. P. Langley on salt-lenses of 75 millim. aperture and 350 millim. focus ; this small ratio of aperture to focus in the lenses being required to economise the feeble heat as much as possible. The prism used with them is first set to minimum deviation on some visible line, and then automatically kept there for the invisible ray under consideration. We owe these specimens of rock-salt to the particular kind- ness of Prof. Hastings, of Yale College, and their extremely exact surfaces to Mr. Brashear, of Allegheny, the maker of the surfaces on which the Rowland gratings are ruled. On this portion of the apparatus alone very great labour has been expended. We have procured, through Mr. Brashear’s skill, by means previously described, a rock-salt prism having a field filled with Fraunhofer lines, and showing distinctly the nickel line between the D’s. This is when first polished as it comes from the maker’s hands, but owing to the deliquescent nature of the material, with the utmost care, the surfaces rapidly deteriorate. As it is necessary for the precision of these re- searches to determine the refracting angle of the prism, and also the indices of refraction of some of the principal lines in its visible and invisible spectrum, with a high degree of accuracy, and as all these labours have to be repeated when the prism is repolished, some idea of the labour in this portion alone may be understood, when it is stated that the prism has been sent to the maker and entirely refigured, and its principal constants redetermined by us no less than thirteen times in the past fourteen months, or since the 1st of January 1885 *. We now give two examples of actual measurement of wave- lengths ; the first, that in sunlight in a flint prism, which we have designated as No. 2; the second in the rock-salt prism just mentioned. First example of Measurement. With Flint-glass prism. (Extract from original record.) Station, Allegheny. Date, March 3rd, 1886. Temperature of apparatus=1°:8 Cent. State of sky, clear overhead, cirrus clouds near the horizon. Aperture of slit §;=2 millim. y Bere Lami Lenses of glass (non achromatic), focal length = 800 millim. for visible rays. * For a full description of the constants of this prism see this Journal for May 1886. og unrecognized Wave-lengths. 161 Grating, Rowland No. 1 (concave). Prism, H No. 2 (glass). Refracting angle= 62° 15’ 03”. Spectrum thrown east. Galvanometer, No. 3, damping magnet at 40 centim. Time of single vibration=21 sec. (with current passing). Bolometer, No. 16 (aperture=1 millim.). Reading on slit (before mounting prism), 0° 00’ 00”. * » Dz, (through prism), 52° 52’ 40”. Current of battery =0°036 ampere™*. Reader at circle, F. W. V. . galvanometer, 8. P. L. Object=measurement of deviation of X=4x2XD, in the spectrum of the flint-glass prism H No. 2, with sunlight. Galvanometer-deflection with arms in line (from the com- bined effect of all the spectra falling on slit 2), a little over 3800 div. ° / je) ! ie) i je} i fe) 4 [e) i Prismatic deviation ......... 49 00|48 58/48 56/48 54) 48 52 | 48 50 Galvanometer-deflections. 11 10 ll 18 19 12 La 2 13 12 15 13 15 a Peiiccos G5) isl Ieee) | aged ias Galvanometer-deflections. — ——_ 13 21 26 32 25 20 Second series ............... 14 22 24 on 30 13 eee Mean deflections ............ 13°5 21°5| 25 31°5| 275} 16°5 Concluded maximum at 48° 54’, Making the galvanometer-deflections ordinates, and the prismatic deviations abscisse, a smooth curve through the points of observation gives in the first case a maximum at 48° 54'. The image of the slit has a certain size, and so has the bolometer-strip. The latter feels the heat before the centre of strip and image coincide ; and it is this point of the coincidence of centres which gives the maximum as denoted by the above figures. * It must be understood that this is the ¢otal current of the battery. The differential current which passes through the galvanometer is of an altogether different order of magnitude, in this case probably not ex~ ceeding 0:000,000,01 ampere. Plhul. Mag. 8. 5. Vol. 22. No. 135. August 1886. M 162 Prof. 8. P. Langley on We now make a Second Series, and though the two series follow each other at a brief interval on a day described as “clear,” the values of the deflections in the second series on the same points indicate nearly twice the heat in the first. The change is due to the altered diathermancy of the apparently clear sky in the brief interval. It is one of the difficulties already signalized by the writer and by others *, and is to be eliminated only by repeated observation. The second series, however, gives the same value as the first, and hence we conclude (as far as this day’s observation goes) that this was the index of refraction, for the wave-length in question and for a certain flint prism. Such observations must generally be repeated on many days before a reliable result is reached; and in the case of the glass prism, which grows rapidly athermanous just beyond the limit of the solar spectrum, they are limited by the nature of the substance to little more than the wave-length in question. We now consider, as our second example, a measurement at 5x2D,, where glass can hardly be used, and where its place is supplied by the rock-salt prism, and that of the sun by the arc. Second example of Measurement. With Rock-salt Prism. (Extract from original record.) Station, Allegheny. Date, April 15th, 1886 (P.m.). Wet bulb at 4% 30™ (inside dark room artificially heated), == 2175C: Dry bulb at 45 30™ (inside dark room artificially heated), =a C. Aperture of slit 8; =4 millim. og = 11645 Prism used, Hastings No. 1 (rock-salt). Lenses used, focus=760 millim. for visible rays. Grating used, Rowland No. 2. Galvanometer used, No. 3, damping magnet at 40 centim. Time of single vibration = 17°5 sec. (with current passing). Bolometer, No. 16 (aperture 1 millim.). Setting on slit=0° 00! 00". D, (spectrum thrown east) =41° 03! 00". ‘i west) =39 57 40. Current of battery = — 0-037 ampere. Reader at circle, J. P. galvanometer, F. W. V. 9) 99 * See Crova, Comptes Rendus, tome ci. p. 418 (August 10, 1885). unrecognized Wave-lengths. 163 Arc managed by A. Object=determination with the great arc, of the deviation in a rock-salt prism corresponding to a wave-length o 5 XAD,= 2945. , SPECTRUM THROWN HAST. 39° 21". 39° 18’. 39° 15', ——$——— | 13 33 5:5 1:3 | 34 |os 13 34 5°4 —1:0) 0-6 75 ir 0-9 | 1-1 | 6°9 | 17 —6°8 16 6°3 EF SPECTRUM THROWN WEST. 39° 12". Deviation...... Ist series ... 2nd series... 3rd series ... 4th series ... \ The sodium-line of the 5th spectrum fell exactly upon the slit S, at the beginning and end of the observation. From smooth curves the following positions were deduced for the maximum :— Spectrum east . 39 15 7 a Pretah= Se ebe N Spectrum west . oe TL6er 0 4 5 au, ay. 4 Mean ad). 1)438 The preceding examples will sufficiently indicate the méans used in making the following measurements of wave-length with the grating, which have been carried on continuous! from December 1885 to April 1886. As a rule, each of the sixty-two determinations in the following Table represents one or more days’ labour, thongh in some cases two or more determinations have been secured in the same day. M 2 164 Prof. 8. P. Langley on They have been taken under the following conditions:— (1) In the case even of the very lowest wave-lengths in the feeblest heat we have been able to use a linear bolometer of not more than 1 millim. width. (2) About an equal number of observations were intended to have been taken with the prism placed so as to throw the spectrum east and west. In doing this a minute systematic error, amounting, at the greatest, to less than i! of arc, was found to be caused by flexure of the arm, due to the weight of the bolometer cable, and a correction for this has been applied. Otherwise the observations are given as originally made ; and as the “ probable error’’ includes all the more or less systematic differences, due to the use of different gratings and different positions of the apparatus, it may be considered to be in this case a fair indication of the amount of error to be actually expected. We do not know of any determination of the change pro- duced in the refractive power of a rock-salt prism by varying temperature. A rough comparison of the deviations of Fraunhofer lines, incidentally measured in the progress of the work at different seasons, during which the temperature has varied nearly 30° C., together with the results of a single day’s measures at temperatures differing by 17° C., have con- curred in indicating a diminution in the deviations through- out the visible spectrum of about 11” for a rise of temperature onl. We do not doubt that a temperature-correction is also required for the invisible spectrum; but not having yet been able to satisfactorily determine any, we think it best to leave all the observations as they stand, uncorrected for tempera- ture, and offer them under this reserve now, with the inten- tion of returning to them hereafter. In the following Table we repeat, for convenience, the results of certain optical measures made on September 14th, 1885, by Mr. J. H. Keeler with the rock-salt prism, whose refracting-angle was on that date 59° 57! 54”, and which we have reduced to 60° by the formula given in this Journal for May 1886. The wave-lengths are those due to Peirce’s and Rowland’s corrections of An gstrom, with which we have been obliged by the authors before their formal publication. In the succeeding portion of the Table we have the re- sults of measures made with the bolometer in the invisible spectrum. The first column gives the source of heat; the second the wave-length selected for measurement; the third the grating employed; the fourth the refracting-angle of the prism on the day of observation; the fifth the temperature unrecognized Wave-lengths. 165 of the apparatus; the sixth the observed deviation; the seventh the mean deviation corrected for flexure error and reduced to a refracting-angle of 60°; the eighth the resulting index of refraction for rock-salt. Source of Are Heai. eaeeee se ueee Grating. | | Refracting- Angle 59 59 Or = 20 45 39 39 39 39 Observed Deviation. — = Es) De- to viation, re- Mean 39 49 18 +12" Index of Re- fraction. 156833 1 55323 1549795 154418 154414 1-54051 153670 1°5301 +0001 1.5272 +0001 15254 +:0001 15243 +-0000 1-5227 +0002 166 heme ° es 25 4s) os ——— Oo vo ag ee a 2) | M 4°1231 PATIO RS. cos SnD — 47121 ASC. 50s ohh — 5°3011 1 Prof. 8. P. Langley on . Table (continued). 59 | Refracting- angle. £ oo8 é ro od ‘ EA een be ha in 5 Ss St os = B oon Ks g 5 2 S.4 30 res ® A oPrmeo S al = al i ° O° i 4 Oo i 4 39 03 42 | 1:5215 lee +:0001 38 56 06 | 1:5201_ +30" +-0002 88 48 06 | 1:5186 +30!" +-0002 Or nS iN bo bo ——~-—O ———_-+-- aS —-—-+~--—_—_“’ In the following brief Table we have summarized the results Our working method gave the index in terms of the wave-length ; but since ordinarily the former is the known and the latter the unknown quantity, we here ive the mean probable error as finally corrected as a function of all this labour. of the latter. Given Indices of Refraction in Rock-salt Prism. 15442 1°5301 1°5272 1°5254 1:5243 1°5227 1°5215 1°5201 15186 Wave-lengths from Direct Observation. (a) by the eye; (6) by the bolometer. _ AD, =0°5890+0:000 (a) 2 x XD.=1:1780+0-002 (b) 3 xX AD.=1:7670+0:005 ay 4 xXD,=2:3560-+0:009 ,, PxiAD, = 29450 +0:0138 43 6 xAD,=3'534140:019 ,, 7x AD, =4:1231 +0029 ,, 8 xrAD,=4:7121 40-043 ,, 9 x AD, =5'3011 40-065 ,, unrecognized Wave-lengths. 167 In Plate VI. we have graphically constructed the relations between n and X for the rock-salt prism, as far as the above wave-length of 5“:3011 or 0:0053 millim. The ordinates are proportional to the indices of refraction given on the axis of Y, the abscissee to the wave-lengths on the axis of X. The two vertical dotted lines carry the eye down to the corre- sponding portion of the spectrum, which is visible. Between these lines lie the points of the visible spectrum observed, and the dotted curves show the results of extrapolations by various formule. The actual points settled by observation are certain mul- tiples of the wave-length D, (0°0005890 millim.), and a small circle whose diameter equals a unit in the third decimal place of the scale of ordinates (indices) gives the position fixed by observation, while the distance from the centre at which the smooth curve cuts the little circle furnishes a graphic representation of its difference from observation. The labours of the past year, then, have enabled us to absolutely and directly measure the index of refraction of rays whose wave- lengths are greater than 0-005 millim., or, more exactly, which reach 53011 of Angstrém’s scale ; and to do so with an error which is probably in most cases confined to the fourth decimal place of the index. As we shall see more clearly by Plate VI., the relation between n and 2 has changed from that appa- rently complex one we see in the visible spectrum, so that n becomes almost a simple linear function of 2, and the results of extrapolation grow to a higher order of trustworthiness than when made from points in the visible spectrum alone. It appears to us that no formula of dispersion with which we ale acquainted * gives entirely correct results on extra- polation, but that among the best are Briot’s and Wiillner’s. We have computed the wave-lengths corresponding to indices of refraction from observed deviations in the visible spectrum according to these formule. The curve from Cauchy’s formula we do not give, because (at least when not more than three terms are taken from observations in the visible part of the spectrum) its results are here of little value, since it declares all the radiations we are now actually dealing with to be impossible of discrimination at all. Redtenbacher’s formula we have also shown in a previous memoir to be - scarcely worth further consideration. The graphically con- structed values are obtained by applying the formula of Briot, | * That proposed by Ketteler has come to the writer’s knowledge too ~ late for trial here. 168 Prof. 8. P. Langley on 1 2 4 rn (greta tha)) to the four points A(A=0+7601, n=1°53670), D,(A=0":5889, n=1°54418), b,(A= 05183, n= 154975), H, (x= 0"-3968, n=1°56833); and the formula of Wiillner, aA (n-1= -Pv+25~), to the three points A(X=0"7601, n= 1°53670), b, (X= 05183, n=1°54975), Hy (X= 03968, n=1-56833). All these are in the visible spectrum, and from them the constants a, b, c, k, P, &. are determined. With their aid we next inquire by the formula what wave-lengths correspond to certain given indices, and the resultant values in the infra- red are then plotted from these computations. It is, however, only just to observe that the wide departure from observation here shown is by no means to be wholly attributed to errer in the formula ; for minute errors of measurement, such as are always present even in the observations in the visible spec- trum, are immensely exaggerated by extending the curve through extrapolation. Wiillner’s formula, for instance, would give a line closely coincident with our curve in the infra-red if we took all our points for computation from that part of the spectrum. A similar remark may be made of Briot’s equations, whose actual tracing, however, with the constants we obtain from the visible spectrum, shows that beyond a certain point the curve, which is its geometrical representation, from being concave to the axis of X, becomes convex, so that the relation between n and A would, according to it, be represented by a sinuous line; and this is not so within the limits of these observations. Its fair agreement with observation, then, within the limits of the visible spectrum and the upper part of the solar infra-red are all that can be claimed for it. Our conclusion is that all theories of disper- sion known to us prove inadequate to predict the relation between wave-length and refraction. The actual relation from direct investigation is here given for the first time from the observations of the past year, which, it will be seen, thus unrecognized Wave-lengths. 169 confirm and greatly extend the results-of 1882 and 1884. Their most salient feature is still perhaps that already noted, 2. eé. while the curvature, as far as we can follow it, grows less and less, at the last point at which we can view it the curve is not only all but sensibly a straight line, but one making a very definite angle with the axis of X. This obviously means that beyond this point 7 is nearly a linear function of A, or that the simple equation n=ad would very closely represent this portion of the curve. It means also that, as far as these observations extend, we find scarcely any limit to the index of the ray which the prism can transmit except from its own absorption. I do not, it will be observed, undertake to advance without limit beyond observation, or to discuss what would happen with wave-lengths so great that the index be- came 0 or negative, as it would with an indefinite prolonga- tion of the curve, if its direction remain unaltered. An intelligible physical meaning might perhaps be attached to these cases; but I here confine myself to the results of direct observation and to the now established fact that the increase of the crowding together of the rays at the red end, which is so conspicuous a feature in the upper prismatic spectrum, has almost wholly ceased, and that the dispersion has become approximately uniform, the action of the prism here being assimilated to that of the diffraction- grating itself. I shall not venture to treat of the theoretical import of this, further than to remark that the ordinary interpretation of Cauchy’s theory will apparently lead us to conclude that dis- persion must sensibly cease at the point where the wave is so long that the size of the components of matter is negligible in comparison. In other theories, also, there appears to be a point below which the index of refraction should never fall ; and we might anticipate that the curve would accordingly tend to become parallel to the axis of X. Of course we cannot assert from observation that it will not finally do so ; but within the very extended limits in which we have fol- lowed it the contrary happens, and the curve presents a continuously increasing angle with that axis. These results, then, in some material points are in contra- diction to what has usually hitherto been believed *. Let me repeat that one consequence of the fact that the * Tam very desirous that they should be verified by physicists, and I have therefore given particulars in some detail of my methods and appa- ratus here. 1 have requested the skilful artists (Mr. William Grunow, Mechanician to the U. 8. Military Academy, West Point, N.Y., and Mr. J. A. Brashear, Allegheny, Penn.) who have so successfully constructed this apparatus to place at the command of physicists all or any details of it. 170 Prof. 8. P. Langley on curve is approaching a straight line is that, unless there is some immediate change in its character, such as we have no right to expect, extrapolation considerably beyond the point to which we have measured will be comparatively easy and safe. Jam aware of the danger attending all extrapolation, but I must insist upon the fact that the old ones, which we have seen falsified by experiment, rested on an extremely limited region of the curve, that namely for the visible region of the spectrum, in which the relation between n and X is also wholly different ; while those on which we now briefly enter depend upon far greater material for induction (about eight times that included in the visible spectrum), which we can also use under much more favourable conditions. Since the curve still presents a slight convexity to the axis of abscissee, unless its character changes in a way which we have no ground to expect, a tangent at any point will meet that axis sooner than the curve itself will. Accordingly, if we now ask what wave-length corresponds to any point in the hitherto unexplored region, for instance the maximum in the spectrum of boiling water, whose index * for the 60° rock-salt prism is 1°5145, or that of melting ice, whose index is 1:5048, we can answer as follows: first, this unknown wave-length is at any rate greater than 5", since to this point we have investigated by direct measurement; second, since the tangent to our curve, even at the point 5", meets the line correspond- ing to the index of the maximum heat in boiling water at over 7“, and a line corresponding to the maximum ordinate in the spectrum of melting ice at over 10 , and since the curve without some change in its essential character cannot meet these lines, save at still greater wave-lengths, it follows that the wave-length of the maximum of the spectrum from boiling water is probably at least ‘0075 millim., and that of the maximum in the spectrum from melting ice is over 0:01 millim. In an article in the Comptes Rendus of the In- stitute of France, Jan. 18, 1886, and in preliminary memoirs, we deferred giving the actual values, but gave (explicitly as minimum values which we believed much within the truth) 5" to 6". That our caution led us to understate the, even then, most probable value, may be seen from the statements just made, which are founded on still later observations. As we proceed further out, extrapolation becomes, of course, more untrustworthy. We can only say that if the curve maintains its present inclination to the axis of X, the wave-lengths of the extreme radiations recognized in the rock-salt prism must indefinitely exceed 0°03 millim. * See this Journal for May 1886, plate iv. fig. 3. unrecognized Wave-lengths. 171 We have shown that the various complex formulz founded on theoretical considerations differ from observation ; and as we have remarked, they have the minor objection also of being extremely difficult of application to practical uses, owing to the inordinately tedious numerical computations involved where many places are to be calculated. ; Struck by the resemblance of the actual curve of observa- tion, as viewed in a large graphical construction, to a hyper- bola, I was therefore led some years ago to use the equation of the hyperbola as an empirical one for interpolations in the infra-red without attaching any physical meaning to it. The further the investigation has been pushed in that part of the spectrum, the more exact the resemblance has become. That it is notable will be seen on consulting Plate VI.*, where the hyperbola does not appear as a distinct curve, because its variation from the smooth curve of observation cannot be recognized on this scale. In obtaining this we have proceeded as follows :—Having five disposable constants, we have taken five nearly equidistant points on the smooth curve of observation, remembering that if the axis of Y is not exactly asymptotic to the curve thus described, we are not necessarily to impute the difference to a fault in the equation chosen, since the condition that the curve of observation shall be rigorously asymptotic to this axis can in any case only be satisfied by infinite exactness of measurement. It will be observed that my estimates of the extreme wave- lengths are in no way founded on the use of this hyperbola, and that I do not assert that it has any physical meaning. I have elsewhere observed that, while Herschel in 1840, Draper in 1842, Fizeau and Foucault in 1846, Lamansky in 1870, with others since, had observed bands in the infra-red prior to 1881, yet that nothing was exactly known as to the wavye-lengths of these bands, even to those who discovered them. It is very likely that the (probably telluric) absorp- tion-band in the solar spectrum placed on our chart (Comptes Rendus, Sept. 11, 1882) at 1"°38 has been recognized by more than one of the above-mentioned observers, yet so little was known as to its actual position even a few years since, that we find the elder Draper, in reviewing these discoveries in 1881, and speaking with the authority of one who was himself a discoverer, expressing his doubt as to the possibility of any wave-length so great as 1":08 haying really been * It should be mentioned that some of the observations on which the computations are founded have been added since this drawing was pre- pared for the engraver. 172 Prof. 8. P. Langley on unrecognized Wave-lengths. Extreme Lengths of Visible and Invisible Etherial Radiations and of Sonorous Waves. Quality of radiations and means of recog- nition. | Invisible Ultra-violet radiations (Photo- graphy). Visible radiations (Hye). Beginning of Infra-red (Photography). (Phosphorescence). Invisible Infra-red radiations from terrestrial sources (Bolometer) Sonorous vibrations (Ear). Wave-lengths Description. Extreme rays of aluminum in the in- duction spark, according to Cornu. Recorded by photography. Extreme limit of solar spectrum at sea- level on best days, according to Cornu. Recorded by photography. Limit of lavender light, visible to nor- Extreme limit of deep red light, visible Supposed extreme possible limit of infra-red wave-lengths in 1881. (According to J. W. Draper.) Wave-lengths assigned by H. Becquerel to lowest absorption-band known to Sensible limit of solar infra-red rays which penetrate our earth’s atmo- sphere. Determined by the grating] and bolometer, 1882. ' Limit of absolute measurement of wave- lengths .corresponding to a given index of refraction in the case ofa rock-salt prism. Determined by the Rowland grating and bolometer; Approximate position of the maximum ordinate in the “‘ heat” spectrum from a lamp-blacked surface at the tem- perature of boiling water (100°); Approximate position of the maximum ordinate in the “ heat” spectrum from a lamp-blacked surface at the tem- perature of melting ice (0°); Alle- gheny, 1886. Approximate estimate of the minimum value assignable to the longest wave recognizable by the bolometer in the “heat” spectrum from a rock-salt in units of one miili- metre. (0:000185 | \ 0000295 0.00086 mal eyes. 0:00081 to normal eyes. (09-0010 \ 0:0015 him in 1884. 0:0027 0:0053 Allegheny, 1886. 0:0075 Allegheny, 1886. 0-011 0-030 prism. 14 Length of shortest sound-wave corre- sponding to highest musical note perceptible by human ear. (Approxi- mately 48,000 s. v. per. sec., Savant.) |_ Theorem relating to Matrices. 173 observed; and M. H. Becquerel (Annales de Chimie et de Physique, 1883, tome 30) gives the wave-length of the longest band known to him as 1"°5. These remarks will not be superfluous as an introduction to the preceding table, which presents a summary view of the advances made beyond the above-named point in the last five years. Broadly speaking, we have learned, through the present measures with certainty, of wave-lengths greater than 0-005 millim., and have grounds for estimating that we have recog- nized radiations whose wave-length exceeds 0:03 millim. ; so that while we have directly measured to nearly 8 times the wave-length known to Newton, we have probable indications of wave-lengths far greater, and the gulf between the shortest vibration of sound and the longest known vibration of the ether is now in some measure bridged over. In closing this memoir I would add that the very consider- able special expenses which have been needed to carry on such a research, have been met by the generosity ofa citizen of Pittsburgh, who in this case, as in others, has been content to promote a useful end, without desiring publicity for his name. I cannot too gratefully acknowledge my constant obliga- tion to the aid of Mr. F. W. Very and Mr. J. E. Keeler of this Observatory, who have laboured with me throughout this long work. In the prolonged numerical and other com- putations rendered necessary, I have been aided by Prof. Hodgkins of Washington, and by Mr. James Page of this Observatory. Allegheny Observatory, May 31, 1886. XX. An Lxtension of a Theorem of Professor Sylvester’s relating to Matrices. By A. BucHHEm, M.A.* CyE of Prof. Sylvester’s fundamental theorems in the theory of matrices is what he calls the interpolation- formula: viz. if m be a matrix of order n, and Ay...A,_ its latent roots, we have, @ being any function, (m—Azy)(mM—Az) «. « (M—An) (Ay —Azg)(Ay—Az) «- «(AL — Ay) pry. This theorem only applies so long as the latent roots are un- equal. In this note I extend it to matrices having equalities dm=> * Communicated by the Author, 174 ‘Theorem relating to Matrices. among their latent roots. In a paper on the theory of ma- trices, published in vol. xvi. of the ‘ Proceedings of the London Mathematical Society, I have given a canonical form to which matrices can be reduced. If we have a matrix m with r distinct latent roots A,...A,, of multiplicities n,...n,, L have shown that there are n; latent points answering to Xj, and that these can be arranged in groups, such that for one of these groups (¢,...é,) we have Nie} Nj eg $ ey oe» Nils + es—1 e €5 es ‘ It can be shown without difficulty that if ¢ is any function, we have : gma Per Pee +P Mie «+ Gio HPA C—1 Ho. + GT Ved ej 2) (ae Now call s the length of the chain (e,...e,), and let s; be the length of the longest chain appertaining to A;; then it is easy to see that the extension of Prof. Sylvester’s theorem amounts to finding a function f such that FOU) =OP RR) ay and that these conditions are satisfied if we take fu=y Xe = i=1 X;A, XK Us ru where Xj = (a — ry)" —Dg)”.. . (@—Ay_1)*-! (@— Aggy) tt... (@ A)” (2@—n) r—l [palt=ynt (o—ayy nt... + Sa pina, and we therefore have gm=y X™ Ne =X; LXMIA; Y eed It is obvious that this reduces to Prof. Sylvester’s formula if the latent roots are all unequal, since in this case s, sy, &e. are all equal to unity. The Grammar School, Manchester, June 25, 1886. bokis .] XXI. Notes on Magnetism.—lI. On the Energy of Magnetized Iron. By Lord Rayuuien, Sec. R.S.* & ans splendid achievements of the last ten years in the practical application of Magnetism have given a re- newed impetus to the study of this subject which is sure to bear valuable fruit. Especially to be noted are two memoirs recently published in the ‘ Philosphical Transactions’ of the Royal Society, by Prof. Ewing}, and by Dr. Hopkinson, in which are detailed very important data derived from labo- rious experiment, accompanied by much interesting and sug- ‘gestive comment. The results of observation are usually expressed, after the example of Rowland and Stoletow, in the form of curves showing the relation between S and §, the magnetic induc- tion and the magnetizing force. It may be well here to recall the convention in accordance with which is measured. At any point in air, the magnetic force is defined in an ele- mentary manner, and without ambiguity, but when we wish to speak of magnetic force in iron, further explanation is needed. ‘The continuity of the iron is supposed to be inter- rupted by an infinitely thin crevasse in the interior of which we imagine the measurement to be effected. If the crevasse is parallel to the direction of magnetization, the force thus found is denoted by §, and is independent of free magnetism on the walls of the crevasse. If, however, the crevasse be perpendicular to the lines of force, there is a full development of free magnetism (3) upon the walls, and the interior force is now %, equal to +473. In the estimation of (as well as of ©) the influence of all free magnetism, not dependent upon the imaginary interruption of continuity, is of course to be included. On this account the value of 9 in the interior, and even at the centre, of a bar of iron placed in an otherwise uniform magnetic field, is greatly reduced, unless the length of the bar be a very large multiple of the diameter. _ Experiment shows that the relation of 8 to D is not of a determinate character. In a cycle of operations, during which § is first increased, and is afterwards brought back to its original value, the induction % is always greater on the descending than on the ascending course. This phenomenon, which is exemplified familiarly by the retention of magnetism in a bar after withdrawal of the magnetizing force, is * Communicated by the Author. T ‘Experimental Researches in Magnetism,” vol, clxxvi. part ii. p. 523. t “ Magnetization of Iron,” zbed. p. 455. 176 Lord Rayleigh on the called by Ewing hysteresis, The accompanying curve ABCDEFGA (fig. 1) is copied Fig. 1. from one given by him as appli- cable to very soft iron, conducted round a cycle from strong nega- tive to strong positive magneti- zation and back again. The “residual magnetism ” or“ re- tentiveness ’’ (OH) amounts to a large fraction (sometimes to 93 per cent.) of the maximum. The work spent in carrying the iron round a magnetic cycle is represented by — \3 dS, as was first shown by Warburg%*, who supposes the magnetic force operative upon the soft iron to be due to permanent magnets, and variable with their position. The work required to carry the permanent magnets through the proposed cycle of motions is then proved to have the above written value, applicable to the unit of volume of the soft iron. If 3 were proportional to §, or even related to it in any determinate manner, the integral would vanish ; but on account of hysteresis it has a finite value. So long as we limit our attention to complete cycles, we may write indifferently —{3 as), (OF =— ra {B dS, since {6 dQ A vanishes. Again, under the same restriction, 1 —|3 d= + {6 dg=z_ J6 ae. When, however, we wish to consider incomplete cycles, espe- cially with reference to the behaviour of soft iron, it is more suitable to take S as independent variable. We are led naturally to this form if we suppose that, as in the more im- portant practical applications, the varying magnetizing force is due to an electric current, upon which the magnetized iron reacts inductively. In order to avoid the question of free polarity, we may * Wied. Ann. xiii. p. 141 (1881). + Hopkinson, J. c. p. 466. Energy of Magnetized Iron. 17% consider, first, a ring electro-magnet with an iron core, of length / and section c. If n be the number of windings, the whole inductive electromotive force is no. dB, and the element. of work is noC.d®, C being the current at the moment in question. But 1S=4anC ;y so that the element of work is, per unit of volume of iron, 1 i ® ad. If we express 8 in terms of § and 3, we have 1 1 1,2 = 7, 24919 d3, of which the latter part is specially due to the iron. In practice the former part is small, and the distinction between 1 a ine and § d3 may often be disregarded. But it is by no means a matter of indifference whether we take SdH or HdB. The difference between the two modes of reckoning may be exemplified in the case of iron already nearly “saturated,” and exposed to an increasing force. Here IB d is large, while is dB is small; so that the latter corresponds better with the changes which we suppose to be taking place in the iron, as well as to the circumstances of ordinary practice. Let us now consider a little more closely the cycle of fig. 1. From A to B, 5 is negative, while dB is positive; so that along AB the inductive electromotive force is in aid of the current, and work is received from the iron of amount repre- sented by the area ABM. From B to D,® is positive as well as d, and work represented by BDNB may be supposed to be put wto the iron. From D to E, work, represented by NED, is received from the iron, and from E to A work, repre- sented by AMH, is expended. From this we see that not only is work, represented by the area ABCDEA, dissipated in the complete cycle, but that at no part of the cycle is there more than an insignificant fraction of work recovered. The case is not one of a storing of energy recoverable with a small rela- tive loss, but rather one of almost continuous dissipation. And here the question is forced upon us, whether it is true, as is usually supposed, that the strong residual magnetism at Phil. Mag. 8. 5. Vol. 22. No. 185. August 1886. N 178 Lord Rayleigh on the E is really a store of energy. From the fact that the mag- netism may be got rid of by very moderate tapping, we may infer, I admit, that some energy is necessarily dissipated in assing from the magnetized to the unmagnetized condition, ae the dissipation may be exceedingly small ; and the argu- ment is not conclusive, since the mechanical energy of the vibrations may be involved in the process. If we attempt to demagnetize the iron in a straightforward manner by the application of a reversed force, following the course indicated by EFGO, then, so far from recovering, we actually expend energy—that, namely, represented by the area HFGOH. For practical purposes, at any rate, it would seem that magnetized iron cannot be regarded as the seat of available energy. The opposite opinion, which is widely entertained, appears to depend upon insufficient observance of the distinction, vital to this subject, between closed and unclosed magnetic circuits, It is not disputed that available energy accompanies the mag- netization of a short bar of iron, but this is in virtue of the free polarity at the ends. The work stored is in fact that which might be obtained, were the bar flexible, by allowing the ends to approach one another, under their mutual attrac- tion. When this operation is finished, so that the bar has become a ring, there is no longer any work to be got out of it, though it remains magnetized. In further illustration of this matter, reference may be made to some interesting observations by Elphinstone and Vincent * on closed magnetic circuits. As is well known, the armature of a horseshoe electromagnet remains strongly attracted after cessation of the battery-current. If, even after a considerable interval of time, the coils of this electromagnet. were connected with those of a second electromagnet also provided with an armature, and the first armature were then violently pulled away, attraction set in and persisted between the second armature and its electromagnet, the magnetism of the original circuit being as it were transferred to the second. Or, if a galvanometer were substituted for the second electro- magnet, a deflection followed the forcible withdrawal of the armature. In these experiments the necessary energy is obtained, not from the magnetism of the closed circuit, but from the work done in opening it, that is in pulling away the armature. | These considerations lead me to differ from Prof. Ewing when he says t :—‘ In connection with ‘secondary generators’ and induction-coils generally, the bearing of the first part of * Proc. Roy. Soc. vol. xxx. p. 287 (1880). tT lc. § 34, p. 554. Energy of Magnetized Iron. 179 this paper should be noted, as showing the enormous advantage which a ring-shaped core, or core forming a complete mag- netic circuit, possesses over a short bar-core with ends. In an ordinary induction-coil, so long as the current in the primary circuit is merely made and broken, a short core is necessary, since a ring-core would lose but a small percentage of its magnetism at each brake, but where reversal of the magnetizing takes place, a core approximating to the con- dition of endlessness has an (advantage in respect of power which fig. 3 makes obvious.” I confess that I do not follow this. It seems to me, on the contrary, thata closed magnetic circuit is above all things to be avoided, as leading to waste of the greater part of the power transferred. A like objection applies to the use of a closed electro- magnet as a ‘‘ throttle” in an alternate-current circuit. When we know, as from Prof. Ewing’s results, the be- haviour of a given sample of iron under the influence of various forces actually operative, we can deduce by means. of Poisson’s theory the magnetism assumed by ellipsoids of any shape in response to any uniform external force D. If 3 be the magnetization parallel to the axis of symmetry (2c), the demagnetizing effect of S is NJ, where N is a numerical constant, a function of the eccentricity (e)*. When the ellipsoid is of the ovary or elongated form, =O / (1—e’)e, N= in(4 -1)(5. ie 1) becoming in the limiting case of the sphere (e=0) Arr N=%; and at the other extreme of cee assuming the form ne dn (log —— us: If the ellipsoid is of the “oo form, ¢ and N=47 n (4- os vC>* sine). * Maxwell’s ‘ Electricity and Magnetism,’ § 438. + There is here a slight variation from Maxwell’s notation. 180 Lord Rayleigh on the _ In the case of a very flattened planetoid (e=1), N becomes in the limit equal to 47r. The force actually operative upon the iron is formed by subtracting NJ from that externally imposed, so that H=H'—-Ns; and if from experiments on very elongated ellipsoids (N=0) we know the relation between ) and 3, then the above equa- tion gives us the relation between §’ and J for any proposed ellipsoid of finite elongation. If we suppose that § is plotted as a function of 3, we have only to add in the ordinates N 3, proper to a straight line, in order to obtain the appropriate curve for §’. As an example, let us apply this method to deduce the behaviour of the soft iron of Ewing’s fig. 2, when made into an ellipsoid whose polar axis is fifty times the equatorial axis, and carried round a cycle through strong positive and strong negative magnetism. We have N= =e flog, 100—1} = 4m x “001442. The curve ABC (fig. 2), traced from Prof. Ewing’s, gives Fig. 2. D Hl c| 8) ida 3 4 5 10 15 20 25 the relation between § and %, the latter of which we may identify with 473° *. The equation of the straight line is H=NS =:001442 x47 3; and with allowance for the different scales adopted for ordi- nates and abscisse, is represented on the diagram by OD. * The curve is symmetrical with respect to O as centre, and § is measured in C.G.S. units. Energy of Magnetized Iron. 181 In order to find the points Q, Q’ appropriate to the ellipsoid (50:1) from P, P’, we have merely to measure PQ, P’Q' equal to RM. We thus obtain the curve AQHQ/FC, on which the points of zero magnetization are the same as on the original curve *. We see that a much stronger field is now required to produce the higher degrees of magnetization, and that there is less hysteresis—the magnetic state is more nearly a definite function of the external field. A similar construction might be used reversely to pass from observed results relative to ellipsoids of moderate elongation to the curve appropriate to ellipsoids of infinite elongation, on which alone we can base our views of the real character of magnetic media. Prof. Ewing has traced by experiment the influence of various degrees of elongation on the magnetism of cylindrical rods. Results of this kind are exhibited in his fig. 3, but they are not strictly comparable with those obtained above, not only because the latter relate to ellipsoids, but also on account of the different character of the magnetic operations represented. His curves begin at a condition of zero field and zero magnetization. The work expended in producing a small change of mag- netization of the ellipsoid, acted upon by a uniform field, is /d3 simply per unit of volume. This we may see, perhaps most easily, by supposing the iron to be replaced by an electric current of equal magnetic moment. The element of work done then depends upon the coefficient of mutual induction M of the two circuits, and M may be regarded as the number of lines of force due to the original current which pass through the fictitious circuit. The whole work is thus {S'd3 = fSd3+ N(3as = | Sd3+ 4N3*, if we reckon from the condition of zero magnetization. The first part is that already considered, and shown to be almost entirely wasted ; the second, which in most cases of open magnetic circuits is much the larger, is completely recovered when the iron is demagnetized. Thus in fig. 2, since QQ’/=PP’, the areas of the two curves are the same, which indicates that the same amount of work is dissipated in a complete cycle. But the work absorbed during one part and restored during the remainder of the * Dr. Hopkinson (loc. cit. p. 465) has already applied this method to the determination of the particular point F, indicative of the residual magnetism in the ellipsoid, when the external force is withdrawn. 182 On the Energy of Magnetized Iron. cycle is much greater in the case of AEC, corresponding to the ellipsoid of moderate elongation. The coefficient N reaches its maximum when the ellipsoid is very oblate. In this case {H/d3=\ H d3+ 279’, which is applicable to large plates magnetized perpendicularly to their surfaces. This is the form to which the iron must be reduced in order that a given magnetization of a given volume may store* the largest amount of energy. In this case the energy is nearly all recoverable; but we must remember that no practicable field would carry the magnetization very far. In the theory of alternating currents the neighbourhood of of iron is often treated as if its only effect were to increase the self and mutual induction of the circuits. A writer con- versant with experiment usually guards himself by a reference to the currents induced in the iron considered as a conductor. The latter effect may be in great measure eliminated by a proper subdivision of the iron, with intervening non-conduct- ing strata; but a glance at fig. 2 shows at once that, apart altogether from internal currents, the influence of the iron is of a more complicated character. If the curve connecting J (or 8) and § were a straight line, the same on the upward as on the downward course, then the presence of iron would ‘simply increase the self-induction. When the iron constitutes a closed magnetic circuit, this is very far from being true. Indeed it would be nearer the mark to say that the iron in- ereases the apparent resistance of the electric circuit, leaving the self-induction unchanged. In so far as the curve of fig. 2 can be identified with an ellipse, the reaction of the iron can be represented as equivalent to a change in the apparent re- sistance and self-induction of the circuit. Which of the two is the more important depends somewhat upon the other cir- cumstances of the case; but with closed electromagnets the magnetic work dissipated during the period (corresponding to increased resistance) is always greater than the work spent during one part and recovered during the remainder of the period (corresponuing to increased self-induction). On this account, the resistarce of an iron wire to variable currents is greater than to steady currents, even though the current be constrained to be uniformly distributed over the section. In the absence of such constraint, the resistance undergoes a further increase in consequence of the tendency of the cur- * It is not meant here to imply that the energy is resident 27 the iron. On the Salts of Tetrethylphosphonium. 183 rent to concentrate itself towards the exterior*. In general both causes must cooperate to produce an apparent increase of resistance to variable currents, When the magnetic circuits are open, as with bars of iron of moderate length, the reaction of the iron manifests itself mainly as increased self-induction. This happens also in the ease of closed magnetic circuits, when the magnetic changes are very small. In general, since the curve of fig. 2 differs widely from an ellipse, the reaction of the iron cannot be fully represented as equivalent to a change in the resistance and self-induction of the magnetizing circuit. In any case of strict periodicity the reaction may, however, be analyzed, in accordance with Fourier’s theorem, into harmonic components with periods which are submultiples of the original period. The neigh- bourhood of iron may thus introduce overtones into what would otherwise be a simple sound. Terling Place, Witham, Essex, July 4. XXII. On the Salts of Tetrethylphosphonium and their De- composition by Heat. By Prof. H. A. Lurts, Ph.D., &c., Queen’s College, Belfast, and Norman Coun, PA.D., Science Lecturer, The Ladies’ College, Cheltenhamf. QE of us (in conjunction with another chemist) has already pointed out the very striking analogies which exist between the elements phosphorus and sulphur and their com- pounds {, and we were anxious to continue our experiments in connection with this question. The action of heat on methyl-sulphine compounds has been studied by Crum Brown and Blaikie §, who have shown that they decompose in a perfectly simple and definite manner. We were accordingly desirous of ascertaining whether the analogous phosphorus derivatives, namely the salts of tetrethylphosphonium, would behave similarly. We had already investigated the action of heat on some of the salts of tetrabenzylphosphonium ||, but the results were * “On the Self-induction and Resistance of Straight Conductors,” Phil, Mag. vol. xxi. (1886) p. 388. + Communicated by the Authors, ¢t Crum Brown and Letts, Trans. Roy. Soc. Edin. vol. xxviii. p. 571; and Letts, Trans. Roy. Soc. Edin. vol. xxviii. p. 583. § Crum Brown and Blaikie, Journal f. prakt. Chemie [2] xxiii. p. 395, || Letts and Collie, Trans. Roy. Soc. Edin. vol. xxx. part i. p. 181. 184 Drs. Letts and Collie on the Salts of not very satisfactory, owing to the tendency of the benzyl radical to split into hydrocarbons at the high temperature at which the decomposition occurred. We thought it probable that the salts of tetrethylphospho- nium would decompose at a lower temperature, and would therefore be better suited for investigation. Several of the salts of tetrethylphosphonium which we investigated had already been described by Hofmann™*; one or two, however, we prepared for the first time. Preparation of Lodide of Tetrethylphosphonium. As this salt was the starting-point for the preparation of the other compounds of the phosphonium, we devoted con- siderable care to its manufacture in the pure state. When triethylphosphine is added to iodide of ethyl, the two combine rather suddenly and with considerable evolution of heat, so that unless care is exercised loss of material may easily occur. Hofmann suggests that the iodide of ethyl should be diluted with ether before the addition of the phosphine, and at first we acted on this suggestion; but subsequently we found that, by carefully cooling the mixture and using excess of iodide of ethyl, the reaction could be kept completely under control, and that the addition of ether was unnecessary. The first quantity of the phosphonium iodide was prepared in etherial solution: 30 grms. of triethylphosphine were added to a mixture of 43 grms. of pure iodide of ethyl and 500 grms. of ether. The mixture was left for a night, when 30 grms. of feathery white crystals separated. These were washed with dry ether, and dried in vacuo over sulphuric acid ; the determination of iodine in them was made volume- trically. 0-600 grm. required 21°9 cubic centim. decinormal nitrate of silver solution =0°27813 grm. iodine=46°36 per cent. From the ether mother liquors a further batch of crystals was obtained. The next quantity of the phosphonium iodide was prepared by the direct addition of 30 grms. of triethylphosphine to a large excess (200 grms.) of iodide of ethyl, which was con- tained in a flask, fitted with an upright condenser and placed in waiter. After a short time violent ebullition occurred, and a considerable quantity of iodide of ethyl volatilized into the condenser. On distilling off the excess of this from a water bath, a snow-white mass of crystals remained. These were recrystallized by dissolving them in alcohol and precipitating with ether. A determination of iodine in the * Hofmann and Cahours, Annalen, vol. civ. p. 1. Tetrethylphosphonium and their Decomposition by Heat. 185 product was made both (a) before, and (0) after recrystalliza- tion. (a) 0:4065 grm. (dried over sulphuric acid) required 14°85 cubic centim. decinormal nitrate of silver solution=0°1886 grm. iodine=46°'40 per cent. (6) 1:085 grm. (dried at 100° C.) required 39°6 cubic centim. decinormal nitrate of silver solution =0'5029 orm. of iodine= 46°35 per cent. A third quantity of the salt made in a similar manner was also analyzed. 0:9485 grm. required 34°7 cubic centim. of decinormal nitrate of silver solution=0°44069 grm. iodine=46°46 per cent. These determinations show that in each case the phospho- nium salt was pure. Obtained. A ~—-—. —— Calculated for 1. 2, 3. 4. (C,H,),PL 46°36 4640 46°35 46°46 46°35 per cent. iodine. As regards the properties of this salt, we have nothing to add to the description given by Hofmann and Cahours *. Action of Heat on the Hydrate of Tetrethylphosphonium. Hofmann and Cahours f had already studied the behaviour of this compound when heated, and had found that it decom- posed in the following manner :— (C,H;),POH= (C.H;);PO + C,H, 3 but as they did not make any exact determination of the quantity of phosphine oxide, nor of the ethane produced, and as we were particularly anxious to ascertain whether the above equation expresses the whole change which occurs, we determined to repeat the experiment. 10 grms, of pure iodide of tetrethylphosphonium were con- verted into the hydrate by adding excess of oxide of silver to its aqueous solution. The liquid filtered from the resulting iodide of silver was transferred to a distilling flask, connected with an apparatus for collecting gas, and heated. At first no change occurred except simple ebullition ; but when the solu- tion grew concentrated effervescence occurred, and eventually from 780-800 cubic centim. of gas were collected. The temperature of distillation slowly rose to 240-248° C., and the distillate crystallized in the condenser. It was identified as triethylphosphine oxide, both by its boiling-point, appear- * Hofmann and Cahours, Annalen, vol. civ. p. 15. t Ibid. 186 Drs. Letts and Collie on the Salts of ance, and by the formation of the characteristic compound with iodide of zinc. The gas, when passed through solution of hydrate of barium, gave no turbidity. It was not affected by bromine, and burnt with a luminous flame. The theoretical yield of ethane from the quantity of base operated upon is 822 cubic centim.—a quantity which is sufficiently near to that found to prove that the base decomposes wholly in the manner indicated by Hofmann and Cahours. We have already shown that the hydrate of tetrabenzyl- phosphonium suffers a similar decomposition under the influ- ence of heat, * (C,H,),POH = (C;H;);P0 + C,Hg. Action of Heat on the Sulphate of the Tetrethylphosphonium. 25 grms. of pure iodide of tetrethylphosphonium were dissolved in \water and converted into the corresponding sulphate by the action of sulphate of silver. The solution was filtered and treated with hydrochloric acid and sulphur- retted hydrogen to remove the dissolved silver. It was again filtered and concentrated to a small bulk over the water-bath. During the concentration, a strong smell of triethylphosphine became manifest, showing that decomposition had commenced, and this increased as the concentration was proceeded with ; and even when the solution was removed from the water-bath and placed in vacuo over phosphoric anhydride, the smell of triethylphosphine was still apparent. After remaining under these conditions for some time the solution solidified to a highly deliquescent crystalline mass, which could not be obtained in a fit state for analysis owing to its hygroscopic nature. It was accordingly transferred to a distilling flask and heated in an oil-bath. In a short time gas was evolved whilst a colourless liquid distilled, and there remained behind in the flask a charred mass. The distillate solidified on cooling, and from its appearance we were led to the conclusion that it contained sulphide of triethylphosphine, in addition to the oxide. A simple method suggested itself for separating the two bodies, depending upon their different solubility in water, the oxide being soluble in all proportions, whilst the sulphide readily crystallizes from a hot saturated solution. The distil- late was accordingly dissolved in hot water, and as the solu- tion cooled, beautiful needle-shaped crystals of considerable length separated. They were collected on a filter, washed with cold water, and recrystallized from boiling water. * Letts and Collie, loc. ct. Tetrethylphosphonium and their Decomposition by Heat. 187 They possessed the following properties ; they easily volati- lized with water vapour, had a melting-point of 94° C., and when heated with sodium yielded triethylphosphine and sulphide of sodium. These properties indicate that the com- pound was sulphide of triethylphosphine, which was further verified by a determination of sulphur : 0°526 grm. gave 0°840 BaSOQ,=0°1106 S=21°03 per cent. Calculated for (C,H;)3PS . . . . . 21°33 per cent. The mother liquors from which the sulphide of triethyl- phosphine had separated were evaporated to a small bulk and caustic soda added, when an oily layer separated, containing only a trace of the sulphide. This was removed by redis- solving the oily layer in a small quantity of water, and filtering. Caustic soda was then added to the solution, and the oily layer which separated was then decanted and submitted to distillation. The temperature rose rapidly to 240° C., between which and 244° C. the distillate solidified. The boiling-point, appear- ance, and other properties left no doubt as to its consisting of oxide of triethylphosphine ; but to be perfectly certain that such was its composition, some of it was converted into the highly characteristic double salt which it forms with zinc iodide, in which a determination of iodine was made. 0°421 grm. required 14°5 cubic centim. of decinormal nitrate of silver solution=0°18415 grm. iodine=43°74 per cent. Calculated for 2{(C,H;);PO}, ZnI, . . . = 43°27 per cent. The gas collected during the decomposition did not turn lime-water milky, nor was it attacked by bromine. It burned with a luminous flame. The wash-water contained traces. of sulphurous acid. Another experiment was performed in order to see whether any other substances besides the oxide and sulphide of triethylphosphine could be isolated. 14 grms. of the iodide of tetrethylphosphonium were con- verted into sulphate, and the latter submitted to the action of heat. 3°5 grms. of sulphide, and 3:0 grms. of oxide of triethylphosphine were obtained ; the theoretical yield [ on the assumption that {(C,H;),P}280, gives (C2H;)3PS, and (C,H;)3PO], is 3°8 grms. of the former and 3°4 grms. of the latter. No other substances could be isolated, so that there can be little doubt that most of the phosphonium salt is decomposed into 188 Drs. Letts and Collie on the Salts of oxide and sulphide of triethylphosphine, gaseous hydrocarbons, and oxidized products. As, however, a strong smell of triethylphosphine became manifest during the concentration of the salt, another reaction probably occurs, though to a very small extent. Its exact nature we had no means of ascer- taining. Action of Heat on the Carbonate of Tetrethylphosphonium. Hofmann and Cahours had already investigated the action of heat on this salt, and state that it decomposes into triethyl- phosphine and ethyl carbonate, C,H; (C, H;)3P<¢: ‘SCO; =2 (C,H s)3ak + ( (C, H;)2CO3. (Cals). PC 2°*5 This reaction appeared to be of peculiar interest: first because of its analogy with the behaviour of the compounds of triethylsulphine when heated*, and, secondly, because it ought to afford an easy and simple method for preparing triethylphosphine from tetrethylphosphonium salts. Unfor- tunately, however, this decomposition of the carbonate by heat does not occur in the simple manner we anticipated, as the following experiments show. A considerable quantity of the rather impure carbonate was submitted to distillation. It yielded a large quantity of the owide of triethylphosphine, but only a small amount of the phosphine itself. In view of this result we determined to investigate the re- action more carefully, and to employ the carbonate in as complete a state of purity as possible. An aqueous solution of 40 grms. of the pure iodide of tetrethylphosphonium was treated with an excess of recently precipitated carbonate of silver. The solution was filtered from the resulting iodide of silver and concentrated on the water-bath, until a slight smell of triethylphosphine became manifest. It was then well shaken with ether to dissolve out any oxide of triethylphos- phine that might be presentf, and placed in vacuo over phos- phoric anhydride, where, after some time, it solidified to a mass of highly deliquescent needle-shaped crystals. This was transferred to an ordinary distilling flask, which was * Crum Brown and Blaikie, Journal f. prak. Chemie [2] xxiii. p. 395. + Ether readily dissolves tiie body, contrary to the statement of another observer. Tetrethylphosphonmum and thew Decomposition by Heat. 189 connected with a Liebig’s condenser, to which an apparatus for collecting gas was attached. On applying heat to the flask the crystals fused, and soon began to decompose with effervescence. A few grammes of liquid passed over at a low temperature, but the thermometer (in the distilling flask) soon rose to 240°, between which temperature and 250° the bulk of the contents of the flask distilled. The distillate consisted of two layers, the lower of which was simply an aqueous solution of oxide of triethylphosphine; the upper layer (which amounted to only a gramme or two) had a strong odour of triethylphosphine. When shaken with hydrochloric acid about two thirds of it was absorbed ; the remaining third was not attacked, even when boiled with strong hydrochloric acid. This latter had a distinct and cha- racteristic odour, and its boiling-point lay between 90-100° C. As it was not attacked when boiled with hydrate of barium or caustic soda, and as carbonate of ethyl boils at 126° C., its identity with that substance was clearly negatived. The gas evolved during the experiment caused a turbidity with lime-water, and was in part absorbable by caustic potash (to the extent of about one third the total volume). It there- fore contained carbonic anhydride. ‘The residue was not acted upon by bromine; and as it burnt with a luminous flame, it probably consisted of a paraffin hydrocarbon, pre- sumably butane. In order to ascertain the nature of the volatile liquid, which accompanied the triethylphosphine, another experiment was performed with a larger quantity of the carbonate. On sub- mitting it to the action of heat the same phenomena were observed as before, and a quantity of the volatile liquid (freed from triethylphosphine), amounting to two grammes, was obtained. After two or three fractionations its boiling-point was found to be between 95°-102° C. Now this boiling-point agrees fairly well with that of diethylketone (101° C), the formation of which, along with the phosphine oxide, could be accounted for by the equation— it C,H; (C.Hs)3P\% = = 190 Drs. Letts and Collie on the Salts of As Schmidt* denies that this ketone yields a crystalline compound with bisulphite of soda, we were not surprised to find that the body in question would not unite with the bi- sulphite. As its quantity was only small we decided to submit it to oxidation. Accordingly it was digested for some time with bichromate of potash and sulphuric acid, and the mixture then submitted to distillation, when an acid distillate was obtained. This was warmed with oxide of silver, filtered and concentrated. On cooling, a white crystalline silver salt separated, in which, when dry, a determination of silver was made. 0-161 grm. gave 0:0975 grm. Ag=60°56 per cent. Calculated for C;H;0,Ag . . =69°66 per cent. The mother liquor yielded on concentration another quan- tity of silver salt, which contained a higher percentage of silver. 0°1885 grm. gave 0'085 grm. Ag. =61°37 per cent. ~ Calculated for C,H;0,Ag . . =64°67 per cent. These results indicate that the product of oxidation con- sisted of a mixture of acetic and propionic acidsf. The formation of which further proves that the volatile liquid was in reality diethylketone. We consider, therefore, that part at least of the Sr nium salt decomposes i in the manner indicated above. But another and totally different reaction also occurs, in which triethylphosphine, carbonic anhydride, and a gaseous hydrocarbon are produced. The following equation probably represents this second change :--- But we could devise no means of proving it absolutely. No trace of carbonate of ethyl could be detected among the pro- ducts of the decomposition of the phosphonium salt; and we * Schmidt, Ber. d. deutsch. Chem. Ges. v. p. 599. + The analytical numbers are not very satisfactory, but the quantity of material at our disposal was so small that we could not purify the two silver salts. Tetrethylphosphonium and their Decomposition by Heat. 191 are thus forced to the conclusion that Hofmann and Cahours were mistaken in their expijanation of the change which it suffers when heated. Action of Heat on the Acid Carbonate of Letrethylphosphonium. When a solution of the hydrate of tetrethylphosphonium is saturated with carbonic anhydride, the solution is faintly acid, and solidifies when evaporated over sulphuric acid in vacuo to a mass of highly deliquescent needle-shaped crystals. These, on being subjected to the action of heat, decomposed at about 140° C., yielding carbonic anhydride gas, and the salt in the distilling flask turned alkaline in its action on litmus paper; on further heating, large quantities of gas were evolved between 140° and 150°C. The thermometer then rapidly rose to 240° C., when it was moved up into the neck | of the flask; and between this temperature and 250° C. the whole of the contents of the flask distilled. 5 grms. of salt yielded when thus distilled 560 cub. centims. of gases, and a distillate consisting of a little free triethyl- phosphine, some ketone insoluble in hydrochloric acid, and with a boiling-point about 100° C.; but the chief product was the oxide of triethylphosphine. As in the case of the normal carbonate, we could not detect any carbonate of ethyl. The gases were composed of equal volumes of carbonic an- hydride, and a hydrocarbon not absorbable by bromine, which burnt with a luminous flame, and was presumably ethane or butane. These results show that the decomposition of the acid carbonate occurs in the same manner as that of the normal carbonate, into which it is probably first converted. Action of Heat on the Acetate of Tetrethylphosphonium. The interesting and unexpected results obtained with the carbonate induced us to study the action of heat on the acetate of tetrethylphosphonium, as we thought it very pro- bable that it would also yield a ketone, as one half of a ketonic group exists ready formed in the acetyl group. The acetate was prepared both by acting on the pure iodide of tetrethylphosphonium with acetate of silver, and by neutralizing the hydrate with acetic acid. In both cases the solution of the salt was first evaporated on the water-bath to a small bulk and then placed over sulphuric acid in vacuo, and eventually yielded a highly deliquescent crystalline mass. Some of the salt was dried in vacuo over sulphuric acid till constant in weight, and then a combustion was made. 192 Drs. Letts and Collie on the Salis of | 0-203 grm. salt gave 0-428 grm. CO., and 0:218 H,0. C=57:49 and H=11:93. Theory for P(C,H,),0,H,0,. Found. C HBS Oa cclaiwesnab 49 HAG. heh a. eae About 24 grms. of the salt were transferred to a distilling flask connected with a Liebig’s condenser and an apparatus for collecting any gases that might be formed. : On applying heat to the crystals they fused, but no appa- rent change took place until the temperature reached 230° C., when an effervescence occurred and liquid began to distil. HKventually 500 cub. centims. of gas were collected and a large quantity of liquid distillate. This distillate when fractionated yielded two products—one boiling at 80-100° C., the other at 230-250° C. The first had a strong odour of triethyl- phosphine. It was therefore shaken with hydrochloric acid to remove that body. There remained a volatile insoluble liquid smelling of acetic ether. This was warmed for some time with caustic potash solution until the odour of the ether disappeared. As the liquid suffered scarcely any change in volume when thus treated, the quantity of acetic ether present could only have been very small. Purified by these processes the liquid now had a slight pleasant odour and a boiling-point which lay between 80-90° C., but as the quantity at our disposal was very small we could not determine it exactly. The boiling-point of ethylmethylketone is 81° C, and yields on oxidation acetic acid only. To establish the composition of the liquid under examination it was submitted to oxidation with bichromate of potash and sulphuric acid, and yielded an acid distillate. The latter was saturated with oxide of silver, and the mixture warmed and filtered ; on cooling, a considerable quantity of a silver salt separated out, which had all the appearance of the acetate. The salt was dried and a determination of silver made. 0-914 grm. gave 0°588 orm. Ag=64°33 per cent. Calculated for C,H;0,Ag .. . =64°67 per cent. The composition of the salt was further verified by warm- ing a portion with sulphuric acid, when the odour of acetic acid became apparent. The other fraction of the distillate boiling from 230-250° C. yielded on redistillation a considerable quantity of oxide of triethylphosphine, which was identified by its boiling-point, Tetrethylphosphonium and their Decomposition by Heat. 198 appearance, and formation of the characteristic double salt with iodide of zinc. There appeared to be present along with the phosphine oxide some substance boiling at a higher temperature than 245° C., but we did not succeed in isolating it in a-fit state for examination. The triethylphosphine was identified in the hydrochloric acid extract of the first fraction (of the original distillate) by the addition of potash in excess, which at once caused the separation of an oily layer. This had the odour of the phos- phine, and when warmed in air inflamed. On mixing a portion with iodide of ethyl, heat was developed, and a crys- talline compound formed, which analysis showed was iodide of tetrethylphosphonium. 0-501 grm. required 18-2 cub. centims. decinormal nitrate of silver solution= _ 0°28114 iodine . . Ye 20 lo percellt. Calculated for (C,H; ),PI em ty) a = AG.) er Cellts In another experiment 10 grms. of the acetate gave, when similarly heated, a distillate composed of 6°5 grms. oxide e triethylphosphine, half a gramme of ketone, and 1—1°2 grn of triethylphosphine, together with 370 cub. centims. of ois which consisted of nearly equal volumes of carbonic anhy- dride (90 cub. centims.), an olefine (100 cub. centims.), and a paraffin (110 cub. centims.); the remaining (70 cub. centims.) gas being the air displaced from the condensing apparatus. In order to ascertain which member of the paraffin series the hydrocarbon was which was not absorbable by bromine, some of the acetate was heated in an apparatus filled with pure carbonic anhydride. The gases evolved during the dis- tillation were therefore free from air; and by subsequent treatment with bromine and caustic potash successively, all the carbonic anhydride and olefine were absorbed, leaving any hydrocarbon of the C_H,,,, series. The gas which remained after this treatment burnt with a faintly luminous flame, and a rough analysis gave the following results :— I. 10% cub. cent. cub. cent. Gas tke Wiles vei ke se, Stal etre eo fa ss Ghavcen anid GaSe 0 ea en i a), me One 44:0 Oy vens cas wand airs)... =|. oo ?y,,) Mone lin de After explosion . . 33'0 44°5 After addition of 2. Peek potash . 26@ aco a Phil. Mag. 8. 5. Vol. 22. No. 1385. August 1886. O 194 Drs. Letts and Collie on the Salts of : Oxygen con- Carbonic anhy- Ses feet. et ‘ ‘ide rprluleadh cub. cent. cub. cent. cub. cent. Upp haaieer Pa 5°6 12:0 6:0 Te ee pei elles 21°9 11°5 Showing that a volume of the gas requires two volumes of oxygen for combustion, and gives one volume of carbonic anhydride. The gas we therefore concluded to be marsh- gas ; for CH, + 20, = 2H,0 + COs. l vol. 2 vols. 1 vol. The results just described indicate that, like the carbonate, the acetate of tetrethylphosphonium, when heated, suffers two distinct changes. | The amount of gases yielded by its decomposition (370 cub. centims. from 10 grms. salt) is very smail, and is probably formed in the same reaction that produces the tri- ethylphosphine ; while the ketone and the phosphine oxide are probably produced by a totally different decomposition. ‘C,H (1) (C2Hy)sPCo,. C O CH= (C,H;);PO+CH;CO. C,H;. CoH, “a (2) (CoH) PC /OG0=: Action of Heat on Benzoate of Tetrethylphosphonium. The benzoate was prepared by neutralizing hydrate of tetrethylphosphonium with benzoic acid. The solution was evaporated in vacuo over sulphuric acid, and eventually gave a solid mass of radiating deliquescent crystals. 10 grms. of the salt were heated. After the crystals had melted at 160° C., white fumes of triethylphosphine appeared in the dis- tilling flask. The thermometer was then raised out of the liquid (when the temperature had risen to 200° C., and de- composition had begun). The temperature of distillation was higher than usual, the thermometer rising ultimately to above 300°, the liquid which condensed in the receiver being of a yellow colour ; slight charring also occurred. 450 cub. centim. of gases were collected, consisting of equal volumes of carbonic anhydride and a hydrocarbon of the olefine series, probably ethylene. The liquid distillate was composed of about 5 to 5°5 grms. of oxide of triethylphosphine, about ‘5 grm. of free triethylphosphine (soluble in hydro- chloric acid), and about 3 grms. of a liquid insoluble in the Tetrethylphosphonium and their Decomposition by Heat. 198 acid, and not changed when warmed with caustic soda. As this liquid seemed to consist of at least two substances, more of the benzoate was decomposed, in order to obtain enough of the mixture to be separated satisfactorily by fractional dis- tillation. This did not prove difficult, as one of the constituents boiled below 100° C., while the other possessed a boiling- point slightly above 200°C. Hventually two liquids were obtained, one boiling from 80°—-85°, and the other boiling from 210°-215° C. The former had an aromatic sweet smell, and was insoluble in water. A combustion gave the following results :— 0°286 grm. gave ‘960 CO, and *213 H,O. C= %la4,) H=8-27. Calculated for C,H, Found. CREE AS 7-560 (0 1 ee tae een A Ba 8 1 pve hs NR enemy Roranye = Thus indicating that the substance was the hydrocarbon benzol. The liquid possessing the high boiling-point of 210°-215° C. had a sweet pleasant odour, and bearing in mind the results obtained when the acetate was heated, we expected to find phenylethyl ketone, which boils at 210° C., and vields on oxidation acetic and benzoic acids. The liquid was therefore boiled for some time with bichromate of potash and sulphuric acid, and then distilled. The first portions of the distillate were strongly acid, and were neutralized with anmonia. The ammonia salt thus obtained gave with nitrate of silver a erystalline salt, which was recrystallized and analyzed. 0°360 grm. gave 0'225 Ag.. =62°5 per cent. Ag. AoC HO 5)... .. (== 640 per cent. Ag. The low percentage of silver found is possibly due to benzoate of silver being present. The chromic-acid solution remaining after the distillation was shaken out with ether, and the etherial solution evaporated. A crystalline acid remained, possessing the characteristic odour of benzoic acid. It was neutralized with ammonia, and precipitated with nitrate of silver solution. The resulting silver salt on analysis gave the following numbers :— 0°345 grm. salt gave 165 Ag. . =47°82 per cent. Ag. | BEG Og hare. 5 Scho, =47°16 per cent. Ag. As the ketone seemed to be formed in fair amount by this reaction, a considerable quantity of the benzoate was prepared and subjected to the action of heat. In one experiment 10 grms. of salt gave 500 cub. centims. of gas (consisting O 2 196 Drs. Letts and Collie on the Salts of — of equal volumes of carbon dioxide and ethylene), showing that the reaction producing these two substances was very limited. Hventually several grammes of the supposed ketone were obtained, boiling from 205°-210° C. On combustion the following numbers were obtained :— I. 0244 grm. gave 0°6785 erm. CO,, and 0:166 HO. TY, *0-1006 59 0°4885 5 071175 H,O. Calculated for Found. ©,H,COC,H;. 0,H,CO00,H,. iy eee rte Ge SO: 5S 72°00 (5°82 75°86 it: 7°46 6°66 1:56 Cai These results show that the supposed ketone is by no means separated from impurities by simple distillation, and the sub- stance most likely to be formed, and lower the percentage of carbon at the same time, is the ethyl benzoate. As the ben- zoate could be removed by decomposing it with caustic potash, the remainder of the liquid (B.P. 205°-210° C.) was boiled with a solution of hydrate of potassium, washed with water, dried, and redistilled. The boiling-point still remained the same (205°-210° C.), but a combustion gave a much larger amount of carbon— 0:240 grm. gave 0°702 grm. CO, and 0:1603 H,0. Calculated for C.H,COC,H,. Found. OR as) cof B0DS 79-76 2 Ue ire aie 25 7-42 Another quantity of the benzoate was prepared by the action of benzoate of silver on the iodide of tetrethylphos- -phonium. On heating the salt, the same phenomena were observed as with the benzoate prepared by the action of benzoic acid on the hydrate. On fractionating, however, the portion of the liquid distillate which was soluble in water, a quantity boiling from 120°-140° C. was obtained. On re- moving the triethylphosphine by shaking with dilute hydro- chloric acid, an aromatic-smelling liquid remained, which had an odour somewhat like that of cymol, and a boiling-point 130°-140° C. Whether this liquid was ethyl benzol or not we were un- able to determine, as the quantity at our disposal was too small. The next fraction collected boiled at 200°-215° C., and was insoluble in water ; and on oxidation with chromic acid yielded acetic and benzoic acids. The last portions of the distillate were pure oxide of triethylphosphine. As this result was slightly different from those at first ob- Tetrethylphosphonium and their Decomposition by Heat. 197 tained, still another experiment with considerable quantities (40-50 grms.) of the benzoate was made. It was prepared by treating a solution of the hydrate with benzoic acid. The solution was evaporated as far as possible on a water-bath, transferred to a distilling flask, and heated. Slight decom- position occurred before 200° C., but it did not become brisk until the thermometer rose above 300°. The products of the reaction consisted of an aqueous solution of oxide of triethyl- phosphine, above which floated a yellow oil. On distilling the former, eight grammes of triethylphosphine oxide were obtained ; the oily liquid weighed about 32 grammes. It was shaken with hydrochloric acid to remove triethylphos- phine (which, when separated by potash, weighed nearly 8 grammes). ‘The oil, after this treatment, was distilled, and yielded 4—-5grammes of benzol boiling from 78°-90° C., and suf- ficiently pure to solidify in a freezing-mixture. The higher boiling portions consisted chiefly of a liquid smelling strongly of benzoic ether, and boiling from 208°-220° C. On treating this with ammonia gas it almost solidified into a crystalline mass of benzamide. This was removed by boiling water, and the oily liquid which remained once more fractionated, when it all passed over between 208°-214° C. A combustion gave the following results :— | 0°3347 grm. gave 0°952 grm. CO, and 0:2235 grm. H,0. Carbon = 77°5 per cent. Hydrogen = 74 per cent. The hydrocarbon ethyl benzol was carefully looked for in this experiment, but could not be detected. The production of triethylphosphine, its oxide, ethylene, benzol, ethylphenyl ketone, and carbonic anhydride, indicate that the benzoate decomposes when heated in a similar manner to the acetate and suffers two distinct changes—the one yield- ing the free phosphine, the other yielding the oxide. The two reactions may he represented by the following equa- tions * :— and ‘C,H ; (C,H;)sPC OCOC,H, =(C,H;);P + C,H;OCOC,H;. * The production of ethyl benzol could be explained by the equation — (OH), 0,+C,H,C-H Jot 53h \ Gee A =(C,H;),P+CO,+ oH, 3 \OCO:C,H, ( 2 )a = 2 etts 198 Drs. Letts and Collie on the Salts of Action of Heat on the Oxalate of Tetrethylphosphonium. _ This salt was prepared by the action of oxalate of silver on a solution of the iodide of tetrethylphosphonium. The filtered solution was evaporated over a water-bath, and the residue dried in vacuo over sulphuric acid. It solidified to a mass of fine crystals. Some of these were allowed to remain in vacuo till of constant weight, and then a combustion was mace. (I.) 0°270 grm. salt gave 0°553 grm. CO,, and 0-253 orm, HO. C=59°35. =e Another quantity of the solution of the oxalate was evapo- rated in a distilling flask, and finally the temperature of the solution was raised to 160° C., when no decomposition oc- curred ; the liquid on cooling solidified to a mass of crystals. Some of these were analyzed. (II.) 0°315 grm. salt gave 0°285 grm. H,O, and 0°658 ‘erm. CO;:) C=56:96." B= iii Found. ~ Calculated for ae —— [P(C2H;)4]2C20,. I. ie Oris ele OOOO Nias Oe ROD 56°96 A OBA ee eee ie Colca 10°05 This salt, therefore, appears to be stable, and can be heated to 160° C. without any change. Decomposition, however, begins slowly at about 200° C., and at 230° C. gas is rapidly evolved. 6:5 grms. salt when subjected to the action of heat gave 650 cub. centims. of gases ; and the whole of the contents of the flask distilled without charring. The distillate was chiefly oxide of triethylphosphine, together with some triethylphos- phine, and a liquid insoluble in water and hydrochloric acid, which possessed a boiling-point 95°-105° C., and which gave on oxidation acetic acid. It resembled in smell and pro- perties the diethyl ketone, obtained when the carbonate was heated, but it was formed only in small quantity. The 650 cub. centims. of gases were first treated with bro- mine, when 100 cub. centims. were absorbed; caustic potash further reduced these by 100 cub. centims., leaving 450 cub. centim. The remaining gases burnt with a bluish and scarcely luminous flame. | - Several analyses were made of the gas evolved by the decomposition of the oxalate, two of which are subjoined. Tetrethylphosphonium and their Decomposition by Heat. 199 Te JOG cub. centims. cub. centims. Graswiaieemer 5-06 fees wwe BY 10-0 Omycen and gas... «ane ie ys BOO 391 After explosion. . . i iol tanoles 34:2 After addition of caustic potash 220 0) 94-5 Which give :— Carbonic Anhy- Gas taken. Oxygen consumed. dride produced. cub. centims. cub. centims. cub, centims. RRR oie, 4°] hes LOO) 4°6 Did Allowing for the roughness of the experiment and the probable slight impurity of the gas experimented upon, the above numbers agree fairly well with the ratio in volumes. Carbonic Anhydride Gas taken. Oxygen consumed. produced. 2 proving that the gas consisted of carbonic oxide ; for 200 +0, = 2CQ,. 2 vols.+1 vol. = 2 vols. From these results we consider it probable that the de- composition of the oxalate can be expressed by the following equations :— (1) 2H; oats Biel ae 2(C,H,);PO + (C,H,),CO+CO. (CoH;)3P< one . SOM ST (2 | / CyH,H (C,H); < Bees oes 0:CO seatbelts tie i Win =(C,H;);PO + (C,H;)3P + 2C,H, Aes +200-+H,0. (C,H;)s Pg anes 1 sens \C, HA 3) [(C,H;)4P ].C.0,= (C.H;)3P 0 + (CoH;)3P + (C.H;)2,CO + CO,. (4) [(C.H;)4P].C.0,=2(C.H;)3P + 2C,H, + H,O +CO,+ CO. Several attempts were made to prove the existence of ethane in the carbonic oxide gas, and numerous analyses of 200 Drs. Letts and Collie on the Salts of the gas were made without success ; thus proving that any - of the ethyl groups which are split off from the phosphorus atom, if they do not form diethyl ketone, lose a hydrogen atom and are evolved as ethylene gas. An interesting reaction was noticed between free sulphur and the oxalate. The salt, when fused and heated gently to about 160° C., gives, with free sulphur, a splendid deep indigo- blue colour, which turns green and finally yellow on cooling. The colour, however, reappears on heating ; but if too strongly heated the sulphide of triethylphosphine is formed, and total decomposition of the oxalate occurs. Many substances con- taining sulphur, such as vulcanized indiarubber, gunpowder, &c. give this blue colour when even the smallest particles are heated with the oxalate on a porcelain crucible lid. Most of the other salts of tetrethylphosphonium give this colour with free sulphur when warmed, but none gave it in such a marked manner as the oxalate. It is remarkable that the oxy-salts of tetrethylphosphonium should so easily give sulphide of triethylphosphine when fused with sulphur, while the oxide of triethylphosphine itself is totally unacted upon by sulphur under similar conditions. Action of Heat on the Cyanide of Tetrethylphosphonium. As all the preceding experiments were performed with salts of the phosphonium derived from oxy-acids, and as in each case part of the oxygen was eliminated in the form of oxide of the tertiary phosphine, we considered that 1¢ would be of interest to study the action of heat on some of the salts of the phosphoniuim which contained no oxygen, and which therefore could not yield that body. We fixed upon the cyanide to commence with as being likely to decompose at a lower temperature than any other haloid compound. The cyanide of tetrethylphosphonium was obtained in solu- tion by double decomposition between the iodide and cyanide of silver. It was found that when this solution was concen- trated on a water-bath partial decomposition occurred, and hydrocyanic acid was evolved together with an inflammable gaseous hydrocarbon. This decomposition being no doubt due to the action of the water on the cyanide, probably in the following manner :— /ONs (CoHs)3P< oy’ + H,O = (C,H5)3PO0 + HON + C.Hg. The solution was transferred to a distilling flask and heated, when hydrocyanic acid, cyanogen gas, a little free triethyl- Tetrethylphosphonium and their Decomposition by Heat. 201 phosphine, and some oxide of triethylphosphine were pro- duced. Probably also a gaseous hydrocarbon was evolved. Mixed with the triethylphosphine and insoluble either in water or hydrochloric acid werea few drops ofa liquid smell-_ ing of acetonitrile, and evolving ammonia when boiled with - caustic potash. The production of free phosphine and aceto- nitrile indicate that part of the cyanide at least decomposes according to the equation H » (C.H5) eee °=(C.H;)3+ C,H;CN. The decomposition, however, is complex, owing to the high temperature at which it occurs. We may mention that the cyanide when heated with cold eaustic-potash solution is not changed, and appears to be quite insoluble if the solution is strong. On warming the mixture a copious evolution of inflammable gas occurs, and the whole of the cyanide decomposes according to the equation. (CoH,).P(GN ° + KOH =(0,H,);PO+ KON + GH. The action of heat on the iodide was also investigated: it first fuses, but does not suffer any further change until the temperature has risen above the boiling-point of mercury. It then turns brown and splits up into hydriodic acid, free phosphorus, and gaseous products, and a charred residue remains. The decomposition is therefore complex, and cannot be ex- pressed by an equation. Although the iodide is not attacked by an aqueous solution of caustic potash, yet when boiled with a concentrated alco- holic solution it slowly decomposes in the same manner as the cyanide. Action of Heat on the Chloride of Tetrethylphosphonium. After finding that neither the cyanide nor the iodide of the phosphonium when heated yield triethylphosphine in large quantities, it was hardly thought probable that by the action of heat on the chloride the tertiary base would be produced. The results obtained were, however, of the most unexpected nature. 13°5 grms. of the phosphonium iodide were con- verted first into hydrate, and then, by careful addition of hydrochloric acid till the solution was neutral, into the chlo- ride. The solution was evaporated on the water-bath until 202 Drs. Letts and Collie on the Salts of it became concentrated, and was then transferred to a desic- cator and allowed to remain there till it became a mass of solid very deliquescent crystals. . A chlorine determination was made with some of these crys- tals, which were dried in vacuo till constant in weight. I. 0°198 grm. took 11:0 cub. centims. of decinormal nitrate of silver solution =19°7 per cent. Cl. II. 0°2935 grm. took 16:1 cub. centims. of decinormal nitrate of silver solution=19-4 per cent. Cl. Found. Calculated for. as (C.Hs),PCl. J. I. per cent. per cent. per cent. Chri 19°7 19-4. 7 grms. of this solid chloride were heated in a small flask connected with the ordinary apparatus used for collecting any gases evolved. The thermometer in the liquid rose to 300° C. before any decomposition of the salt began. . It was then raised into the neck of the flask. As the decomposition continued a white powdery crystalline substance condensed in the neck of the flask, and great difficulty was found in preventing the exit-tube from becoming completely choked by this salt ; for on attempting to melt it with a gas-flame it only sublimed onto another portion of the tube. Hventually the whole of the chloride was distilled out of the flask, and yielded, besides this solid distillate, 800 cub. centim. of gases. This gas was completely absorbed by bromine, yielding a liquid smelling like ethylene bromide and boiling between 130°-140° C. | Some of the white crystals were guickly scraped out of the condenser and weighed and analysed. 0-310 grm. took 21 cub. centims. of decinormal nitrate of silver solution= 22:99 per cent. Cl. The solution of the crystals in water was just acid to lit- mus paper, and when treated with caustic-soda solution gave free triethylphosphine. From the above results and from the above analysis there was, therefore, no doubt that the substance was the hydrochlorate of triethylphosphine which contains 22°98 per cent. of chlorine, and which, when treated with caustic soda, yields the free base. The decomposition therefore is expressed by the equation, . (CH) eC 4 es (0) PHO OLH,, the amount of gas produced in the decomposition, namely Tetrethylphosphonium and their Decomposition by Heat. 203 800 cub. centim., agreeing with the amount demanded by theory, 860 enb. centim. The remainder of the distillate was dissolved in water, and had a faintly acid reaction; on addition of caustic soda an oil separated, which by its properties and boiling-point, 125°-133° C., was triethylphosphine. The amount of free tertiary base obtained was about three grammes. The hydrochlorate of tri- ethylphosphine obtained by this reaction is a white cystalline powdery solid, which scarcely melts when heated to about 210° C., at which temperature it sublimes or probably dis- sociates, for if any air be in the tube dense white clouds are formed of the oxide of triethylphosphine. As this reaction is the only one which will yield considerable quantities of the tertiary base from the phosphonium salt, another experiment was made, with every precaution to see whether the yield was quantitative. A solution of chloride was quickly evaporated, and finally heated in a weighed distilling- flask till the temperature of the liquid had risen to 180° C.; suddenly the whole frothed up and solidified in the flask to a mass of white crystals. The amount of salt thus obtained was 8 grms. This solid mass of crystals did not melt till the temperature was about 270°C., and no decomposition was noticed till the melted salt was at a tem- perature of 340°C., and gas was not rapidly evolved till the liquid was above 360°C.; 910 cub. centims. of gases were evolved, and the same white distillate was obtained. There was no charring, and the whole of the salt was distilled out of the flask. _ The distillate was washed out of the condenser with a little water and transferred to a flask, through which a current of hydrogen was passed; excess of caustic soda was added, and the triethylphosphine which separated was distilled in a current of steam ; 4°3 germs. were obtained, the boiling- point of which lay between 120°-135° C., and when added to ethyl iodide gave the iodide of tetrethylphosphonium. Part of the gases were treated with bromine, when they were completely absorbed, giving dibromide of ethylene, B.P. 130°—135° C. Another portion was analysed :— cub. centim. fe an | A eg ae) ee rR Gare rymrert RO HA TAY li es DRG After explosion . . Jt eal ER) After addition of caustic potash Sa iat! Ox: : Gas taken. Oxygen consumed, oe ed ade A:2 e.e. i277 Ce 204. Drs. Letts and Collie on the Salts of Approximately the ratio i : 3 : 2% and indicating that the gas was ethylene ; for C,H, + a0, = 2C0O,+2H,0 ° 1 vol. +3 volss=2 vols: From these results it will be seen that the chloride of tetrethylphosphonium, when heated, decomposes nearly quantt- tatively into the hydrochlorate of triethylphosphine and ethylene gas. 8 orms. of the salt should yield theoretically 980 cub. centim, of ethylene gas and 5 grms. of triethylphosphine; the amounts obtained were 910 cub. centim. of pure ethylene and 4°3 grms. of the free phosphine. This reaction we consider to be of great interest, as affording a ready method for obtaining a tertiary phosphine from a phosphonium derivative. : Some of the other salts of tetrethylphosphonium with in- organic acids were prepared, and the action of heat on them was also investigated. The sulphide and hydrosulphide of tetrethylphosphonium were obtained by the action of sulphuretted hydrogen on a solution of the base in water. The solutions were strongly alkaline, and when evaporated 7m vacuo over sulphuric acid gave highly deliquescent needles. Their solution in water precipitated metals like potassium sulphide, and the sulphides of arsenic and antimony were soluble in excess of the solution. Sulphur also dissolved easily, with a yellow colour, forming most probably a persulphide. When a concentrated solution of the sulphide was warmed with ethyl iodide, double decom- position occurred, sulphide of ethyl being produced and iodide of the phosphonium. When the aqueous solution was concentrated in a distilling flask, as soon as all the water had distilled, the sulphide for a moment at 150°-160° C. turned a beautiful deep indigo-blue colour. This disappeared on further heating ; at 220°C. gas being evolved, together with triethylphosphine and sulphide of triethylphosphine. Charring also occured. The hyposulphite of tetrethylphosphonium crystallizes well from a concentrated solution in plates, and when heated the same deep indigo-blue colour was noticed as with the sulphide. The decomposition is complex, as much charring occurs: sul- phide and oxide of triethylphosphine are among the products of the distillation. The sulphocyanate of tetrethylphosphonium crystallizes in highly deliquescent needles, and when heated does not decom- Tetrethylphosphonium and their Decomposition by Heat. 205 pose to any extent till the temperature bas risen above 300° C.; it then slowly chars, yielding gases and sulphide of triethyl- phosphine. The nitrate of tetrethylphosphonium, when evaporated in vacuo over sulphuric acid, is deposited from its solution in deliquescent needle-shaped crystals. These, when heated, suddenly puff, yielding large quantities of gases and some oxide of triethylphosphine. Conclusion. From the foregoing experiments it appears that the salts of tetrethylphosphonium with organic oxyacids suffer, as a rule, at least two, and occasionally three, different and distinct changes under the influence of heat. -- In one of these the molecule splits up into three new groups, consisting respectively of carbonic anhydride, a_ paraffin hydrocarbon, and a tertiary phosphine. In another, two hydrocarbons are formed, namely, an olefine and a paraffin, in addition to carbonic anhydride and the tertiary phosphine. Whilst in the third, a totally different change occurs, in which only two products are formed, namely, the oxide of the tertiary phosphine and a ketone. C.H C,H, iH (2) EP OGGR = Ht;P + CO, + C,H, + RH. (CoH It is possible, if not indeed probable, that the third reaction occurs subsequently to the first, and that it really depends upon the reducing action of the triethylphosphine upon the carbonic anhydride, at the high temperature at which the de- composition usually occurs, whereby carbonic oxide is liberated, which combines with the hydrocarbon radicals in statu nas- cendi, forming a ketone :— Kt; + P+CO,+ (Et) +(R)=Et;,PO+EtCOR (or Et;P + (OCO—R) + (Et) =Et;PO+ Et CO R). If we merely consider the third kind of decomposition alone, it appears to be, to a certain extent, analogous to the decomposition which a sulphine compound suffers when heated, the difference depending on the far more powerful attraction which phosphorus has for oxygen, than sulphur has for the same element. In both cases a hydrocarbon group is 206 Mr. J. Aitken on Dew. detached from the molecule, and also the residue of the acid, but whilst with the sulphur compounds these two simply com- bine (forming a compound ether), and leave a hydrocarbon sulphide, in the case of the phosphonium salt the acid residue is reduced by the tertiary phosphine, and the group thus left combines with the hydrocarbon radical, forming a ketone. Thus :— ‘Bt Zi aes ‘Bt BEPC Gog p= HtsPO+Ht OCR, A result of this kind is in perfect harmony with the views already expressed by one of the authors* regarding the analo-' gies and differences existing between phosphorus and sulphur and their compounds. As regards the action of beat on the hydracid salts of tetr- ethylphosphonium, the chloride and cyanide alone afforded in- teresting results. In the case of the cyanide these, however, were complicated by the presence of water, which attacked the compound and converted it into hydrocyanic acid, phosphine oxide, and ethane. A small quantity of it, however, appeared ~ to behave like a sulphur compound, as it was apparently con- verted into cyanide of ethyl and’ triethylphosphine. The action of heat on the chloride seems to be anomalous, and as it is the only method by which the tertiary phosphine can be obtained in any quantity from a phosphonium salt, the decomposition of phosphonium chlorides containing various hydrocarbon radicals has been the subject of a separate re- search, the results of which one of the authors hopes soon to be AGA to publish. XXII. On Dew. By JOHN AITKEN. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, HAVE read the ‘ Remarks on a new Theory of Dew,” by Mr. Charles Tomlinson, F.R.S., in your Journal for June; [ have also read the paper there referred to, and written by the same author in the ‘Hdinburgh New Philosophical Journal,’ “ On the Claim of Dr. Wells to be regarded as the Author of the ‘ Theory of Dew.’”’ In certain respects and up to a certain point, these two papers bear a strong resembiance to * E. A. Letts, Trans. Roy. Soc. Edin. vol. xxx. part 1, p. 285, Mr. J. Aitken on Dew. 207 each other. The aim and object of the greater part of the first of these two papers appears to be to detract from the merit of Dr. Wells’s great work, by endeavouring to show that almost all the facts observed by him had been previously noted by other investigators. As Mr. Tomlinson, however, just at the end of his paper, reinstates Dr. Wells in the position he previously occupied, perhaps nothing further need be said on that part of the subject; but as he has not been equally generous to me, I must undertake the uncongenial task of trying to put myself right with yourreaders. I will take Mr. Tomlinson’s remarks in the order of his paper, and as your space is valuable, will make my reply as short as possible. At the very outset Mr. Tomlinson, by raising a false con- tention, attempts to place me in opposition to recognized authorities, by entitling his paper “ Remarks on a new Theory of Dew.” The results of my investigation are in no sense entitled to be called, and never have been called by me, a “ New Theory;’’ nor was this so-called ‘new Theory of Dew” “ promulgated in opposition to that of Dr. Wells.” My’ work has not resulted in a new theory ; it is only an extension — of the work the foundations of which were laid by Dr. Wells. | The great fundamental principles established by that authority | are unaffected by anything contained in my paper. When Dr. Wells wrote his celebrated “ Essay on Dew,” he must have been quite aware of the uncertainty as to the source of the vapour. Although he appears to have thought that but little of it could rise from the ground while dew was forming, yet with admirable caution he adds, “ He was not acquainted with any means of determining the proportion of this part to the whole,” thus clearly recognizing the soil as a possible source of the vapour. Wells’s theory, however, has principally to do with the condensation of the vapour after it is in the air, and but little with its source. While quite agreeing with the author as to the difference between “literary work” and “scientific work,” and while he in his comparison would extol the literary, I would rather be inclined to consider the difference between literary and scientific work to be so great that they cannot be compared. Both are useful in their way. Mr. Tomlinson no doubt has had sufficient experience as a thinker, a writer, and an in- vestigator to have seen many a fair theory, apparently perfect in all its parts, and consistently thinkable all through, vanish, under the light of experimental test, like the baseless fabric of a dream at the first touch of day. At pages 485-6 Mr. Tomlinson clearly states the chief points, established by Dr. Wells, under six heads, and then ~ 208 Mr. J. Aitken on Dew. with equal clearness gives under eight heads what he calls Mr. Aitken’s theory of dew. If he had entitled these last as eight points contended for in my contribution to the theory of dew, no fault could be found with this part of his paper, as he states the points with clearness and precision. But unfor- tunately he still looks on these eight points as rivals to the previous six of Dr. Wells, and so continues hostile to them. “Such,” he says, “‘is the new theory of dew, which, if ac- cepted, must go far to render nugatory the results obtained by some of the most celebrated observers.” After reading Mr. Tomlinson’s paper, however, I do not find that he adduces any results of previous observers that are in any way ren- dered nugatory by the results set forth in my paper. It has already been stated that the “new theory ” is not in opposition nor are the results contrary to the teaching of Dr. Wells; nor are they, so far as I know, contrary to those of any of the “‘ celebrated observers.”’ In continuation, my critic then, in place of sustaining his accusation by giving examples in which my teaching is at variance with recognized authorities, gives an example in which my observations agree with recognized authority. At the foot of p. 486 he objects to the statement in my paper that “the ground at a short distance below the surface is always hotter than the air over it,” because it is not a new observation, Pictet in 1779 having previously observed the fact. Iam, however, puzzled to understand what bearing this has on the subject, or what conclusion he wishes to be deduced from this statement. To compare small things with great: more than one person had seen an apple fall to the ground before Newton made such good use of the observation. Pictet, when he made this observation, was working at another subject, and did not draw any conclusion which would be rendered nuga- tory by the acceptance of my conclusions. Mr. Tomlinson proceeds to say, “ that dew rose out of the ground is a very old notion.” I was well aware that the idea was an old one, and it is distinctly stated in my paper. It is difficult to understand what impression my critic wishes to convey by this remark. If it is that the idea does not possess the merit of novelty, having been already expressed, my reply is that it has been already said that dew was caused by the moon-beams, that it descended from the stars, &c.; and each and all of these ideas were supported by about the same amount of evidence, namely the ipse diait of the theorist. The same answer applies to the footnote, p. 486, regarding the exudation of moisture by plants. Though “ Muschen- broeck regarded dew as a real perspiration of plants,” yet Mr. J. Aitken on Dew. 209 beyond the bare statement [ am not aware of anything he advanced to support hisidea. His theory thus fails to explain the difference between the dew-drop on plants and true dew. Nor does it account for dew on dead matter. On these his- torical matters I am, however, open to conviction, as it is extremely difficult to get access to all that has been written on this very popular subject. As I nowhere say that vapour may not be condensed out of air under conditions other than those where dew is forming, the two paragraphs at the foot of page 487 have no bearing on the subject. I am unable to see the relation between the observations made by Wells and Six and myself, which Mr. Tomlinson brings together by the footnote at page 488. In the investi- gations made by the two former observers, the difference remarked was between the readings of thermometers placed on the grass and other substances, and thermometers hung above them; while in my experiments some thermometers were placed on and others under the grass. The only like- ness in the two sets of observations is a similarity in the amount of the difference in the readings of the respective sets of thermometers. It is unnecessary to follow Mr. Tomlinson through the natural processes as described in pages 489, 490, further than to say that the cooling he there describes as taking place on grass at night is not correct. At the middle of page 490 he says :—‘‘ We may imagine three layers of air—one in contact with the points of the grass; the second immediately below it, where the blades are more numerous, and more or less ex- posed to the zenith ; and the third entangled in the matted portion, which is entirely sheltered from the sky.’? He then describes the manner in which he supposes these layers to get cooled, one after the other, beginning at the upper, till “in the end the lower stratum will be colder than the first, so that the blades and stems which are least exposed to the aspect of the sky will be colder than the points of the blades, and the thermometer buried in the grass will mark a lower tempera- ture than one in contact with the surface.” If Mr. Tomlinson had read my paper carefully, or practically put the question to nature, he would not have ventured such an opinion, as he would not, [ expect, ever have found the “‘ thermometer buried in the grass mark a lower temperature than the one in contact with the surface.” On the contrary, all through the night the lower thermometer remains warmer than the upper one. Mr. Tomlinson, in describing the progress of cooling, appears to have forgot to take into account the supply of heat given Phil. Mag. 8. 5. Vol. 22. No. 185. August 1886. P 210 Mr. J. Aitken on Dew. off by the earth. This supply is quite sufficient under all conditions, so far as I have observed, to maintain the soil and the thermometer over it at a temperature much above that at the top of the grass; and while the thermometer at the top of the grass falls quickly at night and falls a good deal, the one “ buried in the grass” falls comparatively little, the upper one remaining much the colder throughout the night. It is not till day arrives and the temperature rises that the upper thermometer gets heated and rises to the temperature of the lower one. With regard to the remarks about dew in “ Persia” and “the African desert,” I have nothing to say, as they do not bear on my paper, it being distinctly stated there that my remarks apply only to this climate. I may, however, add here that I wait for further information before forming any opinion as to what takes place in other and unknown con- ditions. At page 493, Mr. Tomlinson says, “There is such a vast consensus of scientific opinion in favour of the received theory of dew, that any attempt to set it aside in favour of another must be supported by the strongest experimental evidence.” Now, as I have already said, I have never made any attempt to set aside Wells’s theory; and as all my work has tended to confirm and extend his, I hope Mr. Tomlinson will, on recon- sideration, give my contribution to this subject a more favour- able reception ; and might I also ask a more careful consider- ation of that part referred to by him in this same paragraph, where he says, ‘‘ Mr. Aitken exposes a turf six inches square to the air in a scale-pan”’!! This misconception of the essen- tial conditions of the experiment has given rise in his mind to the objection to the conclusion I have drawn from the experiment, and has also given rise to the confusion Mr. Tom- linson has made in comparing my experiment with other previous ones. If Mr. Tomlinson had delayed till he had seen the complete paper, he probably would not have found the difficulty he has experienced in reconciling the two sets of observations referred to in the first paragraph, page 494, of his paper. The paper containing the account of my experiments, on which Mr, Tomlinson founds his criticism, being in abstract was made as short as possible, and does not contain sufficient information to make the conditions of the experiments clear to Mr. Tomlinson. The edges of the trays were, as he supposes, in contact with the ground; one slate and one weight were also in contact with the ground. But, in referring to the latter, they were simply described as they appeared without Mr. J. Aitken on Dew. 211 touching them, that is without lifting them from the ground; whereas it is the condition of the undersides or insides of the trays that are specially referred to. The slate and weight resting on the ground were of course quite wet on their under- sides, which were invisible except when moved, while their exposed parts were dry ; whereas the elevated slate and weight were wet all over. With this explanation I have no doubt Mr. Tomlinson will see the importance of the contact with the ground in keeping the temperature of the slate and weight above the dew-point; while the wetness of their undersides, which are invisible, brings these experiments into harmony with the experiments with the trays. In the second last paragraph of Mr. Tomlinson’s paper he seems to object to that part of the “ new theory ” which relates to the excretion of liquid by plants, principally because it is “startling”? and not in keeping with preconceived opinions, and he asks, “ for what purpose are plants endowed with such high radiating powers?” I scarcely think he can expect me to answer that question, as it forms no part of my theory. If any one, surely he is the proper person to give the reply. But are plants really such good radiators? ‘Take even the figures given by Mr. Tomlinson for the different substances named. In none of these is the difference of any importance; indeed the differences are not more than might easily arise from errors of observation by the method used by Melloni, except in the case of vegetable mould. So far as my own measurements go, by the method used by me, I have not found any difference in the radiating powers of grass and soil at night. If there is any difference, it must be very small. One result of recent investigations is to show that plants with leaves like those of grass are very badly adapted for collecting dew, the blades being little narrow strips placed at right angles to the direction of the air-currents ; the slightest movement of the air prevents the surfaces of the blades being cooled by radiation much below the temperature of the air. In illus- tration of this I may mention that narrow strips of glass or other material exposed at night will sometimes collect no dew, while a sheet of the same material will be running with water, the windward edges being of course dry. Plants with large leaves have quite an advantage over grass in dew-collect- ing powers; but, on the other hand, grass has the advantage during sunny days, on account of its narrow blades being but little heated by the sun. This may be one reason, though not ae only one, why grass so seldom flags under a scorching eat. If I might ‘be allowed to make a slight explanation here, BZ 22: , Mr. J. Aitken on Dew. perhaps it would help to remove a difficulty some have felt in accepting some of the conclusions contained in my paper. One of the results contended for is, that dew on grass and on bodies near the ground is formed from vapour rising from the ground during night ; the reason given for this conclusion being that grass-land is always in a condition to give off vapour during dewy nights. The vapour that rises from the ground after sunset will thus displace the vapour that rose during the day, and the latter will diffuse itself into the drier air over the grass. ‘The stems and blades of the grass during night will thus be surrounded by the vapour that has risen most recently from the ground. Another reason given is, that at night the temperature of the air among the stems is much higher than that of the air at the tips of the blades, and being in contact with moist soil is nearly saturated. The vapour-tension of this hot air, rising among the stems, is thus much higher than that of the air over them, and is thus in a much more favourable condition for forming dew than the air higher up. These are the reasons given for concluding that dew on grass is formed from vapour rising at the time from the ground and not from that which rose during the day. The principal difficulty experienced by some in accepting this conclusion, seems to have arisen from extending this con- clusion to the source of the dew deposited on bodies placed some distance from the ground. Now, when we consider what takes place in these higher positions, it is easy to see that the conditions are much more complicated, and we can now say very little about the vapour, either as to the place of its evaporation or as to the time when it changed its state. Whenever the vapour coming from the ground rises above the grass and mixes with the drier air over it, we get an entirely different and much more complicated condition of matters. No doubt some of the vapour molecules in this upper air will have risen from the ground only a short time before, but some of them will certainly, if there is the slightest wind, be mole- cules which have risen during the day, and no doubt some of them will have risen into the air many days previously ; and while some will have risen from the ground immediately underneath, others will have come from lands and oceans far away. But while this may be so, it in no way affects the conclusion that vapour is almost constantly, night and day, given off by the soil, and that dew on grass is part of this rising vapour trapped by the cold blades. Darroch, Falkirk, Yours truly, June 14, 1886, JOHN AITKEN. elas | XXIV. On the Electromotive Force of Voltaic Cells having an Aluminium Plate as one Electrode. By A. P. Laurin, eee SC." | a former number of this Journal f I have given an ac- count of some experiments undertaken with the view of testing whether the E.M.F, ofa cadmium-iodine cell was really due to the combination of cadmium and iodine. As there explained, these experiments were made on account of my feeling an increasing distrust of measurements of H.M.F. made in the usual way with infinitely small currents,—a dis- trust not of the measurements themselves, but of our right to assume that the H.M.F. of a particular cell was due to a particular chemical reaction. This doubtis obviously justified when once we realize that impurities far out of reach of analytic detection could easily produce sufficient current for the measurement of H.M.F. in the usual way. On reading the papers published by Dr. Alder Wright in this Magazine in 1885, I was very much struck by the results of his experiments on aluminium-zince cells, in which he found an H.M.F. actually opposed to that calculated from the thermal data. To quote his figures, given on p. 28, vol. xix. of the Phil. Mag. for 1885, he finds that zinc-aluminium cells, with the metals immersed in sulphate of zinc and potash-alum re- spectively, have an H.M.F’. of 538 volt. According to the thermal data, there should be an E.M.F. of -982 volt in the opposite direction, so that we have a difference of 1:519 volt between the H.M.F. calculated from thermal data and that obtained by actual measurement. Dr. Wright’s experiments with other aluminium cells give similar results. Now observe that the H.M.F. calculated from the thermal data is obtained by taking the heat of formation of sulphate of zinc and sulphate of alumina respectively, and subtracting the comparatively small heat of formation of sulphate of zinc from the large heat of formation of sulphate of alumina ; that, therefore, the assumption is made that this cell is similar to a Daniell cell, and that sulphate of alumina will be formed and sulphate of zinc decomposed, by the passage of the current. The E.M.F. measurement, however, shows an E.M.F. in the opposite direction to that calculated on this assumption. How is this to be accounted for? Dr. Wright accounts for it by assuming a “thermo-voltaic constant” of 1°519 volt. By this I understand him to mean that he believes that sul- * Communicated by the Author, Tt Phil. Mag. May 1886, p. 409. 214 Mr. A. P. Laurie on the Electromotive Force Ofer phate of zinc is formed and sulphate of alumina decomposed during the passage of a current, and that an enormous ab- sorption of heat takes place to enable this reaction to go forward. If we believe that we are dealing only with metallic alu- minium, sulphate of alumina, sulphate of zinc, and metallic zinc, then we must make the above assumption, and un- doubtedly chemical analysis would confirm such a belief. I think, then, I have said sufficient to show that we have here a point of some importance,—that if we have only the above substances to deal with, Dr. Wright has made a surprising discovery. It was to test whether we really had these substances present that I have made the following experiments. Before describing the experiments, however, I must say a few words about aluminium itself. It is a metal with some very peculiar and contradictory properties. Though haying a very high heat of combination with oxygen, it does not tarnish in the air or decompose water at ordinary tempera- tures, nor is it soluble in nitric acid. If used as the electrode at which oxygen is set free by the passage of a current through dilute sulphuric acid, it opposes a very considerable resistance to the passage of the current. This resistance has been called ‘ polarization,” and stated as high as six volts. On amalgamating the metal (which can only be done by immersing first in strong caustic) it becomes capable of de- composing water ; and its oxidation in the air is so rapid that if left at rest it becomes covered with a growth of oxide in a few minutes, resembling some fungus growths in appearance. Now all these properties seem to me to be very easily ex- plained by supposing aluminium to become rapidly coated with a film of oxide, whether in air or water, which, being very insoluble, protects it from all further action, and practi- cally for chemical purposes converts an aluminium plate into an aluminium-oxide plate, unless some solvent of the oxide is present. Starting with this view, I prepared a solution of aluminium sulphate and of zinc sulphate, enclosing one in a porous pot, and immersed a zine rod in the zine sulphate and an alumi- nium wire in the aluminium sulphate. I did not note the strenoth of the solutions, as the experiments were merely qualitative. I obtained a deflection on connecting this cell with the electrometer, amounting to about ‘54 volt in the same direc- tion as that obtained by Alder Wright. On removing the aluminium wire, cleaning it with sandpaper, and immediately Voltaic Cells with Aluminium Plate as one Electrode. 215 plunging it into the liquid and connecting with the electro- meter, a deflection was obtained in the same direction, but only amounting to ‘14 volt. After ten minutes it had risen to ‘35 volt. This seemed to show that the complete forma- tion of the oxide film took some time. The introduction of a little caustic potash also brought the H.M.F. nearly to zero, but I had not as yet obtained an actual reversal of the E.M.F. of the cell. I next amalgamated the wire and plunged it in the solu- tion. I at once obtained a deflection of about °46 volt in the opposite direction. At the same time hydrogen was given off, and the wire above the solution became coated with the growth of oxide already referred to. The next experiment was an obvious one. I plunged two aluminium wires, one cleaned, the other amalgamated, in a solution of sulphate of alumina, and obtained a deflection of 1°08 volt, showing a remarkable difference in chemical pro- perties between the amalgamated and cleaned aluminium wires, In order to test the effect of short-circuiting on the cell, I left a zinc-aluminium cell short-circuited for some days, but the aluminium plate, except for a slight darkening of the surface, remained unchanged, showing that to whatever re- action the H.M.F. of the cell was due, it was of so tempo- rary a kind, or the internal resistance of the cell was so ae, that no perceptible chemical change had been brought about. These experiments are, I think, sufficient to show that there is no evidence that the reactions in the cell are simply the formation and decomposition of the sulphates of zine and aluminium respectively, but that there is strong evidence to show that we are not in the cell dealing with aluminium at all, but with aluminium oxide supported on an aluminium plate. It is therefore more probable that the E.M.F. of the cell is due to the heat of formation of zine sulphate, — the heat of formation of aluminium sulphate,+the heat of for- mation of aluminium oxide, — the heat-formation of water. These reactions would result in an E.M.F. in the direction actually observed. It may be noticed that even the amalga- mated aluminium gives too low a value for the reactions of the formation and decomposition of the sulphates. As, how- ever, the water is being decomposed by the metal, we can hardly tell what the actual source of H.M.F. may be. I do not know if Dr. Wright attached any definite mean- ing to the term “ thermo-yoltaic constant’’ in the case of these particular cells, as the assumption that the reactions are 216 Prof. 8. P. Thompson on a Mode of the formation and decomposition of the sulphates is evidently questionable, even from the results he obtained. This paper is not of course meant as a criticism on Dr. Wright’s work, which I have always admired for its accuracy, but merely gives a new interpretation of certain of his experimental results. XXV. Note on a Mode of maintaining Tuning Forks by Hlectricity. By Prof. 8. P. THompson*. LL who have worked with self-maintained electric tuning- forks have met with the fact that these instruments as ordinarily arranged give a very irregular note, the pitch of which is continually slightly altering, and even those which are fairly constant in pitch are continually changing the phase of their vibrations. These changes of phase and of pitch render the electrically-sustained tuning-fork as usually con- structed almost useless for acoustic work, and diminish the usefulness of the instrument for chronoscopic and electric applications. They appear to be due to a fact which is toler- ably obvious to any one acquainted with the fundamental principles of the vibrations of elastic bodies, namely, that the impetus given by the electromagnet at each vibration is given at the wrong instant of the motion, namely at some other instant than that during which the fork is passing with maxi- mum velocity through the position of zero displacement. In the older forms of electro-diapason constructed by Fessel and by Koenigt, the electromagnet was of horseshoe form, having poles outside the prongs of the fork ; whilst in the more recent instruments by Koenig the electromagnet is of short cylin- drical form, and placed between the prongs of the fork. The latter arrangement, which is preferable for several reasons electrical and mechanical, seems to have been first suggested by Lord Rayleight. In the earlier form contact was made by a stylus, carried by the prong of the fork, dipping into the mercury cup: in the later form the stylus usually makes con- tact against a platinum-headed screw, almost exactly as does the interrupter of Wagner so common in electric bells. In either case contact is made at a very brief interval before the prong of the fork reaches its extreme elongation away from the pole of the electromagnet, and is broken at a slightly longer interval after the prong has passed its extreme elonga- * Communicated by the Physical Society: read June 26, 1886. + Vide Helmholtz, ‘Sensations of Tone,’ Ellis’s edition of 1875, p. 178. ¢ ‘Theory of Sound,’ vol. i. p. 56. Maintaining Tuning-forks by Electricity. 217 tion, the difference arising from the elasticity of the stylus, and the necessary imperfection of the contact until a certain actual contact-pressure has been attained. There is also a certain retardation in the electro-magnetic pull behind the instant of greatest current, owing to the self-induction of the circuit and the mutual induction between the coil and its core. Butif the core be short, and laminated, and of good iron, and if the resistance of the circuit be considerable in proportion to its coefficient of. self-induction, the retardation of phase in the periodic electromagnetic impulses will not be of any great importance. Lord Rayleigh remarks* that if the magnetic force depended only on the position of the fork the phase might be considered to be 180° in advance of that of the fork’s own vibration. That is to say, considering a displacement of the fork toward the electromagnet positive, the maximum force occurs when the displacement is a nega- tive maximum. But, adds Lord Rayleigh, the retardation due to self-induction and imperfect contact reduces this advance. Lord Rayleigh further remarks that if the phase-difference be reduced to 90° the force acts in the most favourable manner, and the greatest possible vibration is produced. He might have added that in this case the tendency to produce phase- change is the least possible. He suggested as a means of producing any desired retardation, the use of a stylus attached not to the prong itself, but to the further end of a light straight spring carried by the fork. It seems to the present writer, that a better way to secure the proper timing of the impulses is to be found in the sug- gestion which he now makes, and which arose in his mind after considering the difference of phase which exists between two dynamo-electric machines associated together. Let two forks in unison with one another be provided, and let each act as interrupter to the other, but not to itself; the electromag- net of each being included in the circuit of the other’s contact- points. One battery will suffice for the two, as they will not both make contact at the same time. Fig. 1 shows the proposed arrangement. The forks when started will settle down to a difference of phase corresponding to an almost exact quarter of a period. Fork B is arranged, as shown in fig. 1, so that it makes contact at the inward stroke at the point when its displace- ment is at the positive maximum; fork A makes its contact at the outward stroke when its displacement is at the negative maximum. * Op. cit. p. 59. 218 Prof. 8. P. Thompson on a Mode of Fig. 1. Reference to figure 2 will elucidate the phase relations. Tn this figure the upper curve relates to fork A and the lower to fork B. Positive values of the ordinates relate to positive displacements in the sense of approach to the electromagnet. Fig. 2. When A has moved through } of its cycle of movements, and the displacement is a negative maximum 4}, it makes contact for B’s electromagnet, which gives a momentary im- maintaining Tuning-forks by Electricity. 219 pulse just as B has phase 0°. A quarter of a period later B makes contact (5,), and gives an impulse to A just when A’s phase is 0°. The arrows indicate the instant and direction of the forces. It is assumed that the retardation due to self- induction is small, as compared with the period of the forks ; _ and this may easily be made so, firstly, by employing only short internal electromagnets with laminated cores, and, secondly, by interposing sufficient inductionless resistance in the circuit. Another point of some importance in the construction of electrically sustained forks is that the contacts should be very firm, and should be made much nearer the hilt of the fork than is usual in these instruments. When the stylus is near the outward end of the prong there is much greater amplitude of motion at the contact than is really requisite. A pin of platinum secured to the prong at about 5 centimetres from the point of bifurcation, making contact against a platinum-faced strip of German silver, half a millimetre in thickness, to serve as a spring, will answer the purpose for forks of ordinary size. A common imperfection in the electro-diapason as usually constructed is the method of mounting the fork. Its shank is held with nut and washer to a block of wood or metal, which is then secured to a stand by a single bolt or screw which runs at right angles to the shank and to the planes of vibration of the prongs. The defect of this mounting is that the fork can shift a little round the bolt, and is lable to _become set with one prong nearer one face of the electromag- net than the other prong is to the other face. This often results in the occurrence of actual rattling contacts between the fork and the electromagnet, as well as in derangements of the adjustment of the working contacts. It also gives rise to another kind of difficulty: if one prong is nearer to the electromagnet than the other is, there will be a tendency for the fork to vibrate as a whole around the bolt or frame upon which it is mounted, and this will give rise to slow alternations of good and bad contacts, producing on the sound an effect not unlike that of beats. Hither the bolt which secures the fork to its mounting should lie in the plane of the vibrations, or else it should be replaced by a more substantial species of mounting. [2200 XXVI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. {Continued from p. 79. ] June 23, 1886.—Prof. J. W. Judd, F.R.S., President, in the Chair. HE following communications were read :—- 1. “On some Perched Blocks and associated Phenomena.” By Prof. T. M*Kenny Hughes, M.A., F.G.S. The Author described certain groups of boulders which occurred on pedestals of limestone rising from 3 to 18 inches above the level of the surrounding rock. The surfaces of these pedestals were stria- ted in the direction of the main ice-flow of the district, while the surrounding lower rock in no case bore traces of glaciation, but showed what is known as a weathered surface. He inferred that the pedestals were portions of the rock protected by the overhanging boulder from the down-pouring rain, which had removed the surrounding exposed parts of the surface. When the pedestals attained a certain height relatively to the surrounding rock the rain would beat in under the boulder, and thus there was a natural limit to their possible height. He referred to the action of vegetation in assisting the decompo- sition of the limestone, and considered that there were so many causes of different rates of waste and so many sources of error, that he distrusted any numerical estimate of the time during which the surrounding limestone had been exposed to denudation. Considering the mode of transport of the boulders, he thought that they could not have been carried by marine currents and coast-ice,: as they had all travelled, in the direction of the furrows on the rock below them, from the parent rock on the north. Moreover, marine currents would have destroyed the glaciation of the rock and filled the hollows with débris. Furthermore, the boulders and strie are found in the same district at such very different levels and in such positions as to preclude the possibility of their being due to icebergs. Nor could the boulders represent the remainder of a mass of drift which had been removed by denudation, for the following reasons :— 1. They were all composed of one rock, and that invariably a rock to be found in place close by. 2. Any denudation which could have removed the clay and smaller stones of the drift would have obliterated the traces of glaciation on the surface of the rock. 3. The boulder which had protected the fine glacial markings below it from the action of the rains would certainly in some cases have preserved a portion of the stiff Boulder-clay. 4, The margin of the Boulder-clay along the flanks of Ingle- borough was generally marked by lines of swallow-holes, into which Geological Society. 221 the water ran off the Boulder-clay ; and when the impervious beds overlying the limestone had been cut back by denudation, a number of lines of swallow-holes marked the successive stages in the process; but there was not such evidence of the former exten- sion of the drift up to the Norber boulders. 5. The boulders themselves were not rounded and glaciated in the same way as the masses of the same rock in the drift, but resembled the pieces now seen broken out by weathering along the outcrop of the rock close by. Having thus shown the improbability of these boulders having been let down out of a mass of drift the finer part of which had been removed by denudation, or of their having been masses floated to their present position on shore-ice, he offered an explanation of their peculiar position, which he thought was not inconsistent with the view that they belong to some part of the age of land-ice. That they were to be referred to some exceptional local circum- stances seemed clear from the rarity of such glaciated pedestals, while boulders and other traces of glaciation were universal over that part of the country. Hetherefore pointed out, in explanation, that they occurred always where there was a great obstacle in the path of the ice :—at Cunswick the mass of Kendal Fell curving round at the south and across the path of the ice; at Farleton the great limestone escarpmert rising abruptly from Crooklands; at Norber the constriction of the Crummack valley near Wharfe and the great mass of Austwich grit running obliquely across its mouth. In all these cases the ice had to force its way up hill; and there would be a time when it would just surmount the obstacle after a season of greater snowfall, and fall back after warm seasons, until it fell back altogether from that part. During the season of recession, boulders would be detached below the ice-foot ; during the seasons of advance they would be pushed forward ; and in those exceptional localities of isolated hills from which the drainage from higher ground was cut off, the boulders were left on a clean furrowed surface of lime- stone, which was then acted upon by rain-water and the vegetation, except where protected by the boulders. 2. “On some derived Fragments in the Longmynd and newer Archean Rocks of Shropshire.” By Dr. Charles Callaway, F.G.S. Further evidence was added to that given in the Author’s previous paper (Q.J.G.S. 1879, p. 661), to show that the Longmynd rocks of Shropshire were chiefly composed of materials derived from the Uriconian series, and that the Uriconian series itself (Newer Archean) was partly formed from the waste of pre-existing rocks. This evidence consisted of (1) the presence, throughout the greatly developed Longmynd conglomerates and grits, of purple rhyolite fragments, recognized by microscopical characters as identical with the Uriconian rhyolites of the Wrekin, and the occurrence of grains, probably derived from the same rhyolites, in the typical green slates of the Longmynd; and (2) the existence of conglomerate beds con- taining rounded fragments of granitoid rock in the core of the 222 Geological Society :— Wrekin itself, whilst the Uriconian beds of other localities, and especially those of Charlton Hill, contained waterworn pebbles, chiefly metamorphic. These pebbles appeared to have been derived from metamorphic rocks of three distinct types. The views put forward were founded on microscopical evidence, of which some details were given in the paper, and were supported by the views of Prof. Bonney, who had furnished notes on the microscopical charac- ters of the rocks. 3. “ Notes on the Relations of the Lincolnshire Carstone.” By A. Strahan, Esq., M.A., F.G.S8. The Lincolnshire Carstone has hitherto been supposed to be correlative with the upper part of the Speeton series, and to be quite unconformably overlain by the Red Chalk (Quart. Journ. Geol. Soc. vol. xxvi. pp. 326-347). But the overlap of the Carstone by the Red Chalk, which seemed to favour this view, is due to the northerly attenuation, which is shared by nearly all the Secondary rocks of Lincolnshire. . Moreover, the Carstone rests on different members of the Tealby group, and presents a strong contrast to them in litho- logical character, and in being, except for the derived fauna, entirely unfossiliferous. It is composed of such materials as would result from the ‘* washing ” of the Tealby beds. In general it is a reddish-brown grit, made up of small quartz- grains, flakes and spherical grains of iron-oxide, with rolled phos- phatic nodules. ‘Towards the south, where it is thick, the nodules are small and sporadic. Northwards, as the Carstone loses in thick- ness, they increase in size and abundance, so as to form a “ coprolite- bed,” and have yielded specimens of Ammonites speetonensis, A. plic- omphalus, Lucina, &e. When the Carstone finally thins out, the conglomeratic character invades the Red Chalk, similar nodules being then found in this rock. The presence of these nodules, with Neocomian species, taken in connexion with the character of the materials of the Carstone, points to considerable erosion of the Tealby beds. On the other hand, there is a passage from the Carstone up into the Ked Chalk. It would seem, then, that the Carstone should be regarded as a ‘“‘ base- ment-bed”’ of the Upper Cretaceous rocks. The Lincolnshire Carstone is probably equivalent to the whole of the Hunstanton Neocomian, the impersistent clay of the latter being a very improbable representative of the Tealby Clay. It therefore follows that the whole Speeton series is absent in Norfolk, and also in Bedfordshire. The unconformity at the base of the Carstone becomes greater southwards, and the nodules have been derived from older rocks. Similarly north of Lincolnshire, where the Spee- ton series is overlapped, the nodules in the Red Chalk, marking the horizon of the Carstone, have been derived from oolitic rocks. In the South of England it would seem that equivalents of the | Speeton series reappear. The Atherfield clay contains an indigenous Upper Speeton fauna, while a pebble-bed near the base of the Folke- stone beds is described by Mr. Meyer as containing derived oolitic On some Well-sections in Middlesex. 223 pebbles, and being probably the representative of the Upware deposit, and presumably, therefore, also of the Lincolnshire Car- stone. 4, “The Geology of Cape-Breton Island, Nova Scotia.” By Edwin Gilpin, Esq., Jun., A.M., F.R.S8.C., Inspector H.M’s. Mines. After referring to previously published descriptions of Cape Breton geology, the author stated that the various formations found in the island had been thus classified by the officers of the Geological Survey :— _Pre-Cambrian (Laurentian) including The Felsite series. | The Crystalline Limestone series. Lower Silurian. Devonian. Carboniferous, including ( Lower Coal-formation. | { Gypsiferous series. <_ | limestones, &c. | Mullstone-Grit. \_ Middle Coal-formation. He then proceeded to give an account of each system and its sub- divisions in order, commencing with the most ancient and adding a few detailed sections of the rocks belonging to some of the principal series. He described the distribution and relations of the several divisions. The paper concluded with a few notes on the superficial geology of the island: There is a general absence of moraines and of the fossiliferous Post-Pliocene marine clays of the Lower St. Lawrence. The older beds are generally exposed, but deeper soils and deposits with erratic boulders are found overlying the Carboniferous beds. Marks of recent ice-action are found on the shores of some of the lakes, and are due to the ice being driven by the wind. 5. “On the Decapod Crustaceans of the Oxford Clay.” By James Carter, Esq., F.G.S., &c. 6. “Some Well-sections in Middlesex.” By W. Whitaker, Esq., B.A. Lond., F.G.S. Accounts of many well-sections and borings having been received since the publication of vol. iv. of the Geological Survey Memoirs, the Author now gave more or less detailed descriptions of fifty-six of these, all in the Metropolitan county, and all either unfinished or, in a few cases, with further information as to published sections. The depths range from 59 to 700 feet, more than half being 300 feet or more deep. Nearly all pass through the Tertiary beds into the Chalk, and most have been carried some way into the Jatter. Papers descriptive of like sections in Essex, Herts, and Surrey have been sent to Societies in those counties. 224 Geological Society :— 7. “On some Cupriferous Shales in the Province of Houpeh, China.” By H. M. Becher, Esq., F.G.S. This communication contained some geological observations made during a visit to a locality on the Yangtse river, near I-chang, about 1000 miles from the sea, for the purpose of examining a spot whence copper-ore (impure oxide with some carbonate and sulphide) had been procured. The principal formations in the neighbourhood of I-chang were said to be Paleozoic (probably Carboniferous) limestones of great thickness, overlain by brecciated calcareous conglomerate and reddish sandstones, which form low hills in the immediate vicinity of the city. About fifty miles further west the limestones pass under a great shale-series with beds of coal, the relations of which to the sandstones are not clearly ascertained. The copper-ore examined by the writer came from the shales, which contained films and specks of malachite and chrysocolla, and in places a siliceous band containing cuprite, besides the oxydized minerals, was interstratified in the beds. Occasionally larger masses of pure copper-ore are found imbedded in the strata. The ground had not been sufficiently explored for the value of the de- posits to be ascertained. 8. “The Cascade Anthracitic Coal-field of the Rocky Mountains, Canada.” By W. Hamilton Merritt, Esq., F.G.S. The coal-field named occurs in the most eastern valley of the Rocky Mountains, that of the Bow river, and, like other coal-fields of the country, consists of Cretaceous rocks, which lie in a synclinal trough at an elevation of about 4300 feet above the sea. The underlying beds, of Lower Carboniferous or, possibly, Devonian age, rise into ranges 3000 feet higher. Further to the eastward the Jurassic and Cretaceous coal contains a large percentage of hygroscopic water and volatile combustible matter, and has the mineral composition of lignite. The average composition is :— f Per cent. Hixedtearbom 672982 hee 42 Volatile combustible matter. 34 Hiyeroscopie waters... 2... 16 Se Nita eee Woke tke 8 100 As the mountains are approached, the amount of hygroscopic water is found to diminish by about one per cent. for every ten miles, and fifteen miles from the range the percentage is about five. In the foot-hills the lignites pass into a true coal, with 1°63 to 6-12 per cent. of hygroscopic water, and 50 to 63 per cent. of fixed carbon. In the Cascade-river Coal-field the average character of the coal is that of a semianthracite, with the following composition :— On certain Eocene Formations of Western Servia. 2.25 Per cent. Brxecu) carbon cee. fo epee: 80°93 Volatile combustible matter.. 10°79 EHysroscopic- water *). 2... 50. aid Sv lees ey cum ar ise Pith) Ae 757 100-00 The coal-seams have been subjected to great pressure, and the change in the quality of the coal appears to be due to metamorphic influence. 9. “On anew Emydine Chelonian from the Pliocene of India.” By R. Lydekker, Esq., B.A., F.G.S. 10. “On certain Eocene Formations of Western Servia.” By Dr. A. B. Griffiths, F.R.S.E., F.C.S. A great thickness of paper-shales containing paraffin occurs near the river Golabara; these extend over 30 square miles of country. Small beds of clay with rock-salt are also found: thé whole is said to resemble the paraffin and salt districts of Galicia. The paraffin shale is free from bituminous impurities. It contains :— Per cent. Parathn Wax. chit Wereoie 1°75 Water of combination.... 3°02 AM ORD Wa Ss. EE 3 1718 The mineral constituents of the shale are: Per cent. 7 9801110101 eee eet me pe oy 32°86 Tram Omen. oc ced «ah a 5:20 Mig EST chs hx) 5th eete sence -26 DS ie PI Np So Wt 2 uk POG iy pit cusses Sh ey ka AWE S(C10 2 ee eee ae PO eae 0-41 SLETC Ss Se aad, Nae prea 56°85 1 Cn ee ge tame eee eee 0:04 100-00 The brown coal of the neighbourhood, whose natural distillation has most probably yielded the hydrocarbon in the shales, contains :— Per cent. Cane Jaca wes te Be 49-2 ER veROteM. Soe. Ck,» ae ge | Water, combined ...... 30°2 Water, hygroscopic .... 19°5 100-00 The beds containing these coals have been invaded by eruptive Phil. Mag. 8. 5. Vol. 22. No. 1385. August 1886. Q 226 ~ Intelligence and Miscellaneous Articles. porphyry and trachytic rocks, of which the former contains 754 and the latter 61 per cent. of silica. The clays from which the shales were originally formed contain abundance of marine Diatomaceze and Foraminifera (chiefly Num- mulites), as also species of Ostrea, Cyrena, Cerithium, Voluta, and Nautilus, together with the remains of Placoid and Teleostean fishes. XXVII. Intelligence and Miscellaneous Articles. ON SOME EXPERIMENTS RELATING TO HALL’S PHENOMENON. BY PROF. BOLTZMANN. peo” the general equations which Maxwell and Rowland had given for the movement of electricity, Lorentz (Wiedemann’s Beiblitier, viii. p. 869) deduced the following equations for the movement of electricity ina plane plate at right angles to the lines of force of a magnetic field, provided that the Hall’s deflection of the current by magnetism is taken into account : wa P hv, v= a A Uu, Vv, p, x are the components of the current, i electrical tension, and specific conductivity, and h a constant which is probably nearly proportional to the strength of the magnetic field. rom these equations I have deduced certain conclusions which seem to me worthy of being experimentally tested. The following integral corresponds to Hall’s observations : p=—au+hay, U=Kd, Die Os Tn an iron strip with the bounding lmes y=0 and y=8, and the thickness 6, a current J=«cabd flows in the direction OX. The north pole is on the positive Zside. The hands of a watch the face of which is turned towards OZ, run from OX towards OY. In a Hall circuit, which from its great resistance does not mate- rially alter the condition of the strip, a current is driven by the electromotive force e=hab, which flows in the iron strip, against the positive y-direction. By rotatory power K Hall understands the quotient ¢d/JM, so that h=RM«. I may observe that the absolute values which Hall gives for R are far too small; possibly owing to a confusion of the ohm with the resistance-unity they should be multiplied by 10°. From the above equations follows: oi A eae u=—k ae ce dx dat dy J’ v—hu= abies! =P +n), dy in which k=«:1-+h?. The equation of continuity, du ate du hy. dx dy Intelligence and Miscellaneous Articles. 227 gives For a circular plate it follows that when one electrode of the primary current is in the centre, and the entire periphery is the second electrode : p=—Alognatr, u=kA(a—hy)|r’, v=kA(y+he)|r?, p=kAlr; the stream-lines are logarithmic spirals with the equation Y=h log natr+const. ; r and 3 are the polar coordinates, p the components of the current in the direction of r. Similarly found stream-lines were observed in Geissler’s and Hittorf’s tubes under the influence of magnets. The flow in an infinite plate with two or more point-shaped electrodes is obtained by superposition. The equation of the stream-lines is then hlog nat (r|r')=3—S3'+ const. for two electrodes to which the polar coordinates r, 3, r’, 3’ refer. Ledue (Journal de Physique, 2nd series, vol. iii. p. 366) has sug- gested the hypothesis that the changes of resistance which mag- netism produces in bismuth are merely apparent, and caused by the fact that the currents are forced into longer paths by Hall’s phenomenon. In this case, from what has been above said, no changes in resistance are to be expected in a rectangular strip in the entire extent of which the currents flow parallel to two sides which lie opposite. For the circular plate the strength of the current 7=2z7pd= 27k Ad, the resistance s is therefore Toe maB Tl del yas eae Pn ote 1 in which r, is the radius of the plate, r, that of the central electrode. Hence it would be increased in the ratio 1+A/?:1 by the mag- netism. If E is the electromotive force of the battery, p the difference of potentials of the electrodes of the plate, then fe i ea Sia Ss in which w is the resistance of the rest of the circuit. Prof. Ettings- hausen found for a bismuth disk corresponding to these conditions, in two experiments with the magnetic fields 6364 and 4810, the values 1-257 and 1°180, which agree tolerably with the formula, since with bismuth from another source it is true R varied from 8 to 10, On the other hand, it seems difficult to explain the great difference in resistance which was found with rectangular bismuth strips as arising from want of homogeneity, or from the fact that the current was led, and taken away from individual points, and not at the entire breadth. The integral p=—ax-+cy corresponds to a rectangular strip of thickness 6, at whose shorter sides 6 the primary current enters and leaves, while the longer sides J are 228 Intelligence and Miscellaneous Articles. entirely covered with numerous Hall’s electrodes, each connected with the one opposite. Let E be the electromotive force of the battery which urges the primary current, J its intensity, r=1| «bd the resistance which it finds in the plate, w the remaining resistance, 2 the entire intensity of the Hall current, p=6| «lo its resistance in the plate, w the remaining resistance, then : Bo Sus ei, ee K UE _ wie olsuy GEIS ay wae, ta eee K Ell= he wv (+f) +9) cE) = piebe. at u= | e+ | in which f=w|r, ¢=w|p. The resistance is therefore increased in the ratio & : 1, since 1 hr peed ——— J | 2a a 7 Sitzungsberichte der kaiserlichen Akademie in Wien, April 8, 1886. ON THE GOLD-LEAF ELECTROSCOPE. BY FR. KOLACEK. In the following way we get at a relation between the angle of divergence ¢ and the difference of potential P between the leaves and the envelope (base-plate metal-case). Let C be the capacity of the electroscope, considered as a condenser, which depends on 4 ; E = PC the quantity of electricity on the inner coating; 1, g the length and weight of one of the leaves, which are assumed to be quite equal; E*/2C the electrostatic, and 27/29(1 — cos /2) the ordinary energy of mechanical origin corresponding to a rise of the centre of gravity of the leaves. If the electroscope has been left to itself and insulated, the variation of the entire energy H?/2C + 9.1(1— cos ¢/2) equal to zero of a virtual deflection dq is imparted to the leaves. If we consider that C, but not E, is affected by the variation, the following equation results between ¢ and P:— For an infinitely small 9, (dC/d9~)¢=0 is to be considered constant, and the deflection in this case indicates magnitudes, which are pro- portional to the square of the above difference of potential. For a greater @ al expansion in series must be made. ‘The formula with two constants ad+b¢?=P? may be experimentally confirmed to a divergence of about 18°. The leaves, 8 centim. in length, were projected in fifteenfold ve i. Intelligence and Miscellaneous Articles. 229 magnification on a scale divided into centimetres. Notwithstanding that the images were imperfect, the error of reading was within the tenth of a division. The reading of the distance was made about twenty seconds after the first impulse; for the leaves were driven beyond the new position of equilibrium, and only attained it after some seconds. It will be seen that this difference of potential measured after twenty seconds, with equally good insulation, is a measure of this magnitude at the time of the impulse. The insu- lation was sufficiently good, for in 100 seconds the deflection sank for instance only from 10°3 to 8-9. In order to test the above formula, one of the poles of one, two, three Daniells was connected with a condenser-plate, the other with the earth and the second movable plate. It is clear that the charge arising from the dif- ference of potential of 1, 2,3 If Daniells is again a measure for the magnitude P. Hence we have to test the formula ad+ 60°=TI’, if 6 is the deflection determined in double centimetres. The actual distance of the leaves #, expressed in centimetres, is then 20: 15. ; The condenser consisted of two copper plates, 7 centim. in dia- meter, in which only that screwed on the electroscope was varnished. Each experiment was repeated ten times and the mean taken. Probably owing to the formation of a residue in the shellac, the first deflections were a few percentages too small compared with the later very regular values. » Measurement oa for 6 the value 1°91, 5°46, 9°48, corresponding to the value =1, 2, 3 Daniells. With the latter two values of 6 and I], a=0°488, poe In this case the potential-difference corresponding to 6 = 1:91 represents 1°016 Daniell instead of 1 Daniell. _6=5°'13 corresponds to a zinc-carbon element in potassic bichro- mate to which some sulphuric acid was added; hence IT=1:914 Daniell compared with 1°86 Daniell with the electrometer. According to the formula the deflection 17:5 represented this and the three Daniell elements, and thus Il =4°914, which measure- ment gave 18:3. The unvarnished copper plate was replaced by a zine plate of the same size cleaned with emery-paper, the metallic connection established with the condenser, and a deflection of 0°95 obtained from two distinct series of observations, each containing ten readings. With 6=0-94 the formula gives Zu/Cu=0°677 Daniell. If @ is the constant ratio between II and P, and if in the above formula 6=15.4/2 and Il!=P/g, it passes into the formula P=6V 3825 +3-032 2%, which is independent of the magnitude and choice of the condenser. @ was determined by connecting the electroscope with a Beetz’s dry pile of 144 elements. Of course the second pile was put to earth. It is quite essential that the resistance of the battery be infinitely small in comparison with the resistance of the insulating parts of 230 Intelligence and Miscellaneous Articles. the electroscope, for otherwise the entire amount of the difference of potential due to 144 elements is not produced in the electroscope. When no current could be ascertained in the battery, even with a reflecting-galvanometer which gave ten-millionths of an am- pere, the above condition seems to have been fulfilled. For if one pole was insulated, and the other connected with an electroscope, and if it was charged with an ebonite rod, an instantaneous dis- charge took place, when the insulated pole was put to earth. The small divergence of the gold leaves was determined with sufficient certainty by viewing it at a distance of 1-605 metre with a powerfully magnifying spectrometer-telescope, the micrometer- screw of which was provided with a milled head divided into 100 parts, in which six parts represented a minute. As a mean of several values, the smallest of which differed from the largest by about 4 per cent, the number of divisions was found to be 12:9; so that #=0:1004 centim., and thus ¢=240-0. By the aid of the electroscope used and of the formula P= 3:285x4+3:032 2%, or, still better, of a curve on coordinate paper, differences of poten- tial of considerable amount may be determined with an error amounting to perhaps 5 per cent., in which, instead of reading by projection, the far simpler method of reading by a glass micrometer in the eyepiece of the telescope may be used. The upper limit of the differences of potential to be ascertained may be determined by means of an electroscope with heavy leaves. I might mention, in conclusion, that in the absence of a Beetz’s dry pile, ¢ might be ascertained if the deflection of an elec- troscope connected with a Leyden jar is noted, when the jar is discharged with a given length of spark. or this the knowledge of the difference of potential corresponding to a given striking dis- tance is required. An experiment gave avery close agreement fur the value 9.—Wiedemann’s Annalen, No. 7, 1886. APPLICATION OF THERMODYNAMICS TO CAPILLARY PHENOMENA. BY P. DUHEM. Gauss and Laplace have founded a theory of capillary phenomena on molecular considerations, in which they assume that the forces with which the smallest parts act on each other possess a function of force in the ordinary mechanical sense. Poisson and several others have shown that this theory holds also when the density near the surface varies; the first supposition has been generally adopted. This, however, is no longer admissible if we are compelled to take into account virtual displacements which changes of tempera- ture bring about, and such changes of temperature, as Thomson has shown, are in general necessarily connected with changes of the capillary surface. Intelligence and Miscellaneous Articles. 231 Properly to treat this subject, and to bring Thomson’s investiga- tions into connection with the older ones, we must no longer adopt the ordinary mechanical treatment, but must have recourse to general thermodynamical methods. The considerations arising from this point of view form the contents of the present research. It is first of all shown, that for a system of bodies touching each other we are not to seek the potential at a fixed temperature, but the thermodynamical potential, which contains the changes of energy for varying temperature. The only assumption made is that the densities and the actions of the molecular forces of bodies vary only in infinitely thin layers at the surfaces. It is seen that this supposition is sufficient to prove that the thermodynamical potential consists of two parts, one of which is a linear function of the content of the various bodies, the other a lmear function of the surface in contact. ‘This easily furnishes a proof of the admissibility of the older view. From the formule obtained, the laws of Gauss and Laplace for the shape of the sur- faces are explicitly deduced; the same formule render it possible to investigate the capillary changes which occur in thermal changes. New results are thus not obtained, but the old ones are brought into connection with each other. The general equations are then applied to the two special pro- . cesses of evaporation and supersaturation.—Ann. Keole Normale [3] 1. p. 217 (1885); Beiblatter der Physik, vol. x. p. 330. OF PELTIER’S PHENOMENON IN LIQUIDS. BY E. NACCARI AND A. BATTELLI. Two glass cylinders, 16 centim. in width, were placed near each other in a vessel of water, and a paper disk fastened in. each half way up. At the bottom of each cylinder was a copper disk-13 cen- tim. in diameter, Solution of blue vitriol was poured in up to the disk, and on this solution of zine sulphate; in each of these solu- tions a zine plate perforated in the centre was suspended. Both zinc plates were connected by a copper wire and a current passed through the apparatus, the intensity of which was determined by a reflecting-galvanometer, one division of the scale of which repre- sented an ampere. A very thin perforated glass plate was brought in the centre of each of the paper disks. In this aperture was accurately fitted the bulb of a thermometer, which was surrounded by a caoutchouc tube as far as the part in the aperture. The cur- rent of one or two Bunsen elements was sent through the apparatus in either direction, and the course of the thermometer observed every minute. If 7 and 2, are the intensities of the current, y and y, the corresponding thermal effects, the magnitude of the Peltier phenomenon is given by the formula h= (yr —y’)/ (22, - av’). Only those experiments were taken into consideration in which 2 and i, were not greatly different from each other. With the 282 Intelligence and Miscellaneous Articles. following intensities were obtained :— BA ERE IGS 123 148 187 VC) akon Rrra fl 64 72 67 The value of h is therefore almost independent of 7, and thus Peltier’s phenomenon proportional to the intensity of the current. If when solutions of Glauber’s salts of various concentration were superposed on each other, and the bulb of the thermometer was not at the surface of separation of the liquids, or if the same liquid was above and below the diaphragm, no appreciable value was obtained for h, so that the above results are not limited by other influences of heat. From the experiments as a whole, the value of h is obtained when solutions of the following sulphates are combined with solu- tion of CuSO, of specific gravity s, and in like manner the solutions of the chlorides with solution of cupric chloride of sp. gr. 1°10. The values of h are positive, when the greater heating effect takes place, 7. e. when the current passes from the lower and more con- centrated solution to the one above it. Formula. iS: h. Formula. Ss. h. 1 eee Con Sie NIG eaiwenetiws ae 1115 —60 IND Ojceeenceet sued Teds} —A49 TC eee 1:029 —46 Ni(NH,),SO, 1:07 i MeO Seat 12 —36 CuSOjewe ols 13 29 cll aL as. estes seh eee 119 Je (NIE, )5SO)j wcnn-tte8 1:06 29 MGCL becker 1:29 se 4: DSO) ite ctnse one 1:057 52 KGlihg rae, £42 See. 1°08 24 Mini SO Geet Theil G4 ANAC] tea eae 1:068 27 NTE SO) ees 109 74 CAC LEAY Soh te 1:15 38 KeSOg ehddesee 1:07 91 MIAN. . ck netese 1:167 38 PSOE Lschecvnieee 1:137 101 NEL ICL 4 ce cdcbaetce 1:026 52 WAS Op i srcaeveecee hele 106 EL SO) hs camcnke wee 1:05 120 The series do not agree with each other. With solutions of various strengths, apart from H,SO, and HCl, # is always posi- tive. If the entire apparatus was filled with water, and the thermo- meter was surrounded by a semicircle which was half of iron and half of zine wire, and the current was passed through, the mean of ten experiments was found to be h=18; so that the Peltier pheno- menon between liquids is not of aless order of magnitude than between metals. As according to Bellati the electromotive force between zinc and iron is 0:0024 volt, the absolute value of Pel- tier’s phenomenon may be calculated from this for the liquids investigated.— Atti della R. Acc. di Torino, vol. xx. 1885; Bei- blatter der Physik, vol. x. p. 118. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIES. ] SEPTEMBER 1886. XXVIII. On the Physical Structure of the Earth. By Henry Hennessy, /.R.S., Professor of Applied Mathematics in the Royal College of Science for Ireland.* HE structure of the Earth as a mechanical and physical question is closely connected with the origin and for- mation of its satellite, and of the planets and satellites belong- ing to the same solar system. The brilliant results obtained during the present and preceding century by the aid of mathematical analysis, whereby the motions of these bodies have been brought within the grasp of dynamical laws, may have led to the notion that by similar methods many obscure problems relating to the planet we inhabit might be accurately solved. But, although the general configuration of the Harth and planets has been treated mathematically with results which leave little to be desired, when applications of analy- tical methods are attempted to questions of detail in terrestrial structure, the complication of the conditions is so great as to impose the necessity on some investigators of so altering these conditions as to make their results perfectly inapplicable to the real state of the Earth. Physical Geology presents problems the solution of which undoubtedly calls for mecha- nical and physical considerations; but these may in general, under the complex nature of the phenomena, be often better reasoned out without the employment of the symbolical methods of analysis. In most cases the conditions are totally unlike those above alluded to, which admit of precise nume- * Communicated by the Author. Phil. Mag. 8. 5. Vol. 22. No. 186. Sept. 1886. R 234 Prof. H. Hennessy on the rical computations. The heterogeneous character of the rocks composing the Earth’s crust, and the probably varied nature of the matter composing its interior, render mathematical applications rarely possible, and sometimes misleading. Such views seem to be gradually gaining strength among geologists who pay attention to questions of a general nature, and no one has better expressed them in recent times than Professor M. E. Wadsworth*. The principle upon which I have ventured to found all my researches on terrestrial physics is this: to reason on the matter composing the globe from our knowledge of the phy- sical and mechanical properties of its materials which come under our notice. Of these properties the most important are density, compressibility, contraction or dilatation from changes of temperature. Newton and other philosophers have already adopted the same principle to a limited extent, when assuming for the mass of fluid composing the Harth in its primitive condition those specific properties which have been assigned to all kinds of fluids observed at the surface. . Itis impossible to frame any statement more erroneous and misleading than that I have endeavoured to render the ques- tion more hypothetical than it was. On the contrary, I have discarded the invariable assumption of mathematicians who treated the question, namely, the hypothesis of the invaria- bility of positions of the particles composing the solidifying earth. The speculations of all rational inquirers. upon the Harth’s internal structure must necessarily start from the same general principle as above. Some investigators have disre- garded that principle, and made the problem thereby a purely matliematical exercise. In order to reason upon the Harth’s figure, we must assume that the laws of fluid equilibrium apply to the inner portions of the fluid as well as the outer. There is nothing hypothe- tical in reasonings as to the formation of the solid shell and the law of increase of ellipticity of its inner surface as a re- sult of the transition of the formerly fluid matter to the state of solidity. On the contrary, the assumptions of Mr. Hopkins and other mathematicians, that this transition created no change in the law of density of the matter composing the Harth, and in the ellipticity of the strata of equal pressure, are not merely hypothetical ; they are directly opposed to well established physical and mechanical laws. On the other hand, those who have concluded that nothing can be known of the form of the fluid nucleus, seem to deny * “VL ithological Studies.” Memoirs of Harvard College Museum, vol. i. p. 3; and ‘American Naturalist,’ June 1884, p. 587. Physical Structure of the Earth. 235 that the recognized laws of matter apply to the internal con- dition of the Harth. The shape of the nucleus and the figures of its strata of equal density follow from physical and me- chanical laws, just as the forms of the isothermal surfaces within the spheroid follow from the known laws of conduction of heat. Some of the mechanical reasonings regarding the strata of the nucleus and the structure of the solid shell can be presented without employing mathematical symbols, and in what follows I have as far as possible avoided the use of such symbols. This course, moreover, possesses the advantage of making many parts of the reasonings more clear to geologists and observers of the stratigraphical features of the Earth, who are in reality the ultimate judges of the matter, and not mathe- maticians. ‘The necessity under which the latter are con- strained when dealing with problems, of throwing the pre- liminary propositions into simple well-defined shapes, admitting of definite deductions, obliges them to overlook the most essential conditions of the very questions at issue, and they thus arrive at results which may be precise, but which are totally inconclusive with reference to the Harth’s structure. The Mechanical and Physical Properties of the Matter composing the Earth. (1) The materials of the Harth must manifestly influence its general structure, and no inquiries with this structure can be usefully made if the physical properties of these materials are not kept in view. If the interior of the Earth is ina fluid state it is reasonable to believe that the fluid is not the ideal substance called by mathematicians a perfect liquid, namely a substance not only endowed with perfect mobility among its particles but also absolutely incompressible. It is more reasonable to believe that the fluid in question resembles the liquid outpourings of volcanoes, or at least some real and tangible liquid whose properties have been experimentally studied. I have already shown that by overlooking this simple principle certain untenable conclusions, which assert the exclusively solid character of the Harth, have been deduced. Here I propose to develop some additional arguments re- lative to one of the properties of liquids which has an essential bearing upon the internal structure of the Earth. (2) In a former paper, on the limits of hypotheses regard- ing the properties of matter composing the Harth’s interior*, I find that, having referred to published statements where the * Philosophical Magazine for October 1878, p. 265. R2 236 Prof. H. Hennessy on the facts were not clearly put forward, I underrated the com- pressibility of liquids as compared with solids. The influence of the imperfect experiments of the Academia del Cimento: has long injuriously operated in defining liquid and solid matter, and has produced a remarkable conflict of opinions. On taking the results of the best experimental investiga- tions, it appears that, although liquids are but slightly com- pressible as compared with gases, they are highly compres- sible as compared with solids. In many treatises on Physics and Mechanics which have a high reputation matter is divided into solids, elastic fluids or gases, and incompressible fluids or liquids. Hence the erroneous inference seems to have arisen that liquids are incompressible, not only in comparison with gases, but also in comparison with solid bodies. I was surprised to find this remarkably misleading proposition for- mally stated long after the decisive experiments of Oersted, Colladon and Sturm, Regnault, Wertheim, and Grassi, in such a work as Pouillet’s Eléments de Physique, and also in the German translation by Miller. The greater compressibility of liquids as compared with solids is seldom affirmed as a dis- tinct general proposition in books on Physics. It occurs, however, in Deschanel’s treatise, both in the original and in the English edition. Daguin states, in vol. i. of his Traité de Physique, 2nd edition, p. 40, that the compressibility of liquids was long considered doubtful, but nevertheless they are more compressible than solids. Lamé also pointed out the great compressibility of liquids as compared to solids. I have before now referred to the statement of the same proposition in the comprehensive work of the late Professor C. F. Naumann, the Lehrbuch der Geognosie, vol. i. p. 269, 2nd edition*. Although in many physical questions the compressibility of liquids may be neglected as well as the compressibility of solids, we are not entitled to assume at any time that the latter are relatively more compressible than the former. In questions where the pressure of columns of liquid of great magnitude comes under consideration, we can no longer treat the liquid as incompressible. In the problem of oceanic tides the incompressibility of the water has been assumed, but if a planet were covered with water to a depth of one hundred miles it would be scarcely correct to make such an assump- tion. The compressibility is negligible in a small mass of water, but it cannot be neglected in a large mass. Such an assumption is equally unwarrantable with regard to properties * “ Flussige Korper sind aber mit einer weit starkeren Compressibilitat begabt, als starre Korper.” Physical Structure of the Earth. 237 of matter which, though negligible in some problems, are not in others. Thus in the common hydraulic questions, liquids are assumed to be incompressible ; it would be more correct to say the compressibility is neglected. In small problems connected with limited portions of the atmosphere, the compressibility of air may be also neglected, but we could not neglect it for a high column of the atmosphere. If, as before remarked, the Earth were surrounded with an ocean 100 miles deep, the compressibility of the water could not be well overlooked in tidal questions; then, a fortiori, compres- sibility cannot be neglected in such a problem as the tides of a liquid spheroid having a radius nearly equal to that of the Harth. ‘This isimmediately made manifest by expressing the compressibilities of liquids, not in terms of the amount due to a single atmosphere of pressure, as is done in most tabu- lated groups of resulis, but by some very much greater standard, such as one or two thousand atmospheres. In the experiments of Perkins* the highest pressure employed was two thousand atmospheres, and with this he reduced a column of water by nearly =, of its volume. The results of experi- ments with great pressures such as this are highly illustrative of the force by which a fluid may be compressed in the Earth’s interior. The actual coefficients of cubical compressibility, on which calculations could be based, may be partly obtained from the more exact researches of Regnault, Grassi, and other recent experiments, or from special investigations on fluid matter conducted with precautions such as these ob- servers have employed. By then comparing the moduli of compressibilities calculated from pressures of 1000 or 10,000 atmospheres, there could be no possibility of overlooking the consequences as to the relations of liquid and solid bodies in any case where they would be subjected to pressures of ab- normal magnitude. (3) The propagation of sound in solids and liquids gives further proof of the greater compressibility of liquids. The rate v of transmission of sound in solids and liquids is a function of their compressibilities. In solids, Uae fe where H is the modulus of elasticity and p the density. In liquids, dhe gia pr * Phil. Trans. 1826, p. 541, VYy= 238 Prof. H. Hennessy on the where y# is the coefficient of cubic compressibility, H the pressure of the atmosphere, and a the density of mercury. But, as in solids, the modulus of elasticity is inversely as the compressibility k, we have Pex Mh MP, v) koHa Both in solids and liquids the velocity of sound is inversely as the square roots of the densities and compressibilities. Although such solids as metals and rocks are denser than most liquids, the limits of their elastic compressibility are so much less, that sound is propagated far more quickly through such solids than through liquids. In steel and metals gene- rally this has been long since established. In rocks the velocity of sound has been computed from direct experiment by Mallet, and has been found to be greater in continuous homogeneous rock than the velocities observed in liquids*. (4) If we had not the results of direct experiment on the compressibilities of liquids and solids to assure us that these properties in liquids are in excess of those obtained for solids, we might fairly infer this conclusion from the relative dilata- bility of such substances under differences of temperaturef. The construction of our common thermometers is based on the greatly superior dilatability of the liquids enclosed in the thermometer-tube over the material of the tube itself. The dynamical theory of heat clearly establishes that the expan- sion of solids and liquids is a mechanical action as much as their compression under the action of force, and the substances which contract least by cooling are precisely those which con- tract least under pressure. (Gases which contract most by pressure are also the most dilatable by heat. Liquids occupy * See ‘ Philosophical Transactions’ for 1861 and 1862. + Expansions of Metals and Glass for 1° Centigrade, according to Dulong and Petit, at different Temperatures T. Sots. LiQvuID. Platinum. | T. Tron. T. | Copper.| TT. Glass. | T. |Mercury. (e) fe) oO fe) 1 1 1 1 1 sop |) 1) Seag0 | | teao0 | | Ba700 11°" sean0 ee py) 36,800 54DB 1 1 1 1 1 33300 | 211) as700| 3” | Tr700| 228} 32,000 | 23) 5300 Physical Structure of the Earth. 239 an intermediate place between solids and gases, in relation both to the dynamical effect of pressure and the action of loss of heat. If, instead of the experiment of the Academia del Cimento, with globes of porous metals, an experiment with equally strong but impervious vessels had been made, the de- formation of each globe would have been unaccompanied by the exudation of the liquid, and the totally false statement that solids are more compressible than liquids would not have so long injuriously influenced physical science. The Rotation of the Earth considered as partly Fluid and partly Solid. (1) The problem of the precessional motion of the Earth, considered as a solid shell filled with liquid devoid of viscidity and friction, has been elaborately investigated by Mr. Hopkins in his ‘ Researches of Physical Geology’ in the ‘ Philosophical Transactions’ for 1839, 1840, and 1842, and the result ob- tained by him has been often quoted as extremely remarkable. Before treating the same question, it may be necessary to state that on the continent of Europe the application made by Mr. Hopkins of his result to Geology is not generally admitted, and views such as I have always firmly upheld seem to be more generally adopted ; but some confusion appears to exist as to Mr. Hopkins’s results and those to which I have been led. Thus in a recent treatise on Systematic Geology, the author says, with reference to the thickness of the solid crust of the Earth, there are plainly only four possibilities to be thought of :— 1. The Harth is through and through solid. 2. The Earth is through and through fluid, with a solid crust. 3. The Earth has a solid nucleus and a solid crust, with a fluid stratum lying between. 4, The Harth is solid but furnished with cavities which are filled with fluid. The first and last of those possibilities are not admissible, according to astronomical observations. According to the investigations of Hopkins the action exercised by the Sun and Moon on the position of the Earth’s axis in space, by which Precession and Nutation are produced, would be different ac- cording to the structure we attribute to the Harth. The values established by observation compel us to regard the Earth as for the most part in a fluid state, in order that the results may harmonize with calculation (Pfaff, Grundriss der Geologie). This is the reverse of what Hopkins has concluded, and is precisely what I have long since enunciated, which I have 240 Prof. H. Hennessy on the always continued to maintain, and which forms the cumulative result of the investigations in the text of this paper. Ina Report to the Royal Irish Academy on ‘ Experiments on the Influence of the Molecular Influence of Fluids on their Motion when in Rotation,” p. 57*, I referred to a proof obtained by me of the result alluded to, and I now may be allowed to submit this proof to those interested in the question. (2) Let us suppose the earth to consist of a solid spheroidal shell, composed of nearly similar spheroidal strata of equal density, and having the ellipticities of the inner and outer surfaces small and nearly equal. The shell is supposed to be full of liquid and to rotate around its polar axis. Under these conditions the attraction of an exterior body would tend to produce pressure between the fluid nucleus and the inner surface of the shell. Whatever may be the direction of this pressure, it can be resolved into a force normal to the shell’s surface and into forces in its tangent plane. The normal force might be effective in causing a deformation of the shell, or, if the latter were rigid, it would be destroyed by the shell’s re- sistance. If friction existed between the materials of the shell and the fluid of the nucleus, the resolved forces in the tangent plane would tend to change the motion of the shell from the motion it would have if empty. But if no friction and no adhesion existed between the particles of the liquid and the shell’s nearly spherical surface, and if the particles of the liquid are free from viscidity and internal friction among themselves, this purely tangential component could exercise no influence on the motions of the shell. If the solid envelope containing fluid was bounded by planes such as a prismatic vessel or box, it is manifest that unequal normal pressures on the faces of such prism would tend to produce couples, and thus possibly rotations. Such a case has been considered by Professor Stokes, and he has shown that a rectangular prism filled with fluid will have the same motion as if the fluid was replaced by a solid having the same mass, centre of gravity, and principal axes, but with much smaller moments of inertia corresponding to these axes. But ina continuously curved and nearly spherical vessel, the normal pressure arising from disturbance of the liquid could not pro- duce the same results. The tangential components of the forces acting at the surface of the liquid could, in this case, be alone etfective, and if no friction or viscidity existed at this surface such tangential action would totally disappear. The conclusion of Mr. Hopkins’s first memoir is, that if the ellip- ticity of the inner and outer surfaces of the solid shell were * Proceedings of R. I. A., 2nd series, vol. iii, Science. Physical Structure of the Earth. 241 the same, the precession would be unaffected by the fluid, and any small inequality of nutation would be totally inappreciable to observation, p. 423, Phil. Trans. 1839. This may be ren- dered more manifest by recalling the general equations for the surface of a fluid obtained by Poisson, Navier, Meyer, and other mathematicians, when the internal friction of the fluid is taken into account. If a,8,y be the angles made by the normal to the curved surface of the fluid, X, Y, Z the compo- nents parallel to the rectangular axes of x, y, and z, it appears that we shall have at the fluid surface, when nearly spherical, du dv ‘du ee 2 —_— X=hk os poet (T+ ia cos B+ as a cos y |, - pe dv ¥=h/’ “ae cosa +2 7 cos B+(7 + i 7) c087 | dw oe dw dv 2 eae L=hk ie, cos a+ di pala cos 8-425 ~ cosy |, where u, v, w are components of velocity parallel to the coordinate axes, and where & is a coefficient depending on friction and viscidity. If no viscidity and no friction exists we must have £=0, and hence also Zoe 2 Now as X, Y, and Z are the effective components with which the nearly spherical mass of fluid acts at its surface when each of them is separately equal to zero, it follows that the fluid can do no work at the surface, and the motions of the shell would take place quite independently of the contained mass of fluid when the latter is totally devoid of friction and viscidity. (3) It has long since been clearly shown that the motion of the axis of the earth, considered as a solid body, may be de- termined by the differential equations devs gil ond¥ dt Cnsin@ d@’ dé dy dt Cnsinddy V is the potential of the rotating solid, C its maximum moment ‘ of inertia, 0 and yf direction-angles of the axis of rotation. In the case of the Harth @ has a particular value when it becomes the obliquity of the ecliptic, and ~ the longitude of the first point of Aries. It follows that the determination of W and @ at any time depends upon C and VY. 242 Prof. H. Hennessy on the By analytical transformations, which are fully given by Poisson in his memoir, Sur la rotation de la Terre autour de son centre de Gravité, and other writers, it finally appears that the variations of @ and yy depend on equations in which a factor enters of the form 2C—A—B ad pina? where A, B, C are the three principal moments of inertia of the earth. In a spheroid of revolution A=B, and the factor 2(C—A) becomes Sgt ee As precession depends essentially on the variation of the angle yy, it follows that the complete expres- sion of the factor ae is of primary importance. (4) Mathematicians, during the past two centuries, have de- voted much attention to the question of the figure of a rota- ting mass of fluid, with especial reference to the explanation of the spheroidal figures of the Harth and her sister planets. Solutions of this problem have been presented, especially by Clairaut, Legendre, Laplace, Gauss, Ivory, Jacobi, and Airy; and it is not a little remarkable that in applying these solu- tions to the case of the Karth every one of these investigators has not only supposed the Harth to have been originally in a fluid state, but that the particles of the mass retained the same positions after solidification had taken place. This tacit or openly expressed assumption of the unchangeable position of the particles of the original fluid mass on their passage to a complete or partial state of solidity lies at the root of the whole question of the Earth’s structure. Jor the first time in the treatment of the physico-mathematical problem, I distinctly discarded this assumption, and I affirmed that the position of the particles of matter, on passing from the state of fluidity to solidity, must assume positions in conformity with mechanical and physical laws. In this way the hypothesis of the Harth’s primitive fluidity became more simple and much more rational ; for it was as manifestly absurd to assume that the particles of the fluid mass, on passing into a solid state of con- sistence, retained their original positions, as it would be to assume that if the whole Earth became liquified the positions of its particles would be unchanged. The corrected and sim- plified hypothesis is also fruitful in important results; but itis singular that, as far as ] am aware, no mathematician seems to have understood or appreciated its bearing on the physical structure of the Earth, except M. Plana, by a remark in a memoir published by him towards the close of his career. Physical Structure of the Earth. 243 (5) Before presenting my conclusions on the shape of the inner surface of the solidified shell and Plana’s remark relative to the same subject, it is necessary to recall some results esta- blished by Clairaut, and frequently put forward by mathema- tical investigators of the Earth’s figure. It seems to be uni- versally admitted that if a mass of heterogeneous fluid com- posed of strata of equal density, each increasing in density from the surface of the mass to its centre, is set in rotation, the several strata will be spheroidal, but their ellipticities will not be equal. The ellipticities will decrease from the outer surface towards the centre. This law of decrease of ellip- ticity towards the centre is not a hypothetical result, but a necessary deduction from the properties of fluids. As all known fluids are compressible, such an arrangement of strata of equal density as that referred to must follow from the supposition of the existence of any mass of fluid of such magnitude as the whole Harth. The increase of the Harth’s density from its surface to its centre is, moreover, a fact clearly revealed by the mean density of the Harth being double that of the materials composing the outside of its solid shell. If the increase of density, in going from the surface to the centre of a large mass of fluid, is due to compression exer- cised by the outer upon the inner strata, it follows that the greater the total quantity of fluid the greater will be the difference between the density at its surface and its centre, and the less the quantity of fluid the less will be this dif- ference. With a small spheroid of compressible fluid, the variation of density might be neglected, and the mass re- garded as homogeneous. Suppose such a small mass of fluid to be set in rotation, its surface will become spheroidal, and it will have the well-known ellipticity 3 m, where m is the ratio of centrifugal force to gravity at the equator of the spheroid. If, now, this original spheroid be supposed to be overlaid with masses of the fluid, one after another, the inner portions will be sensibly compressed, and the whole mass will begin to vary in density in going from centre tosurface. The outer surface will now present an ellipticity less than 3 m. If fresh layers of fluid are continually applied to the outer surface, the variation of density will continue, and the differ- ence between the density at the centre and surface will increase. The ellipticity of the outer stratum of fluid will at the same time diminish to a value corresponding to the law of density. Let us now reverse this operation, and suppose a great mass of liquid in rotation, its outer stratum will be less dense than those beneath, and its greatest density must be at the centre. 244 Prof. H. Hennessy on the Let the outer strata of equal density be successively removed, so as to leave a succession of free fluid surfaces, until a spheroid is reached in which the difference of density is insensible. It is manifest that, with each successive removal of the upper stratum of liquid, the compression in the remaining strata be- comes reduced, and also the variation in density from surface to centre, until this variation becomes altogether extinguished. With the same velocity of rotation, the ellipticities of the sur- faces of liquid thus successively exposed would increase up to the limiting value, > m. If at any time of the Harth’s solidification we suppose a nucleus of fluid to be enclosed within the solid shell, the successive increasing of thickness of the shell, from the congelation of the fluid matter of the nucleus, must be accom- panied by the removal of successive outer strata from the nucleus. From what has been seen already, the nucleus will tend to acquire an increase of ellipticity, and therefore to mould the semifluid pasty matter about’ to pass into a solid state into a shape different from what it would have if no change what- ever in the positions of the particles had taken place. As the nucleus is supposed to be in a state of fusion from heat, the successive additions to the inner surface of the shell from the matter of the nucleus must proceed at a very slow rate. The congelation of the surface-stratum of the nucleus must be a process of the same order of slowness as the flow of heat through the shell; and the mathematical theory of conduc- tion established by Fourier shows that this cannot proceed otherwise than slowly. The changes in shape of the surface of the nucleus would be correspondingly slow and gradual. When once a comparatively rigid outer crust had been formed, the process of moulding additional strata of solidified matter against the inner surface of the crust from the nucleus would proceed in a slow and gradual order, so that the resulting solid strata would conform to the shape impressed upon them by the moulding forces. A remarkable illustration of the way in which fused matter ejected from the Earth’s interior may, while turning on its centre and at the same time cooling, mould itself against a solid crust formed upon it has been ad- duced by Charles Darwin, and has been already quoted by me ona former occasion. From these considerations I have been led to conclude, that the ellipticity of the shell’s inner surface may exceed, but cannot be less than, the ellipticity of its outer surface *; and, referring to the same question, Plana used the words, “ La loi des ellipticités a subi dans le passage de |’état * See the subjoined representation of a section of the shell and nucleus, Physical Structure of the Earth. 245 liquide & l’état solide une alteration sensible par laquelle toutes les couches se sont constitueés de maniere a avoir un méme applatissement et plus grand que le précédent.” M. Plana has further stated his views in the same volume of the Astronomische Nachrichten for 1852, thus :—“ Il est permis de penser que ces couches (de la fluide intérieuse) en se consoli- dant, ont subi des modifications a la verité fort petites, mais assez grandes pour nous empécher de pouvoir dériver, avec tout l’exactitude que l’on pourrait souhaiter, l’état de la Terre solide de son état antérieure de fluidité.” This paragraph gives a distinct adhesion to the improved form of the hypothesis of the original fluidity of the HKarth; and this concurrence on the part of M. Plana is the more im- portant, as it is possible that he had formed his conclusions independently. He refers to a letter written by him on the subject to Humboldt; and it is remarkable that, in the fifth and last volume of ‘Cosmos,’ published not long before the author’s death, some adjacent notes allude to Plana’s views, and contain references to the investigations of Mr. Hopkins and to my early researches. At this period Humboldt could scarcely have had time to examine the mechanical and physical reasonings, and he merely quoted the papers in the ‘ Philo- phical Transactions’ as if he had seen them for the first time. I am not aware of any evidence as to whether Plana had known their contents; and it is possible that his conclusions as to the forms of the strata of the shell and nucleus had been formed independently, though published a short time after my investigations. The annexed figure may assist in making clear the results of the preceding paragraphs. The outer ellipse represents the out- line of the exterior surface of the Earth’s crust, which is shaded and bounded inwardly by a sur- - face slightly more elliptical. The fluid nucleus included within the shell is represented with strata decreasing in ellipticity towards the centre. This arrangement is necessarily followed by a mass of fluid under such conditions as the nucleus, or under the conditions of the entirely fluid Earth. If the matter composing the Harth underwent no change in passing from the fluid to the solid state, instead of the arrangement here represented, the inner surface of the 246 Prof. H. Hennessy on the shell would have a smaller ellipticity than its outer surface, and the strata of the shell, as well as those of the nucleus, would be less oblate in going from the outer surface. (6) It is important to distinctly bear in mind that the con- stitution of the shell and nucleus indicated by the foregoing reasonings is not based on any hypothesis of a specific law of density of the interior strata of the Earth. It is a deduction from the established properties of fluids quite as rigorous as the conclusions regarding the spheroidal shape of a mass of rotating liquid. On the other hand, the supposition tacitly or openly made by Mr. Hopkins and his followers, that the ellipticity of the inner stratum of the solid shell is precisely the same as that which this stratum had when fluid, is not merely a hypothesis—it is an assumption which is directly contradicted by the recognized physical properties of all known liquids, and even contradicted by the fundamental principles of hydrodynamics. Upon this assumption was based the calculation of the ratios of the inner and outer ellipticities of the shell which would correspond to the observed value of the precession of the Harth’s axis, and hence the limiting value of the thickness of the shell. But when the fundamental assumption on which this ratio is calculated is shown to bein contradiction to physical and mechanical laws, the whole of the conclusions drawn from such a calculation must fall to the ground. In the Mécanique Celeste, Laplace, following Clairaut, proved that if the density in a fluid spheroid decreases from the centre to the surface, the ellipticity of the strata of equal density must decrease from the surface towards the centre. This result forms the groundwork of some of the arguments employed in the present inquiry. Legendre and Laplace also deduced a law of density from the properties of compressible fluids, and from this law the latter unfolded a law of ellip- ticity of the strata of equaldensity. ‘The results arrived at in my present inquiry are manifestly totally independent of the law of density p ee deduced by Legendre and Laplace. In order to apply this law to the strata of the solidified shell, the assumption must necessarily be made that the particles of the fluid underwent no change in position on passing to the solid state. This was assumed by Mr. Hopkins and Arch- deacon Pratt; and, as we have seen, such an assumption is not only unwarranted, but is absolutely contradicted by the established laws of hydrodynamics. My conclusions are not only in harmony with those laws, but necessarily require them to be kept constantly in view throughout the whole investi- gation, Physical Structure of the Earth. 247 (7) The result obtained in section (8) allows of an imme- diate and easy application to the inquiry before us, if we admit that the strata of equal density in the shell have all equal ellipticities—an admission which has been already shown to be a particular case of a rigorous and exact deduction from hydrodynamical principles. In this case let us consider the ratio of the difference of the moments of inertia of any sphe- roidal stratum to its greatest moment of inertia. It will readilv appear that the difference of the greatest and least moments of inertia, of all the strata divided by the sum of the greatest moments of inertia will be the same as that for a homogeneous shell whose inner and outer ellipticities are equal. If p be the density of any spheroidal stratum of equal den- sity, then for that stratum Ci—A,__ {p(a? +y?)da dy dz —|p (2? + y?)da dy dz Gigs f(a? +y")dxdy dz ; and as p may be placed outside the sign of integration, it disappears both from numerator and denominator. As we shall presently see, Ci= Ay 1 (1 te C, ye me) d where 6, and a, are the semiaxes of the stratum; and for all other strata of equal density we would have Op Ay) 1 oe OU, =5( ass : Cs 1) be) Ca Ay 1, be? Ce aed |e ina OS =a a3) Now if these strata are all similar, and have equal ellipticities, ENE BORN and hence WAY Gah)" Gn, Sia Ur Ay C3 As, ce eee) Ci C, + A; aes, ©, re 2 a? : where 6 and a are the outer semiaxes of the shell composed of all the strata of equal density. But a( 3 \= ge --+C,—(A,+A,+...+A,) 2 are Crm Q,4+0,+...+0, This is the symbolical form of the proposition just stated. In a homogeneous solid of revolution the general expres- sion for the moment of inertia is a\yade : 248 Prof. H. Hennessy on the and from the ordinary treatises on Mechanics it readily appears that for a spheroid, C =F a? b, A=B=A mad(a? +0), where 6 is the semipolar and a the semiequatorial axis. Hence we have CSA 200-000) Gb a0 alae ab C.s a ab. cits Bach aaa =" a1 ~) se Og no ay’ 2(C—A) =(1 Ha C In a spheroidal shell for whose inner surface the semiaxes are b, and a,, we have the moments of inertia with respect to the axes by taking the moments for the inner spheroid bounded by 0, and a, from those of the outer spheroid. Calling the former ©, and Ay, we have, as before, C,= and 4 Naan TRE "by (a, + b,’). Calling C, and A, the moments of inertia of the shell, we have, therefore, ap May *b,, A,= C= Sn(at b—a,'* by), A,= an [a’b (a? + b?) = ay"b,(a;” +6/ jd P and hence C,— Ay _a’b(a? —b") —a,"b,(a,’— b,’) OF ae 2(a*hb —ay*b,) b? ore _ato(1—3) —0’) Ay(A—A,).5J 7 ~ = If, again, for shortness, Ha 2 A, mm we get by substituting the values (12) in (7) and (8), dividing these equations by 2m and 2m, respectively, and subtracting, A A-A\=B(a,-0) 5°, . 3) ay” {m(a? — a") + m(a,?—6")} = x s/(13) where eo LOH == a aos 2m 2m, Thus from (7) and (8), finally, we obtain A=-—B | o +(o,—¢) — an?(a* ay") Ay }, c= iy (15) a —B {a = (o,—= A.ds+ As As and (22) ee of varying Elasticity. 265 At each of the 7 surfaces of separation there are two equa- tions of the types (21) and (22). At the outmost surface r=a, neglecting atmospheric pressure as before, we have (m;—n,)B+2m:A;— “2 As=0; . . (23) and at the inmost surface r=0 a similar equation, writing b for a and dropping the suffix. If the cylinder be not hollow this last equation does not exist, but in its place we obviously have A’=0. Thus there are 22+ 2 or 27+ 1 equations accord- ing as the cylinder is hollow or not, to determine the constants of types A and A’ in terms of B. Of these constants each medium possesses two, except when the cylinder is solid, when the central medium has only one constant, viz. A. Thus, whether the cylinder be hollow or not the equations are the same in number as the constants, which thus may all be directly expressed in termsof B. The values of the constants can be at once written down under the form of determinants. To determine B, we have F=a[(a2—0)R+ (a2—a2)R, +... (a2—a2)R;). For the special case = = constant for all values of s, of which we have previously ‘seen the importance, a very simple solution holds. In fact it is obvious from (21), (22), and (23) that all our conditions are then satisfied by es at » (24 Ja apne cme ek ee, 2m in our previous notation, o being, in accordance with this hypothesis, the same for all the media. In this case the equa- tion for B is simply F=7B[(a"—0?)M + (a.”—a,”)M, +... + (a@—a,?)Mj],... (25) where M, M,, &c. denote Young’s moduli for the several media. Also the solution (19) applies to all the media. The equations (21) and (22) have been established inde- pendently of the absolute thickness of the layers: thus, under certain limitations to be presently considered, they may be supposed to hold when the thickness is indefinitely reduced, and thus in the limit to apply to a continuously varying medium. Thus, dropping the suffixes and writing r for as;4,, we get in place of the constants of types A and A’ certain functions of Phil. Mag. 8. 5. Vol. 22. No. 186. Sept 1886. fs 266 Mr. C. Chree on Bars and Wires r determined by the equations d d 2d 2—. (mA) + Be (m—n)— 3 FZ (nA!) =0, Mae 4.53) el C1) Being given the law of variation of m and n, we can from these equations get a differential equation in A or in A’ as is desired. To determine the constants of the solution, we have A‘ 2mA + (m—n)B—2n =0; when r=a, and also when r=6 if the cylinder be hollow ; if it be solid, instead of the latter equation we have A’=0 when r=0. In obtaining these equations we have, it seems to me, tacitly assumed that the force at any point is the same as if all the neighbouring material, at least on one side of a plane through the point, within the distance at which molecular forces are sensible, were the same as at the point considered. Thus, if the variation in the material were very rapid, the validity of deductions from these equations might be questioned. As an example of the use of (26) and (27), let us consider the case of a solid cylinder of which the material has elastic constants given by (28) where 7, %, p, g are absolute constants; while pa:1 and ga:1 are so small that terms containing their squares or pro- ducts may be neglected. The form of (26) and (27), then, suggests m=m(1+ pr) H n =m (1+qr); A= Ag(L i 0r)5 ye) erie ot) aie ae where Ay and ¢ are constants, the latter being of the same order of quantities as p and g. Then from (27) we get cr? ee no constant being required as A’ vanishes when r=0. Sub- stituting (29) and (30) in (26) and retaining only the principal terms, we get Bmp — Ng) + 2m pAo Bran His 2a) (31) The surface-condition P=0 when 7=a gives, when the above of varying Elasticity. 267 values of m, n, A, and A’ are substituted in (3), 2A [ms(1 + pa) + ca(m, +47) |= —Blm—n E(nppecnaele <2 v- AO2) The first approximation gives Ap=—aB. Writing this in (81), we get Bn, ( p—- Age — See. aan DNB aR Substituting this value of c in (82), we get, after reduction, for a second approximation, ia Ny (p—q)a | A\= Bl e+ SeCAG a) Te ee Introducing these values of Ao, c, A’ inj(1), we get, after reduction, u=—o,Br [1+ 3 th P= 9) (nya + mgr) | oo. (eo) One while throughout, 1p 1z, It should be noticed that correct values of P and R are to be obtained only from (8) and (4), substituting therein the above values of A, A’, m, and n. If, as previously, F denote the total traction over the terminal section, we get P= {2nrtar, 0 where R has the value (4), when m, n, and A are regarded as variables given by (28), (29), (33), and (84). This equation easily leads to 2 2 n pamvitai ders fotev sattg] «00 where M, is the value of Young’s modulus for the material at the axis. When - is constant, p=q; and the above value of B obviously agrees with that derived from (25), noticing that then M=M)(1+¢@7). The torsion of a cylinder, hollow or solid, formed of differ- ent materials or of one continuously varying material, as in the cases just considered, presents no difficulty. Regarding T2 268 Mr. C. Chree on Bars and Wires first a succession of materials in contact, we see from (29a) and (30a) that, supposing the end z=0 fixed, a possible solution for any medium is v=Hr. .. . .) ere Further, this solution, regarding H as an absolute constant, will apply to all the media. For from (9a) we see that the only stress existing will be 8, and at all the surfaces of se- paration, as well as at the bounding surfaces of the cylinder, #@=v=0; thus all the surface-equations (10a) are identically satisfied. Also the value of v is the same for any two adja- cent media at their common surface. If G be the couple of torsion applied at the end of the cylinder, and if q...a;, n, m...7; have their previous meaning, we get,to determine H, G=27H [ | meas +f marae. oot (" sar |, . (38) 6 a ea; ] ene: G= ge [2(ay*—b*) + 2y4(aq*—ay*) +... +ni(a*—a*)]. (39) The values of a and 0 and the number of the media may be any whatever. With limitations similar to the case of longitudinal traction this solution may be supposed to apply to a continuously varying medium, and the value of E will then be given by G=2nb | “nrar. Oe ais 6 This last expression obviously includes (38), treating n as a discontinuous function. The chief use of the preceding investigations would pro- bably be in assigning the limit to the traction, pressure, or torsion which could be applied with safety to a structure of the kind considered. Unfortunately there seems no general agreement among practical men as to how the limits of safety may be fixed for a material when exposed to any system of force, except perhaps longitudinal traction. One theory that seems to meet with considerable approval is that, whatever the system of forces may be, the structure is safe so long as the greatest positive strain does not exceed a certain limit, to be determined experimentally for each separate material, pre- sumably by uniform traction. As to the correctness of this limit for the case of uniform traction no doubt need be entertained. It does not follow that rupture will ensue as soon as this limit is passed; but the nature of the material itself will be altered, and rupture will follow sooner or later. of varying Elasticity. 269 Tn the case of traction the only positive strain is a or B, which is the same for all the media of which the bar may consist. If, then, the traction F be increased till B exceed the limit of safety of any one material, that material will finally give way. Theoretically, of course, this material might give way uniformly all round, with the result that the traction F would then have to be supported by the remaining mate- rials. This would lead to increased strain in all these; but the structure as a whole would still be safe if this new strain were less than the limit of safety for each of the materials left. If the increased strain exceeded this limit then a second rup- ture would occur, andsoon. In practice, owing to some want of symmetry in the distribution of the traction, or to slight inequality in the material, the first yielding material would probably crack and give way only in the neighbourhood of one point. This would alter the distribution of the traction, and might bring it to bear most largely on the strongest materials. If, however, the result were that the line of action of the resultant of the tractional forces got displaced to a finite distance from the axis of the cylinder, the strain would be considerably lessened at some points and considerably increased at others. It is obvious that such local increase of strain would be extremely dangerous. Thus the traction to be ap- plied with safety to a composite bar of this kind should be calculated on the basis of the resultant strain not exceeding the limit of safety of the material for which the limit is least. In the case of longitudinal pressure B is negative, and for a bar of one isotropic material the other two principal strains du (viz. rm and -) are each equal to —cB. In accordance with the theory recently referred to, —cB should not exceed the limit of safety as determined for the material by longitudinal ‘traction. Others hold that the compression (7. e. B taken numerically) should not exceed a certain independent limit -obtained from pressure experiments. The results we have obtained will enable the greatest pressure to be calculated which can be applied to a composite bar, without passing the limit of safety, for any one of the materials, determined in accordance with either of the above theories. In the case of torsion the only existent strain is the shear dv dz equivalent to an equal extension and compression in directions making each an angle of 45° with the direction of shear, the = Er, which is shown in any treatise on elastic solids to be 270 Mr. C. Tomlinson’s Further Remarks numerical measure of either being Hr. Thus, according to the theory first mentioned, in the case of torsion of a compo- site circular bar, if the couple of torsion be such that }Hr exceed at any distance from the axis the limit of safety, as determined by tractional experiments, for the material at that distance rupture will ensue. The rupture would at first be limited to the single material ; and if it proceeded right round would merely produce an increased strain in the remaining media, which might or might not, according to circumstances, produce further rupture. In practice, rupture would in all probability be at first limited to a small region, and the bar would undoubtedly tend to become warped. Direct experi- ments on the rupture of isotropic bars by torsion may disprove the above theory, but itis fairly obvious that the true law must depend on the state of strain and stress in the material. Thus the preceding solution will in any case supply the data that may be necessary in determining the limit of safety of a composite bar under torsion. XXXII. Further Remarks on Mr. Aitken’s Theory of Dew. By Cuarwes Tomuinson, /.R.S.* iT HAD no idea that the innocent title of my paper, “Remarks on a New Theory of Dew,’ had a guilty meaning ; but according to Mr. Aitken I ‘“ was raising a false contention,’ and so attempting to place the author “ in oppo- sition to recognized authorities”; that the results of his investigation “are in no sense entitled to be called new ;” and he repeatedly states that his investigation was not promul- gated in opposition to the theory of Dr. Wells, but “in extension of the work, the foundations of which were laid by Dr. Wells.” Again, he says the new theory “is not in oppo- sition, nor are the results contrary to the teaching of Dr. Wells.” Once more, the author “ never made any attempt to set aside Wells’s theory.” : And yet it is curious to notice that ‘Chambers’s Journal ? for May 29th last contains an article headed “ A New Theory of Dew,” in which the writer, after giving an accurate outline of Wells’s theory, goes on to say that Mr. Aitken “ has brought forward many observations, and the results of numerous experiments, which appear to prove that Dr. Wells’ theory of dew is not, after all, correct.” We are further informed that * Communicated by the Author. on Mr. Aitken’s Theory of Dew. 271 “‘ the essential difference between the old and the new theories is as to the source of the moisture which forms the dew. Instead of being condensed from the air above by the cool vegetation, Mr. Aitken maintains that it comes from the ground.” Again the Chambers’s article, referring to Wells, says :— “The points of the grass, small twigs, and all other good radiating surfaces are cooled the most; and accordingly we find the dew-drops most abundant on these bodies ; whilst on metal, or hard stone surfaces, which are poor radiators, we seldom or never find any dew.” This is the Wells picture ; the writer now turns to the Aitken picture. ‘ A closer observation reveals the fact that these so-called ‘ dewdrops’ are formed at the end of the minute veins of the leaves and grass, and are not now recognised as dew at all, but moisture exuded from the interior of the plants themselves.” And yet Mr. Aitken is angry with me for calling his theory new, and for asserting that, if true, it will supersede the labours of previous observers. He says :—“I do not find that he [ Mr. Tomlinson] adduces any results of previous observers that are in any way rendered nugatory by the results set forth in my paper.” If Mr. Aitken would condescend to study the classical memoir of Melloni (an abstract of which occupies the greater part of my paper), I should be much surprised if he did not become a convert to its experimental methods and conclusions. But at present he sees through the spectacles of his own theory, and therefore cannot appreciate the force of Melloni’s singular care with which he protected his thermometers from sharing in the radiation of the surrounding bodies, whose temperature they had to indicate ; for while other observers get differences of temperature between their two thermometers, amounting in some cases to as much as 16° or 18° F., Melloni is satisfied with a difference of only 2° or 3° C., and his theory of convec- tion justifies this modest difference, and also accounts for many other phenomena, including the inverted trays and other objects which Mr. Aitken found wetted only on their under surfaces. In lke manner Mr. Aitken does not see the force of the observations made in Persia and the African Desert, seeing that his remarks “apply only to this climate.” Surely the great forces of Nature rule as impartially in Persia and in Africa as in Scotland ; and where no aerial vapour exists, there is no deposit of dew. The cases given were intended to show that in the arid regions there was no dew ; but that long ee a: ee ans oe RE en rs y * 272 Further Remarks on Mr. Aitken’s Theory of Dew. before the travellers reached any considerable body of water, nocturnal dews were abundant, and they were deposited from the air, and did not rise out of the ground. Mr. Aitken also remarks that my notice of the Florentine Academicians, of Robert Boyle, and Le Roi have no bearing on the subject. The bearing is that these early observers proved that the moisture which forms dew and hoar frost exists in the air, and does not exhale from the ground. Mr. Aitken is also ‘‘ puzzled to understand”? what bearing Pictet’s observation has on the subject. In the abstract in ‘Nature’ of Mr. Aitken’s memoir, it appears as an original discovery that “ these observations made at night showed the ground at a short distance below the surface to be always hotter than the air over it.” Pictet observed the same fact in 1779. So also in my account of the weighed turf, I certainly did not wilfully form a ‘‘ misconception of the essential features of the experiment,’’ when I compared it to objects which, when exposed on Patrick Wilson’s scale-board, gained weight, while in Mr. Aitken’s case the turf lost weight. It is true that my observations were founded on the abstract of the memoir contained in ‘ Nature.’ In January last I wrote for a copy of the memoir, which was promised as soon as the ‘ Edinburgh Transactions’ were published. I waited until May and did not receive it. I inquired for it at the Royal Society in June, but it had not arrived, nor have I yet had the privilege of reading it. Mr. Aitken is therefore entitled to any advantage that may arise from my use of the abstract instead of the original memoir. As I do not intend to write again on this subject, I conclude by assuring Mr. Aitken that 1 have no unfriendly feeling towards him; but on the contrary freely admit that he has achieved much good scientific work, which I cannot but admire ; but as regards his new theory of Dew I think he has gone astray, and in the interests of scientific truth I have ventured to criticise it. The subject is one that has occupied a great deal of my attention, and there is no doubt in my mind that, if this theory be accepted, a large amount of excellent work on the part of first-rate observers must be set aside as false. Highgate, N., August 9, 1886, prraes> ty XXXII. On the Self-induction of Wires.—Part II. By OutveR HEAVISIDE*. N. Part I. (p. 118) the inner conductor was solid. Let now the central portion be removed, making it a hollow tube of outer radius a, andinnera,. ‘The reason for this modification is that the theory of a tube is not the same when the return- conductor is outside as when it is inside it; that is to say, it depends upon the position of the dielectric, the primary seat of the transfer of energy. The expression for H,, the mag- netic force at distance 7 from the axis, will now be Hy= {J1(s17) —(J1/Ky) (S149) Ky (sur) $ Ay; . (49) instead of the former A,Jj(sy7), of the first of equations (18); if we impose the condition H,;=0 at the inner boundary of the wire (as we may still call the inner tube). This means that there is to be no current from r=0 to r=a,; we there- fore ignore the minute longitudinal dielectric current in this space, just as we ignored that beyond r=as3 previously. If we wish to necessitate that this shall be rigidly true, we may suppose that within r=a, and beyond r=a; we have not merely k=0, but also c=0, thus preventing current, either conducting or dielectric. In any case, with only £=0, the dielectric disturbance must be exceedingly small. On this point I may mention that my brother, Mr. A. W. Heaviside, experimenting with a wire and outer tube for the return, © using a (for telegraphic purposes) very strong current, rapidly interrupted, and a sensitive telephone in circuit with a parallel outer wire, could not detect the least sign of any inductive action outside the tube, at least when the source of energy (the battery) was kept at a distance from the telephone. In explanation of the last remark, we need only consider that, although the transfer of energy is from the battery along the tubular space between the wire and return, yet, before getting to this confined space, there is a spreading out of the disturb- ances, so that in the neighbourhood of the battery the disk of a telephone may be strongly influenced by the variations of the magnetic field. On the other hand, the induction between parallel wires whose circuits are completed through the earth, is perceptible with the telephone at hundreds of miles distance, or practically at any distance, if the proper means be taken which theory points out. His direct experiments have, so far, only gone as far as forty miles, quite recently ; but this may easily be extended. » * Communicated by the Author. 274 Mr. O. Heaviside on the Corresponding to (49) we shall have 4nT y= 8, {Jo(s17) — (Ji /K1)(s14,) K, (217) } Ay 5 eee (50) omitting, in both, the z and ¢ factors. Now, to obtain the corresponding development of the general equation (22), we have only to change the J,(s,a,) in it to the quantity in the {} in (50) and the J,(s,a,) to that in the {} in (49), with . 7 =a, in both cases. The method by which (22) was got was the simplest pos- sible, reducing to mere algebra the work that would otherwise involve much thinking out ; and, in particular, avoiding some extremely difficult reasoning relatively to potentials, scalar and vector, that would occur were they considered ab initio. But, having got (22), the interpretation is comparatively easy. Starting with the inner tube, (49) is the general solution of (14), with the limitation H,=0 at r=a,, if, in s, given by — si=4Arpyky p+m’, we let » mean d/dt and m? mean —d?/dz’, instead of the con- stants in a normal system of subsidence, and let A, be an arbitrary function of z and ¢. Similarly, (50) gives us the connection between I’ and A,. From it we may see what A, means. For, put r=a, in (50); then, since (J,K:—J,K,)(2)=—a-, we see that A, = —47ra)K,(s,a,)I,, if U, is the current-density at r=a,. When the tube is solid, A; =47I,/s,. But, without knowing A,, (49) and (50) connect H, and I’, directly, when A, is eliminated by division. Also H,=C, x (2/r), if C; be the total longitudinal current from r=a, to r; hence pa 8 FMT (ora) K y(t) ggg BT Ome Vive aia aaa: ena KG connects the current-density and the integral current. Now pass to the outer tube. Quite similarly, remembering that H;=0 at »=as3, we shall arrive at Ppa ft Salen) = FUR oe Kaleo) gy, DE lg ob. peak eee aes . (52) connecting [;, the longitudinal current-density at distance r in the outer tube, with C3, the current through the circle of radius 7 in the plane perpendicular to the axis. Next, let there be longitudinal impressed electric forces in the wire and return, of uniform intensities e, and é,, over the sections of the two conductors. We shall have el =a4+ Kh, p3l'3=e3+ Hs; we) @ 5 ° (53) Self-induction of Wires. 275 if H, and E; are the longitudinal electric forces “of the field.” Therefore é — ¢s-=e=p,1,—p31'3—(H,—E;), area (54) where e is the impressed force per unit length in the circuit at the place considered ; the positive direction in the circuit being along the wire in the direction of increasing z, and oppositely in the return. If in (51) we take r=a,, and r=a, in (52), and use them in (54), then, since C, becomes C, the wire current, and C, becomes the same plus the longitudinal dielectric current, if we agree to ignore the latter, and can put E,—E; in terms of C, (54) will become an equation betweeneandC. = r=a, To obtain the required H,—H3, consider raat a rectangular circuit in a plane through the axis, two of whose sides are of unit length parallel to z at distances a, and ag from the axis, and the other two sides parallel to 7, and calculate the H.M.F. of the field in this circuit in the direction of r=4, the circular arrow. If z be positive from areal left to right, the positive direction of the magnetic force through the circuit is upward through the paper. Therefore, if V be the line integral of the radial electric force from r= ay to r=az, so that dV/dz is the part of the E.M.F. in the rect- angular circuit due to the radial force, we shall have d Wats K,—E;+ = = -{ PaHodr, by the Faraday law, or equation (7) ; H, being the magnetic force in the dielectric. This being 2C/r, on account of our neglect of I',, we get, on performing the integration, —L,C, on the right side, where L, is the previously used inductance of the dielectric per unit length. This brings (54) to G4 Piss Sal sits) —(Ir/Ky (sao) Kus) ¢ ay dV er =Lyp 27a, Jy ahi sih er siaregi oMwlelesba) vei biel vehic ie Ki, neces Jo( sade) — (Sif Rr) (8548) Ko(s302) (55 21dgJ, +... — ChE ote PeNOkIOeG Weave , ) which, for brevity, write thus, e—~N =L,pC+RIC+R0, . . . (56) where R," and R," define themselves in (55). They are gene- ralized resistances of wire and return respectively, per unit length. But of their structure, later. Equation (56) is what 276 Mr. O. Heaviside on the we get from (22) by treating sor as a small quantity and using (26); remembering also the extension from a solid to a hollow wire. By more complex reasoning we may similarly put the right member of (54) in terms of C without the neglect of P,, and arrive at (22) itself, in a form similar to (55) or (56). But we may get it from (22) at once by a proper arrangement of the terms, and introducing e. It becomes e= (RN + RIGS, SE Fell a ) Co eae Here Ry’ and R,! are as before, whilst Ro,’ and Ro,” are similar expressions for the dielectric, on the assumption that H=0 at r=q, or at r=azy respectively ; thus, = aS P22 J o(Soaq) — (J ,/Ky )(soa, aay 277A, Jy ei eee rans Senet velebe tears K, 5 (ei 282 Jo( S21) — Bn) K Ct) vr Brad... — Dahh, pete eee Ky aate Ro;!’ has a different structure, being given by Rod — PB HE er a K Parad Chik eth tee : In these take s,r small ; they will become oe paths P2 . Ro" = Ros! = “(pa that is, if pp be imagined to be resistivity, the steady flow resistance per unit length of the dielectric tube (fully, py is the reciprocal of ky +c, p/47); and, with seat 2 s Re = Binet 2, log A = Lip + ee if S is the electrostatic capacity per unit length, such that L,S=/e¢.. Then (57) reduces to 7 e=(L,pt+m/Sp+R"+R")0, . . . (58) which is really the same as (56). For, by continuity, or by the second of (11), ie =2nra, po=SpV = We = 20a = 27a, po = ) ds, Ona ie ee (59) if o is the time-integral of the radial current at r=qa,, or, in other words, the electrification surface- density there, when the conductors are non-dielectric. (There is equal —o at the Self-induction of Wires. 277 r=dy, surface). Therefore a Ca dV Sa ga ett yt fy tay (iene ais (OO Sp.dz* Sp de” Ro) which establishes the equivalence. Particular attention to the meaning of the quantity V is needed. It is the line-integral of the radial force in the dielectric from r=a, to r=a,. Or it may be defined by SV =21ra,o=Q, if Q be the charge per unit length of wire. But it is not the electric potential at the surface of the wire. It is not even the excess of the potential at the wire boundary over that at the inner boundary of the return. For, as it is the line- integral of the electric force from end to end of the tubes of displacement, it includes the line-integral of the electric force of inertia. It has, however, the obvious property of allowing us to express the electric energy in the dielectric in the form of a surface-integral, thus, $ Vo per unit area of wire surface, or £VQ per unit length of wire, instead of by a volume integra- tion throughout the dielectric. Hence the utility of V. The possibility of this property depends upon the comparative insignificance of the longitudinal current in the dielectric, which we ignore. It may happen, however, that the longi- tudinal displacement is far greater than the radial; but then it will be of so little moment that the problem could be taken to be a purely electromagnetic one. We need not use V at all, (58) being the equation between e and C without it. It is, however, useful in electrostatic problems, for the above- mentioned reason. Again, instead of V, we may use o or Q, which are definitely localized. The physical interpretation of the force —dV/dz, in terms of Maxwell’s inimitable dielectric theory, a theory which is spoiled by the least amount of tinkering, confusion and bemuddlement immediately arising, is sufficiently clear, espe- cially when we assist ourselves by imagining the dielectric displacement to be a real displacement, elastically resisted, or any similar elastically resisted generalized displacement of a vector character. When there is current from the wire into the dielectric there is necessarily a back electric force in it due to the elastic displacement ; and if it vary in amount along the wire, its variation constitutes a longitudinal electric force. (58) being a differential equation previously, let in it m? be a constant. Then R,’ and R," may be thus expressed :— R," a R’ —_ Ly'p, Bal = ikea + Lp, “ - 4 (61) 278 Mr. O. Heaviside on the where R,! and R,!, L,' and L,! are functions of p’. The utility of this notation arises from R,' &. becoming mere constants in simple-harmonically vibrating systems. Let em, Vm, and C,, be the corresponding quantities for the particular m ; then, by (56), em— Me = per a5 (Rint Lim P) Cn + (Rom ar Lom P) Cm: (62) Or 2 Vm (RL, ELE p)Ons. ck ne Em dz R'n= Rim +f R'm 5 ID = Jk, a Lim 7 Mon . (64) R’,, and L',, are functions of p’. Therefore, by (62), sum- ming up e~ SE Up) m ee Now, although HR, and L’,, are really different functions of p’ for every different value of m, since they contain m’, yet if, in changing from one m to another, through a great many ~ ms, from m=O upward, they should not materially change, we may regard RH’, and L’,, as having the m=0 expressions, as in the purely electromagnetic case, and denote them by R’ and L' simply. Then (65) becomes eX (ep). |. Bey simply. The equation of V is now CZ UE Seta, pair 4 67 dz de? PILE Y 7 IS See ( ) and that of C,, being ém= (Ba + Uinp+m/Sp)Cn -. . -. 468) in the m case, that of C becomes now simply 2 Spot Se = (B+ Lip) SpC. cn. eli GOR The assumption above made is, in general, justifiable. Let us now compare these equations with the principal ways that have been previously employed to express the con- ditions of propagation of signals along wires. for simplicity, leave out the impressed force e. First, we have Ohm’s system, which may be thus written :— dV dC av Gruden Ps | acca Binet? Here the first equation expresses Ohm’s law. C is the wire =RSpV. . (70) Self-induction of Wires. 279 current, R the resistance per unit length, and V is a quantity whose meaning is rather indistinct in Ohm’s memoir, but which would be now called the potential. The second equation is of continuity. Misled by an entirely erroneous analogy, Ohm supposed electricity could accumulate in the wire ina manner expressed by the second of (70), wherein 8 therefore depends upon a specific quality of the conductor. The third equation results from the two previous, and shows that V, or C, or Q=SV diffuse themselves through the wire as heat does by differences of temperature when there is no surface loss. This system has at present only historical interest. The most remarkable thing about it is the getting of equations correct in form, at least approximately, by entirely erroneous reasoning. The matter was not set straight till a generation later, when Sir W. Thomson arrived at a system which is formally the same as (70), but in which V is precisely defined, whilst 8 changes its meaning entirely. V is now to be the electro- static potential, and 8 is the electrostatic capacity of the con- denser formed by the opposed surfaces of the wire and return with dielectric between. The continuity of the current in the wire is asserted ; but it can be discontinuous at its surface, where electricity accumulates and charges the condenser. In short, we simply unite Ohm’s law (with continuity of current in the conductor) and the similar condenser law. The return is supposed to be of no resistance, and V=Oatits boundary. The next obvious step is to bring the electric force of inertia into the Ohm’s law equation, and make the corresponding change in that of V ; thatis, if we decide to accept the law of guasi-incompressibility of electricity in the conductor, which is implied by the second of (70), when Sir W. Thomson’s mean- ings of S and V are accepted. Kirchhoff seems to have been the first to take inertia into account, arriving at an equation of the form EV /dz22=(R+ Lp)SpV. Tam, unfortunately, not acquainted with his views regarding the continuity of the current, so that, translated into physical ideas, his equation may not be conformable to Maxwell’s ideas, even as regards the conductor. Also, as his estimation of the quantity L was founded upon Weber’s hypothesis, it may possibly turn out to be different in value from that in the next following system. In ignorance of Kirchhoff’s investigation, I made the necessary change of bringing in the electric force of inertia in a paper “On the Hxtra Current”? (Phil. Mag. August 1876), getting this system, dC V dV —g ~(R+I)G, -Z =SpV, de Zz =(R+ Lp)SpV, (71) 280 Mr. O. Heaviside on the wherein everything is the same as in Sir W. Thomson’s system, with the addition of the electric force of inertia —LpC, where L is the coefficient of self-induction, or, as I now prefer to call it, for brevity, the inductance, per unit length of the wire, according to Maxwell’s system, being numerically equal to twice the energy, per unit length of wire, of the unit current in the wire, uniformly distributed. Coming after Maxwell’s treatise, there is of course no question of any important step in advance here, except perhaps in the clearing away of hypotheses involved in Kirchhoff’s investigation. The system (71) is amply sufficient for all ordinary pur- poses, with exceptions to be later mentioned. It applies to short lines as well as to long ones ; whereas the omission of L, reducing (71) to (70), renders the system quite inapplicable to lines of moderate length, as the influence of 8 tends to diminish as the line is shortened, relatively to that of L. An easily made extension of (71) is to regard R as the sum of the steady-flow resistances of wire and return, and V as the quantity Q/S, Q being the charge per unit length of wire. Nor are we, in this approximate system (71), obliged to have the return equidistant from the wire. It may, for instance, be the earth, or a parallel wire, with the corresponding changes in the formule for the electrostatic capacity and inductance. But there are extreme cases when (71) is not sufficient. For example, an iron wire, unless very fine, by reason of its high inductivity ; a very thick copper wire, by reason of thickness and high conductivity ; or, a very close return current, in which case, no matter how fine a wire may be, there is extreme departure from uniformity of current dis- tribution in the variable period ; or, extremely rapid reversals of current, for, no matter what the conductors may be, by sufficiently increasing the frequency we approximate to surface conduction. We must then, in the system (71), with the extension of meaning of R and V just mentioned, change R and L to R’ and L’, as in (67), and other equations. In a §.H. problem, this simply changes R and L from certain constants to others, depending on the frequency. But, in general, it would I imagine be of no use developing R," &c. in powers of p, so that we must regard (R,!+L,'p) &. merely as a convenient abbreviation for the R," &c. defined by (56) and (55). A further refinement is to recognise the differences between R’ and L/ in one m system and another, instead of assuming m=0 in R"”. And lastly, to obtain a complete development, and exact solutions of Maxwell’s equations, so as to be able to fully trace the transfer of energy from source to sink, fall Self-induction of Wires. 281 back upon (57), or (22), and the normal systems (18) of Fart: J, Now, as regards our obtaining the expansions of Ri), &c. in powers of p*, we have to expand the numerators and the denominators of R,” and R,” in powers of p, perform the divisions, and then separate into odd and even powers. When the wire is solid, the division is merely of $zJo(#) by J;(@), a comparatively easy matter. The solid wire R’ and L’ expansions were given by Lord Rayleigh (Phil. Mag. May 1886). I should mention that my abbreviated notation was suggested by his. But in the tubular case, the work is very heavy, so, on account of possible mistakes, I go only as far as p”, or three terms in the quotient. The work does not need to be done separately for the inner and the outer tube, as a simple change converts one R’ or L’ into the other. Thus, in the case of the inner tube; we shall have _ 2 2)2 iaat 7% _ 229 log (an/a0) Ri =H) 1 +n (uk, 7ra") 12 3 OP == 12 a a2( a2 — a?) 4ayilog (ay/ao) }? } , (72) ————— ai(aj— a5)" L/=R (unbimaty | 3—§%% + Dog 4. _ (73) 1 Wi 7a, U2 2B 5 a a2 (a2 — a2) vu where n’ is written for —p’, for the 8.H. application. As for L', it is simply the inductance of the tube per unit length (of the tube only), as may be at once verified by the square of force method. ‘The first correction depends upon p But R' gives us the first correction to R,, which is the steady-flow resistance, so it is of some use. To obtain R,/ and L,' from these, change R, to Ry, w, and hk, to ws and kes, a) to a3, and a, to dz. Or, more simply, (72) and (73) being the tube-formulze when the return is outside it, if we simply exchange ay and a, we shall get the formule for the same tube when the return is inside it. If the tube is thin, there is little change made by thus shifting the locality of the return. But if a,/a) be large, there is a large change. This will be readily understood by considering the case of a wire whose return is outside it, and of great bulk. Although the steady resistance of the return may be very low, yet the percentage correction will be very large, compared with that for the wire. Taking a,/a)=2 only, we shall find Ry! = Ry[1 + (whya2uyn)? x 012) Phil. Mag. 8. 5. Vol. 22. No. 136. Sept. 1886. U 282 Mr. O. Heaviside on the when the return is outside, and | Ry! = Ry [1+ (whya2uyn)? x *503 | = Ry[ 1+ (rhyo?u,n)? x 031] when the return is inside. In the case of a solid wire, the decimals are 083, so that whilst the correction is reduced, in this a,/a)=2 example, the reduction is far greater when the return is outside than when it is inside. The high-speed tube formule are readily obtained. Those for the inner tube are the same as for a solid wire, and that for the outer tube depends not on its bulk, but on its inner radius. ‘That is, in both cases it is the extent of surface that is in question, next the dielectric, from which the current is transmitted into the conductors. Let Go(v)=(2/r)K,(z), and G,(«) = (2/7) K(x) ; then, when 2 is very large, Jo(v) = aa Gi(2) a (sin 2+cos L) ik Ah (74) Ji(@)= Go(v)=(sin e—cos 2) + (wx)? | Use these in the Ry” fraction, and put in the exponential form. We shall obtain R's (p,812)/(27ra,). But 351 t= (why p)?m, therefore Ry" = (Mp p/7a})?. But Be a ae therefore p= (any +i) = (dn) +p an), so that, finally, (ZC ee icra fon L/=—, e e e (75) q=n/2mr=the frequency. To get R,! and L,’, change the # and p of course, and also ay to ap. It is clear that the thinner the tube, the greater must be the frequency before these formule can be applicable. For the steady-flow resistance is increased indefinitely by reducing the thickness of the tube, whilst the high-speed resistance is independent of the steady-flow resistance, and must be much greater than it. In (75) then, g must be great enough to make R’ several times R, itself very large when the tube is very thin. Consequently thin tubes, as is otherwise clear, Self-induction of Wires. 283 may be treated as linear conductors, subject to the equations (71), with no corrections, except under extreme circumstances. The L may be taken as L,, except in the case of iron. I will now give the 8.H. solution in the general case, subject to (58). Let there be any distribution of e (longitu- dinal, and of uniform intensity over cross sections). Hxpand it in the Fourier series appropriate to the terminal conditions at z=Qand/. For definiteness, let wire and return be joined direct, without any terminal resistances. Then, e sin nt being e at distance z, the proper expansion is Co = C00 + Coy COS 1242 + Cog COS MpZ+ 2+, where m=7/l, m,=27/l, &e. [It should be remembered that e is the e,—e, of (54) and (53). Shifting impressed force from the wire to the return, with a simultaneous reversal of its direction, makes no difference in e. Thus two e’s directed the same way in space, of equal amounts, and in the same plane z = constant, one in the inner, the other in the outer conductor, cancel. This will clearly become departed from as the distance of the return from the wire isincreased.| Then, in the equation n= (Bln + Ln) C+ (2/Sp) Cn =FBnCn+ (Un—m?/8n?)pCn, we know eé» 3; whilst R/,, and L’,, are constants. The com- plete solution is obtained by adding together the separate solutions for eq, é,, &c., and is 1 f eosin (né—4)) Com Sin (nt —O,,) Cos Mz C=7{ OS ON 8 (76 l (R? + L?n?)? + > TRE + (Li, me Sn?Pn » ( ) where the summation includes all the m’s, and tan 0,,= (L',—m?/Sn”)n— Bn. A practical case is, no impressed force anywhere except at z=0, one end of the line, where it is Vosinnt. Then, imagining it to be Vo/% from z=0 to z=2,, and zero else- where, and diminishing 2, indefinitely, the expansion required is Vo a=(V,/l)(1 +22 cos jrrz/L), j going from 1, 2,...to 0. This makes the current solu- tion become we sf (ne ORE Ar C= H (R? + L?n?)? +25 fRF,+ (Un 2/Qn2\2n2\2 ( ) If the line is short, neglect the summation altogether, unless U 284 Mr. O. Heaviside on the the speed is excessive. Now (77) may perhaps be put ina finite form when R’,, is allowed to be different from BR, though I do not see how to do it. But when R,=R’ and L',=L! it can of course be done, for we may then use the finite solutions of (66) and (67). Thus, given V=Vosin né at z=0, and no impressed force elsewhere, find V and C everywhere subject to (66) and (67) with e=0, and V=0 at ot Let | P=(i Sn)? {(R? + L?n?)? — Ln}? leh | ae C= Sareea PFs Sabie ene tan 6,=sin 2Q/— (e—??’—cos 2QJ), | (79) tan 0, =(LinP—R’/Q)+(R/P + L'nQ) ; then the finite V and C solutions are V=Voe7* sin (nt —Qz) Pz gj 6,) —e-F* sin (nt —Qz—8,) Vv ev sin (nt + Qe + @) —e7™* sin (nt — Qe =O) | 80 + Vo EF (e2Pl 4 @—2Pl__ 2 cos 2Q1)? ee Sn)? : C=V, Toe |e“? sin (nt — Qz—0,) _ Sisin(ntt Qe= 0409) +e-Fesin w= QE— AFB]. a Fl (e2Pl 4 ¢—2P!__9 cos 2QI)? ; If we expand this last in cosines of mz we shall obtain (77), with R,,=R’. There are three waves; the first is what would represent the solution if the line were of infinite length ; being of finite length there is a reflected wave (the ef“ term), and another reflected at <=0, the third and least important. The amplitude of C anywhere is (Sn)? pa + e~2P-2) + 2 cos 2Q(1—z) 3 °(R? + LPn?)? PPL 4 e—2F! __ 2 cos 2Q)/ At the distant z=/ end it is ; Sn)? } C= 2Vo rat pans (C+ eA —B.c082QN)-% (62) I have already spoken of the apparent resistance of a line as its impedance (from impede). ‘The steady flow impedance is the resistance. The short line impedance is (R?+ L’n’)2l Self-induction of Wires. 285 or (R? + L?n*):1, at the frequency n/2z7, according as current density differences are, or are not, ignorable. The impedance according to the latter formula increases with the speed, but is greater or less than that of the former formula (linear theory) according as the speed is below or above a certain speed. "But if the speed is sufficiently increased, even on a short line, the formula ceases to represent the impedance, whilst, if the line be long it will not do so at any speed except zero. According to (82) we have ( R24 Ln?) 2(Sn)? as the distant end impedance of the line. That is, we have extended the meaning of impedance, as we must (or else have a new word), since the current-amplitude varies as we pass from beginning to end of the line. (83) will, roughly speaking, on the average, give the greatest value of the impedance. It is what the resistance of the line would have to be in order that when an S.H. impressed force acts at one end, the current-amplitude at the distant end should be, without any electromagnetic and electrostatic induction, what it really is. The distant end impedance may easily be less than the impedance according to the electromagnetic reckon- ing. What is more remarkable, however, is that it may be much less than the steady-flow resistance of the line. This is due to the to-and-fro reflection of the dielectric waves, which is a phenomenon similar to resonance. To show this, take R/=0 in the first place, which requires the conductors to be of infinite conductivity. Then L/=L,, the dielectric inductance. We shall have, by (83) and (78), Vi /Oo= (e7F! + ¢—2P!__2 cos 2QI)*, (83) V,/C, = b,esin (ale) os iene rei (84) where v=(L,S)-?=(jscy)-?, the speed of waves through the dielectric when undissipated. The sine is to be taken positive always. If nljw=, 27, &e., the impedance is zero, and the current-amplitude infinite. Here nl/v=a means that the period of a wave equals the time taken to travel to the distant end and back again, which accounts for the infinite accumu- lation, which is, of course, quite unrealizable. Now, giving resistance to the line, it is clear that although the impedance can never vanish, it will be subject to maxima and minima values as the speed increases continuously, itself increasing, on the whole. We may transform (83) to 286 Mr. O. Heaviside on the SRW ACD Whi) a) nl! h nl\? V/0,= (H+ Ln] (7) sint(%)+(T7) go HG) nl\4 4. nl 6 +rbs(@) (+ ie4)— ap5-gg (7) + se) 10 nl\s % + rospoot(y) Atioh tet). ]. 8 where o=(LIS)-}, and h=(R!/L'n). The factor outside the [ | is the electromagnetic impedance ; and, if we take only the first term within the [ |, we shall obtain the former infinite conductivity formula (84). The effect of resistance is shown by the terms containing h. With this v' and / notation (83) becomes Vo/Co= 4h Lie! (1 +h)*{e2P! + e2F’—2 cos 2Q/H ; (86) where : i: Ql=(nl/v')( V1 +441)? V2, Pl=(nl/v')( /1+h—1)? + V2. Choose @ so that 2Q/=27, and let h=1. This requires nl/yl=2°85. Then V,/C,=s Lie’. 9? [@'8284r 4 @—.- -— 27, = 60°6 L’ ohms, if we take v=30!° cm.=30 ohms. This implies L'=I, and the dielectric air. Without making use of current-density differences, we may suppose that the conductors are thin tubes. Therefore, Impedance _ 60°6 L’. 10° Resistance _ RI by making use of the above values of h and nl/v’. But take 2Q/=47, or one fourth of the above value. Then V,/Co=28 L! ohms, =about 292, and Impedance : =about 4. Resistance 2 Thus the amplitude of the current, from being less than the steady-flow strength in the last case, becomes 42 per cent. greater than the steady-flow current by quadrupling nl/v', and keeping h=1. We have evidently ranged from some- where near the first maximum to the first minimum value of the impedance. These figures suit lines of any length, if we Self-induction of Wires. 287 choose the resistances &c. properly. The following will show how the above apply practically. Remember that 1 ohm per kilom.=10* per cm. Then, if 4, =length of line in kilom., ieee and Pie 1 y.:.’a= 10%, and 4= 856, foi ee nO, ye, = 8568, eee te a, ION, oo, Boy Pee io eo AiG tT bee per et E00 e107, nn) 18568, » H=10?, , U= 1,,, n=10, ,, h=85,: obey — es Oa LO Lee BD mero 10 E005. w= 10", * Yop, eee) te =e LO) ara 10?Y If oa Sd. The resistances vary from ;/, to 100 ohms per kilom., the inductances from 1 to 100 per cm., the frequencies from 10?/2a to 10°/27, and the lengths from 8°5 to 8568 kilom. In all cases 2 is the ratio of the distant end impedance to the resistance. The common value of nl, is 856800. In the other case, n//v' has one fourth of the value just used, so that, with the same R/ and L’, J, has values one fourth of those in the above series. Telephonic currents are so rapidly undulatory (it is the upper tones that go to make good articulation, and convert mumblings and murmurs into something like human speech) that it is evident there must be a considerable amount of this dielectric resonance, if a tone last through the time of several wave periods. Having got the solution for C, the wire current, we may obtain those for H, IT, and y from it. Thus, H, being the same as (2/r)C,, where C,1is the longitudinal current through the circle of radius 7, we may first derive C, or H, from ©, and then derive TI and y from either by (11). Thus, make use of (49) and (50), and the value of A, there given. Then we shall obtain oth Ji(syr) — (Jy/Ky)(s140) K, (817) = Cone a CD (sya) Ky (sya) 7 © i ae where, in the s;, p and m? are to be d/dt and —d*/d2’. Similarly for the return tube. In a comprehensive investigation, the C solution would be only a special result ; as this special result 1s more easily got by itself, it might appear that there would be some saving of labour by first getting the C solution and then deriving from 288 Dr. 8. P. Thompson on the Formule of the it the general. But this does not stand examination ; the work has to be done, whether we derive the special results from the general, or conversely. In the solid wire case Goes Ji(s17) C, oF "0d (8%) 2 C= {14 H(mmhip + ene )(0? a2) 1 1 +4 spa (munik, p + bm?)?(r? —a2) (0° — 2a) 1 a 429232 (m@uykyp + gm’ )*(r° — az) (1° —Sr°ay + Tay) 5 +o } C. Or, use the M and N functions of Part I., equations (42). For we have Jo(s;r) = (M +iN)(s,r#), where s,ri? takes the place of the y in those equations. M contains the even and N the odd powers of (p+ m?/4arpyh). We have also Ly = Jo (syr) T “1 Jo (s1”) 0 Ora, J1(s1a) T, being I at r=0 ; and, since by the first of these Q,,=Jd o(s1%) 1 connects the boundary and axial current-densities, we see that the ratio of their amplitudes in the 8.H. case is (M?+N?)’, using the r=a, expressions, with m=0. I hope to be able to conclude this paper in a third part. XXXIV. Further Notes on the Formule of the Electromagnet and the Equations of the Dynamo. By Professor SILVANUS P. Tuompson, D.Sc., B.A.* 1. The Lamont-Frélich Formula. R. O. FROLICH has done me the honour of replyingt to a certain point in my former communication to the Physical Society, “On the Law of the Electromagnet and the Law of the Dynamo”. In that communication I pointed * Communicated by the Physical Society: read June 26, 1886. + Elektrotechnische Zeitschrift, vii. p. 168, May 1886. t Phil. Mag. vol. xxi. p. 1, January 1886, ~ Electromagnet and the Equations of the Dynamo. 289 out that Lamont had in 1867 published a rational theory of the electromagnet, based upon the assumption that the per- meability of the iron was at every stage of the magnetization proportional to the deficit of saturation, leading him to an exponential expression, m=M(1—e-*), where m is the magnetism present at any stage, M its maxi- mum value, & the ratio of the permeability to the deficit of saturation, and 2 the magnetizing force proportional (approxi- mately) to the number of ampere-turns of the magnetizing current. This formula more correctly expressed the facts than either of the commoner formule of Lenz and Jacobi and of Miler. I further pointed out that Lamont had himself* given, as a sufficient approximation to the formula, the simpler expression, pt aMe ~ M+ae’ which formula is mathematically identical with that now commonly attributed to Dr. Frolich. For, writing a=kM, we get at once ka L+ke which is the formula claimed by Frolich. Lamont having developed his exponential expression in a series of ascending powers of ka, I did the same for the simpler formula for the purpose of comparison, and showed that, neglecting the fourth and higher terms of each series, the expansions are very nearly equal for all values of ka except for very large ones, and are identical for the value kr=3. Dr. Frélich, overlooking the words I have above italicized, commits the mistake of supposing that I had said that La- mont’s exponential expression is identical in value with the simpler formula when kz=%. I have said nothing of the kind. Further, when Dr. Frolich says, “ Hiernach ist die Aussicht vorhanden dass nicht die Lamontsche sondern die von mir benutzte Formel die wahre Gesetz der Hlektromagnete ent- halt,” he is forgetting that the formula used by him is also Lamont’s. He has proved, in his most recent communication, that the differences between the calculated and the observed values are about half as great when calculated by the simpler formula. The second and simpler formula suggested by Lamont appears therefore to be better than the first and more m=M * Lamont, Magnetismus, p. 41. 290 Dr. 8. P. Thompson on the Formule of the complex formula which he suggested. It has been recently shown by Mr. Bosanquet* that if we take as expressive of the permeability, not the instantaneous value dm/dzx, but the integral value m/x, and treat this on Lamont’s plan as pro- portional to the deficit of saturation, m - =k(M—m), we deduce at once the formula in question. Dr. Frélich’s researches upon the dynamo have given us the most complete and perfect proofs of the adequacy of the formula to represent the facts of the electromagnet as it is used in practice. My former communication was indeed mainly written to point out the extreme value and interest of Dr. Froélich’s work from this point of view. 2. Frélich’s smplified Formula of the Hlectromagnet. Since my former communication to the Physical Society was made, a further work on the theory of the dynamo by Dr. Froélich has appeared}. In this work he carries the sim- plification of the formule of magnet and of dynamo one stage further by introducing considerations somewhat closely con- nected with those which entered into my own work of 1883-4. The form adopted by me in 1883 for the Lamont-Frélich formula was ape GKeSi a3 1+o8%° where H is the resulting average intensity of the magnetic field (in which the armature rotates), « the initial value of the magnetic permeability, G a coefficient depending upon such purely geometrical quantities as the form and size of core, pole-pieces, and coils, S the number of windings of the mag- netizing coil, and o the “saturation-coefficient.”” This can be transformed at once to Froélich’s form by writing M=Gx«/e; k=o; «=Si. In 1884 I pointed outt the nature of this saturation-constant and its importance in the resulting equa- tions of the dynamo. It is the reciprocal of that number of ampere-turns which, to mark its importance, I ventured to term “ diacritical,’’ namely that number of ampere-turns which will reduce the instantaneous value of the magnetic permeability to half its initial value, or which, in the formula used, will give * ‘Electrician,’ vol. xvi. p. 247, February 1886. | + Die dynamoelektrische Maschine. Eine phystkalische Beschreibung fiir den technischen Gebrauch, von Dr. O. Frolich (Berlin, 1886). { ‘Dynamoelectric Machinery,’ first edition, p. 221; also Report Brit. Assoc., Montreal Meeting, 1884. Ellectromagnet and the Equations of the Dynamo. 291 to the magnet exactly half its maximum magnetism. 1 further pointed out in my lectures on the dynamo that year, that, if the number of windings of the coil 8 is given, there will be a “ dia- critical” current, namely a particular value of current which will exactly half-saturate the magnet. Dr. Frélich has inde- pendently made use of this conception, and has applied it to the formula of the electromagnet. The argument is his, but I retain the notation I have used. Writing (S:)’ for the diacritical number of ampere-turns, we have (as I showed in 1884) (Si)/=1/c. Taking the expression Gksi Gk Sz H= ae — 1+ocsi Co L 13; oO and writing 5 y= GF, oO we have Sz Oo" Sie Bi where Y is obviously the limiting maximum value of H when the excitement is infinitely great. If 58 is given, then 7’ is the diacritical current, and the expression becomes a which is true for every electromagnet excited by a single current. Two observations made on any electromagnet will determine the two constants Y and 7’. Further, if r be the resistance of the magnetizing coil, since ir=e (the potential requisite to send the current through the coil), we may obviously write the equation € Ee ates j where ¢’ is the diacritical difference of potential, namely that difference of potential which, applied to the coil of resistance ry and of 8 convolutions, will half-saturate the core. The extreme convenience of this form of the Jaw of the electromagnet must be at once apparent, since it enables the equation of a given magnet to be instantly adapted to the case of any given current or potential, and is equally applicable to express either the intensity of the field or the magnetic moment of the magnet. To put the matter in a more general way, let yr represent current, or potential, or ampere-turns, and let a’ 292 Dr. 8. P. Thompson on the Formule of the be the diacritical value of the same for the given magnet ; let ¢ be the intensity of the field, or the strength of the pole, or the magnetic moment, or the integral of the magnetic induc- tion, and ® its maximum value; then gee ai ‘ame apy’ This being the general equation of the electromagnet, it remains to be shown how excessively simple become the equations of the various kinds of dynamo. 3. Hquations of the Series-wound Dynamo. If A is the ‘equivalent area”’ of the coils of the armature, and H the average strength of the field in which it turns, the number of lines of force cut in each quadrant is AH ; hence the average electromotive force at the speed n is H=4nAGN... 1). But and, writing B=4AY, and remembering that, if }R be the sum of the resistances in the circuit, H/>R=7 (by Ohm’s law), we get Sy Oe ary = spit But Y being the maximum value of H, it is obvious that nB is the maximum value that E could possibly have (at that speed) even if the magnets were separately excited to satura- tion. Hence nB/>R is the maximum value that 7 could have if the magnets were thus separately saturated and the arma- ture, driven at speed n, were in a circuit the total resistance of which was equal to >R. Adopting Frélich’s notation here, we will write as 7 this current; and as it is important to distinguish the current generated under such conditions, I propose to call it the “maximal” current*. The equation of the series dynamo now becomes | Pavesi, Laie! le oo ee or, multiplying each term by =R, Fee ee) iN ea eae * The maximal current must not be confused with the maximum cur- rent. The latter would be obtained by rotating the armature in the saturated field at a very high speed in a circuit of resistance so small that the current did just not fuse the conductors. The maximal current is that obtained at speed m in circuit of resistance 2R when the magnets are separately excited to saturation. _ Ellectromagnet and the Equations of the Dynamo. 298 where is the “‘ maximal” value of H at that speed with saturated magnets. And, again, writing e for the differ- ence of potentials at the terminals of the machine, since e, multiplied by the external resistance R=7 under all circum- stances, we have Bee IC Ueno itee he) It appears, then, that in the case of the series-wound dynamo, each of the single electrical quantities is equal to the difference between the “ maximal”’ value which that quantity could have at that speed if the field-magnet were separately saturated and the “diacritical’’ value of the same quantity. This important result was announced by Frélich* in 1885. It may be remarked that, since we may write aD) Ee Y ea a we may deduce from (1) the result H=nB—L’, and from this derive equation (8). It may be noted, in passing, that B is the electromotive forces that would be generated in the armature at speed = 1 if the field-magnets were separately excited to absolute satu- ration. It is the maximal value of H at unit speed. 4, Expressions for the “Dead Turns.” It is known that in every dynamo the current (witha given resistance) is not proportional to the speed, but is proportional to the speed less a certain number of revolutions per second. This latter number is known familiarly as the “ dead turns.” It is also known that (with given resistance) there is a certain speed below which the dynamo does not excite itself. This least speed of excitement (with given resistance) is the same as the “dead turns.” It is called by some the “ critical” speed ; though that name is preferably reserved for the speed that is critical for self-regulation, and which is (unlike the least speed of excitation) independent of the resistance. We can find an expression for the dead turns as follows :— Taking the expression for the current, j= Bg =sR-") equate it to zero; the current of the machine (series-wound) being nil when is reduced sufficiently to n’ (= the dead * Elektrotechnische Zeitschrift, vi. p. 133, March 1885, 294 Dr. 8. P. Thompson on the Formule of the turns), Then 7 WB FT aR and ; | (ae It appears, then, that the dead turns are proportional to the diacritical current and to the resistance ; and inversely pro- portional to A and Y, the factors of B. It will be noted that VTR=H’, the diacritical electromotive force. Again, we have found K=nB—EL’ ; and as E’=n’B, it follows that H=(n—0') BL. |. a) ee This result is interesting in itself, and might have been used as a starting-point for the equations of the dynamo, inasmuch as it is readily found by experiment as a fundamental relation between speed and electromotive force. Following the ana- logy and the nomenclature adopted, we may regard n’ (the dead turns) as the diacritical speed. In other words, it is that speed at which, with magnets separately excited to saturation, the induced electromotive force will be diacritical, and will, with the given resistances >R, give the diacritical current. This equation (6) gives by far the best method of determining the important constant B. ‘Two experiments to observe a pair of values of H and n will suffice to determine both 7’ and B. 5. Equations of the Shunt Dynamo. Here we use 7; and 7, for the resistance and current of the shunt-coil, and 7, and 7, for those of the armature. We may then calculate the potential at terminals as follows. Writins 4 for the resistance of the whole system of machine and its circuits, as measured from brush to brush’, H=e+reis=e ge =4nAH, +e ne e egg eb ge whence nBR Sa OR Macaca ee. ee * See my ‘ Dynamoelectric Machinery,’ second edition, p. 298. Electromagnet and the Equations of the Dynamo. 295 But es which the machine would give, if the magnets were sepa- rately saturated, when working at the speed through resist- ances as given; it may therefore be written as e. Whenee, finally, is the same thing as the “maximal’’ value of e Cra GO AUT BL 10 ¢beaGS) From which we get at once also for the shunt-current, WAT Os el KIMOTIUO 0s, LGD) and for the main-circuit current, aaah aed hentiie thq, 21110) 6. General Equation of the Self-exciting Dynamo, Let yy be any one of the currents or potentials of the dynamo, its “ maximal”’ value, that is to say the value it has when the magnet is separately saturated, and w’ its “ diacritical value.” p=f(a, A, Y, [R]) 3 and, by the very nature of the case, whatever the form of the function /, p=f(n, A, H, [R]); and sos pT. aoe eae whence = Ae ; vay and, finally, oh Ne Ue ie 1s ek at ae CER) which is the general equation of the self-exciting dynamo. It may here be pointed out that the two terms on the right- hand side of this equation—the “ maximal” and “ diacritical ”’ values of the quantity on the left—possess certain properties. In a given dynamo the “ diacritical ”’ term is a constant, whilst the “maximal” term is a variable which increases with the speed of driving. The maximal term when representing a current varies also with changes of resistance, in an inverse way, but differently in shunt dynamos from series dynamos: when it represents an electromotive force it does not vary with changes of resistance. A year ago Dr. Frolich sought to divide the equation of the dynamo into two parts—an 296 Dr. 8. P. Thompson on the Formule of the armature part and a field-magnet part. This does not quite correspond to the case. The “ diacritical” term appertains to field-magnet primarily ; but since the number of ampere- turns of current that will produce a half-saturated magnetic field depends also on the quantity and quality of the iron in the machine, it is impossible to regard it as independent of the iron masses of the armature. The “ maximal” term is proportional both to A, the total effective area of the armature-coils, and to Y, the maximum value of the magnetic field. The “ diacritical” term for currents is lowered by increase in the number of magnetizing coils, and for potentials is raised by increase in the resistance of the magnetizing coils. The “maximal” term is not altered by altering the magnetizing coils, but increases with an increase in the number of coils of the arma- ture, and, for currents, decreases with increasing resistance. If the iron parts of a dynamo be given of a certain form, size, and quality, then it may in general be said that yy’ depends only on the windings (and, for currents, on the resistance) of the field-magnet, and 1 depends only on the windings of the armature (and, for currents, on its resistance) and not upon the windings of the field-magnet, or on their resistance except (and this only for currents) so far as their resistance contri- butes to the resistance of the whole circuit through which the current generated in the armature flows. 7. Conditions of Self-regulation for Compound-wound Dynamos. In this case we write the formula of the electromagnet in terms of the ampere-turns ; Sz, for the excitation due to the coils in the armature part of the circuit, Zz, for that due to the shunt, and ¢’ as the diacritical number of ampere-turns. Then, writing H=4nAH ; e=H —(?at?m)ta 3 hy Li, + Sz, Lis + Sty oe i H=Y Zi,+8i,+ 6! = Tar eile ; by equation (11) ante 5 B=4AY : we have ep! =1,Z(nB—e) +¢,{8(nB)—e)—(ra timp}. . (12) The condition sought is to be such as to make e constant. Now ¢’ is a constant, so is 7, if e is, and m may be made con- stant at any value we please. Hence, as 2, is a variable, the condition of constancy can only be attained by giving the Electromagnet and the Equations of the Dynamo. 297 dynamo such a speed n, that the coefficient of 7, shall be zero, or that eS Nb =Hey— (Paty gee Sern Ma ELD) But ¢, the maximal number of ampere-turns, is not itself a con- stant, since it contains as one of the three terms in its sum the quantity S¢,. Hence we gather that absolute self-regulation is physically impossible ; and it approaches to perfection as Zi,+’ are great as compared with Sz, In other words, there must be so much iron in the machine that the diacritical excitement is very great, and it must have a small armature- resistance ; otherwise S (and 8z,) cannot be small as compared with Z(and Z7,). This is known already to electric engineers. Assuming that the dynamo is well designed in these respects, @ will be very nearly constant, and the equation of condition may be accepted as adequately true. ‘This leaves equation (12) in the form ed’ =1,Z(n,B—e), ~~ =n, B—e, s 7 =mB —e. Putting in this value of »,B—e into equation (13), we have ~ rs Peay Ge a ea ga eR) aa) Now ¢’ may be written either as Sz.’ or as Zi,’, and @ may be written either as S7, or as Zi,. Choosing the second form in the first case and the first form in the second case, we may obtain Tots =(%atTm) ba 3 or, finally, — P= Cd tas pareve odie ti) soy: es eee QED or, the diacritical value of the potential at terminals for the shunt-wound part of the circuit must be equal to the maximal value of the potential at terminals for the armature and series- wound part. ‘The equation (14) also gives obs PUREE: 49 1064 Phil. Mag. 8. 5. Vol. 22. No. 136. Sept. 1886. Xx 298 Mr. R. H. M. Bosanquet on Electromagnets. which is more correct than the formula usually given hitherto for the ratio of the shunt- and series-windings, and which assumes absence of saturation terms. The simplicity of these results, no less than that of the pro- cesses by which they are derived, lends additional value to the new formula of Dr. Frélich, whose work deserves to be more widely known and recognized than it now is. City and Guilds of London Technical College, Finsbury, June 1886. XXXV. Electromagnets—V. The Law of similar Electro- magnets. Saturation, &c. By R. H. M. Bosanquzet, St. John’s College, Oxford. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, if previous papers on Hlectromagnets I have (1)* indicated the general nature of a formula for the moment and in- duction in electromagnets, and in subsequent papers f given the details of the complete examination of the permeabilities of many specimens of iron and steel, together with an attempt at a molecular formula for these permeabilities worked out in some detail. In the present communication I propose to give as shortly as possible an abstract of the results of a great number of experiments having special reference to the mag- netic resistance of cylindrical bars of length equal to twenty times their diameter, with and without pole-pieces. The datum in question (magnetic resistance) is that needed to define the magnetism under given electromagnetic exci- tation. The experiments cover the whole region from small mag- netic inductions up to saturation, or what would be commonly called so. _ The experiments have been made on bars of different sizes of the proportions in question, so as to furnish for the first time an experimental examination of the law of the magnetism of similar solids. It appears to be of great interest to ascer- tain how far this law can be depended upon in practice. The result is that, while in the main the law is conformed to, the irregularity shown by different specimens, especially in the extreme regions of small inductions and saturation, is very great. General deductions therefore, such as have been recently published, depending on the behaviour of single ‘* Phil. Mag. xvii. p. 581. + Ibid. xix. pp. 73, 388; xx. p. 318. Mr. R. H. M. Bosanquet on Electromagnets. 299 specimens, cannot be accepted as having more demonstrative force than belongs to preliminary investigations. So far as work of this description goes, my own paper in Phil. Mag. June 1884 (xvii.) p. 531, showed in detail the properties of bars ; and specially pointed out that the course of the values did not at al indicate any definite limit to the magnetism, and that the existing idea of saturation was a complete mis- take (p.535). This is now being published as a new discovery. In subsequent papers I have shown that rings do not ex- hibit the same behaviour ; and though some ring specimens admit of the magnetism being forced up very high, yet others do not. Further, where the magnetism is forced up very high in rings, the whole magnetization function appears to change under the influence; and the values thus cbtained are not generally capable of being satisfied by the same expression, which represents the behaviour of the specimen under induc- tions less than $3 =18,000. The difference in this respect between rings and bars is already indicated in my paper, Phil. Mag. 1885 (xix.) p. 79 ; and it is there pointed out that the cause is probably the dif- ference in distribution, 7. e. that in bars the lines of force are crowded closely only at the equatorial section. The law of similar solids is easily deduced from the fact that magnetic resistance (quotient of potential by induction) is of linear dimensions. Hence, in similar electromagnets, where the inductions are the same, the number of current turns required is proportional to the linear dimension*. The present two series of experiments were made on cylin- drical bars having the following dimensions:—The pole-pieces of the second set were circular, had a diameter five times that of the bar, and thickness equal to the diameter of the bar. The object of the experiments on the bars with pole-pieces was mainly to obtain numbers which might assist to serve as bases for a knowledge of the behaviour of the field-magnets of dynamos. * I have long been under the impression that this law was enunciated in substance by Sir W. Thomson. But after carefully going through the reprinted papers, the nearest I could find to it is at p. 485 (‘ Electrostatics and Magnetism’). But this statement does not include the proposition that the number of current-turns is proportional to the linear dimensions. And I am therefore doubtful whether the law is really due to Sir W. Thomson. X 2 300 Mr. R. H. M. Bosanquet on Electromagnets. The linear scale is varied in proportions up to 1: 9d. The. largest of the masses weighed considerably over a ewt., and taxed our resources to handle it. Dimensions of Bars. Puiain Bars. Bars wity soup P.P. (pole-pieces). Bars. Length, Radius, n coils. || Bars. Length, el. n coils, centim. | centim. centim. | centim. I, 20 0-5 1092 I.| 206 0: 1121 1. 20 0°5 1109 Diy 20% 0-5 1110 IV. 40 10 1933 Tif. | 38°08 1:0 2024 V. 40 10 1931 VI. | 38:08 1-0 2134 VII. 60 Ves) 1951 XVII.|, 57:5 15 28387 VIII. 60 15 1909 ||XVIITI.| 57:0 15 2813 IX. 80 2°0 2557 KV a9 204 3046 X. 80 2:0 2543 XVL| 73:9 2:04 3074 XE OG 2°5 3187 XIII. | 94-85 25025 | 3041 XII. | 100 2:5 3172 XIV.| 94:8 2°525 3028 The determinations of bar IT. with pole-pieces were puzzling and anomalous, and were excluded. Determinations for small inductions were in all cases made in the first instance by reversing the effect of the horizontal component of the earth’s magnetism. ‘The bars were then wound uniformly from end to end, and the inductions determined by the effect on coils placed equatorially. In many cases great discrepancies occurred even between bars of the same measurement (they were all in pairs). In these cases the determinations were repeated, and in several cases the whole of the magnetizing coils were rewound. The results were then plotted on a large figure for each bar, and a curve was drawn by hand to represent the values as well as possible; 9% was abscissa, and p (magnetic resistance) ordinate. It was at once evident that in all cases, almost without ex- ception, the curve did not show p=co for any finite value of $3, but rather indicated the existence of an asymptote inclined at a moderate angle to the axis of p. The values of p corresponding to B=0, 1000, 2000, &e. were then measured off, and all divided by the lengths of the bars, so as to reduce them to data for bars 1 centim. long, according to the above Jaw of similarity. These values, with the resulting means, form the two fol- lowing Tables, 301 ug S| 33= | 0,000 = S T. | 01200 8 II. | 00800 = IV. | 01000 j=) 6 V. | 01100 = & | VII. |-00905 2 | VIIT. | 01000 ro . | IX.|-0098 = X. | 00975 Si alo 7 | 00770 os 2 | XII. |-01203 fee toes Means.| ‘00990 1,000 | 2,000 ‘00610 ‘00510 ‘00735) 00680 00650} -00592 ‘00650} ‘00570 00590} ‘00550 00595) ‘00533 00687) 00625 ‘00627 ‘00589 ‘00706 ‘00621 ‘00755} ‘00576 | —_ ‘00660 00592 3,000 | 4,000 | 5,000 ‘00570} 00560} 00555 00645) 00615} ‘00590 00587 00560) 00545 00550} ‘00547 ‘00517 ‘00616 ‘00513 ‘00617 ‘00625) 00625 00590) :00588 00568) ‘00567 00582) 00576 00580) :00575 Plain Bars. Table of p for length = 1 centim. 00555 ‘00570 ‘00575 00545) ‘00545 00547) 00547 00520} ‘00530 00615)-00615 00625) ‘00625 00588 00567) ‘00564 00573 00555 00560 (00570 00545 00547 00538 00615 (00625 00588 00565 00571} :00570} 00571) 00572 6,000 | 7,000} 8,000 | 9,000 00555) : 00560) - 00565) * 00545) ° ‘00550) 00550 00542 ‘00615 ‘00547 ‘00620 00625) 00625 00590) ‘00593 00566) 00568 10,000 ‘00555 ‘00560 ‘00570 ‘00545 00560 ‘00552 ‘00622 ‘00631 -00600 ‘00572 ‘00577 11,000) 12,000 ‘00555 00565) 00565 00570, ‘09575 00545) ‘00550 ‘00570 00567 ‘00625 ‘00562 00568 00625 00662 ‘00617 ‘00580 ‘00641 ‘00602 ‘00574 00580) 00587 13,000 00555 ‘00565 ‘00590 ‘00555 ‘00577 ‘00577 ‘00654 ‘00715 00634) - ‘00587 ‘00601 14,000 ‘00555 ‘00570 ‘00608 00562 ‘00583 ‘00588 ‘00720 ‘00766 ‘00621 15,000) 16,000 ‘00565} ‘00595 00570} ‘00575 00675} ‘00780 00582) ‘00610 00612) 00705 ‘00605} :00647 00831) 00975 ‘01056 ‘00712 ‘00875 | 17,000} 18,000 00640} 00700 00615] 00685 / 00970} 01840 0650} -00750 00957 00790} 01050 01162] -01375 01800) 01552 ‘00660 —_ | ————_ ‘00661 ‘00740 . ‘00866) (01064 302 Mr. R. H. M. Bosanquet on Electromagnets. ¢TOTO- SGOTO- 68600: VGOTO- Grr LO: 000‘8T |000‘LT |000‘9T |000‘ET |000‘FT |000‘ST |000‘ZT \000‘TT [00001 | 0006 | 0008 | 000‘L | 000‘9 | 000°E | 000‘ | 000'E | 000° | 0D0'T | 000'0 | = 9E 9900: G8200- 61200: 0€900- |9EF00: 06900: |F6E00- G6600-.|68900- 6GE00- |60600- O&E00- |9TE00: 69¢00: |VEE00- - |GOE00- \60E00- - SPE00- |L6800- L600: |91800- [6G00- |88c00- - 1GLE00- |GOE00- L6600- |68600- O&&00- | TT800- * 08600: |08c00- 10800: | T0800: 60800: GOS00- 61800: 68600: F8c00: GO0E00- GTS00- 96600: 78c00: 16600: |68200- 08d00- 96600: 96600: 90600: GOSO00: ITS00: 88600: P8c00: 68600- 08600. G6c00- ¥0800- 60800: TL800- 68600: ¥8c00- {6600- ¥0800- 60800: T1800: 16c00- 98200. 68Z00- |6800- 08200: |08c00- 16600: |16200: ‘TIYUOO [T = YSuey oz d jo o[quy, "(saoard-aj0d) "7" pyos yum sung ‘peydope saequinu|zeeoo- |69800- S600: SFL00- | SUP 6600: |96c00- POE00: |\F0800- GOE00- TT800- 96600: 88Z00: FOE00- F100. c0&00- 66600: 68600: |68600- 16600: |STE00- 16600: 10800- |86Z00- |G6Z00- |F6Z00- |F6Z00- |F6E00: |L6G00- |10&00- |S TE00- SS OE | ee ee ey ey 90800: |86200- |46600- 66600: |S1S00- 20800: 40600: LIE00: GG00- 8T800: G0800- G0&00- vgsoo- GVE00: /ETFO0: G6E00: |69600: &1800- |66E00: OFE00- |O8800: GPS00- |ELE00- O&E00: |TSE00- €G600: |L600- 0Gr00- |19L00- €8£00: |8Gr00: 16200: |90&00- |OSE00- GLLOO- | AIX G8900- | “TITX FOLOO: | TAX 60900: | AX ¢L¢c00- |TITAX 8cr00: | ITAX OFSTO- | TA 88200. | IIT 68400: | I Mr. R. H. M. Bosanquet on Electromagnets. 303 It will be noticed that the numbers of windings are not exactly proportional to the linear dimensions. It was not practicable to carry this out ; and of course it does nut matter whether a given number of current-turns is made up one way or the other (by current or turns). In the first three entries of the bars with pole-pieces, the initial discrepancies being very great, the means of selected values were adopted. The discrepancies between the numbers in any one column are those in which any deviation from the law of similars would appear. While the values (particularly at beginning and end) are capricious in the extreme, it is certainly not possible to detect any systematic difference between the numbers belonging to the large and small bars. Before proceeding to calculate with these numbers, I will now exhibit in a small figure the nature of the relation between the magnetic resistances due to (1) the permeability of the metal, as derived from rings ; (2) the bars with pole-pieces (P.P.); and (3) plain cylindrical bars. poke hestktadbec lo ength = xh | | agnetic resistance \of ben imal cdc MR Vea | te PY p oy a. oe poe I 1000 5000 10,0L0 15,000 20,000 The resistance in each case is supposed to be due to a length of metal = 1 centim., and p is expressed in decimals of a centimetre, though drawn to a ditferent scale for convenience. The lowest curve is made up of the values from rings H, F 304 Mr. R. H. M. Bosanquet on Electromagnets. (Phil. Mag. xix. p. 76), so as to show the wide limits within which specimens vary. The middle curve is that of the bars with pole-pieces. The initial mean and adopted value are both traced. The upper curve is that of the plain bars. This figure illustrates, and indeed proves, for approximate purposes the rule I enunciated (Phil. Mag. xvii. p. 534), and shows how the rule includes bars with pole-pieces, and where it fails, viz. in the region spoken of as in the neighbourhood of saturation. The rule is :—“ The magnetic resistance of any bar can be expressed as the sum of a resistance due to its form, and a quantity formed by dividing the length by yp, the per- meability of the metal.” In ‘the figure the lowest curve represents the rosishanes due to the imperfect permeability of the metal. The two upper curves obviously admit of derivation from the lower one by addition of constants due to the respective shapes of the bars, allowing for the capriciousness of the values in different specimens, and for the difference in the region of saturation (% =18,000 and thereabouts). As to the amount:—The analogy of the resistance to a fluid flowing from the end of a pipe would give 2 x ‘6 of the radius, or in the present case ‘012.centim., for the shape-resistance. The fact that magnetism issues all aloug the bar, more or less, diminishes this resistance ; so that we find in fact that in the plain bar the addition for shape is somewhat over ‘005 centim., as we can see in the figure. Again, the pole-pieces still further facilitate the flux of the magnetic induction through the ends, and the shape-resistance in this case is somewhat over ‘002 centim. I proceed to a more detailed examination of the numbers. For the plain bars, the shape-resistance calculated from the formula at Phil. Mag. xvii. p. 534 exactly satisfies the require- ments of the numbers. Hence the first step is to subtract the value thus calculated from the number to be dealt with ; so l that 377 10-7 = -00528 is subtracted, and p—-00528= . gives the calculated values of w. ‘These are then fitted to formule in accordance with the theory of Phil. Mag, xix. p. 92. With one exception, which is as follows :—Since the saturation-curve approximates to an inclined asymptote, it is impossible that % can have finite maximum values ; and the best way of making allowance for this has appeared to be, to substitute for the definite maximum %,, a sum made up of a constant B and a term proportional to the magnetizing Mr. R. H. M. Bosanquet on Electromagnets. d05 force, whose coefficient was determined from the observed values. In the figure which follows, the crosses show the observed values ; the dotted line shows the best representation by means of a maximum value of the induction B,, fitted to the highest value ; and the dashed line shows the representation by means of the constant B and the term proportional to the magne- tizing force. | z agnetic- resistance Pian Bars 4 1000 5000 10,¢00 15,000 20,000 The two following Tables give the complete numerical results of the representation of the numbers for plain bars ; first, according to the calculation with a fixed maximum induc- tion, and, secondly, where the maximum induction is made to contain a term pr ‘oportional to the magnetizing force. The extraordinary gain in the representation of the saturation region in the latter case will be noticed. — is the magnetizing force. Its coefficient m in the second iv table is 18°5 nearly. 306 Mr. R. H. M. Bosanquet on Electromagnets. Plain Bars without Magnetizing-Force term. picale=00598%. = (=i wena log © =2-34670, pe p p-='895 (18593 — 98) cos 6, log f= ‘16896. 6=fo, —* Ue, sin 8 pe 0. T5 EDRs: | eS a Cale. Obs. Cale. Obs. 0,000 188 188 0 ‘01060 | 01061 |—-00001 1,000 845 806 | + 39 00646 | -00652 |—-00006 2,000 1366 1562 | —196 00601 | 00592 |+-00009 3,000 1766 1852 | — 86 00585 | 00582 |+-00003 4,000 2052 2083 | — 31 00577 =| -00576 |+:00001 5,000 2239 2222 6,000 2343 2326 7,000 2376 2381 ale 00573 | :00578 “00000 + 8,000 2346 2326 | + + + + 17 17 || 00571 | -00571 00000 5 || 00570 | :00570 | -00000 20 || 00571 | -00571 ‘(0000 9 || 00572 | -00572 | -00000 98 || 00575 | :00577 |—-00002 11,000 | 1979 1923 56 |} 00579 | -00580 |—-00001 12,000 | 1785 1695 90 || 00584 | -00587 |—-00003 13,000 | 1566 1370 | +196 |] 00592 | -00601 |—-00009 | 14,000 | 1324 1075 | +249 |) 00604 | -00621 |—-00017 15,000 | 1062 752 | +310 || -00622 | -00661 |—-00039 16,000 784 495 | +289 || -00656 | -00730 |—-00074 17,000 491 296 | +195 || 00732 | -00866 |— 00134 18,000 186 186 0 ‘01065 | 01064 |+-00001 9,000 2264 2273 10,000 2139 2041 Plain Bars with Magnetizing-Force term. aeale, =00526-4.-,. I=1-centim., -tdlon = 2 n4iaey ph p , ni log f= "17126, — 65("— +16,640—38 ) cos 6, log n=1°266685. op sin@é pL. : P- : 1G. bs EN hye eee ee Diffs. Cale. Obs. Cale. Obs. 0,000 189 188 |} + 1 01057 | -OL061 | —-00004 1,000 813 806 | + 7 ‘00651 | ‘00652 | —-OOCOL 2,000 1326 1562 | —236 ‘00608 | -00592 | + 00011 3,000 1731 1852 | —121 00586 | 00582 | +-COU04 4,000 2028 2083 | — 55 00577 = | 00576 | +:00001 5,000 2235 2222 | + 13 60573 | *00573 ‘OCO0O 6,000 2355 2326 | + 29 ‘00570 | ‘00571 | —-OCO001 7,000 2394 0381 13 ‘00570. | ‘00570 ‘O000CO 8,000 2391 2326 65 ‘00570 | 00571 | —-O00001 9,000 2271 2273 2 00572 | C0572 ‘U0C000 10,000 2123 2041 82 00575 =| -00577 | —-COCO2 11,000 1924 1923 1 00580 | ‘00580 “00000 12,000 1683 1695 12 ‘00587 | °00587 ‘00000 13,000 1409 1370 ‘00599 | 00601 | —-00002 14,000 1108 1075 33 00618 | 00621 | —-C0003 15,000 801 752 9 ‘00653 | -CO661 | — 00008 16,000 509 495 4 ‘00724 | 00730 | —-00006 17,000 297 296 1 ‘00864 | ‘00866 | —-00002 18,000 186 186 0 ‘OL065 | 01064 | + 00001 $+t4++ 144144 ey) lo) Mr. R. H. M. Bosanquet on Electromagnets. 307 The representation of the numbers derived from the bars with pole-pieces affords results of a similar character. It is not necessary to present the figure showing the nature of the gain in accuracy by introduction of the term proportional to the magnetizing force, as it is precisely similar in character to that already given for plain bars. The two following Tables give the representation of these numbers in the same way as the two last. The coefficient of the magnetizing force in the second table is 1271 nearly. Bars with P.P. (pole-pieces) without Magnetizing-Force term. K A +:415 (18366 —38) cos 8, Sg los fa= 1 GOOU: p. p. 9.0 (———_____—__| Diffs. ||——_____—_| Diffs. Cale. Obs. Cale. Obs 0,000 290 293 | — 3 00596 | 00593 |+-00003 1,000 905 855 | + 50 ‘00362 | -00369 |—-00007 2,000 1397 1250 | +147 00324 | :00332 |—-00008 3,000 1772 1639 | +133 ‘00808 | 00313 |—-00005 4,000 2046 2041 | + 5 00301 | :00301 ‘00000 5,000 2225 2222 | + 3 00297 | -00297 00000 6,000 | 2323 | 2381 | — 58 || -00295 | -00294 |4-00001 7,000 | 2352 | 2381 | — 29 || -00294 | -00294 | -o0000 8,000 | 2323 | 2381 | — 58 || 00295 | -00294 |+-00001 9,000 | 2240 | 2326 | — 86 || -00297 | -00295 |+4-c0002 10,000 | 2115 | 2174 | — 59 || -o0299 | -o0298 14-0001 11,000 | 1952 | 2041 | — 89 || -00303 | 00801 |4-00002 12,000 1756 1754 | + 2 ‘C0309 | -00309 ‘00000 13,000 1532 1408 | +124 00317 | °00323 |—-00006 14,000 1285 1031 | +254 00330 | :00349 |—-00019 15,000 1051 595 | +456 00347 | 00420 |—-00073 16,000 733 300 | +433 00388 | °00585 |—-00197 |. 17,000 432 207 | +225 00484 | 00734 |—-00250 18,000 118 118 0 01100 | -01103 |—-00003 308 Mr. R. H. M. Bosanquet on Electromagnets. Bars with P.P. (pole-pieces) with Magnetizing-Force term. log n=1-08160, Veet VE mes 69 (“2 + B—38) cos8, logs =2°57346, log f= -16878. | fh. 0. Bb. Diffs. Diffs. Cale. Obs. Cale. Obs. —_— ——. 0,000 297 Zoe ha: 00589 | °00593 |—-00004 1,000 894 855 | + 39 00564 | -00369 |—-00005 2,000 1384 1250 | +134 00324 | 00332 |—-00008 3,000 1767 1639 | +128 00309 | 00313 |—-00004 4,000 2056 2041 + Lo 00301 | 00301 00000 5,000 2250 2222 | + 28 00296 | 00297 |—-00001 6,000 2358 2381 — 23 00294 | -00294 ‘06000 7,000 2389 2381 SE ine. 00234 | :00294 ‘00000 8,000 2351 2381 — 30 00294 | 00294 ‘00000 9,000 2249 2326 | — 77 00296 | 00295 |+°00001 10,000 2091 2174 | — 83 ‘00300 | 00298 |+:00002 11,000 1882 2041 —159 ‘00305 | -00301 |+-00004 12,000 1627 1754 | —127 00313 | 00309 |+-00004 13,000 1339 1408 | — 69 ‘00327 | 00323 |+-00004 14,000 1021 1031 | — 10 00350 | °00349 |+-00001 15,000 707 095 | +112 00393 | -00420 |—-00027 16,000 451 300 | +151 00473 | 00585 |—-00112 17,000 179 207 | — 28 ‘00810 | 00784 |+-00076 18,000 118 118 0 ‘01102 | 01103 |—-00001 Many rings have been completely examined in my various papers. It is a matter of interest to what extent these sup- port the position that there is an absolute saturation limit, and to what extent they oppose it. First, as to Rowland’s own measures of the one ring which he examined completely. 1 have reduced them according to my formula at Phil. Mag. xix. p. 340. Although the magnetism is not carried very high, and the fitting has not been as perfectly accomplished as in my more récent calcu- lations, yet it is clear that there is no indication of variation of the saturation limit. This generally shows itself in an excess of the calculated values of » in the region %=15,000 —16,000. In all my own determinations the magnetism has been carried up much higher; and I think in some of the rings discussed at Phil. Mag. xix. pp. 76-79 there is some appearance of this kind. But I have repeatedly submitted nearly all of these to calculation, with the result that the introduction of the variable saturation limit, so as to fit the region in question, invariably Intelligence and Miscellaneous Articles. 309 unsettles the general representation of the values. So that in every case, without exception, the general representation of the behaviour of rings, furnished by calculation on the basis of a fixed saturation limit, has been far better than any ob- tained by using a variable saturation limit. At the same time it must be admitted that experiment sometimes shows a heightening of the apparent saturation limit as it is approached. The change, however, resembles rather a change in the pro- perties of the metal than a continuation according to the same laws which successfully represent the general course of the values. I must reserve for another occasion some considerations as to dynamo machines, founded on the numbers here obtained for bars with pole pieces. It must be noticed that the bars here dealt with are only of one definite shape, viz. diameter : length :: 1: 20 or there- abouts. And the object of the investigation was to get a series of reliable data with respect to this one shape, which might afford a sound starting-point for the investigation of other proportions. XXXVI. Intelligence and Miscellaneous Articles. MEASUREMENT OF PITCH BY MANOMETRIC FLAMES. BY M. DOUMER. N ANOMETRIC flames have hitherto only been used as a means of demonstration, and for studying the quality of vowel sounds. They are, however, susceptible of more varied applications, and may in particular vie with the graphicai method in studying the height of sounds. For this purpose two adjacent manometric flames are taken, one of which is due to a sound of known height, the other to a sound the height of which is to be determined, and then by means of a rotating mirror it is ascertained how many vibrations of the sound under investigation correspond to a known number of vi- brations of the chronograph. A simple proportion gives the desired height. This method, which is very simple in theory, is in practice com- plicated, and almost impossible from the want of centering of the mirror. But it retains all its simplicity, and a certain elegance moreover, if we substitute a sensitive plate for the rotating mirror ; in other words, if the two manometric flames are simultaneously photographed on the same plate. The camera obscura used is a broad one, provided with a lens of 310 — Intelligence and Miscellaneous Articles. very short focus, and with a suitable shutter devised by M. Duboscq. The focussing is easily effected, either by moving the object-glass, or, better, by moving the manometric flames. The motion of the flames to be photographed is so rapid that we should have recourse to the most sensitive plates. Those of Monck- hoven have given very good results; I prefer, however, the very sensitive plates prepared with silver iodide by Frank’s formula. But whatever be the plates used, the negatives will always be too weak if we do not take the precaution of using a lens of short-focus, and of making the flames as bright as possible. This is attained if we carbonize the gas by passing it through pumicestone impregnated with benzole, and burning it in pure oxygen. By suitably regulating the supply of gas and oxygen, we obtain a flame of great lustre. The plates are developed by the ordinary photographic methods ; they have then two rows of parallel dentations, one of which cor- responds to the vibrations of the chronographic flame, and the other to the vibrations of the flame worked by the sound whose height is to be measured. The comparison of ihe two flames is then an easy matter. It may be made in two ways; either by the measurement of the number of vibrations and fractions of vibrations comprised within equal lengths, or by the determination of the space occupied by known whole numbers of vibrations. If the height of the flame has been suitably adjusted, so as to give images of 1:5 to 2 millim., these measurements are made with great facility and remarkable precision. This method, devised for the purpose of lengthened researches on the vowel sounds, has been verified with great care for the sounds corresponding to the scales Ut,, Ut,, Ut,, aud Ut, by the aid of the chronograph Ut,, of sliding diapasons, and of open pipes constructed by M. Konig with his usual care. The following table shows the certainty and accuracy of the method, since the distances between the heights found and the heights indicated do not exceed a double vibration :— Heights. ate . Note. ispeaved. indicated JE lg oamenuees Re, 287-88 288 Tipe eg Mi. 1280-00 1280 tos Pn aenre Ut, 256-20 256 TOO? hee ee eee Ut, 1022-50 1024 SRR ade: Sol, 767-10 768 To measure very high or very low sounds, it is good to have recourse to two chronographic diapasons, one giving 100 vibrations per second, and the other about 2000. In this case in fact, where the difference of height between the two sounds is too considerable, Intelligence and Miscellaneous Articles. 3l1 the velocity to be given to the plate to photograph the most acute sound spreads out the image of the deeper one, and the measure- ment becomes very difficult—Comptes Rendus, August 2, 1886. ON A NEW METHOD FOR DETERMINING THE VERTICAL INTENSITY OF A MAGNETIC FIELD. BY R. KRUGER. The methods hitherto used for measuring the vertical intensity of a magnetic field depend on the electromotive action which it exerts on a spiral which rotates about a horizontal axis. If, more- over, the spiral turns about a vertical axis, the ratio of the currents induced in the two rotations gives the inclination of the magnetic lines of force to the horizon. If, again, the current induced by a rotation about a horizontal axis be determined in absolute measure by a galvanometer, the vertical intensity of the field is determined from the contents of the surface enclosed by the windings of the spiral, and from the absolute resistance of the circuit formed by the spiral and the galvanometer. Ina field of small extent the rotation must be replaced by a parallel displacement of the spiral in its plane. In any case the determination of a vertical intensity by these means is a difficult and tedious operation. In contrast with this, the deflection which a disk suspended horizontally by means of a vertical wire in a solution of copper sulphate experiences when traversed in a radial direction by a cur- rent, forms a very convenient means of determining the vertical magnetic force which produces that deflection. In order to test the capability of this method, it was used to de- termine the vertical intensity of the terrestrial magnetism, or rather the magnetic inclination. The value of the vertical intensity was thus found to be V=2°2903 x dip in which T is the horizontal intensity. Simultaneous observations with the terrestrial inductor gave while the vertical intensity deduced from the formula for variations given by Prof. Schering* would be V=2°2895T. Taking as a mean, V=2:2899T, * Gott. Nach. 1882, p. 388. 312 Intelligence and Miscellaneous Articles. it will be seen that the divergence obtained by this method does not exceed =755. The rest of the paper contains the establishment of the formula, and the experimental details and data.—Wiedemann’s Annalen, vol. xxvill. p. 613 (1886). ON THE CONSTANT OF THE SUN’S HEAT. BY M. MAURER. The author compares the values for this constant, obtained pre- viously by Pouillet, Hagen, Crova, Violle, and Langley ; those given by Violle and Langley appear too great. This seems to arise from the fact that Langley and Violle have found too high a value for the amount of the solar radiation on the Earth’s surface, while the methods of other observers, to which those of Rontgen and Exner, and Desains must be added, agree very well with each other. Actinometric measurements have recently been made with a new apparatus of F. Weber’s, under remarkably good atmospheric conditions. Six of them were made on the terrace of the Polytech- nicum at Ziirich, two on the top of the St. Gothard pass (2100 metres), and one on the Pizzo Centrale (8000 metres). - According to these, the maximum heat from the Sun at midday, which a surface of a square centimetre receives in a minute under perpen- dicular radiation, is :— im ‘Aumnich civsiats ae A’ 1-10—1°32 thermal unit. On the St. Gothard .... 1:°38—1°41 __,, 3 ,», Pizzo Centrale. . 2 thts ” These agree very well with the above observations, excepting for Violle’s and Langley’s.—Zeitschrift der Oesterr. Gesellschaft. ftir Meteorologie, vol. xx. p. 296, 1885 ; Beiblatter der Physik, vol. x. p. 182. ON THE INCREASE OF TEMPERATURE PRODUCED BY A WATERFALL. BY M. KELLER. The slight increase of temperature in a waterfall is powerfully affected by the sources of error which the author discusses. His observations at the waterfall at Terni should have given a differ- ence of temperature of 0°37; the measured ones varied between 0°-08 and 0°-73. The author considers these numbers an evidence of the transformation of vis viva into heat in this waterfall.— Adz della R Acc. dei Lincei [4] i. pp. 671-676 (1885); Berblatier der Physik, vol. x. p. 333. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIKES.] OCTOBER 1886. XXXVII. Turbines. By J. LESTER Woopsrince, B.S., M.E.* PROPOSE to discuss the action of turbines in general. Consider first the water after it has entered the wheel and is passing along its vanes. Conceive the water to be divided into an infinite number of filaments by vanes similar to those of the wheel, but subjected to the condition that, at each point, their width, ac (fig. 1), measured on the are whose centre is O, shall subtend at the centre a constant angle dé. Conceive each filament to be divided into small prisms whose bases are represented by the shaded areas a’b'c'd', d'c'c'd", and abed, by vertical planes normal to the vanes, making the divisions ae, ef, intercepted on the radius by circles passing through the consecutive vertices on the same vane a’, d', d'’, &e., equal. The variable height of a prism represent by 2, and let p be the variable distance from the centre. _ ‘Then dp=ae, ef, Ke. pd@dp=abcd, &c.=area of the base of an infinitesimal prism ; xpd@dp=volume of infinitesimal prism ; xdpd@dp=m=the mass of prism, 6 being its density ; y=san=angle between the normal to the vane at any point p, and the radius Oa prolonged through that point ; v=the velocity of a particle along the vane at p; w= the uniform angular velocity of the wheel ; and p=the pressure of the water at the point p. * Communicated by De Volson Wood, Professor of Engineering in Stevens Institute of Technology, Hoboken, N.J. Phil. Mag. 8. 5. Vol. 22. No. 137. Oct. 1886. Y Mr. J. L. Woodbridge on Turbines. 315 For the element of time, dt, we will take the time occupied by one of the liquid prisms in passing along the vane through a distance equal to its own length, or outwardly a distance dp, thus making é a function of p, the latter being considered the independent variable. We have a 98) dp): > dp dp — dp vsing dt In the figure let aa! be the distance passed through by the turbine in the time dt, then aa! = wpdt=wp = dp. The mass m will have two motions; one along the vane, the other with the wheel perpendicular to the radius. By changing its position successively in each of these directions, both its velocity with the wheel and its velocity along the vane may suffer changes both in amount and direction, and will give rise to the following eight pessible reactions :-— I. By moving from a to a’ in the arc of a circle. 1. wp may be increased or diminished ; 2. wp may be changed in direction ; 3. v may be increased or diminished ; 4. v may be changed in direction. II. By moving from a to d’ along the vane. 5. wp may be increased or diminished ; 6. wp may be changed in direction ; 7. v may be increased or diminished ; 8. v may be changed in direction. By the conditions imposed, 1 and 3 are zero. For the others we have :— No. 2. By moving from a toa’, the velocity wp is changed in direction from ak to ak! in the time dt. The momentum is mp, and the rate of angular change is kak _ @dt _ ea and hence the reaction will be mw’p in a direction radially outwards. This is the centrifugal force as designated by most writers. Resolving into two components, we have mw’ psiny along the vane, mw’p cos y normal to the vane. No. 4. In moving from a to a’, the velocity along the vane, v, is changed in direction from at to a't' at the rate as ¥2 316 Mr. J. L. Woodbridge on Turbines. in No. 3. The momentum is mv, and the force will be mva, which acts in the direction na, and being resolved gives __ 0 along the vane, —mvq normal to the vane. No. 5. In passing from a! to d’ the increase of wp will be edp in a time dé, and the reaction will be mw , in a direction tangential to the motion of the wheel but backwards, and its components will be mo cosy along the vane, — mot sin y normal to the vane. No. 6. In passing from a! to d’, wp will be changed in : direction by the angle a!Od! = oT — oe, and the rate of angular change will be cola; ap Sina and the momentum will be MOP ; hence the reaction will be me cot y up dt which acts radially inward, and its components are — Mo COS ee along the vane, —mo coty cosy S normal to the vane. No. 7. By moving from a’ to d’, v will be increased by an = dp, in the time dé, and the reaction will be m — . us e which acts backward along the vane, and its components are _ 22, dp " dp’ dt 0 normal to the vane. amount along the vane, No. 8. In passing from a! to d', v is changed in direction by two amounts :—Ist, the angle y changes an amount d = ip dp, and 2nd, the radius changes in direction an amount Mr. J. L. Woodbridge on Turbines. 317 BE oh Y as in No. 6; hence the total change will be the sum of these, and the rate of change will be the sum divided by dé, which result multiplied by the momentum, mv, will give the reaction, the components of which will be : 0 along the vane, mv = ” = Sue ae ali normal to the vane. This completes the reactions. Next consider the pressure in the wheel. The intensity of the pressure on the two sides ab and cd differs by an amount dp= dp. The area of the face is dex e=apd@ sin y, and the force due to the difference of pressures will be xpdé sin ry de dp. dp If dp is positive, which will be the case when the pressure on de exceeds that on ab, the force acts backwards, and the preceding expression will be minus along the vane. In regard to the pressure normal to the vane, if a uniform pressure p existed from one end of the vane V W to the other, the resultant effect would be zero, since the pressure in one direction on VW would equal the opposite pressure on XY. If, however, the pressure at a exceeds that at d by an amount —dp, since Va is longer than Xd, the pressure on Va, due to this —dp, will exceed that on Xd by an amount —dp.«xah=—dp.x.pd0 cos y= —2p 003 dO dp. Collecting these several reactions, we have Normal to the vane. Along the vane. (2) + mw’ pcos ¥. : + mo’ p sin y. 4) —mov. 0. (4) +) dp dp (5) —mq@ siny ae + mo cos y 5. dp dp (6) —ma coty cosy i’ — Mw COS Y FF. dp dv (7) 0. Se ° dp coty dp dy dp (8) +mv a [oer vat 0. dp yO PF (9) —zpcos ae dpdé. | xp sin y dp dpdé. 318 Mr. J. L. Woodbridge on Turbines. : The sum of the quantities in the second column will be zero ; hence mo” p sin y— —m2&. o — ap siny dpd@=0. . (I) Substituting ap i _m ‘rimkae and xpdédp= > and dividing by msin y, we have w*pdp— = dp=vdv. . 6, + ern limit limit per {i-fI --. 0 limit Timi! The sum of the quantities in the first column gives the pressure normal to the vane, which multiplied by psiny gives the moment. This done, and substituting as before, we have Integrating, 2 p mi ee dy ydp dM= mv | op(2 @ COs 2) pv siny 7 +0 cosy — pase = Fe sin ¥. e e éQ e e Putting mw sin Wigs _d8.dp, where Q is the quantity of water flowing through the “rie per second, and integrating in reference to 0 between 0 and 2ar, we have dy cos y dp dM= 5Q| wp(? » cos y—2)— pu siny teh ae Ps dp dp. Multiplying (2) by - cosy, we have ae Bee a cos ydp 5 Tag d mtn g Figg which, substituted above, gives aM=8Q| — 2wpdp +p cos ‘ap dp +vcosydp—pvsiny a and integrating, M=6Q[ —op’+ pv cosy] =—8Qelopvcosylin + +. - + () But wp—vcos y is the circumferential velocity in space of Mr. J. L. Woodbridge on Turbines. 319 the water at any point,.and d6Qp[@p—v cos x] is the moment of the momentum ; hence, integrating between limits for the inner and outer rims, the moment exerted by the water on the wheel equals the difference in its moment of momentum on entering and leaving the wheel. Thus we have deduced an expression which some writers have made the basis of their investigations. Let the values of the variables at the entrance of the wheel be Pir Yi» Vly Piy and at exit be P2, Y2) V2, Poe Then equations (3) and (5) become 407 (0,2 —p;?) = hv)? 02). oe te oe (6) a dQ @( 1” — ps”) — P11 COS Y1 + poe cos ¥2 | ° (7) “. U=Me=sQo| @(p,? — p2”) —p1v, cos ¥1 + pots cos y2|- (8) Hguation (8) gives the work per second in terms of the known quantities 6, @, p1, ¥1, P2, ¥2, and the three quantities. Q, v1, ve as yet unknown. ‘These three quantities are, how- ever, connected by the condition that the quantity of water flowing through all the sections radially is constant. Calling a, the entire area of all the orifices at the entrance of the wheel (= 27rp,2,), and ag those at exit (=2p,x,), we have Q=a,r, sin yj=ayvysinye, .« . » « (9) which reduces this number of unknown quantities to one. Hquation (6) is the equation of the motion of the water in the wheel. Besides the velocities vj and v,, it contains p, and po. Let a=the atmospheric pressure, h =the mean depth of the wheel below the surface of the tail-race, pe=Ogh=the pressure due to flooding in the tail-race ; then P2=Pat pe The pressure p, where the water passes from the guide- plates into the wheel is unknown. Another condition is necessary, which may be found by considering the passage of the water from the guide-plates into the wheel. In fig. 2 let A C be the tangent to the guide-plate at its extremity, V the actual velocity of the water on leaving the guide-plate, wp,=AD the velocity of the initial rim of the wheel; then 320 Mr. J. L. Woodbridge on Turbines. will A B be the velocity of the stream relative to the wheel. Now if A B does not coincide with the tangent to the vane at A, the stream cannot suddenly be made to change its direction into that of the vane, or float ; and the water, by cushioning in the angles, will make its own angles, as roughly shown in fig. 3. Fig. 2. Fig. 3. Yphiae tif iC { - Bil! c WWD TSS ~ It is impossible, either practically or theoretically, to determine the new angles, and probably they are not constant ; neither is it possible to determine the loss of energy due to eddying ; we therefore make the hypothesis that the final direction of the guide-plates, the initial direction of the vanes, the angular velocity of the wheel, and the velocity of the flow, are so related that the water on leaving the guide-plates shall coincide in direction with the initial elements of the vanes. Any three of the four quantities above mentioned being fixed, the fourth becomes known by this relation. We will leave the angle of the guide-plates to be determined later. Let V =the actual velocity of the water on leaving the guide-plates. v, =the velocity relative to the vane, as before. Then V must be the resultant of wp, Fig. 4. and v,, and we have VY? = te te wp — 20, 0p COS Yj. (10) Y 1 From Bernouilli’s theorem we have for wr, the flow of water in the head-race the equation ; Pasi! mapas Me 59 + O59 t 29° eee) }) being the mean height of the surface of the water in the reservoir above the wheel. We thus have two equations, (10) and (11), intro- ducing one new unknown quantity. Mr. J. L. Woodbridge on Turbines. Eliminating V in (10) and (11), we have v2 + wp? —2v,0p, Cosy, = 29h) — =P ra pe Substituting in equation (6), po==pat oe sate 2 20a w*p,?—w*p.” 1 + = + 2gh=v,?—v,”. Adding (12) and (13), we have 200° p;? — wp. = 29() —h) + 2v ap; cos y1— 23°. 321 (12) (13) (14) From (9) and (14), H being substituted for (}—h), we find 2 ain? a, sin? ¥2 — —2 ——_*= wp, cos 1 a,? sin? y; pe 4 ee en Gora ay? sin? y, a,* sin4 A ag sin Y2 = SES =a siny, ay? sin? y 7 +/w%p:?— —2w*p,?+2 gH + poor w*p,” cos? ¥,. 2 in? az SIN” Ye = wp, ———— cos ae a, sin; M1 ay? sin? y> +4, 81D ¥2 a 07 p22 —2w?p,? + 29H + easneiy: w*p,? cos? yy. . The efficiency will be, from equation (8), U — 7SQH = 5H sei which, substituting the values of v, and v2, gives E 07 pz” — Py COS 91 + Pave COS Yo |, = al COs n | 1 Baa | opt —o'p? +0 ps ee a, sin y, (15) (15a) (16) (17) A, Sin a ges) ao SIN ey De @2 S10 Y2 a2 yp, cos nta/o «wp, 2 Peta? 49H + 22 7 sin? + wp? bs a] }. (18) a Sin y, ay? sin? y, To find the angular velocity that will give a maximum efficiency, make = =0, in (18). 322 Mr. J. L. Woodbridge on Turbines. For brevity make hi @y sin ¥. N= pe COS ng a0 ¥ PiCOS iy... eee __ a9 sin Ye ‘ ~ a siny, sin "1 pA COs Vy ® tC) ® e . e e (20) 2 Pi ay? sin? y, *! Mic ae A, Sin ef=N 2? prcos y+ pi*— ps", ... .. ee ay sin V1 adi scams then a will give —ws? / wl? + 2g H=n?ol?+ngH, . . . (23) a BSE eel eee PL Vs eez and this substituted in (18) will give the maximum efficiency for any turbine, and in (16) will give the quantity of water discharged. It would be simpler to find » numerically for any particular case before making the substitution. We will now use these equations to show some errors made by Rankine and Weisbach. Rankine, in his ‘ Steam-Hngine and other Prime Movers,’ discusses a special wheel of the Fourneyron type, in which he assumes dg = a andy; = 90°. These in equation (15) give v,=sin Yo Nw*p.2—2w*p,2+2gH. . . . (25) This velocity is radial along the vane, and is called by Rankine the “ velocity of flow.” The tangential component of the actual velocity must, in this case, be wp , and this, by Rankine, is called the velocity of whirl (v) ; and his value of v on page 196, equation (9) of the ‘Steam-Engine’ is not only incorrect but meaningless, even for the turbine he is con- sidering. From equation (15a) we also have tg= Vowp.*—20*p,*+2gH. . . . (26) The expression for the efficiency becomes 2 ort [ @p,* — wp,” + pz COS Y3 V wp? — 2w*py? + 29H | - (27) Mr. J. L. Woodbridge on Turbines. 323 This expression differs entirely from Rankine’s equation (10), page 196. But running this wheel at the same speed as Rankine does his, Art. 175, that is, making the final velocity of whirl zero, we shall have U2 COS Y2=@Pa, and equation (26) becomes @)9= COS Yq V wpa” —2wp,? + 29H ; from which we find a 29H on Vr? «08 29H . OP) = Sa aan 2+ eas tan? yo P1 which is the same as Rankine’s equation (3), Art.175. Sub- stituting these in (27), we have | esi od cae 2p2? + po tan? y2’ which is Rankine’s equation (4), Art. 175. We thus see that Rankine’s equations not only do not. fit any wheel except the one he is considering, but they apply to that only at one particular speed. These conclusions agree with those in an article on Turbines, by Professor Wood, in the Journal of the Franklin Institute for June 1884. In regard to the speed for maximum efficiency, Rankine, in the ‘ Steam-Engine,’ Art. 173, says, “In order that the water may work to the best advantage, it should leave the wheel without whirling motion, for which purpose the velocity of whirl relative to the wheel should be equal and contrary to that of the second circumference of the wheel.’’ Plausible as this appears, it is true only for special cases even for his wheel. Also Weisbach makes the erroneous statement that the velocity of the second rim of the wheel should equal the relative velocity of discharge. Thus in ‘The Mechanics of Engineering and of Machinery,’ vol. ii. page 400 (Wiley and Sons) he says (substituting my notation for his) w= Vw" ps? +02? — 2v.m@p2 COSY. = a/ (w@p2—,)*® + 4ap2v, sin? Ya r 324 Mr. J. L. Woodbridge on Turbines. in regard to which he states that for w a minimum @p2 must equal v,, which is not generally true, and is true only when Y2=0, or when v,wp, and (w@p2,— v2)” happen to be a minimum together. The value of » in equation (24) will not in general satisfy Rankine’s condition WPo=Up COSYn, = + 1s a) eee nor Weisbach’s OD GAS UGs* nels ieee Lar eae (29a) Substituting in equation (24) Rankine’s condition y,=90°, and making pal P2 oa. gH fl—7?— V1 —71?)?—cos? yo(1— 27?) ae BORD 2)\2 2 Q r VW (1 —1?)? —cos? yo(1 — 27?) Substituting this in (15a) gives 0 COS Y¥2=@p2| 1—1r? + oV (1—1?)?—cos* y,(1—27?) ]. (81) , we find |. @0) This satisfies equation (29) only when 7?=4, and (29a) only when y,=0 or r=1. The latter condition is that of a parallel flow wheel, or of an infinitely narrow wheel, in which case we have poe WP, = V2 = n/ gH. These in (17) give for the efficiency EH =cos 72, which always exceeds the value given by Rankine, when r=1, __ 2cos* yo Fe i ae cos? Yo except when y,=0, when both become unity, but the work done will be zero. The pressure in the wheel may be found by integrating equation (3) between initial and general limits, and elimi- nating p, and v by means of equations (12) and (9), giving Oe sin2 V1 yy 2 a@ ¢ ¥ = 0%? —20%p,°+ eo + gb +2v,p, COS %—2;? az sin? vy ) (32) which may be discussed for the various conditions to which the wheel is subjected. Rig S250] XXXVI. The Expansion of Mercury between 0° C. and —39° C. By Professors W. E. Ayrton, £.2.S., and JoHN PERRY. tS. T a meeting of the Physical Society in November 1885, Mr. G. Whipple gave the Society the results of the examination of thermometers down to the melting-point of mercury. ‘There was, however, no evidence as to whether the contraction of the mercury was uniform, as the thermometers were only compared with mercurial ones, and as, in addition, we were not able to find the results of any experiments made on the expansion of mercury between 0° and —39° C., its temperature of solidification, we thought it desirable to make a series of comparisons of a mercury-thermometer, the stem of which had been accurately subdivided into equal volumes, with an air-thermometer, both immersed in a bath of frozen mer- eury which was allowed to gradually become warm. For this purpose we borrowed a mercury-thermometer from Mr. Whipple, which he was so kind as to lend us, and one of our assistants (Mr. Mather) constructed a very simple form of constant-volume air-thermometer, shown in the diagram. B B is a wooden box, at the bottom of which is a hole closed * Communicated by the Physical Society : read March 27, 1886. 326 Expansion of Mercury between 0° C. and —39° C. by an india-rubber stopper through which passes tightly the glass stem of an air-thermometer, AA. The bottom of the air-thermometer is attached by a piece of india-rubber tubing, II, to a vertical glass tube, TT; and the thermometer being filled with dry air, some mercury is introduced into the tube so as to stand at about the same height in the two limbs when the air in the bulb 6 is at atmospheric pressure. The height of the level of the mercury can be varied by turning the nut, N, which causes the clamp, C, turning on the hinge H, to squeeze the india-rubber tube more or less tightly, and so to alter its internal capacity ; and in this way the level of the mercury in the left-hand tube can be kept quite fixed, and therefore the volume of air in the bulb 6 quite constant, while its temperature is altered, the corresponding pressure being of course measured by the difference between the levels of the mercury columns in the two limbs. To perform the experiment, the box B B was first filled with mercury and frozen by stirring carbonic-acid snow and ether up with it ; and when sufficient of the mercury was frozen, the bulb of the mercurial thermometer, ¢t, was introduced into the pasty mercury, and the thermometer fixed in position by means of the clamp, C’. The nut N was now turned by one observer until the level of the mercury in the left-hand limb came opposite a jized mark on the tube which is only just below the bottom of the box, when the height of the mercury in the right-hand tube was read by a cathetometer made by the Cambridge Instrument Company, and the position of the mercury in the mercury-thermometer was read by a third observer. In this way several series of simultaneous observa- tions were taken, during the course of some weeks, of the pressure to which the air in the air-thermometer had to be subjected to keep its volume constant, as the mercury in the box BB varied in temperature from about — 39° C. to 0° C. Plotting these results, it was found that they lay in so nearly a straight line that we may conclude that mercury expands regularly below 0° C. as it is known to do above 0° C.; and that there is no critical point for mercury, as there is for water, above the freezing-point. When the mercury freezes it contracts still further, as may be seen from the following extract from page 6 of the second volume of Nordenskiéld’s ‘ Voyage of the Vega,’ which Mr. Whipple has kindly looked up for us :— “ When mercury freezes in a common thermometer, it con- tracts so much that the column of mercury suddenly sinks in the tube, or, if it is short, goes wholly into the ball. The position of the column is therefore no measure of the actual degree of cold when the freezing takes place.” On the Expansion produced by Amalgamation. 327 We have to express our thanks to Messrs. Chatterton, Humphrey, and Martin, three of the students of the Central Institution, for assistance rendered in the carrying out of this experiment. XXXIX. On the Expansion produced by Amalgamation. By Professors W.H.Ayrton, F.2.S., and Joun Perry, F.A.S.* ( N amalgamating the edge of a brass bar, nearly three quarters of an inch thick and about a foot long, for the purpose of enabling the edge to make good electric contact with a plate, we were surprised to find that the bar rapidly curved, the amalgamated edge becoming convex, exactly as happens when one side of a piece of paper is wetted. On hammering the bar to straighten it, the curvature became instead greater. Seeing that to bend a short brass bar more than half an inch in thickness to the extent produced by the amalgamation of the edge requires the exertion of very con- siderable stresses, it follows that very great forces must be produced by amalgamation. We think it possible that this bending by amalgamation may be an important cause in the production of the Japanese “magic mirrors,” the reflecting surface of which is polished with a mercury amalgam. Japanese mirrors are made of bronze and have a raised pattern cast on their backs ; and although the eye can detect no trace of the pattern on looking at the polished reflecting surface, yet when certain of these mirrors are used to reflect a divergent beam on to a screen, the pattern at the back can be seen as a bright image on a dark ground. In a paper communicated, some years ago, to the Royal Society, we showed that this peculiar effect arose from the fact that, while the reflecting surface was generally convex, the portions corresponding with the pattern or thicker parts were less convex (that is, more concave) than the rest ; and this conclusion we verified by finding that when a conver- gent, instead of a divergent, beam of light was allowed to fall on the mirror the image on the screen was reversed ;_ that is, the pattern was seen as a dark image on a bright ground. This inequality of curvature we considered at that time was due partly to the pressure of the “ distorting-rod”’ used to make the surface convex, and partly to the pressure exercised on the subsequent polishing ; but we now think that, in ad- dition, the action of the mercury-amalgam employed by the polisher may assist in making the thin portions of the mirror more concave than the thicker. * Communicated by the Physical Society: read March 27, 1886. [ 328 ] XL. Note on the Annual Precession calculated on the Hypo- thesis of the Earth’s Solidity. By Hunry Hennessy, /.R.S., Professor of Applied Mathematies in the Royal College of Science for Ireland *. | N discussing the influence of the areal structure of the Karth upon precession, it has been frequently assumed that with the ellipticity a 0 the annual precession of a homo- geneous solid shell or completely solid spheroid would be 57”. This was the resuit of Mr. Hopkins’s calculations ; and the difference,amounting to between six and seven seconds, between it and the observed value, formed the basis of all ‘his con- clusions relative to the Harth’s internal condition. Hitherto I have not seen any reason for doubting the above numerical result; but on looking more closely into the question, it appears probable that we must reduce the precession for the hypo- thetical solid spheroid to about 55”. If the Harth werea spheroid perfectly rigid, the amount of precession can be calculated from formule given in Airy’s ‘ Tracts,’ Pratt’s ‘Mechanical Philosophy, Pontécoulant’s Théorie Analytique du Systeme du Monde, or Resal’s Traité de Mécanique Celeste. In the two latter works, Poisson’s memoir on the rotation of the Harth about its centre of gravity is very closely followed, and the formulz are those which I have generally employed. From these wr ae gs we find 3m? (20 — A—B) =~ C where I is the inclination of the equator to the ecliptic, y the ratio of the Moon’s action on the Harth compared to that of (1+y¥) cos I; the Sun, m the Harth’s mean motion around the Sun, ~ the ratio of this mean motion to the Harth’s rotation, and A, B, C the three principal movements of inertia of the Earth. When the Harth is supposed to be a spheroid of revolution A=B, and the above becomes 2 (1) PHS =—. (1+y) cos I Pratt gives the formula Oye ies (ae 8 ee ASE SE*} 180°; * Communicated by the Author. Precession on the Hypothesis of the Earth’s Solidity. 329 where ¢ is the inclination of the Moon’s orbit to the ecliptic, y the ratio of the Earth’s mass to that of the Moon. In all these formule, or in any others by which the pre- cession can be calculated, the Moon’s mass enters directly or indirectly. When Mr. Hopkins made his calculation, more than forty years ago, he appears to have taken the value of the Moon’s mass and all his other numerical data from the early editions of Airy’s ‘Tracts.’ He uses 366:26 for the Harth’s period, 27°32 for the Moon’s. He makes [= 23° 28’, i=5° 8’ 50", and the Moon’s mass = a of the Harth’s mass. All of these values require revision; and it may be remarked that Sir George Airy has more recently expressed the opinion that = may be taken as the value of the Moon’s mass. (Monthly Notices of the Royal Astronomical Society, Decem- ber 1878, p. 140). On this question, I may be permitted to remark that there are three different phenomena from which the Moon’s mass has been determined :—1, the perturbations of the Harth’s motion in its orbit around the Sun by the action of the Moon ; 2, the Tides; and 3, the Nutation of the HKarth’s axis. The largest mass, or 70 nearly, has been ob- tained from the first, and the smallest from Nutation. But the values obtained from Nutation are not very accordant, and, moreover, the close connection between Nutation and Preces- sion makes it a doubtful matter to calculate the amount of one from a quantity depending on the other. The Moon’s mass obtained from the Tides is that which has been employed by Laplace, Poisson, and other mathematicians as the most probable. It appears that a recent discussion of the Tides in the United States, made by Mr. Ferrel, has given the same value as that found by Laplace. This circumstance, as well as the fact that the value so obtained lies between the values found by the other methods, give us reason to place much confidence in the result. If we call P, the precession for a homogeneous spheroid whose ellipticity is H, then from (1) 3m Pi=5—8 (1+) cos I. If we take the value of the Moon’s mass given by the tides or rather the ratio of the Moon’s action to that of the Sun thus given, we shall use the value of y employed by Poisson Pontécoulant, and Resal: if we also employ for FE’ the value which Colonel Clarke shows good grounds for deeming the Phil. Mag. 8. 5. Vol. 22. No. 1387. Oct. 1886, Z 330 Prof. H. Hennessy on the Precession calculated 1 e 1 * 7 —— : most probable 3 that is 9346 instead. of 300’ or even smaller fractions hitherto accepted, I find that P, becomes 5605. By Pratt’s formula and the numerical values he employs, except for H, I find Pie ones If we take a for the Moon’s mass in Poisson’s formula, y becomes 2°2062, and P,=53'574, If we change y to 80 in Pratt’s formula with i 2 OHI B= 593-40) Ei 92) “7: The value for the observed precession now generally ad- mitted is 50/37. It is therefore manifest that the difference between this and the precession of a homogeneous equi- elliptic spheroid cannot be admitted to be as great as Mr. Hopkins has declared it to be. From the values of P, which I have calculated we should have P,—P=5’"68 and 4-507, with the Moon’s mass = 753 ll. ; Pi—Pasi yz and 2”-58, if we take the Moon’s mass = On calculating P with the Moon’s mass =s) Sun’s mass 354936, y is 2°25395. If we take for I its value in 1852, or 23° 27’ 32”, and make m i em on Lif ae : = m=359°9931, —=0027303, E=s55-7, the following calculations can be made :— * See Colonel Clarke’s paper in the Philosophical Magazine for August 1878, where he maintains that recent geodesical results tend to increase the value of the Earth’s ellipticity and to make the measured value approach to that obtained from pendulum observations. on the Hypothesis of the Harth’s Solidity. 331 log m = 2:5562965, log (1+¥) =0°5124109, log cos I = 9°9625322, i __ —38+4862104 °8 7 ‘4674500’ 3 SoZ log 5) [ 60 x 60] =4-1998437’ 2:°4675489 log Fa =7-7399948 08 53/988, or P,=54” nearly, P} —P=3''617. Consequently, instead of admitting Mr. Hopkins’s result of 7” for the difference between the precession of a homogeneous spheroid with the Earth’s ellipticity and the precession actu- ally observed, we may affirm that this difference is probably not more than 4” or 5”. With the best values for the numerical elements the dif- ference is, however, too well ascertained to be overlooked, and it leads to the conclusion that the Earth cannot consist of an entirely solid mass composed of equielliptic strata, and that it is therefore partly composed of a solid shell bounded by surfaces such as I have elsewhere indicated, with an interior mass of viscid liquid, such as is seen flowing from the volcanic openings of the shell, arranged in strata conforming to the laws of hydrostatics, or, in other words, with strata of equal density decreasing in ellipticity towards the Earth’s centre. Notr.—tThe section shown in the engraving at p. 245 of the paper in the Number for September should be placed thus, with the longer axis parallel to the lines of the page. el MN ts! Erratum in same Paper. Line 16 from top of p. 247, for —{p (x?+y?) dx dydz read — Sp(z? +y?) dx dy dz. Z 2 [ 332 J XLI. On the Self-induction of Wires.—Part III. By OutverR HEAVIsIDE*. TFX\HE subject of the decomposition of an arbitrary function into the sum of functions of special types has many fascinations. No student of mathematical physics, if he possess any soul at all, can fail to recognize the poetry that pervades this branch of mathematics. The great work of Fourier is full of it, although there only the mere fringe of the subject is reached. Tor that very reason, and because the solutions can be fully realized, the poetry is more plainly evident than in cases of greater complexity. Another remarkable thing to be observed is the way the principle of conservation of energy and its transfer, or the equation of activity, governs the whole subject, in dynamical applications, as regards the possibility of effecting certain expansions, the forms of the functions in- volved, the manner of effecting the expansions, and the possible nature of the “‘ terminal conditions ’’ which may be imposed. Special proofs of the possibility of certain expansions are sometimes very vexatious. They are frequently long, com- plex, difficult to follow, unconvincing, and, after all, quite special ; whilst there are infinite numbers of functions equally deserving. Something of a quite general nature is clearly wanted, and simple in its generality, to cover the whole field. This will, I believe, be ultimately found in the principle of energy, at least as regards the functions of mathematical physics. But in the present place only a smali part of the question will be touched upon, with special reference to the physical problem of the propagation of electromagnetic dis- turbances through a dielectric tube, bounded by conductors. It will be, perhaps, in the recollection of some readers that Professor Sylvester, a few years since, in the course of his learned paper on the Bipotential, poked fun at Professor Maxwell for having, in kis investigation of the conjugate pro- perties possessed by complete spherical-surface harmonics, made use of Green’s Theorem concerning the mutual energy of two electrified systems. He said (in effect, for the quota- tion is from memory) that one might as well prove the rule of three by the laws of hydrostatics, or something similar to that. In the second edition of his treatise, Prof. Maxwell made some remarks that appear to be meant for a reply to this ; to the effect that although names, involving physical ideas, are given to certain quantities, yet as the reasoning is purely mathematical, the physicist has a right to assist himself by the physical ideas. * Communicated by the Author. On the Self-induction of Wires. t 333 Certainly ; but there is much more in it than that. For not only the conjugate properties of spherical harmonics, but those of all other functions of the fluctuating character, which present themselves in physical problems, including the infi- nitely undiscoverable, are involved in the principle of energy, and are most simply and immediately proved by it, and pre- dicted beforehand. We may indeed get rid: of the principle of energy, and treat the matter as a question of the properties of quadratic functions ; a method which may commend itself to the pure mathematician. But by the use of the principle of energy, and assisted by the physical ideas involved, we are enabled to go straight to the mark at once, and avoid the un- necessary complexities connected with the use of the special functions in question, which may be so great as to wholly prevent the recognition of the properties which, through the principle of energy, are necessitated. Considering only adynamical system in which the forces of reaction are proportional to displacements, and the forces of resistance to velocities, there are three important quantities— the potential energy, the kinetic energy, and the dissipativity, say U, T, and Q, which are quadratic functions of the variables or their velocities. When there is no kinetic energy, the conjugate properties of normal systems are U;,=0 and Q,,=03 these standing for the mutual potential energy and the mutual dissipativity ofa pair of normal systems. When there is no potential energy, we have T;,=0 and Q,,=0. When there is no dissipation of energy, U;,.=0 and T,,=0. And in general, U;,.= Die, which covers all cases, and has two equiva- lents, 4 Qio+ Uyp=0, and 2 1Q..+T».=0; for, as the mutual potential and kinetic ener gies are equal, the mutual dissipa- tivity is derived half from each. ; Let the variables be 2, x.,..., their velocities v,=2,,..., and the equations of motion Fy =(An + By p+Cy p*)art (Ait Bi pt Cy p”)aet..., F,=(Agi + Bupt On PB) a5 Cnt oT ae ay Cas Py 28 ; (88) where F’,, Fy,..., are Lente ee forces, and p ho for d/dt. Forming he equation of total activity we obtain Spgs Pry ows relioheggg) pi Aya? + 2 Apo Ue + Agoa? aeooe ) Q = Byv? + 2Bis VzV2 + Bogv? + eee? @ Paces (90) 2T= C,,0? + 2C190,U2 + Coov? ees where 334 Mr. O. Heaviside on the So far will define in the briefest manner, U, T, Q, and activity. Now let the F’s vanish, so that no energy can be communi- cated to the system, whilst it can only leave it irreversibly, through Q. Then let p,, p, be any two values of p satisfying (38) regarded as algebraic. Let Q,, U,, T, belong to the system p, existing alone; then, by (89) and (90), 0=Q,+U, +1, or 0=Q,+2p,(U,4+T)) ; 0=Q,.4U,+T,, 9) 0=Q.+ 2p,(U,+T.). But when existing simultaneously, so that Q=0:4+Q:4+Q2, U=U,;+U,+Uyn, T=T,+7T,4+Ty, where Uj, Ty2, Qi2 depend upon products from both systems, thus :— Qie= 2 {Byywyry! + Bogvyre! + Byo(v,ve! + v9%/) +--+, Uyg= Agta! + Agototte! + Ayo (ayo +.2,2)/) +..., Tye Cyyuyoy! + Cog ve v9! + Cro (vy 4! + 0907’) eee the accents distinguishing one system from the other, we shall find, by forming the equations of mutual activity }Fo’=..., and }H’y=..., that is, with the F’s of one system, and the v’s of the other, in turn, 0O=4$ Qt poUiet+ pT, y 0O=4Qi+ 7,012.4 pele; adding which, there results the equation of mutual activity, O=Qr + (pitpos)(Uigt Tis), or O= Qu + Ui + Tiss and, on subtraction, there results O=(pi-p2)(Ur—T), - - » + (91) giving Uj,=Ty,, if the p’s are unequal. But this property is true whether the p’s be equal or not; that is, Uj,;='T,; when p, is a repeated root. Various cases of the above are discussed in ‘The Electrician,’ November 27 and December 11, 1885, with special reference to the dynamical system expressed by Maxwell’s electromagnetic equations. The following applies to Maxwell’s system, using the equa- tions (4) to (10) of Part I. (Phil. Mag. August 1886). A comparison with the above is instructive. Let E,, H, and E,, H, be any two systems satisfying these equations, with no impressed ‘forces, or e=0,h=0. ‘Then the energy entering the unit volume per second by the action of the first system Self-induction of Wires. 339 on the second is cony. VE,H,/47= (E, curl H,—H, curl E,) /47, = EP,+H,G,, = E,C,+E,D,+H,B/47. . . . (92) Similarly, by the action of the second system on the first, conv. VE,H,/47=E,C, +E,D,+H,B,/47. . . (98) Addition gives the equation of mutual activity. And, sub- tracting (93) from (92), we find } cony. (VE,H,—VE,H,) /47 =(E,D,—E,D,) —(H,B,—H,B,)/47; . (94) since E,C,=E,kE, = E,kE, = E,C,, if there be no rotatory power, or C be a symmetrical linear function of E. Similarly for Dand E, and BandH. Hence, if the systems are normal, making d/dt=p, in one, and py in the other, (94) becomes cony. (VE,H, = VE,H,)/47r = (po —p1)(E,D, —= H,B,/47). (95) Therefore, by the well-known theorem of Convergence, if we integrate through any region, and Uj, Ty, be the mutual electric energy and the mutual magnetic energy of the two systems in that region, we obtain =N (VE.H, — VE,H,) [Acar : : (96) Pi—Pfe2 where N is the unit normal drawn inward from the boundary of the region, over which the summation extends. And if the region include the whole space through which the systems extend, the right member will vanish, giving U;,=Ty,, when these are complete. From (96) we obtain, by differentiation, the value of twice the excess of the electric over the magnetic energy ofa single normal system in any region ; thus xU—T)=SN (veD -Ve H) fm. . (97) Uy —Typ = This formula, or special representatives of the same, is very useful in saving labour in investigations relating to normal systems of subsidence. The quantity that appears in the numerator in (96) is the excess of the energy entering the region through its boundary per second by the action of the second system on the first, over that similarly entering due to the action of the first on 336 Mr. O. Heaviside on the the second system. Bearing this in mind, we can easily form the corresponding formula in a less general case. Suppose, for example, we have two fine wire terminals, a and 6, that are joined through any electromagnetic and electrostatic combination which does not contain impressed forces, nor receive energy from without except by means of the current, say C, entering it at a and leaving it at 6b. Let also V be the excess of the potential of a over that of b.. Then VC is the energy-current, or the amount of energy added per second to the combination through the terminal connections with, ne- cessarily, some other combination. (In the previous thick- letter vector investigation V was the symbol of vector product. There will, however, be no confusion with the following use of V, as in Part II., to express the line-integral of an electric force. One of the awkward things about the notation in Prof. Tait’s ‘ Quaternions’ is the employment of a number of most useful letters, as 8,.T, U, V, wanted for other purposes, as mere symbols of operations, putting another barrier in the way of practically combining vector methods with ordinary scalar methods, besides the perpetual negative sign before scalar products.) ‘The combination need not be of mere linear circuits, in which differences of current-density are insen- sible ; there may, for example, be induction of currents in a mass of metal not connected conductively with a and b, or the same mass may be in connection ; but in any case it is necessary that the arrangement should terminate in fine wires at a ard 6, in order that the two quantities V and C may suffice to specify, by their product, the energy-current at the terminals. Hven in this we completely ignore the dielectric currents and also the displacement, in the neighbourhood of the terminals, 7. e. we assume c=0, to stop displacement. This is, of course, what is always done, unless specially allowed for. Now supposing the structure of the combination to be given, we can always, by writing out the equations of its different parts, arrive at the characteristic equation connecting the terminal V and C. For instance, MeaZOp ws 0. a where Z is a function of d/dt. In the simplest case Z is a mere resistance. A common form of this equation is fVtAVthV+t...=9,CtmOtgG+..., where the /’s and g’s are constants. But there is no restriction to such simple forms. All that is necessary is that the equa- - Self-induction of Wires. 337 tion should be linear, so that Z may be a function of p. If, for example, (dC/dt)? occurred, we could not do it. Now this combination must necessarily be joined on to an- other, however elementary, to make a complete system, unless V is to be zero always. The complete system, without im- pressed forces in it, has its proper normal modes of subsidence, corresponding to definite values of p. Consequently, by (96), Uy,— Tie = (VC, —V,C,) (1 —Do), Piao (99) if V,, O; belong to p,, and Vo, C, to ps, whilst the left member refers to the combination given by V=ZO. Or Vuow V5 LZ, —Zz Up,—Ty=C,C, C7) =) =A0s Pa—Py (100) and the value of 2(U—T) in asingle normal system is rd OMe ng Ve Oe er ml doe =—C wor C ape (101) In a. similar manner we can write down the energy- differences for the complementary combination, whose equation is, say, V=YC; remembering that —VC is the energy entering it per second, we get AG 7 Y, CC; dY and ©? 6 respectively. By addition, the complete U,,—T), is ig es a Aa Ze yg, 1 . (102) Pi—P2 Pi—P2 and the complete 2(U—T) is =, (108) where 6=0, or Y—Z=0, is the determinantal equation of the complete system (both combinations which join on at a and } where V and C are reckoned), expressed in such a form that every term in ¢ is of the dimensions of a resistance. If the complete system depends only upon a finite number of variables, it is clear that the number of independent normal systems is also finite, and there is no difficulty whatever in understanding how any possible initial state is decomposable into the finite number of normal states; nor is any proof needed that it is possible to do it. The constant Aj, fixing d d OF (X—D), or or? 338 Mr. O. Heaviside on the the size of a particular normal system ,, will be given by pee VYo=Toa Un-To — Ya =To Du-Tn = 2(0i:—Th) C2 agp ‘dp by the previous, if Uo, be the mutual electric energy of the given initial state and the normal system, and To; similarly the mutual magnetic energy. And when we increase the number of variables infinitely, and pass to partial differential equations and continuously varying normal functions, it is, by continuity, equally clear that the decomposition of the initial state into the now infinite series of normal functions is not only possible, but necessary. Provided always that we have the whole series of normal functions at command. Therein lies the difficulty, when there is any. In such a case as the system (71) of Part II., involving the partial differential equation ON aN = Py Ze ae ene wherein R, 8, and L are constants, to hold good between the limits z=0 and z=1, subject to V=Z,C at #=0, and V=Z,C at «=, there is no possible missing of the true normal functions which arise by treating d/dt as a constant ; so that we can be sure of the possibility of the expansions. Thus, denoting RSp+ LSp? by —m’, we may take the normal V function as w= sin (mz+0), 2 «. «is ap ap (104) 4+L8 (105) and the corresponding normal C function as ren ae eg BPs (mz+0). . ., G07) m Here 6 will be determined by the terminal conditions u u 7 =I), bee =f, atz=l, , « . . (108) and the complete V and C solutions are VSs Anes OOS sare ek ne aay) at time ¢ ; where any A is to be found from the initial state, Self-induction of Wires. } 339 say Vo, Co, functions of z, by { epee rere: ere a ea CH ETO) [ea G-2)), provided there be no energy initially in the terminal arrange- ments. If there be, we must make corresponding additions to the numerator, without changing the denominator of A. The expression to be used for u/w is, by (106) and (107), “um oe eae (mZeeO se Base wd e (LD) remembering that m is a function of p. There are four com- ponents in the denominator of (110), as there are three elec- trical systems ; viz. the terminal arrangements, which can only receive energy from the “ line,” and the line itself, which can receive or part with energy at both ends. In a similar manner, if we make R, 8, and L any single- valued functions of z, subject to the elementary relations of fLePart 11... or getting this characteristic equation of O, Ge OY. d\dC “8 =)= (R+LZ)5;, 2 eisai A and, putting w for C and p for aS this equation for the - current function, 3 d dw 5,(8" 3) =(R+ Lp pu, A) Pesto lg dk 3 and finding the u functions by the second of (112), giving —Spu= ww es (AIS) we see that the expansions of the initial states V) and OC, can be effected, subject to the terminal conditions (108). For the normal potential and current functions will be perfectly definite (singularities, of course, to receive special attention), given by (113) and (114), as each the sum of two indepen- dent functions, and the terminal conditions will settle in what ratio they must be taken. (109) and (110) will constitute 340 Mr. O. Heaviside on the the solution, except as regards the initial energy beyond the terminals. It is, however, remarkable, that we can often, perhaps uni- versally, find the expression for the part of the numerator of (110) to be added for the terminal arrangements, except as regards arbitrary multipliers, from the mere form of the Z functions, without knowing in detail what electrical combina- tions they represent. This is to be done by first decomposing the expression for C*(dZ/dp) into the sum of squares, for instance, adZ Op ee 00s where 71, ao ee are constants. The terminal arbitraries are then ZAfi(p), 2Af.(p), &e.: calling these H,, Hy,..., the additions to the numerator of (110) are — {Eni fi(p) + Bare fe(p) +e+- 5 +» CLD) wherein the H’s may have any values. This must be done separately for each terminal arrangement. The matter is best studied in the concrete application, which I may consider under a separate heading. It is also remarkable that, as regards the obtaining of cor- rect expansions of functions, there is no occasion to impose upon R, 8, and L the physical necessity of being positive quantities, or real. This will be understandable by going back to a finite number of variables, and then passing to continuous functions. Let us now proceed to the far more difficult problems con- nected with propagation along a dielectric tube bounded by concentric conducting tubes, and examine how the preceding results apply, and in what cases we can be sure of getting correct solutions. Start with the general system, equations (11) to (14), Part I., with the extension mentioned at the commencement of Part II. from a solid to a tubular inner conductor. Suppose that the initial state is of purely longi- tudinal electric force, independent of z, so that the longitudinal K and circular H are functions of r only. How can we secure that they shall, in subsiding, remain functions of 7 only, so that any short length is representative of the whole? Since Hi is to be longitudinal, there must be no longitudinal energy- current, or it must be entirely radial. Therefore no energy must be communicated to the system at z=0 or z=1, or leave it at those places. This seems to be securable in only five cases. Put infinitely conducting plates across the section at either or both ends of the line. ‘This will make V=0O there, Self-induction of Wires. 341 if V is the line-integral of the radial electric force across the dielectric. Or put non-conducting and non-dielectric plates there similarly. This will make C=0. Or, which is the fifth case, let the inner and the outer conductors be closed upon themselves. In any of these cases, the electric force will remain longitudinal during the subsidence, which will take place similarly all along the line. By (14), the equation of H will be did : : ae rH=4rkyvH + cH ; and it is clear that the normal functions are quite definite, so that the expansion of the initial state of HE and H can be truly effected. In the already given normal functions take m=0. But if we were to join the conductors at one end of the line through a resistance, we should, to some extent, upset this regular subsidence everywhere alike. Tor energy would leave the line; this would cause radial displacement, first at the end where the resistance was attached, and later all along the line. (By “the line ” is meant, for brevity, the system of tubes extending from z=0 to z=/.) Now in short-wire problems the electric energy is of insig- nificant importance, as compared with the magnetic. It is usual to ignore it altogether. This we can do by assuming c=(Q. This necessitates equality of wire and return current, for one thing; but, more importantly, it prevents current leaving the conductors, so that C and H and I the current- density, are independent of z. There will be no radial electric force in the conductors, in which therefore the energy-current will be radial. But there will be radial force in the dielectric, and therefore longitudinal energy-current. Since the radial electric force and also the magnetic force in the dielectric vary inversely as the distance from the axis, the longitudinal energy-¢urrent density will vary inversely as the square of the distance. But, on account of symmetry, we are only concerned with its total amount over the complete section of the dielectric. This is 1 (@%2C aa ie 2rr dr=VC, ori) 54 he (118) if V is the line-integral of E, the radial force, and C the wire- current. It is clear, then, that we can now allow terminal connections of the form V/C=Z before used, and still have correct expansions of the initial magnetic field, giving correct subsidence solutions. But it is simpler to ignore V altogether. For the equation 342 Mr. O. Heaviside on the of H.M.F. will be . eo=(Z,+Z,+lLop + UR" +7R")C, . . (119) if e, is the total impressed force in the circuit, R,"” and R,! the wire and sheath functions of equations (55) and (56), Part II., on the assumption m=O, and Z), Z, the terminal functions, such that V/C=Z, at z=l, and =—Z,atz=0. It does not matter how e, is distributed so far as the magnetic field and the current is concerned. Let it then be distributed in such a way as to do away with the radial electric field, for simplicity of reasoning. The simple-harmonic solution of (119) is obviously to be got by expanding Z, and Z, in the form R+ Lp, where R and L are functions of p”, and adding them on to the /(R’+ L’p) equivalent of J(Lop + Ry" + R,"), as In equation (66), Part II. Regarding the free subsidence, putting ¢—=0 in (119) gives us the determinantal equation of the p’s; and as the normal H functions are definitely known, the expansion of the mag- netic field can be effected. The influence of the terminal arrangements must not be forgotten in reckoning A. In coming, next, to the more general case of equation (56), but without restriction to exactly longitudinal current in the conductors, it is necessary to consider the transfer of energy more fully. In the dielectric the longitudinal energy-current is still VC. The rate of decrease of this quantity with z is to be accounted for by increase of electric and magnetic energy in the dielectric, and by the transfer of energy into the con- ductors which bound it. Thus, ad xrey dV~ aC SEH SEM dz ie But here, dC : dV : sahees = and = de =—L,C+H-—F, s (120) by (59) and (56), Part II., E and F being the longitudinal electric forces at the inner and outer boundaries of the dielec- tric (when there is no impressed force). So — £ vo=S8VV+1,00+EC-FC. bo ihe The first term on the right side is the rate of increase of the electric energy, the second term the rate of increase of the magnetic energy in the dielectric, the third is the energy entering the inner conductor per second, the fourth that entering the outer conductor ; all per unit length. If the electric current in the conductors were exactly lon- Self-induction of Wires. 343 gitudinal, the energy-transfer in them would be exactly radial, and HC and —FC would be precisely equal to the Joule heat per second plus the rate of increase of the magnetic energy, in the inner and the outer conductor, respectively. But as there is a small radial current, there is also a small longitu- dinal transfer of energy in the conductors. Thus, H, and H, being the radial and longitudinal components of the electric force, in the inner conductor, for example, the longitudinal and the radial components of the energy-current per unit area are E,H/4mr and H,H/4z, the latter being inward. ‘Their convergences are d E,H Redes Wei ae ee ae ae? or KH, aH H d&#, E,H H,dH . H dH, Beer ge) aa ae Oe aa a ae ie ae or H di, H dH, E,0,— AG vy, and EU ,+ eS ipa if T, and [’, are the components of the electric current-density. The sum of the first terms is clearly the dissipativity per unit volume; and that of the second terms is, by equation (18), Part I., HwH/4z,, the rate of increase of the magnetic energy. The longitudinal transfer of energy in either conductor per unit area is also expressed by —(47k)—1H(dH /dz) ; or, by — (Aakp)—1 (dT, /dz) across the complete section, if T, tempo- rarily denote the magnetic energy in the conductor per unit length. | Now let H,, Fi, C,, V;, and E,, F,, C,, V. refer to two dis- tinct normal systems. Then, if we could neglect the longi- tudinal transfer in the conductors, we should have d Up—-Ty= ae (V,C,—V2C,)+(pi-pe), « «+ (122) the left side referring to unit length of line; and, in the whole line, U,.—Ty= [V,0,—V.C,]h+ (1 Pa) = 49 ot os RB) Similarly, for a single normal system, d Ce ae eR aa ape tee es ov 24) 344 Mr. O. Heaviside on the per unit length ; and, in the whole line “u—n=[0s GI. Nicdint af) che We have to see how far these are affected by the longitudinal transfer. We have d - = qe V1U2 = SViV. =F IoC .C, fe (Hi, = 1) Co, d | : : — Ge V.C, = SV.V, + LO ORGs, + (KE, —F,)C, 3 therefore, if the systems are normal, d GE (ViC2—V201) =(p,— p2) SViV2—L,01 C2) — (Hy — F,)C, + (H,—F,)C;. It will be found that we cannot make the parts depending upon Hf and F exactly represent the U,.—T,,. in the conduc- tors except when m? is the same in both systems p, and In that case, the parts (H,),(H), and (E,).(H), of the longi- tudinal transfer of energy in the conductors, depending upon the mutual action of the two systems, are equal; (H,), and (E,). being proportional to sin mz, and H, and H, propor- tional to cos mz. Bo, j in case p, and p, are values of p belong- ing to the same m’, the influence of the longitudinal energy- transfer in the conductors goes out from (122) and (128), which are therefore true in spite of it. Similarly, provided the m’s can be settled independently of the p’s, equations (124) and (125) are true. Now the normal V and C functions, say wu and w, as before, ma 2 be taken to be “= Sp v =u ay 4 (81a) —ta, (J,/Ky) (s S14) \Ky (s,a@,) Ky (s,a,) sin (mz+ 0) De Pep ee ew ee ee le wl so that V=Aue??, C= Awe”; and , vest Vek an ' wy Gee ey ig aa and the complete equations for the determination of m, 0, and p are gp tan (ml +0)=Z,, 2 ; 0= 5+ Bint Lnps S tan =n (128) Self-induction of Wires. 345 the first two of these being the terminal conditions, and RB’, + L'np being merely a convenient way of writing the real complex expressions; (equation (68), with e,,=0). It is clear that the only cases in which the m’s become clear of the p’s are the before-mentioned five cases, equivalent to Z and Z, being zero or infinite, and the line closed upon itself, which is a sort of combination of both. Considering only the four, they are summed up in this, VC=0 at the terminals, or the line cut off from receiving or losing energy at the ends. We have then the series of m’s, 0, 7/1, 27/l, &e.; or $.2/l, 3 7/1, 87/1, &c. ; and every m? has its own infinite series of p’s throug the third equation (128). These, though very special, are certainly important cases, as well as being the most simple. We can definitely effect the expansions of the initial states in the normal functions, and obtain the complete solutions in every particular. Although rather laborious, it is well to verify the above results by direct integration of the proper expressions for the electric and magnetic energies of normal systems throughout the whole line. ‘Thus, let did dr rdr a tid Tew gp? tat GHa=0, where —s}=4amyhy py +m, rH, +s{H,;=0, where —s?=4apyhkyp,+m?, in the inner conductor. We shall find (s -2) “H,Hyrdr = 82(C,P,—C.I}), a as H,=0=H, at r=a,; T, and I, being the longitudinal current-densities at r=a,. Similarly for the outer conductor, Re —#) ("Hr Hl ordr= —82r( GC, P'2—CeI";) ag if C,, C, still be the currents in the inner conductor ; the accents merely meaning changes produced by the altered pu and k in the outer conductor. H,’/=0=H,! at r=ag in this case. Then, thirdly, for the intermediate space, (HEyrar= C,C, x 4 log 2 ay J Therefore the total mutual magnetic energy of the two distri- Phil. Mag. 8. 5. Vol. 22. No. 137. Oct. 1886. 2A 346 Mr. O. Heaviside on the butions per unit length is ba (% , Be fy i a H, H, . 2arrdr + > H,"H,! . 2arrdr + fe "ay H,! . 2arrdr, which, by using the above expressions, becomes, provided m=me, C(E-F,) , C.-K) Pi—P2 Pi-p2 ” HK and F being I'/é or the longitudinal electric forces at r=a, OF 74.06 lt Lec _. (1262) H—F=R'C, where R!= the R,’+R," of equation (56), Part II. ; and 2 Fae O= Ss +L,p+R’= S +R/+L'p, so (126) becomes ; dV» a z ae : mC,C, (C. a —O, rs rad (2, Pa) y or Sp, ps shpee (127) The mutual electric energy is obviously SV, V2 per unit length. By summation with respect to z from 0 to /, subject to VC=0 at both ends, we verify that the total mutual magnetic energy equals the total mutual electric energy. The value of 2T in a single normal system is, by (126a), and the next equation, // 10+ 0S = 5 +Lp) meee per unit length; and that of 2U is SV*. Hence, per unit length, d / BL ata cae (B’+1/p). tie In this use V=w and C=w, equations (126), and we shall obtain, for the complete energy-difference in the whole line, a 271 d (mv 3 — 1B AI(oyu)—..t 5 ap Se +B/+Lp)=M say, (130) which is the expanded form of ; du dw]! HO et )} me 7 | cr u ap, or E apts ? as may be verified by performing the differentiations, using the expression for u/w in (127), remembering that m? in it is Self-induction of Wires. 347 a function of p; or more explicitly, put »/ —Sp(R’+ Lp) for m, and then differentiate to p. Given, then, the initial state to be V=V,, a function of z, and H=H,, in the inner conductor, Ho, in the dielectric, and Hy; in the outer conductor, fanichions of r and gz, aaa that this system is left without impressed force, subject to VC=0 at both ends, the state at time ¢ later will be given by V= Aue, C= Awe? ; the summations to include every p, with similar expressions for H, I’, y, &., the magnetic force and two components of current, by substituting for w or w the proper corresponding normal "functions ; the coefficient A being given by the frac- tion whose denominator is the expression M in (130), and whose numerator is the excess of the mutual electric energy of the initial and the normal system over their mutual mag- netic energy, expressed by ~ me mo’ Vo sin (mz+ @)dz =f cos (mz == 0) de} (" by Hy C,'dr == (“1H..04 0 a9 a - { ns ROR } Hapa aes) CO! = F $F, (s101) — (Fs/Ei) (8149) Ka (syn) where and C,' is the same with r put for a,, and C,! is the same with 7 put Ge ay, a3 for a), and s3 for s,. It should not be forgotten that in the case m=0, the denominator (130) requires to be doubled, 41 becoming 1. Also that R", or R’+ L'p, contains m’, and must not be the m= 0 expressions for the same. To check, take the initial state to be e,(1—2/l), with no magnetic force, and that V=0 at both ends. We find imme- diately, by (180) and (1381), that at time ¢, 2 2 t v— Pai >, ee mz > Al GS “in (bos) ae ~5(< +R+Lp) dp\Sp “ where the m’s are to be z/l, 27/1, 37/1, &e. ; the first summa- tion being with respect to m, and the second for the p’s of a particular m. But, Syeay / 348 Mr. O. Heaviside on the Therefore we must have 2 oe - agp tB’+L) Simplified, it makes this theorem af LE ( Be ~ FO" ap) ? if the p’s are the roots of 6()=0. This is correct. To determine the effect of longitudinal impressed force, keeping to the case of uniform intensity over the cross section of either conductor. Let a steady impressed force of integral amount e, be introduced in the line at distance z, ; it may be partly in one and partly in the other conductor, as in Part II. By elementary methods, we can find the steady state of V, C it will set up. If, then, we remove eé,, we can, by the prece- ding, find the transient state that will result. Let V, be the steady state of V set up, and V, what it becomes at time ¢ after removal of e,; then V)— V, represents the state at time t after & is put on. So, if }Aw represent the V set up by the unit impressed force at 2, V=V.—¢,2Auert will give the distribution of V at time ¢ after e, is put on, being zero when ¢=0, and V, when t=oo. No zero value of p is admissible here. From this we deduce that the effect of e, lasting from t= to t=¢,+dt, at the later time ¢ is —TAupedtyert- ; therefore, by time integration, the effect due to an impressed force é at one spot, variable with the time, starting at time é, 18 t iS —SAnper { ee Pdt,, to in which e, is a function of ¢). By integrating along the line, we find the effect of a con- tinuously distributed impressed force, e per unit length, to be l(t V= — Super ( Age Padardt,.)\ 3" 2 (say a 0 a t0 wherein ¢ is a function of both z, and ¢, and starts at time ¢); whilst A is a function of z,, the position of the elementary impressed force edz,. Self-induction of Wires. 349 To find A as a function of 2, we might, since SAw is the V set up by unit eat 2, expand this state by the former process of integration. But the following method, though unnecessary for the present purpose, has the advantage of being applicable to cases in which VC is not zero ai the ter- minals, but V=ZC instead. It is clear that the integration process, including the energy in the terminal apparatus, would be very lengthy, and would require a detailed knowledge of the terminal combinations. This is avoided by replacing the impressed force at z, by a charged condenser; when, clearly, the integration is confined to one spot. Let 8, be the capacity and V, the difference of potential of a condenser inserted at 7. If we increase 8, infinitely it becomes mathematically equiva- lent to an impressed force Vo, without the condenser. Suppose {Av ’e?? is the current at z at time ¢ after the intro- duction of the condenser, of finite capacity ; then, since —8,V is the current leaving the condenser, or the current at z,, we have : —§,V= sAw,e, w,! being the value of w! at z;. The expansion of V, is there- fore Vij =—2Aw//8: p, initially ; and the mutual potential energy of the initial charge of the condenser and of the normal w’ corresponding to w’ must be SiVo(—w1/8, p) = — Vowr' /p- But since there is, initially, electric energy only at z,, and magnetic energy nowhere at all, the only term in the nume- rator of A will be that due to the condenser, or this — Vyw,!/p; hence a — Vow,/pM, where M is the 27;U—T) of the complete normal system, as modified by the presence of the condenser, is the value of A in V=2Avw’e?’, making V=—V,2(wl/pM)u'e", expressing the effect at time ¢ after the introduction of the condenser, and due to its initial charge. So far 8, has been finite, and consequently wu’, w’, M, and p depend on its capacity as well as on the line and terminal conditions. But on infinitely increasing its capacity, wu’ and w’ become wu and w, the same as if the condenser were non- existent. Therefore V=—EV,(wi/pM)uert . . . . (184) 350 Mr. O. Heaviside on the expresses the effect due to the steady impressed force V, at 2 at time ¢ after it was started. This will have a term corre- sponding to a zero p (due to the infinite increase of 8, in the previous problem), expressing the final state. Hence, leaving out this term, the summation (134), with sign changed, and ¢=(, expresses the final state itself. Thus, taking V,=1, YAu=w,u/pM is the expansion required to be applied to (133). Put A=w,/pM in it, and it becomes Let V=3 (uae ( wyeeP'idz,dt,, . . . (135) 0 Jt fully expressing the effect at z,¢, due to the impressed force e, a,function of z, and 4, starting at time ¢,. To obtain the current, change wu to w outside the double integral. The M, when the condition VC=0 at the ends is imposed, is that of (180) ; the wu and w expressions those of (126), But if we regard 8, R’, and L’ as constants (or functions of z), then (135) holds good when terminal conditions V=ZC are im- posed, provided the impressed force be in the line only, as supposed in (135). When the impressed force is steady, and is confined to the place <=0, and is of integral amount eé, (185) gives V=e,2uw,/pM—e,suw "pM, . . . (186) w, being the value of w at z=0, as the effect at time ¢ after starting ¢,. The first summation expresses the state finally arrived at. Again, in (135) let the impressed force be a simple har- monic function of the time. I have already given the solution in this case, so far as the formula for C is concerned, in the case V=0 at both ends, in equation (76), Part II., which may be derived from (135), by using in it w instead of wu at its commencement, putting e=e sin nt, and effecting some reduc- tions. The V formula may be got in a similar manner to that used in getting (76), but it is instructive to derive it from (135), as showing the inner meaning of that formula. Let in it e=e sin (nt+a), where é is a function of z. Effect the é, integration, with t)=0 for simplicity. The result is uePe sina + ncosa\ ("/ e ui} Sg (YA ade, 87 Self-induction of Wires. 301 The first summation cancels the second at the first moment, and ultimately vanishes, leaving the second part to represent the final periodic solution. Take «=0; and use the u, w, M expressions of (126) and (1380), and let Pm stand for m “+ Sp(R'n + Linp), so that d,=0 gives the p’s for a par- ticular m?. Then we obtain, (with V=0 at both ends), Z cos el COS Mz, . 6,42, . (p sin nt +n cos nt) — d 0 = de 11 Pn ( 2 4, 1”) 2 dp P I d 2. © me ( COS MZ, . é) Sin nt . dz, (<- ro de... =, because d@?/d?=—n”. But, if uk the equation of V,, is ratte dz (by (60) and (63), Part IT.), so that 2 Em Sin nt dd’ (ie -?) ap by a well-known algebraical theorem, the summation being with respect to the p’s, which are the roots of ¢,=0, con- sidered as algebraic. We have also $3 F8=— 53 (139) 1 a : > cos me cosmz, edz, . . . (140) the summation being with respect to m. Uniting (139) and (140), there results the previous equation (138), in which the summation is with respect to all the p’s belonging to all the m’s. In the case m=O, the 2/1 must be halved. In the form of a summation with respect to m, similar to (77) for C, the corresponding V solution is ae a 3@ sin mz 4(U',—m?/Sn’)n sin nt + Rl, cos ade Bal R2 + (Lin —178/Sn2)'n?, the impressed force being V,sinnt, at z=0. This, on the assumption R’,=R!, L',=L, "will be found to be the expan- sion of the form (80), Part IL. 352 On the Self-induction of Wires. Now to make some remarks on the impossibility of joining on terminal apparatus without altering the normal functions, the terminal arrangements being made to impose conditions of the form V=ZC. It is clear, in the first place, that if the quantity VC at z=0 and z=/ really represents the energy- transfer in or out of the line at those places, then the equation Do cE To | “Sfo2 ae will be valid, provided u and w be the correct normal func- tions. But to make VC be the energy-transfer at the ends requires us to stop the longitudinal transfer in the conductors there, or make the current in the conductors longitudinal. This condition is violated when the current function w is pro- portional to cos(mz+@), as in the previous, except in the special cases, because the radial current y in the conductors is proportional to sin (mz+@), and y has to vanish. Not in the dielectric, but merely in the conductors. We can ensure that VC is the energy-transfer at the ends by coating the conductors over their exposed sections with infinitely conducting material and joining the terminal appa- ratus on to the latter. The current in the conductors will be made strictly longitudinal, close up to the infinitely conducting material, and y will vanish in the conductors. But y in the dielectric at the same place will be continuous with the radial surface-current on the infinitely conducting ends, due to the sudden discontinuity in the magnetic force. Thus the energy- transfer, at the ends, is confined to the dielectric. It is clear, however, that the normal current-functions in the two conductors must be such as to have no radial compo- nents at the terminals, so that they cannot be what have been used, such that d’/dz*= constant. They require alteration, of sensible amount may be, only near the terminals, but theore- tically, all along the line. It would therefore appear that only the five cases of V=O at either or both ends, or C=0 ditto, or the line closed upon itself, admit of full solution in the above manner. The only practical way out of the difficulty is to abolish the radial electric current in the conductors, making (66) the equation of V, and VC the longitudinal energy-transfer, with full applicability of the V=ZC terminal conditions. With a further consideration of this system, and some solutions relating to it, I propose to conclude this paper. f 358 -.] XLII. On Stationary Waves in Flowing Water.—Part I. | By Sir Witt1am THomson *. peas subject includes the beautiful wave-group produced by a ship propelled uniformly through previously still water, but the present communication} is limited to two- dimensional motion. Imagine frictionless water flowing in uniform regime through an infinitely long canal with vertical sides; and bottom horizontal except where modified by transverse ridges or hollows, or slopes between portions of horizontal bottom at different levels. Included among such inequalities we may suppose bars above the bottom, fixed perpendicularly between the sides. Let these inequalities be all within a finite por- tion, AB, of the length, and let f denote the difference of levels of the bottom on the two sides of this position, positive if the bottom beyond A is higher than the bottom beyond B. Now, iet the water be given at an infinite, or very great, distance beyond A, perpetually flowing towards A with any prescribed constant velocity w, and filling up the canal to a prescribed constant depth a. It is required to find the motion of the water towards A, through AB, and beyond B as disturbed by the inequalities between A and B. This problem is essentially determinate; and it has only one solu- tion if we confine it to cases in which the vertical component of the water’s velocity is everywhere small in comparison with the velocity acquired by a falling body falling from a height equal to half the depth. Let 6 be the mean depth, and v the mean horizontal velocity at very great distances beyond B; and (to have w to denote wave-energy) let w be such that Sra MENT) (es ok RN ar Wie eterna LF Pail) is the whole energy, kinetic and potential, per unit of the canal’s breadth and per unit of its length. In cases in which the water flows away unrufiled at great distances from B, w is zero. But,in general, the surface is ruffled, and the water flows “ steadily’ between the plane bottom and a corrugated free surface, as in the well-known appearance of water flow- ing in a mill-lead, or Highland burn, or in the clear rivulet * Communicated by the Author, having been read before Section A of the British Association, Birmingham, Sept. 7, 1886. T I have since found, in a sufficiently practical form, the solution for the wave-group produced by the ship, which I hope to communicate to the Philosophical Magazine for publication in the November number.— W. T., September 18, 1886, 304 Sir William Thomson on Stationary on the east side of Trumpington Street, Cambridge, or in the race of Portland or Islay overfalls. The train of diminishing waves which we see in the wake of each little irregularity of the bottom would, of course, extend to infinity if the stream were infinitely long, and the water absolutely inviscid (frictionless); and a single inequality, or group of inequalities, in any part AB of the stream would give rise to corrugation in the whole of the flow after passing the inequalities, more and more nearly uniform, and with ridges and hollows more and more nearly perpendicular to the sides of the canal, the farther we are from the last of the inequalities. Observation, with a little common sense of the mathematical kind, shows that at a distance of two or three wave-lengths from the last of the irregularities if the breadth of the canal. is small in comparison with the wave-length, or at a distance of nine or ten breadths of the canal if the breadth is large in comparison with the wave-length, the condition of uniform corrugations with straight ridges per- pendicular to the sides of the canal, would be fairly well approximated to; even though the irregularity were a single projection or hollow in the middle of the stream. But the subject of the present communication is simpler, as it is limited to two-dimensional motion; and our inequalities are bars, or ridges, or hollows, perpendicular to the sides of the canal. Thus, in our present case, we see that the con- dition of ultimate uniformity of the standing waves in the wake of the irregularities is closely approximated to at a distance of two or three wave-lengths from the last of the inequalities. Let SA, SB denote two fixed vertical sections of the canal at infinitely great distances beyond A and beyond B. It will simplify considerations and formulas if we take SB at a node (or place where the depth is equal to 6, the mean depth), and we therefore take it so; although this is not necessary for the following kinematic and dynamical statements:— I. The volumes of fluid crossing SA and SB in the same or equal times are equal; or, in symbols, au=bu Me ee where M denotes the volume of water passing per unit of time. II. The excess (positive or negative) of the work done by p on any volume of the water entering across SA, above the work done by g on an equal volume of the water passing away across SB is equal to the excess of the energy, potential and kinetic, of the water passing away above that of the water entering. Hence, and by (1), taking the volume of water Waves in Flowing Water. 395 unity, we have p—g=H( +90) + 7— [het g(ftha)] » (8). Now, calling the pressure at the free surface zero, we have ae (4); p=iga; and g=hgb+ B w denoting a quantity depending on wave-disturbance. Hence, and by (2), Ee —w! ge 9 bt/)+—Z—=0 . . (8), Now, put 1 \ —— and M=VD ° ° ° ry (6). Thus D will denote a mean depth (intermediate between a and 6and approximately equal to their arithmetic mean, when their difference is small in comparison with either) ; and V will denote a corresponding mean velocity of flow (intermediate between wu and v, and approximately equal to their arithmetic mean, when their difference is small in comparison with either). With this notation, (5) gives b b—a=——— e ° ° aie Cia): If b—a were exactly equal tof, and if there were no beruffle- ment of the water beyond B, the mean level of the water would be the same in the entering and leaving water at great distances on the two sides of AB; but this is not generally the case, and there is a (positive or negative) rise of level, given by the formula Vv? w—w b go! *" 9 y= eh Te eT ae eet tate (8). ~ ID Consider now the case of no corrugation (that is to say, of plane free surface and uniform flow) at great distances beyond B. We have w—w'=0; and therefore V2 gD NRG gee) ae is at ea GLE C2) ‘~9D 396 Sir William Thomson on Stationary or, with V? replaced by M/D?, ae og DP y=b—a—f= M. ° ° e ° (10), '~ 9D3 where, as above, ab? Tae (ia The elimination of b and D between these three equations gives y as a function of f. It is clear that the change of level of the bottom may be sufficiently gradual to obviate any of the corrugational effect ; and when this is the case, the equa- tion of the free surface will be found from y in terms of /; / being a given function of the horizontal coordinate, «. If f is everywhere small in comparison with a, D is approxi- mately constant [much more approximately equal to $(a+)) |, and y is approximately in constant proportion to f. When the flow is so gentle that V is small in comparison 2 = ae with /9D, 7D mately equal to this fraction of /. Generally, in every case when V < “/ 9D the upper surface of the water rises, when the bottom falls, and the water falls, when the bottom rises. On the other hand, when V> 9D, the water surface rises convex over every projection of the bottom, and falls concave over hollows of the bottom ; and the rise and fall of the water are each greater in amount than the rise and fall of the bottom ; so that the water is deeper over elevations of the bottom, and is shallower over depressions of the bottom. Returning now to the subject of standing waves (or cor- rugations of the surface) of frictionless water flowing over a horizontal bottom of a canal with vertical sides, I shall not at present enter on the mathematical analysis by which the effect of a given set of inequalities within a limited space AB of the canal’s length, in producing such corrugations in the water after passing such inequalities, can be calculated, provided the slopes of the inequalities and of the surface corrugations are everywhere very small fractions of a radian. I hope before long to communicate a paper to the Philosophical Magazine on this subject for publication. I shall only just now make the following remarks :— 1. Any set of inequalities large or small must in general is a small proper fraction, and y is approxi- Waves in Flowing Water. 307 give rise to stationary corrugations large or small, but per- fectly stationary, however large, short of the limit that would produce infinite convex curvature (according to Stokes’s theory an obtuse angle of 120°) at any transverse line of the water surface. 2. But in particular cases the water flowing away from the inequalities may be perfectly smooth and horizontal. This is obvious because of the following reasons :— Gi.) If water is flowing over plane bottom with infinitesimal corrugations, an inequality which could produce such corru- gations may be placed on the bottom so as either to double those previously existing corrugations of the surface or to annul them. (ii.) The wave-length (that is to say the length from crest to crest) is a determinate function of the mean depth of the water and of the height of the corrugations above it, and of the volume of water flowing per unit of time. This function is determined graphically in Stokes’s theory of finite waves. It is independent of the height, and is given by the well-known formula when the height is infinitesimal. (iii.) From No. ii. it follows that, as it is always possible to diminish the height of the corrugations by properly adjusted obstacles in the bottom, it is always possible to annul them. 3. The fundamental principle in this mode of considering the subject is that whatever disturbance there may be in a perpet- ually sustained stream, the motion becomes ultimately steady, all agitations being carried away down stream, because the velocity of propagation, relatively to the water, of waves of less than the critical length, is less than the velocity of flow of the water relatively to the canal. In Part II., to be published in the November number of the Magazine, the integral horizontal component of fluid pressure on any number of inequalities in the bottom, or bars, will be found from consideration of the work done in genera- ting stationary waves, and the obvious application to the work done by wave-making in towing a boat through a canal will be considered. The definitive investigation of the wave- making effect when the inequalities in the bottom are geo- metrically defined, to which I have just now referred, will follow ; and I hope to include in Part II., or at all events in Part III. to be published in December, a complete investiga- tion, illustrated by drawings, of the beautiful pattern of waves produced by a ship propelled uniformly through calm deep water. ee ane XLII. On the Electrical Resistance of Soft Carbon under Pressure. By T. C. MENDENHALL.* A PAPHR by the writer on “The Influence of Time on the Change in the Resistance of the Carbon Disk of Hidison’s Tasimeter,” was published in this Journal in July 1882 | Phil. Mag. for August, p. 115]. The object of the paper, as its title indicated, was to present the results of some experiments with the carbon disk which appeared to show that, when pressure was applied, the entire diminution of resistance did not take place at once, but that the reduction continued with diminished rapidity through a considerable period of time. At the conclusion of the paper brief refer- ence was made to investigations of the same subject by Mr. Herbert Tomlinson and by Professors Sylvanus P. Thompson and W. F. Barrett. Only the conclusion reached by some of these physicists was at that time known to the writer, their verdict being that the observed diminution of resistance was really due to the better surface-contact of the electrodes, and not to any actual change in the specific resistance of the carbon itself. The last paragraph in the paper contains the following :— ‘¢ Without knowing anything about the nature of these experi- ments, the writer desires to record his belief that this theory does not entirely account for the facts stated above.” This, certainly not too rash, declaration of belief in a true pressure-effect was the subject of decidedly unfavourable criti- cism in the columns of one or two Huropean scientific journals ; and in this Journal of December 1882, Professor Sylvanus P. Thompson published an article entitled “ Note on the alleged Change in the Resistance of Carbon due to Change of Pres- sure,” which was an exceptionally severe criticism of the pre- vious paper by the writer. In this article Professor Thompson refers to the investigations of Mr. Tomlinson, Prof. Barrett, and himself, and also to experiments made by Professors Naccari and Paglianiand Mr. Conrad W. Cooke, and he declares that, with the exception of Professor Mendenhall, all who have in- vestigated the point are agreed in their verdict ‘that this alleged effect was due not to any change in the specific resis- tance of carbon, but to better external contact between the piece or pieces of carbon and the conductors in contact with them.” The truth of this statement is the question at issue. It may be well to remark, however, that although Professor Thompson makes this assertion in December, Mr. Tomlinson had shown, nearly a year earlier, in a paper presented to the Royal Society, on the 26th of the previous January, that the * From Silliman’s American Journal for September, 1886. Electrical Resistance of Soft Carbon under Pressure. 359 electrical resistance of hard carbon was diminished by pres- sure. The amount of the diminution is small, however, and he afterwards expresses the opinion that in such instruments as the microphone transmitter, the greater portion of the ob- served diminution in resistance is due to variation in surface contact.* Mr. Tomlinson’s experiments were made with hard carbon, similar in character to that made use of in experi- ments to be described presently. In the summer of 1884, the writer communicated to the American Association for the Advancement of Science a brief account of experiments which satisfied him that the opinion which he had previously expressed concerning the nature of the phenomenon was unquestionably correct. Within. the past year the subject has been taken up again, and by means of improved methods and instruments all doubts seem to have been removed. Innumerable experiments made by physicists of many coun- tries have established, beyond question, the fact that the elec- trical properties of matter are modified by stress and strain. In carbon the effect of pressure is to diminish resistance. For hard carbon this was established by the investigation of Mr. Tomlimson. In compressed lampblack, as seen in Hdison’s disks, the effect is very great, and that this is for the most part a true pressure-effect is proved, it is believed, by the experiments about to be described. In the beginning it was desirable to determine, roughly, the magitude of this effect in the case of hard carbon. For this purpose a copper-plated rod, such as is used in the arc lamp, about 12 centim. in length and 1°5 centim. in diameter, was selected and its ends were ground flat at right angles to its axis. The plating was then removed, except that a band about *5 centim. in width was left near each end of the rod. Two cork rings 1°5 centim. thick were fitted to the rod, after which they were tunnelled out on the inside, and a hole was made in each so that when they were in place over the copper bands, and mercury was poured in, it would flow around the ring tunnel and make a contact with the carbon as satisfactory as could be desired. The ends of the rod were protected by thin plates of vulcanite, and it was placed between the jaws of a vice. The current from a battery of two or three gravity- cells was passed through the rod by plunging wires into the mercury cups formed by the corks. By this arrangement it was possible to apply pressure at the ends of the rod without in any way influencing the contacts through which the cur- rent passed. * “Nature, March 16, 1882. 360 Mr. T. C. Mendenhall on the Electrical Resistance The terminals of a reflecting-galvanometer whose resistance was about 5000 ohms were also introduced into these mercury cups, and enough additional resistance was introduced to make a convenient deflection of the spot of light upon the scale. When all was adjusted and the spot of light was at rest, the pressure was applied by turning the handle of the vice. In every instance the deflection decreased, showing diminished resistance. This effect was not due to the heat produced by compression, as experiment proved that cause to be inadequate. It was found to be necessary to make the carbon rod decidedly warm to the touch in order to lower the resistance by the same amount; besides the effect was not transient, as would have been the case if it had been due to the change in temperature. It was also found that compression at right angles to the direction of the current produced a similar effect, but less in magnitude. These facts had been already announced by Mr. Tomlinson. These experiments with hard carbon or with other rigid bodies are comparatively easy, as there is no difficulty in applying the pressure independent of the contact surfaces, so that possible variation of the latter need not be considered. Unfortunately it appears to be quite impossible to secure this arrangement in the examination of soft carbon. It cannot readily be obtained in forms different from the small disk or button in which it originally appeared, and it is so fragile that it requires the most careful manipulation. Under these circumstances, the only thing to do is to secure the best possible surface-contact of the poles to begin with. Perhaps the ideal arrangement would be a disk with its two opposite faces electroplated with copper, through which a contact with mercury can be secured. The electroplating of two opposite faces of a disk of compressed lampblack is a work of extreme difficulty, and so far as known to the writer has not yet been accomplished, although he is greatly indebted to Mr. Edison for a serious and persistent effort to secure this resuit, none the less appreciated because, owing to the extremely fragile character of the disk, it proved to be unsuccessful. It was therefore necessary to depend upon the contact of mercury with the surface of the carbon itself. As this was the contact employed by Professor Barrett in the experiment which Professor Thompson considered ‘ crucial,” its use can hardly be objected to in this instance. The arrangements for the test of the soft carbon were as follows :—two glass tubes about 20 centim. in length were bent at one end into a quarter of a circumference, so that when the two were joined and the straight branches of the tube were of Soft Carbon under Pressure. 361 in a vertical position the appearance was that of the letter “U,” the height being about 15 centim. Near the lower end of each a short tube was sealed in, over which rubber tubing could be passed, and at the lower part of the curve, in each, a platinum wire was passed through and sealed. The ends of the tubes were ground flat, and they were mounted in such a way that while one was fixed in position, the other could be moved toward or away from it in one plane, and so that the ground ends of the curved parts were always exactly opposite to each other. The movable tube was then taken from its place, the ground edge of its curved end was coated with glue, and it was carefully brought down upon the upper surface of . a carbon disk which rested in a horizontal plane. The glue causing the disk to adhere to the tube, the latter could then be secured to its sliding stand, ready to move into place. The ground edge of the fixed tube was now coated with glue, after which the movable tube holding the carbon disk was gently moved up until the disk pressed against the end of the other tube, the glue forming the junction. In this way a carbon wall or partition was formed between the two halves of a “U” tube. When the glue had hardened, mercury was introduced on both sides to a height sufficient to entirely cover the faces of the carbon disk. The current was introduced through the platinum electrodes, which plunged into mercury cups on either side. In some of the earlier experiments variations of pressure were produced by the addition of mercury to the two branches of the tube, but vastly better than this was the method latterly used, in which the pressure of air was substituted for that of mercury. Glass plates were sealed on the open ends of the two upright branches, thus enclosing a space on each side, except at the small side tubes, to which short pieces of rubber tubing were attached. These were joined by means of a T- tube, so that equality of pressure on both faces of the disk was secured. The circuit consisted of the battery, the disk, and an addi- tional resistance varying from 3 ohms to 10 ohms for purposes of comparison. The electric ends of the disk and of the resis- tance were joined to a specially arranged key, by means of which either could be connected with the terminals ofa reflect- ing galvanometer whose resistance was about 7000 ohms. By means of the deflections of the needle of this galvanometer, the resistances were compared and variations noted, the arrangement being substantially the same as that previously used in the experiments with hard carbon. A pressure-gauge, sometimes of water, sometimes of mercury, was attached to Phlail. Mag. 8. 5. Vol. 22. No. 137. Oct. 1886. 2B 362 Electrical Resistance of Soft Carbon under Pressure. the apparatus to indicate variations in pressure, and these variations were generally produced by blowing from the mouth into a rubber tube about two metres in length. Very many experiments were made, all without exception showing great diminution in the resistance of the disk by increase of pressure ; and it will be sufficient to quote a few of the results. The disk is sufficiently sensitive to show very slight changes in atmospheric pressure. On closing the open end of the rubber tube, and slightly pressing any part of it between the thumb and finger, the spot of light instantly moved, showing decrease of resistance. A pressure measured by 5 millims. of water produced a decided effect. The resistance of the disk, with its mercury and platinum wire connections, under ordi- nary conditions was slightly greater than 6 ohms. A pressure measured by 5 centim. of mercury instantly reduced it to less than 3 ohms. If the pressure was maintained, a slow fall of resistance continued for a long time, as found in the previous investigation of the subject. If the initial pressure was small the recovery would be instantaneous on its removal ; but if it was large, so as to greatly reduce the resistance, it was found that the recovery would not be complete on the withdrawal of the pressure, sometimes falling short by as much as ten per cent., after which a slow rise would take place. This result is not quite in agreement with the statement made in the first paper upon this subject, which was based, however, upon a much less satisfactory series of experiments. An examination was made of the effect of the strength of the current upon the resistance of the disk. The weakest current used was a little less than ‘001 ampere, and the strongest was about 37 ampere, so that one was approximately 400 times the other. Throughout this range no sensible differences in the resistance of the disk was observed, the agreement at the two extremes being within the errors of measurement. Under all conditions the effect of variations of pressure was the same. ; | The faces of a soft carbon disk are always smooth and polished ; the surface of hard carbon, on the contrary, is gen- erally more or less rough and irregular. It would appear, therefore, that, if the reduction of the resistance of soft carbon by increase of pressure is due to better surface-contact, this reduction of resistance should be much more marked with hard than with soft carbon. Experiments already described showed that the effect of pressure on hard carbon was very small; so small, in fact, that the pressure of a few centimetres of mer- cury would hardly produce a sensible effect. Mr. John Aitken on Dew. 363 The substitution of a disk of hard carbon for the soft, in the apparatus described, ought to show, then, whether any con- siderable part of the resistance-variations observed could be attributed to variation of contact between mercury and carbon. A disk of hard carbon similar in dimensions to the soft disk previously employed was accordingly inserted between two similarly arranged tubes. ‘The result of this experiment was to show, as had been anticipated, a small decrease of resistance when the pressure was increased. A pressure of about 7 centim. of mercury reduced the galvanometer-deflection from 36 to 35 divisions of the scale. ‘This indicates a change of less than 3 per cent., resulting from a pressure which with the soft disk lowered the resistance by more than 60 per cent. There can be little doubt that this small reduction is due almost entirely to better surface-contact produced by pressure. Throughout all of the experiments with soft carbon, it ex- hibited more or less irregularity in its behaviour. The appli- cation of a pressure very largely in excess of the maximum referred to above would sometimes result in a permanent reduction of the resistance of the disk, indicating that a per- manent set had taken place. By the exercise of care, however, what may be called the “normal” resistance may be main- tained fairly constant for a considerable length of time. Conclustons.— When carbon is prepared in the form of com- pressed lampblack, its electrical resistance varies greatly with the pressure to which it is subjected. A small part of this variation is doubtless to be attributed to change in sur- face-contact between the carbon and the electrodes through which the current is introduced, but by far the larger part (provided any effort is made to secure good surface-contact) is due to a real change in the resistance of the carbon itself. The resistance of carbon in this condition is fluctuating and uncertain to a degree that seems to prevent its use as a factor in any device for the accurate measure of pressure. XLIV. On Dew. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, | the paper by Mr. Charles Tomlinson, F.R.S., in the September number of your Journal, entitled “Further Remarks on Mr. Aitken’s Theory of Dew,” there is little of scientific interest, the points advanced being mostly of a con- troversial character. As, however, the whole tone of his 2B2 364 Mr. John Aitken on Dew. remarks is antagonistic to my work, as he candidly admits at the end of his paper, where he says, “ As regards his new theory of Dew I think he has gone astray,” it is therefore necessary that I reply to his criticisms. I sincerely trust Mr. Tomlinson does not think that I accuse him of intentionally raising a false contention by entitling his first paper “‘ Remarks on a New Theory of Dew.” The title, however, indicated the attitude of the writer’s mind towards my conclusions, and it could not be left unchallenged, as it struck at the very root of the matter. Mr. Tomlinson thinks it curious that I should in my last letter have so frequently repeated, in different forms, the statement that my results are not contrary to the teaching of Dr. Wells. If he will refer to his “ Remarks” he will find that these repeated statements are all replies to the contents of different paragraphs in his own letter. Mr. Tomlinson attempts to justify the title of his paper by saying that the author of an article on the same subject in ‘Chambers’s Journal’ used the same words. That we are right because we think with the majority is an argument which is generally supposed to be.a monopoly of the political mind ; to the scientific mind, I venture to say, it carries no weight. Besides, in an article in a popular journal like ‘Chambers’ the writer may be pardoned for selecting a catching title, and we do not expect from him the scientific accuracy of language we do from a writer in the ‘ Philoso- phical Magazine.’ Though the writer in ‘ Chambers’s Journal ’ gives a misleading title, yet he very fairly states the position. For instance, he says :— “The essential difference between the old and the new theories is as to the source of the moisture which forms the dew. Instead of being condensed from the air above by the cooled vegetation, Mr. Aitken maintains that it comes from the ground. The author of the original theory admitted that some of the dew might come from below, but affirmed that it must be an exceedingly small proportion. Mr. Aitken’s ex- periments, on the contrary, seem to prove that most, if not the whole, comes from the ground.” From the above it will be seen that, as 1 have so frequently stated, my results do not touch on the teaching of Dr. Wells. As every one knows, he concerned himself principally with the condensation of the vapour after it is in the air, and but little with its source, as he distinctly states in his Hssay that he had no means of investigating this latter point. At page 271 Mr. Tomlinson says:—“Again the Chambers’s article, referring to Wells, says:—‘ The points of the grass, 5a Mr. John Aitken on Dew. 365 small twigs, and all other good radiating surfaces are cooled the most ; and accordingly we find the dew-drops most abundant on these bodies ; whilst on metal, or hard stone surfaces, which are poor radiators, we seldom or never find any dew.’ This is the Wells picture; the writer now turns to the Aitken pic- ture. ‘A closer observation reveals the fact that these so-called ‘dew-drops’ are formed at the end of the minute veins of the leaves and grass, and are not now recognised as dew at all, but moisture exuded from the interior of the plants them- selves.’ And yet Mr. Aitken is angry with me for calling his theory new, and for asserting that, if true, it will supersede the labours of previous observers.”” My critic here shifts his ground ; when he finds the contents of the first part of my paper, regarding the rising of vapour at night, does not justify his title of New Theory, he cleverly seems to put me in oppo- sition to recognized authority on another point. The above quotations certainly are in opposition to each other, and great credit is due for the very ingenious manner in which the case is put. But, unfortunately for the critic, the words are not the words of Dr. Wells, but are those of the writer in ‘ Cham- bers’s Journal, and can scarcely even be said to be founded on the teaching of Wells; so that, further than affording Mr. Tomlinson a little mental gymnastics, his efforts are here entirely thrown away. It does seem strange, considering how much is indirectly attributed to Dr. Wells regarding the deposition of dew on grass, that his ‘Hissay’ really contains very little that is definite either about the radiation from grass, compared with that from other bodies, or about the amount of dew deposited upon it. Towards the end of the first part of his ‘ Essay,’ where he gives ‘‘the results of some experiments which were made for the purpose of ascertaining the tendencies of various bodies to become cold upon exposure to the sky at night,’ he says :—‘‘In the observations hitherto given by me on the cold connected with dew, the temperature of the grass has been chiefly considered, partly because my first experiments had been made upon it, and partly from a wish, which arose after- wards, to compair my own experiments with those of M. Six, which had been confined to that substance. I found it, how- ever, very unfit to furnish the means of compairing the degrees of cold produced at night on the surface of the earth, at dif- ferent times and places ; as its state on different nights, on the same parts of the plat I commonly made use of, and in different parts of the plat on the same nights, was often very unequal in point of height, thickness, and fineness ; all of which circumstances influenced the degree of cold produced by it.”’ 366 Mr. John Aitken on Dew. Near the end of the second part of his ‘ Hssay’ he refers to the dew on plants, but it is simply for the purpose of refuting the opinion that “the dew found on growing vegetables is the condensed vapour of the very plants on which it appears.” He says :—‘ This seems to be erroneous, for several reasons. (1) Dew forms as copiously upon dead as upon living vege- table substances. (2) The transpired humour of plants will be carried away by the air which passes over them when they are not sufficiently cold to condense the watery vapour con- tained: mm lit:7?). 0). 4. I draw attention to these passages to show that the distinction between true dew and the dew-drop was not recognized at the time Wells wrote his ‘ Essay.’ All his arguments are directed against the opinion that the dew formed over the whole surface of the blades of plants was produced by the condensation of the vapour transpired by the plants themselves, which he pointed out would be a small quantity at night on account of the absence of light. He makes no reference, so far as I am aware, to the dew-drop, which my investigations tend to show is the result of exuded liquid, and not of the transpired vapour to which Wells di- rected his criticisms; so that my observations are in a different field from those of Dr. Wells. But even supposing my conclusions regarding the source ~ of the dew-drop to be correct and to be in opposition to recog- nized authority, still, if we wish to be accurate, we shall not be entitled to call it a new theory of dew, as it is a theory of the dew-drop as distinct from dew. If Mr. Tomlinson will change his style of criticism, and will explain to us whence came the drops which formed on the plants experimented on when they were isolated in dry air, and all supply of moisture cut off except that which came up through the tissues of the plants ; and if he will show us that we have misinterpreted the teaching of the experiments made by weighing turfs and others, which we have adduced to show that vapour is given off from the ground while dew is forming at night, he will be entitled to be listened to; but purely literary criticism is a mere jangling of words, and seldom leads to satisfactory conclusions. I fear Mr. Tomlinson has misunderstood the bearing of my remarks with regard to what takes place in Persia and the African Desert. The impression I wished to convey was, that we cannot conclude from experiments made in this climate as to what takes place in arid regions. Jam happy to say I can assure my critic that ‘the great forces of Nature rule as im- partially in Persia and in Africa as in Scotland;” igs it not for this very reason that we cannot conclude from what Mr. John Aitken on Dew. 367 takes place in one country what will take place in the other, unless the conditions are alike? All my experiments show that in our climate dew never falls on the earth—though of course Mr. Tomlinson, if he is consistent, must think otherwise—and dew is only deposited on plants and other bodies not in good heat communication with the ground. But it would be rash from this to conclude that in arid countries, where the air is highly diathermatous, and the ground dry and probably a bad conductor of heat, the surface of the earth is never cooled by radiation below the dew-point. But on this point I repeat, “I wait for further information before forming any opinion as to what takes place in other and unknown conditions.” I observe that Mr. Tomlinson does not observe the same caution, as he states distinctly that, after passing through arid regions, “long before the travellers reached any considerable body of water, nocturnal dews were abundant, and they were de- posited from the air, and did not rise out of the ground.” Now on what does he found this last statement which I have put in italics? The fact that the dews were deposited out of the air in no way proves that vapour did not rise from the ground while the dews were being deposited. With regard to the bearing of the experiments of the Flo- rentine Academicians, of Robert Boyle, and Le Roi, my critic says, “‘ These early observers proved that the moisture which forms dew and hoar-frost exists in the air, and does not exhale from the ground.” It is a self-evident fact that the vapour must be in the air before it condenses on the different surfaces ; but, as has been already said in my previous letter, this fact has ‘no bearing on the subject,” as it neither proves nor dis- proves that vapour “does not exhale from the ground.” Mr. Tomlinson is welcome to any consolation he can get by shielding himself behind the word “ abstract’ to account for his misconception of the essential conditions of the crucial experiment made by weighing the turfs before and after “dew- fall.” As, however, my last letter gave no further information on the subject, and as Mr. Tomlinson seems now to under- stand the conditions of this test, it may be presumed that the abstract was complete enough on that point and did not give rise to the misunderstanding. I may say I have carefully considered the observations of Melloni, referred to by Mr. Tomlinson, and do not find any- thing in them that affect the conclusions I have arrived at. To enter, however, into a detailed examination of his work would extend the limits of this letter to an undue length. To many this discussion must have appeared extremely 368. Mr. Thomas Gray on a New unsatisfactory and unreal, too much attention having been given to purely literary points, to the almost entire exclusion of the realities connected with the phenomena of dew, and the interpretation to be put on them. No one, I am sure, regrets the unfortunate direction the discussion has taken more than Ido. It could not, however, be avoided, owing to the direc- tion from which the attack was delivered. After all this difference of opinion it is a comfort to find one point on which we are agreed, namely “ not to write again on this subject ;’? and in closing the discussion I wish to thank Mr. Tomlinson for his criticisms, because I am sure his objections will be the objections of many others to my conclu- sions ; and though I fear I have not succeeded in making a convert of him, I may perhaps have been more successful with those whose ideas are not so defined and stereotyped by frequent writing on the subject. I think, however, I can assure Mr. Tomlinson that when he candidly and calmly con- siders the result of my investigations, he will not find them so heterodox as he at present seems to think; nor do I think that my observations will entirely “supersede the labours of previous observers ’’ for whose work he has so much respect, though some of their views will require modification to meet the present state of our knowledge. | Yours truly, Darroch, Falkirk, JOHN AITKEN. September 11, 1886. XLV. On a new Standard Sine- Galvanometer. By Toomas Gray, B.Se., PRS E.* HE standard galvanometers commonly employed for the determination of currents in absolute measure consist either of one bobbin of large radius, having a groove of rela- tively small breadth and depth filled with wire, or of two such bobbins mounted with their planes parallel and at a distance apart equal to the radius of either coil. These coils are suitably mounted for use either as sine- or as tangent- galvanometers. The object of making the coils of large radius is twofold—first, in order that it may be possible to measure it with sufficient accuracy, and second, in order that the magnetic field, produced by the current, may be of nearly uniform strength near the centre of the coil. So far as uni- formity of field is concerned, the double coil, or Helmholtz arrangement, is all that can be desired; but it introduces a multiplicity of measurements, one at least of which, namely. * Communicated by the Author, Standard Sine-Galvanometer. 369 the distance between the mean planes of the coils, is difficult to make with sufficient accuracy. The arrangement here proposed, and illustrated in figures 1 and 2, consists of one layer of wire wound on a tube of com- paratively small diameter, say 10 centimetres or less, and of great length. The force a the centre of such a tube, produced by unit current, is given by the equation A4trnl re JS r+l? where / is half the length of the coil, v its radius, and n the number of turns per centimetre of its length. When 7 is very great compared with r the force becomes 47rn, that is to say, it depends wholly upon the number of turns 7 per unit-length of the coil. The correction for the length of the tube is shown clearly oe expanding ton 1 +7 B We. which gives , ir 1 fodan( 13 a7 er. ic the.) When / is ten times the radius, this series becomes 1 1 fame 1-95 + sqq9&) which shows that for this moderate length the correction is very small, and is amply obtained by taking in the second term of the series. Any small error in the measurement of the diameter of the coil is thus of little importance, and hence the chief question becomes the degree of accuracy which can be attained in the estimation of n, the number of turns per centimetre. The total number of turns, or nl, can of course be obtained with absolute accuracy, and the length, /, can be easily measured to the tenth of a millimetre, which, on the assumption of uniformity of winding, gives n to one part in 10,000. Providing that there exists no want of uniformity which will affect the average value of n, perfect uniformity is only important near the central part. Now an irregularity to the extent of one hundredth part of n, or one tenth of a milli- metre, can be readily detected by simple measurement ; and supposing that this was due to an opening between the wind- ings in the same planeas the needle (or at the most advantageous position), and that there was no compensation due to denser winding near the middle position, the effect would only amount to about one in 1200 of the total force. The determination of the value of n does not, however, depend altogether on the measurements here described; the operation of winding a 370 On a new Standard Sine- Galvanometer. coil uniformly is one to which mechanical appliances of great precision can be readily applied. The wire can, for instance, be laid on by means of a self-feeding lathe with a pitch which is not only perfectly uniform, but which is also known with minute accuracy from the pitch of the feed-screw. Any slight irregularity which might remain can, if it be within the sensibility of the instrument, be detected by moving the needle along the axis by withdrawing the tube T, and obsery- ing the change of sensibility. The great advantage of the arrangement is the simplicity and possible accuracy of the measurements and the great uniformity of the field all round the central point. The resistance of the coil is, when thin wire is used, somewhat higher for the same sensibility than it is in the ordinary form, but for a standard instrument this is of little importance. AR OLA LLL LSTA OLA ss ss SSS N SSS === = ————| Veen EAE = =< Referring now to the sketches, the tube T, which carries the coil, is mounted on a platform P, furnished with three levelling screws L, L, L, and can be turned round a vertical axis V, sliding at the same time on two feet Fy Fx (ihe angle through which the coil is turned is measured by a scale 8. In the centre of the tube a small plane mirror m is suspended, and at one end of the tube a short scale s, illu- minated by means of an inclined mirror or prism, which receives light through a small hole h, is fixed, and immedi- ately above it a plane mirror M. The light from the scale s is reflected from the suspended mirror to the fixed mirror M, and then through the telescope ¢ fixed in the end of the tube T,. The scale s here shown may be replaced by a narrow slit, or a round hole with a wire across it, and the telescope by a sheet of obscure glass. A lens placed in front of the opening may then be used to focus an image of the slit or wire Mr. F. Y. Edgeworth’s Problems in Probabilities. 371 on the obscure glass, the position of which can be read ona scale fixed to the glass precisely as in Jacob’s well-known galvano- meter arrangement. ‘The lens may be dispensed with, and an ordinary Thomson’s spherical mirror substituted for the sus- pended mirror by fixing the obscured glass in the end of a third tube, which can be telescoped out of and into the tube T, so as to adjust the focus. One end of the coil is attached to the pin p, and the other to the pin ps, while the wire w conducts the current back parallel to the axis of the coil to a third pin p3, fixed close to p,. From p, and pz the current is conducted, by means of a pair of flexible electrodes twisted together, to proper terminals on the platform P. In using the instrument, place it ona table in a well- lighted room and level the platform P. Then turn the coil until the central division of the scale s coincides with the cross wires of the telescope, and take the reading on the scale S. Pass a steady current through the coil and note how far the tube has to be turned to bring the central division of s again to the cross wire of the telescope. Repeat this reading with the current reversed, and move the scale S if necessary, until the angles on the two sides of zero are equal. The cur- rent flowing through the coil is then given by the equation Gs H sin @ where @ is the angle through which the tube is turned from the zero position to bring the central division of s to the cross- wire of the telescope. The degree of accuracy attainable in the determination of @ is evidently very great, and can be pushed to almost any extent by fine division of 8 and micro- scope reading. XLVI. Problems in Probabilities. By F. Y. EDGEWoRTH, F.S.S., Lecturer at King’s College, London*. SOME interesting problems in the Calculus of Probabilities are presented by the business of Banking{. ‘The profits of the Banker depend upon the probability that he will not be called upon to meet at once more than a certain amount of his liabilities. Assuming that the demands made upon him fluctuate, like so many other phenomena, according to the exponential law of frequency, we may employ the Theory of * Communicated by the Author. + See the writer’s paper “On the Mathematical Theory of Banking,” read before the British Association, September 1886, 372 «Mr. F. Y. Edgeworth’s Problems in Probabilities. Errors to determine the probability that the demand will not exceed any proposed limit. The assumption just made has been defended elsewhere* ; the mathematical constructions which rest upon that foundation are the subject of this paper. I. The first and main problem is: Given a series of Bank- ing returns (e.g. of Notes in the hands of the Public, or of the Reserve), to find the probability that the next return, or the returns in the proximate future, will not exceed certain limits. The general method is to find the Meanf of the given series and the Modulus ; to put T for the ratio of the distance between the Mean and the proposed limit to the T Modulus; to find =. ( e~“dt from the usual tables, and put T 9 the value so found for the probability that the next observa- tion on the same side of the Mean as the proposed limit will not exceed the limit, or put half that value for the probability that the next observation unconditionally will not exceed the limit. The question here arises, What method is to be adopted in discovering the Mean and Modulus? I have elsewheret considered generally the relative advantages and disadvantages of the different methods. Here it will be sufficient to take account of what is special to the statistics under consideration. It is a peculiarity of Banking returns (as of some meteorolo- gical records) that they cannot be regarded as so many inde- pendent observations, like the different measurements of the same object, or like the heights of different individuals. Con- sider, for instance, the weekly returns of Bank-of-Hngland Notes in the hands of the Public between the years 1833 and 1844. very single return from January to September 1834 is above the Mean of the whole series, which is about £18,400,000. Every single return from August 1839 to June 1841 is below that Mean. Such sequences are incon- sistent with the hypothesis of independent observations. We must liken the series of returns, not to a series of numbers each one of which is the sum of, say, twenty digits taken at random (from pages of mathematical tables), but rather to an entangled series constituted in the following manner. Form a first term in the manner just described. Then, for the next term (instead of taking twenty fresh digits), remove one of the constituent digits from the first number and add a fresh digit taken at random. Repeat the process continually. Then you will have an entangled series like the following :— * Ibid. + Supposing that there is no secular variation. t “Observations and Statistics,’ Camb, Phil. Trans. 1885, Mr. F. Y. Edgeworth’s Problems in Probabilities. 378 95, 99, 99, 92, 95, 94, 94, 99, 101, 100, 108, 108, 106, Ho, 108; 20a, 107/112, 105; £05, 101, ...’; where the first term and the last term are the only indepen- dent observations; and where there occur twenty observations running above the Mean of the series (supposed indefinitely prolonged). In case of such entangled series, I submit that the advantage of accuracy generally attaching to the Arith- metic Mean and the determination of Modulus by way of mean square of error disappears. There is no set-off against their disadvantage of inconvenience. Accordingly the proper mode of reducing such observations is what may be called the Galton-Quetelet method, by way of Median and “ Quartile.” I have applied this method to find what was the probability in 1844 that the Notes in the hands of the public would not in the proximate future fall below the limit fixed in that year for uncovered Notes. The returns upon which the calculation is based™ are the monthly returns from 1833-1884, and cer- tain biennial returns from 1826-1844. I find that these returns conform pretty accurately to a probability-curve, whose Median is £18,400,000, and whose probable-error is £1,000,000. The Modulus then is about £2,000,000. And the proposed limit is £15,500,0007; that is, at a distance from the Median of about 14 the Modulus. The probability, then, of the Notes not falling below the regulation-limit in the proxi- mate future is about ‘98t. II. The problem becomes complicated if we suppose the number of (independent) observations to be not large, and seek to determine, by way of Inverse Probability§, the error due to that limitation. The error incurred by the Galton-Quetelet method depends upon the errors attaching to the assumptions that the point dividing the given set of observations into two equal groups is the real Median, and thut the points dividing the given set of observations in the ratio of three to one are the real “ Quar- tiles.”” The errors of these assumptions may be investigated by the following method, which was suggested by Laplace’s determination of the error incurred by his “‘ Method of Situa- tion” (Théorie Analytique, Supplement 2, Sect. 2). * See ‘Mathematical Theory of Banking.’ + £14,000,000+-1,500,000 (Bills and Lost Notes), ibid. { More exactly ‘98 is the probability of the next observation falling below the limit. Within what time the next observation must take place depends upon the number of (independent) observations assigned to the series. It would be safe, I think, to say—within six months, § Problem I. belongs to the class termed d in my ‘Observations and Statistics ;’ Problem II. to d. 374. Mr. F. Y. Edgeworth’s Problems in Probabilities. Let n be the number of observations supposed to range under one and the same curve y=¢(x); whereof P is the (real) ordinate at the Median point ; which point may be taken as the origin. The probability of an error z being committed by the apparent Median is equal to the proba- bility of half of the n observations falling on one side and half on the other side of z. Now the probability of a single observation filling outside z is G i dz), And the probability ofa single observation falling inside z is (3 +f ae dz). Hence the probability of z being the apparent Median is «(3-(“glode)’ (3+ “#@ae) o 20 0 if z is small (a a y (3+ zs 2) (4!(0) being =0) «4(1— aie) The probability of the error z is therefore proportioned to 1 2, ot (by a step* frequent in this region of mathe- matics, and which success and the authority of Laplace —2P? 2 sanction) e ~ . Nowif And the law of facility for the error of the Median is 2n nae re , multiplied by a proper coefficient. The error incurred by assuming that the point which di- vides the given observations in two groups numbering respec- tively 2n and in is (the extremity of) the real Quartile, may be similarly calculated. Let Q be the real Quartile, and let us consider the probability of the point (Q+z) dividing the observations in the ratio of 2? to +. The probability of an observation falling outside (Q+2z) is ; Py) [3—-(<4(Q) + 5 #'(Q))]. * T desiderate a more rigid proof of this step; such as I have attempted to give for the general case of the law of error; “Observations and Statistics,’ Camb. Phil. Trans, 1885, p. 142. Mr. F. Y. Edgeworth’s Problems in Probabilities. 375 And the probability of an observation falling inside (Q+2z) is 2 2+29(Q+F4(Q)]: z, as before, being small. Hence the probability of the error z being committed is proportioned to | ae nae el Caesar aera My 4 nr ote H 48 OY x [ie 20H i b og [1-29 HN] oe ee n n = Now == este n n =—__— 2 = — e— 227 = — ‘8. $(Q) /1r¢ /1¢ /11e Whence, as the law of facility for the error of the Quartile point, we have —1'Tn.9 veaie where J is taken so that \ydz between extreme limits =1. _ Thus we have found the error incident to both extremities of the line which we assume as the “ probable error”’ of the curve under consideration. And it appears at first sight that we might calculate the error in the length of the line by adding together the Modulus-squared for each of the two errors on which the error of the line depends. There occurs, however, the difficulty that these errors are not independent *. A drift which shifts the Median in either direction is apt to shift the Quartile in the same direction, and vice versd. In view of this difficulty, I see no way but to be contented with the sum of the Moduli-squared as a superior limit to the sought Modu- lus-squared of error; and with the smaller of them as an inferior limit. The superior limit of the Modulus-squared 2 (for the error of the Quartile) is then nearly = The (supe- rior limit of the) Modulus for the same error is \/ = 0. The error incurred in determining the Modulus (of the given set of observations) follows from the proposition that the Modulus is 2°097 x probable-error. The Modulus for the error in the determination of the Modulus of the given observations is found to be <3°7 FF ; where for c we may put the apparent Modulus (in the example above given, 2,000,000). * The same difficulty attends the use of the ‘ Octiles ” and “ Deciles.” 3876 =Mr. F. Y. Edgeworth’s Problems in Probabilities. This error may be somewhat reduced, if we determine the probable-error (for the given observations) not by the dis- tance between the apparent Median and the apparent Quartile point, but by the distance between the two Quartile points. The Modulus for the error of¢ is in this case = nN We have so far been investigating the error incident to ~ reasoning up to the character of the curve from a not very large number of observations. Let us now consider the error incident to the complete process of reasoning up from the given observations and down again to a new observation. First, let us suppose the Mean given, and calculate the error incident to the assumption that Q’, the apparent Quartile, is the real one. Let it be required to determine the probability that a subsequent observation will not exceed aQ’. Put O(a) for Fah erat Then, if Q’+< be the real probable error, 0 the probability that an observation being in excess of the Mean ‘ fe a()’ ) Z will not exceed aQ’ is Ie CED Now ¢ ceents with a frequency expressed by the facility-curve y= =e FB, kala where : Ware 13/0 nearly. We have therefore as the probability of the next observation above the Mean falling within «Q!, the expression ‘fe ‘ATT a Negi sxe Gerd oc oO Put -477«a=8, and expand @ in ascending powers of z. The first term of the expanded expression is fe ea (een at 06) | ae Bae= 08), the uncorrected value. The second term vanishes. The third term is af nies die 2s Eve I a oy es 3 at So Hoteles made k2 et (—26 Bret fight aad Tae 2 = PPB) x G eee Vn Sea 17x °227xn’ Mr. F. Y. Edgeworth’s Problems in Probabilities. 377 since b= Tae and §=Q!=-477c¢ approximately. The odd terms above the third vanish, and the even terms may be neglected, involving the higher powers of k”, i.e. of es nr In this reasoning we have supposed the Quartile to be determined by taking the point which divides the whole set of given observations into two groups whose numbers are to each other as three to one. If we determined the Quartile by taking the point which divided : observations above the Median into two groups each numbering 4) the conclusion would be much the same except that, instead of n, we must 5 and put 1:7 for 2 in the value of &. As an example of the last case, let us take n=80; that being, I think, the greatest number of independent observa- tions to which the series of returns of Notes in the hands of the public between 1826-1844 can be regarded as equivalent. And let «=4. Then the correction which is to be made upon the primd facie solution 6(4x°477) or :993 is 008, or about ‘O01. This conclusion may be roughly verified by the following table, in which the first column represents several hypotheses as to the relation between Q’ the length of the (apparent) Quartile measured from the Mean (supposed given) and ¢ the Modulus of the given observations. These hypotheses are thus constituted. The central hypo- thesis corresponds to the case in which a single observation is as likely to fall outside as inside Q’. According to the hypo- thesis immediately above the central one (Q'="60c), the probability of a single observation falling inside Q’ is “6. According to the hypothesis next above, the corresponding probability is ‘7. Conversely, below the centre the corre- sponding probabilities are -4,°3, &c. According to received principles * it is allowable to regard each of these hypotheses as @ priort equally probable. * See my paper on @ priori Probabilities (Phil. Mag. 1884, Sept.) ; also that on “Observations and Statistics” (Camb. Phil. Trans. 1885). It might have seemed more natural, though not really, I think, preferable, to take as the equi-probable hypotheses the equations of Q’ to equicrescent values of ¢ (or of ¢ to equicrescent values of Q'). I have done so in con- structing analogous tables for the solution of problem iii. (below, p. 381). Phil. Mag. 8. 5. Vol. 22. No. 187. Oct. 1886. 20 put 878 Mr. F. Y. Edgeworth’s Problems in Probabilities. 0000- SUIpaeoxe WOT}BATEsqO guonbasqns Jo Pies 2torwagsod OF GE IT: 6960: 400: 2000- £0000: 00000- OF Uey} 109v018 uolVeIAEp Jo Aqrtiqeqoad jeuoyipucg 8h 16-1 VG 96-6 b9- "sisoqjodAy yore uodn sn[npou 04 OP JO Oey 0000: 910: Tg. SG 910: 0000: *SISOT]} -odéy yore Jo Ayrrqeqoad 2towlagsod F 1000: 180-91 1000: 180-91 a x wuInyoo ‘Z WoINnTOD 0} sutpuods SOT -Q1100 S10Q UN Ny oz(8-) X ot(Z-) ov(L-) X oe(E-) oz(9-) X ox(F-) or(@-) X oe(G-) oe(F-) X or(9-) or(S-) X oz L-) oe(Z-) X o(8+) "1OJOVJ wom. B X sisoqjodéy Jo Aqrtqeqoad 2602099800 FF eeocee IQI. ‘6 Seg eae 9 QS. 66 eeesee 2 18. sé coves oLLP. 6c eevece 209. ¢c ceecee 2p), $6 SIG00C 5) 16-=,0 ee *% 04 9 JO O19BI OT} 04 se sosaujodATy Mr. F. Y. Edgeworth’s Problems in Probabilities. 379 The fifth column is obtained by dividing each entry in the fourth column by the sum of them all. The seventh column gives the probability that, if any hypothesis is true, the effect under consideration would follow from it. The eighth column is obtained by multiplying each entry in the seventh column by the corresponding entry in the fifth column. The sum of the entries in the eighth column gives, very roughly of course, the probability that the next observation above the Mean will exceed 4Q!. The probability that the next observation above the Mean will fall within that limit is -987, corresponding to °985 *, as found by the approximative method. So far we have been supposing the Mean given; but we must now investigate the error incurred in assigning the probability that a subsequent observation will not exceed a certain multiple of the probable-error measured not from the real, but the apparent, Mean. Let O be the real, O! the O Q’ Q Q’ apparent Median. Let Q be the real Quartile point, Q! the apparent one, namely, that which separates the given obser- vations into two groups numbering respectively 22 and in. Let O’P=e0'Q’. Let us find the probability that the (n+1)th observation f will fall outside P. Let z=O00’, €=QQ’. And let & be the Modulus for the error of the Median, « for that of the Quartile point. It is shown above that cae fe a ean / Sa 2n lin For any particular values of z and & the probability that a subsequent observation will exceed P is proportioned to + [1-9(->)]} 2 x @ -2{ a alts a a ee Now OP=00! + O'P=z+20'Q’. * The method of quasi-integration by means of a table appears theo- retically safer, however practically rough. t It is a little awkward here to introduce the condition “being above the mean,” as in the preceding problems, on account of our uncertainty where the real mean is. where, as before, 380 Mr. F. Y. Edgeworth’s Problems in Probabilities. And ¢ (the real Modulus) =2-096... x OQ (the real probable-error) | = 2-096 (O'Q!-+2—2). The probability of the (n+1)th observation exceeding P is therefore proportional to l+— 7 | This expression is to be multiplied by 1 ay 54 “Bx Was “a tae, and integrated through the whole range of zand ¢ The ~ first term of the expanded expression is £[1—0(-477)], which is the uncorrected value of the sought probability. The second term vanishes. For the third term, the correction, I find (putting, as before, 6 for :477 a) Lis /aat V0 e~F? (B? + 248) x Wf a ee vee re Ogg The second portion of this addendum is the correction due to the error of putting the apparent Mean for the real Mean. The first part of the expression corresponds to the correction which we obtained when we supposed the real Mean known. The present result differs from the former one in that its sign is negative, and its absolute quantity less by half; as it ought to do in view of the different enunciation of the problem. It should be added that in so far as 2 and ¢* tend to vary in the same direction, the correction is not accurate but of the nature of a superior limit. I have applied this correction to the question. discussed under Problem I., namely, what was the probability that the limit fixed in 1844 for the uncovered Notes would not be passed in the proximate future by the Notes in the hands of the public. The value found by Problem I., viz. -98, is reduced by Problem II. to :95. III. A third problem is suggested by the procedure of the * Above, p. 379. Mr. F. Y. Edgeworth’s Problems in Probabilities. 381 legislators in 1844, when they fixed the limit of uncovered Bank of England notes at £14,000,000; upon the ground that the Notes in the hands of the public (less* by the Bank Post Bills together with the lost notes) had never fallen below that figure. What is the probability that if M is the maxi- mum or minimum (measured from the Mean) of 5 observa- tions, a subsequent observation, the (5+1)th in excess (or defect), will not exceed M, or more generally gM. If, like the legislators, we are to ignore the grouping of the n obser- vations, and to utilize only the datum that M is the maximum of n observations on one side of the Mean; then a priori one ratio of the given maximum to the unknown modulus may be regarded as about as likely as another. Consider the parti- cular ratio . The probability that a single observation in excess of the Mean will fall within M is 0(7); where, as before, O(a) is identical with Se ‘ e-"dt. The probability that n T Jo observations being in excess of the Mean should not exceed M is [@(r) |". Hence the 4 posterior probability that the par- ticular hypothesis considered is the true one is cay] =f“ Laeyr The probability that if the particular hypothesis is true the next observation above the mean will not exceed gM is @(qM). Hence the & posteriort probability that the next observation will not exceed gM is i, [O(r) |" A(gr) dr a [O(r) ]*. I have attempted roughly to evaluatef this expression for cer- tain interesting values of g and n. First, let 5740 and g=1. We have then a problem much the same as that which the legislators in 1844 set to themselves. I find that the odds against the regulation limit £14,000,000t being passed in the proximate future were below a hundred to one. A safer limit would have been constructed by putting g=14. The odds against this limit (£12,500,000$) being passed are some hundreds to one. Again, suppose “=20, To obtain the 2 * Above, p. 375. + By Tables analogous to that given above. t¢ £15,000,000 with Bills and lost Notes; see above. § £14,000,000 with Bills and lost Notes, 382 Mr. F. Y. Edgeworth’s Problems in Probabilities. degree of probability just mentioned it would be necessary to make the limit 2M. These and similar results may be used to find in a rough and ready fashion a superior (or inferior) limit to the fluctuations of any phenomenon; the Mean and the Maximum of a certain number of observations being given. ‘Thus, consider the Registrar-General’s returns of the proportions of male to female children for the different counties and for several consecutive years, e.g. Report 46, table 16. Omitting Lancashire, West Riding, Huntingdon- shire, and Rutlandshire, we have 40 observations of pretty much the same weight in each of the first ten columns in the table referred to. Take any one of these, e.g. the fourth, and observe the difference between the maximum return in that column and either its own or the general Mean. The Mean + twice that difference constitutes a fairly safe superior limit. In the case selected, the maximum return is 1089. And the Mean (both of this column and all the columns) is 1038. Hence for the superior limit we have 1140, which is not I think reached by any of the four hundred returns in the table. I have tested the proposed canons in the Art of Conjecturing by similarly applying them to various phenomena more or less obeying the law of error; such as* a series of figures, each of which is formed by adding together a certain number of digits taken at random from mathematical tables; recordsf of temperature; statistics{t of the dactyls in Virgilian lines, c. IV. My fourth problem is: Given the Modulus of the law of error for a certain species of demand made upon a bank, to find what the Modulus becomes when the liabilities are increased in any proportion ceteris paribus. The general answer is that the Modulus increases, not in the ratio of the volume of business, but in the square root of that ratio. The general rule and the exceptions may be illustrated by ex- amples taken from Vital Statistics. Consider§ a Table of Proportions of Male to Female children born in Registration Counties. Form the sums ofthe columns. Compare the fluc- tuation of the fringe-row thus formed with the fluctuation of the figures in the ordinary rows. It will be found that the Modulus for the former is not n times, but about /n times, * See my paper on “ Methods of Statistics,” Journal of the Statistical Society, Jubilee volume. t Such as Mr, Glaisher’s in ‘ Philosophical Transactions,’ and in the ‘ British Meteorological Journal’; where the means and the maxima are given ready to hand. { “On Rates,” Journal of the Statistical Society, Dec. 1885. § See ‘ Methods of Statistics,’ p. 198. Mr. F. Y. Edgeworth’s Problems in Probabilities. 383 that of the latter. But the general rule is not so well ful- filled when we similarly consider the Death-rates* in Regis- tration Counties. The death-rates in different counties for the same year are not, as the theory requires, independent, but tend to increase or decrease all together. Accordingly the Modulus for the sums does not shrink in the regulation fashion. It will be less indeed than n times, but greater than /n times, the Modulus for the ordinary rows. A converse f exception may be illustrated by the following supposition. Suppose that, when there is a high death-rate from a particular disease in one year, there is apt to be a low death-rate in the following year or years. Form a table of the rates of deaths from such a disease for a set of counties (like the agricultural) not differing much in respect of healthi- ness. Consider the fluctuation (not now of the horizontal but) of the vertical fringe of sums in such a table. In virtue of the compensatory action between the years the Modulus for the vertical fringe of sums (or means) will shrink more than the general rule requires. Both these kinds of exception occur in Banking, as will be shown elsewhere{. Here it need only be explained that in order to verify the rule, or to establish an exception, there is required a great number of observations. Suppose that C,? is a certain Modulus-squared, and that C,” is another, double C,’. In order to prove that relation, it may fairly be required that C,? should be determined so accurately that its error should not exceed 3, or at most 3, C,”. In order that the error of C,? should not exceed 3 C,’, the Modulus for C,? should not exceed 3, 0,?.. But the Modulus for C,” as deter- mined by 7x observations is = §. Hence “x must be 2 greater 16 eS 10. And n must be greater than 100. aE The following figures illustrate this theory. Hvery term in the first series stands for one and the same quantity, a certain Modulus-squared. And similarly every term in the second series stands for another Modulus-squared. (1) 162, 18, 200, 162, 2, 8, 288, 32, 8, 32, 50, 162, 32, Bee 2) £285 1250, 1250, 2......008 (2) 32, 200, 8, 128, 338, 8, 382, 578, 648, 32, 32, 200, 8, ee DIG, Op D2vvaaddece * See my paper “On Methods of ascertaining Rates,” Journal of the Statistical Society, Dec. 1885. ; Tt See the discussion of Virgilian statistics in the paper just referred 0 + Journal of the Statistical Society, 1886. § By Laplace’s formula for the error of the mean-square-of-error. 384 Intelligence and Miscellaneous Articles. - It would be impossible to determine by inspection of these measurements what is the real relation of the objects measured. ‘The Mean of the first series being 210, while that of the second series is 162, those who look only to averages without testing their significance will conclude offhand that the first object is greater than the second. But in fact the real value under- lying the first series is only half the corresponding value for the second series ; the former being the twice-mean-square- of-error for aggregates of ten digits taken at random, the latter for aggregates of twenty digits. The real relation would come out if we went on long enough. I have gone on long enough, in the case of the first series, to get within a thirty-fifth part. of its theoretical value, namely 165. I obtain 160 as the mean of two hundred and eighty terms of the first series—-a number of observations which corresponds to a pro- bable errror of about 5, the error which I have incurred. If I went on long enough with the second series I should, doubtless, get equally near its real value, which is 330. But _ to the unitiated such statistics are hopelessly misleading. Blind palpation is sure to err. XLVII. Intelligence and Miscellaneous Articles. ON THE MEASUREMENT OF VERY HIGH PRESSURES, AND THE COMPRESSIBILITY OF LIQUIDS. BY M. F. AMAGAT, io measuring very high pressures I have adopted the principle of the differential manometer; the conditions to be realized for obtaining exact measurements is that the pistons be completely mobile and at the same time perfectly tight. M. Marcel Deprez had the idea of dispensing with the leather of the small piston, and making the escape extremely small by a con- venient adjustment. This device is insufficient for very high pressures, especially in the conditions of my experiments; this is also the case with the use of goldbeater’s skin, adopted by M. Cailletet. On the other hand, numerous experiments have shown me that the membrane, on which the large piston rests, introduces several sources of error. I have altogether dispensed with the leather and the membrane, and have solved the difficulty by using a viscous body suitably chosen. The large piston, which only receives the reduced pressure, rests on a cushion of castor-oil, which transmits the pres- sure to the mercury; the small piston, which receives all the pressure at the top, becomes quite tight if, after being soaked in oil, and put in its place, it is wetted on its base with a sufficiently viscous liquid, such as molasses, which answers perfectly. In these con- ditions, the pistons being even somewhat free, there is no real leak,. but only an extremely slow oozing, which does not affect the mea- surements, and this up to pressures higher than 3000 atmospheres. Intelligence and Miscellaneous Articles. 385 Nevertheless in these conditions the mercurial column rises with starts which cause considerable errors. They are completely de- stroyed by giving a rotatory motion to the pistons, which is easily obtained. I have hitherto investigated only the compressibility of water and that of ordinary ether. ‘The piece in which the piezometer is compressed is a steel cylinder 1:20 metre in height; it is hooped for its entire length except part of the breech; its internal dia- meter is 3 centim., and its sides are 8 centim. in thickness. It was cast and hooped at the cannon foundry of Firminy. This cylinder is fixed vertically in a large copper reservoir, so that it can be worked with in melting ice, or in a current of water at a constant temperature. The reading of the volumes of the compressed liquid was made by the following method, which was pointed out to me by Prof. Tait. A series of platinum wires are soldered in the stem of the piezometer, as in fire-alarm thermometers; these wires are con- nected by a metal spiral with a resistance of two ohms between each wire, and the prolongation of which passes through the short cylinder by means of a special insulated joint. The current of a battery reaches the mercury in which the stem is immersed through the steel cylinder. It will thus be seen how galvanometric indica- tions may give the precise moment at which the mercury rising in the stem, owing to the compression of the liquid, successively reaches each platinum wire. The liquid of the piezometer and the liquid which transmits the pressure in which it is immersed become considerably heated by the pressure; this makes the experiments very long; a considerable time is required to counterbalance the mass, which is a bad con- ductor; the readings must be repeated until the indications of the manometer are constant in contact. The series of observations made with decreasing pressures produce the same effect in the oppo- site direction; the mean of the results is taken, and their general asreement shows that the whole method leaves really little to be desired. We see by this what gross errors may have been committed with the other devices hitherto used for measuring volumes in analogous conditions. Ether and water have been studied at zero and at two adjacent temperatures, the one of 20° and the other of 40°. For both liquids the direction of the variation of the coefficient of contractibility with the temperature is the same under very strong pressures as under very weak. Water continues to form an exception; its compressibility diminishes as the temperature in- creases, in the above limits; the variation seems, however, to diminish at the highest pressures. The coefficient of the variation with the pressure, as was easily to be foreseen, gradually diminishes as the pressure increases ; and this is the case throughout the entire scale of pressures, contrary Phil. Mag. 8. 5. Vol. 22. No. 137. Oct. 1886. 2D 386 —— Intelligence and Miscellaneous Articles. to what is thought by many physicists. This I arrived at in my memoir of 1877, for pressures below 40 atmospheres (Ann. de Chim. et de Phys.), and which long before had been found by Colladon and Sturm in their classical work Sur la Compressibilité des Liquides. I shall only give here the results of two series, one on water and the other on ether. Water at 17°°6. Ether at 17°°4. ——- ws —— ae A—_—__—__—_ —-., Pressures, in Coefficients of Pressures, in Coeflicients of atmospheres. compressibility. atmospheres. compressibility. atm. atm. atm, atm. Between land 262 00000429 Between land 154 0:000156 A 20d ae) SUS 0:0000379 up 154 ,, 487 0:000107 Fin CUD ny PLO 0:0000332 t 487 ,, 870 0:000083 ww. leat, 1784 - -0-00003802" ° 7 = 870 ,, 1243 0:000063 » 1784 ,, 2202 00000276 ,, 1243 ,, 1623 ~~ GaGnnny » 2202 ,, 2590 0:0000257 ,, 1623 ,, 2002 0-000045 » 2590 , 2981 0:0000238 At 3000 atmospheres the volume of water is reduced by one tenth, and its coefficient of compressibility by one half. The study of ether will be resumed and carried as far as 3000 atmospheres.—Comptes Rendus, August 25, 1886. ON THE SPECIFIC INDUCTION CONSTANTS OF MAGNETS IN MAG- NETIC FIELDS OF DIFFERENT STRENGTHS, BY HILMAR SACK. Lamont concluded from his experiments on the changes effected by the earth’s magnetism on the magnetism of steel bars that the change is greater when the force acts in opposition to the pre- vious magnetization than when it strengthens it. Kohlrausch has recently shown (Wiedemann’s Annalen, vol. xxi. p. 415, 1884) that such a difference does not exist, at any rate not for a field of the strength of the earth’s magnetism. The object of the present investigation is to ascertain within what limits this equality of the specific induction constants holds. The investigation in question was made at the invitation and under the direction of Prof. Kohlrausch in the Physical Laboratory at Wiirzbureg. After describing in detail the method of the investigation and the data obtained, the author summarizes the results in the following terms :— As respects hardened and powerfully magnetized steel bars,— 1. The coefficients of strengthening and of enfeeblement as found by F. Kohlrausch were appreciably the same for fields of the same strength as the horizontal component of the earth’s magne- tism, that is 0-2, Intelligence and Miscellaneous Articles. 387 2. This principle holds also if the magnetic field does not exceed 1:2. 3. If the magnetic field is stronger, then the first coefficient of enfeeblement which is due to the current on opening, first of all exceeds the following ones obtained in the same way by but little, but afterwards by 5 or 6 per cent., if the magnetic field is 3 or 4. Hence in order to obtain a stable value for the strength- ening and the enfeeblement with these high values, a bar magnet should first of all be subjected to the same strengthenings and enfeeblements. 4, With powerful magnetic fields the first closing of the current, if this strengthens the magnetic moment of the bar, has a greater constant than the following ones; but it does not attain the magnitude of the constant of enfeeblement produced by the mag- netic field on closing. 5. Magnetizing forces, even when they are ten to twelve times as much as the earth’s magnetism, produced no considerable durable changes of the permanent state. Forces which were twenty times as strong as the horizontal intensity produced un- doubted changes in the moment of the bar—Wiedemann’s Annalen, No. 9, 1886. ON THE ELECTRICAL CONDUCTIVITY OF GASES AND VAPOURS. BY M. JEAN LUVINI. It follows from the experiments of Becquerel, Grove, Matteucci, Marangoni, Agostini, and others, that gases and vapours are very bad conductors of electricity. Grove demonstrated this proposition for air at very high pressures; Becquerel and Matteucci for very low pressures (1 to 3 millim.). MM. Mascart and Joubert place air and vapours and generally all gases in the class of bad conductors ; and Sir W. Thomson has stated that aqueous vapour is an excellent insulator. Notwithstanding this we still read in treatises on Physics, and it is repeated in lectures, that moist air and vapours conduct electricity ; and this very serious error is the base of several theories. I have made several sets of experiments on this subject, the results of which, combined with those of other experimenters, have led me to conclude that gases and vapours, whatever be the pressure and temperatures, are perfect insulators, and cannot be electrified either by friction with each other or with solids or liquids. I arrange the experiment so that the fluids in which the electri- fiéd bodies are introduced cannot be deposited as liquid on the whole length of the insulating supports. In a large room a long thread consisting of seven cocoon fibres, without torsion and without joints, is stretched. In its centre is suspended a hollow 388 Intelligence and Miscellaneous Articles. brass sphere 5 inches in diameter. A secorid thread, similar and parallel to the first, supports a pith ball pendulum, which, when not electrified, is in contact with the brass sphere. IT usually electrify the sphere with the conducting die of an electrophorus. The extent of the room and the purity of the air enable me to fill the space about the electrified bodies with a con- siderable quantity of the gas or vapour which I am investigating, without these fluids settling for a time on the long thread. In this way, apart from the small conductivity of the wires, and the losses due to dust in the air, any diminution of the electrical tension observed must have been due to the conductivity of the fluids experimented on. I always electrified the sphere almost to the same tension; I then observed the time in which the deflection fell a certain number of degrees, first in air and then by surrounding the sphere and the pendulum by a dense atmosphere of the gas to be studied. I worked thus with air saturated with aqueous vapour at various temperatures from 16° to 100°; hydrogen and carbonic acid not dried, but as they emerge from the bath in which they are pro- duced ; air heated by charcoal or by the flame of a candle, the Sacks. of an extinguished candle, the fumes of burnt sugar, incense, &c. None of these substances showed any trace of conductivity. In one set of experiments, instead of cocoon-threads I used ordinary sewing-thread stretched horizontally, and with a double pendulum in the centre provided with pith balls. The results did not change, but when I worked with aqueous vapour at high tempe- ratures, the divergence diminished rapidly by a certain quantity and then became again almost constant. This effect is due to the vapour which is deposited on the larger and less insulating wire, and to the fact that the small quantity of electricity of the balls becomes divided between the thread and the balls themselves.—Comptes Rendus, Sept. 13. AN ELECTRICAL EXPERIMENT. BY M. BUSCH. The author.calls attention to the curious figures formed when electricity diffuses on a plate previously dusted with lycopodium powder. The two balls of a Henley’s electrometer are brought in contact with ’ two sides of a dusted glass plate, and a large Leyden jar is dis- charged through the bails. The figure resulting from the discharge has the appearance of a lightning discharge as seen in the photo- graphs of lightning.—Berblatter der Physik, vol. x. p. 302. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIES.] NOVEMBER 1886. XLVI. On the Electrolysis of Silver and of Copper, and the Application of Electrolysis to the Standardizing of Electric Current- and Potential-meters. By Tomas Gray, B.Sc., TOP Dia [Plate VII.] tee following paper contains an account of a large num- ber of experiments on Hlectrolysis, and on its application to the standardizing of electrical measuring instruments, which have been made during the past year in the Physical Labora- tory of Glasgow University: it forms a summary of the reports which have been from time to time made to Sir William Thomson on the subject. The primary object of most of the earlier experiments was to obtain the value, in absolute measure, of the indications of ampere- and volt-meters ; but these ex- periments were always taken advantage of for the purpose of investigating the reliability of the method under various cir- cumstances as to treatment, size of plates, density of solution, and soon. Many of the later experiments were made specially for the purpose of investigating points of interest suggested by the earlier experiments. It should be borne in mind, with regard to the conclusions put forward in this paper, that they have not been arrived at simply as the result of the ex- periments made personally by the writer, but that he has had the advantage of seeing the methods applied by several inde- pendent experimenters in this laboratory and of comparing the results they obtained. The electrolysis of copper has for several years been occa- * Communicated by Sir William Thomson, F.R.S. Phil. Mag. 8. 5. Vol. 22. No. 188. Nov. 1886. 24H 390 Mr. T. Gray on the Electrolysis sionally used by Sir William Thomson as a check on the graduation of galvanometers; but the question of its reliability had not been systematically attacked before the series of expe- riments here described were taken up. From a number of experiments made in the above laboratory about four years ago, I inferred that copper was capable of giving fairly uniform results; and I obtained as the electrochemical equivalent of copper ‘003307, or as the amount of copper deposited by a coulomb of electricity ‘0003307. This value has since been found to be too high; and recent experience indicates that there are at least two ways of explaining the discrepance— namely, either that the measurement of the current was in error, or that the solutions were wrongly treated. The first of these needs no remark, it is impossible to tell now whether the current was measured with the needful accuracy or not; but the second will be referred to at some length in this paper. The solutions were in that case very carefully saturated with copper by shaking them up with, and filtering them through, copper oxide ; and I bring the matter forward now because I believe that treatment was a serious mistake. It is easy to make a solution of copper sulphate which will give as high, and even a much higher, value for the electrochemical equiva- lent than that above stated; but once the cause of this is known, it is equally easy to avoid the error. In order to obtain good results it is necessary that the solution be distinctly acid. The ordinary commercial copper sulphate, or the sulphate ordinarily sold as pure, usually contains enough of acid; but the same solution cannot be repeatedly used with safety. Following the results of the investigations of Kohlrausch, and of Lord Rayleigh and Mrs. Sidgwick, the substances used in the earlier of the present series of experiments were pure silver and pure silver nitrate. Sheet silver was used both for the gain and for the loss plates ; it was supplied by Messrs. Johnson and Mathey as pure silver. The form of the cell used for the silver is illustrated in fig. 1 (Plate VII.); it con- sists of a glass vessel partially filled with a solution of silver nitrate, in which three plates of silver, arranged with their planes parallel, are suspended from spring clips of the form illustrated in figs. 2 and 38. This form of cell offers, in com- parison with the platinum-bowl method recommended by Poggendorff and adopted by Lord Rayleigh, some advantages which, in my estimation, outweigh its disadvantages. The total weight of the plate is very small, and hence a light and very delicate balance can be used. The thorough polishing and cleaning of the plates is easier than when a bowl is used. The cleaning of the platinum bowls by dissolving off the of Silver and of Copper. 391 deposit in nitric acid and the satisfactory washing of a bowl generally is troublesome ; indeed the time spent in this ope- ration is often more valuable than if the gain plate, deposit, and all were thrown away ; while, since the deposit is pure silver, the depreciation in value due to cutting up the sheet is small. When the deposit is good it may be rolled, beaten, or burnished down sufficiently to allow the plate to be again used, and thus the operation simply means a transfer of silver from the anode to the cathode. Another and perhaps more important reason for preferring the vertical plates is the fact that, if the plates be properly proportioned as to size and properly prepared, the loss plate can be used with perfect ease as a check on the result of the gain plate. This is a point of some importance in the case of silver, where the deposit is apt to be of such a nature that there is considerable risk of loss in the operation of washing. There is of course the objection to the use of vertical plates, that the density of the solution is apt to decrease at the cathode and increase at the anode. Such an action does take place, and the result is a slightly thicker deposit on the lower part of the plate, thus changing to a small extent the effective area; but no difficulty has been experienced from this cause, even when the current is allowed to flow for three or four hours, when the current and the solution have the densities stated below. In the later experiments copper has been almost exclusively used. It is found to be very much more easily managed than silver, and gives, over a wide range of size of plate and den- sity of solution, deposits which are perfectly adherent and homogeneous, thus rendering it more generally convenient where currents of a considerable variety of strength are used. Besides, for large currents such as from 10 to 100 amperes and upwards, the use of silver is almost excluded on account of the expense of the necessary materials. When the highest accuracy is required, however, and when it is used in experi- enced hands, silver is decidedly superior to copper. It pre- sents hardly any of the uncertainties which, as will be pointed out below, it is absolutely necessary to guard against in the case of copper ; but the fact must not be lost sight of that an expertness in manipulation and a degree of care are required which cannot always be obtained. The objections which have been brought against the use of copper are its readiness to oxidize in the air, especially if moisture be present, and the fact that it loses weight in the liquid*. Both of these are of course legitimate objections, * See a paper by G. Gore, LL.D., F.R.S., ‘Nature,’ March 16, 1882 ; and Lord Rayleigh and Mrs. meee Trans. Roy. Soc. 1884, pp. 411-460. 2 yy 392 Mr. T. Gray on the Electrolysis and more might easily be added ; hence the point to be de- cided is whether the oxidation and loss in the solution can be avoided or their effects eliminated. As the result of a good deal of experience, it appears to me that the first objection is of very little weight indeed, as the plate need not be sensibly oxidized in the process of washing and drying if even ordinary care is taken in the operation. The loss by corrosion in the liquid leads to results which depend to some extent on the size of the plate and the temperature of the solution ; and a great many experiments have been made on the effect this may have on the apparent value of the electrochemical equivalent in dif- ferent cases. Examples of such experiments are given below, and they are there discussed more in detail ; but it may be stated here that the effect can be fairly well allowed for if the size of the plate and the strength of the current, as well as the state of the solution, are known. ‘The error due to this cause need never exceed a tenth per cent., and will generally be considerably less than that. It appears from experiments on the loss of copper in solu- tions of sulphate of copper, that a strong solution of normal pure sulphate is at ordinary atmospheric temperature even more active in dissolving copper than the same solution when as much as 5 per cent. of acid has been added ; but that in the use of the normal solution there is a danger that the presence of the copper plates in the cell may render the solution satu- rated with copper, after which a rapid oxidizing action takes place which invariably causes the gain of weight in an electro- lysis experiment to be too great. The deposit in such a case is of a darker colour than when the solution is kept sufficiently acid ; and indeed a bad plate can generally be detected by simple inspection, but it is well always to add such a quantity of acid (a very little is sufficient) as will prevent any un- certainty on the subject. Electrolytic Cells —Some of the arrangements which have been found convenient for the electrolytic cells are illustrated in figs. 1 10 6. They consist, in the smaller sizes, of round glasses into which three plates, arranged with their planes parallel and about one centimetre apart, are held by means of clips of stiff platinoid or brass wire of the form shown in figs. 2 and 3. ‘The outer clips are connected together by a cross piece, a (fig. 1), at the top, but are insulated from the middle clip by a block of vulcanite, fixed in the cross piece, 5, into which they are all tightly fitted. These clips are very con- venient, and are easily made by taking a piece of stiff wire, bending it nearly close at its middle point, then winding each half two or three times round a rod of metal of suitable size of Silver and of Copper. 393 so as to form the springs, and after bending the two ends together soldering them to a stiff back-piece, as shown ; care must be taken that the spring front presses firmly on the back before the two are soldered together. Several sets of clips are fitted into one cross piece 0, so that one frame may serve for one or more cells, the double clips of any one set being connected permanently to the single clip of the next set. When the plates are being placed in the clips, the cross piece which holds them is lifted off the frame and the plates placed in position, the jaws of the clip being opened before the plate is introduced. Care is also taken to open the jaws of the clip before removing the plate, so as to obviate any risk of loss of metal by friction between the holding-points and the plate. This arrangement has been found very convenient for small cells, as it allows the plates to be quickly and almost simulta- neously placed in, and removed from, the liquid, and avoids all risk of the plates touching each other or the sides of the vessel; but for cells to be used with currents of over 10 am- peres the plates become rather too large, and the arrangement shown in figs. 4 to 6 is then found more convenient. In this form a frame of insulating material is fitted to the top of the cell, and two sets of spring-contact clips, one set on each of two opposite sides of the cell, are fixed to it. These clips may be of the form just described, or they may be simply flat strips of springy metal soldered to a stiff base piece, as in fig. 5, in such a way that they press firmly against each other. In the use of this form of cell the anode plates are placed in contact with one set of clips and the cathode plates with the other set, the number of anodes being always one greater than the number of cathodes. The form of the plates and the mode of placing them in the cells are illustrated in figs, 4 and 6, from which it will be seen that the surface of the plate above the liquid is made as small as possible by cutting away the plate so as to leave two narrow strips connected to a cross piece, c,d.- The end c of the cross piece is held in the con- tact clip, while the other end, d, is kept in position by a shallow notch in the insulating rim. The insulating frame and attached clips are simply laid on the top of the vessel, and can therefore be lifted off and cleaned so as to insure perfect insulation. Sizes of Plates and Densities of Solutions.—The size of the plate can be varied within moderate limits in the case of silver, and within very wide limits in the case of copper, without greatly interfering with the quality of the deposit. In the case of silver and silver nitrate the effect of making the plate either too small or too large is, as has been pointed 394 Mr. T, Gray on the Electrolysis out by Lord Rayleigh, to make the deposit less adherent and more roughly crystalline. There is then a tendency for the deposit to grow out from the cathode towards the anode in long branch-like crystals. This tendency is increased by any sharp corner on the cathode-plates, and hence care must be taken to round the corners and smooth the edges thoroughly. The deposit also deteriorates as time goes on in consequence of the crystals presenting sharp protuberances, which tend to grow and become more and more prominent. It follows that a somewhat smaller plate or greater density of current can be used if the time be short; but I have generally found that when the density of the current is such that the deposit would deteriorate greatly during the first two or three hours, the adhesion to the plate is likely to be uncertain. The best results seem to be obtained with a solution contain- ing about five per cent. by weight of nitrate of silver, and cathode-plates which present an area of not less than 200 nor more than 600 square centimetres per ampere of current. If this strength of solution and size of plate be used and the plates be properly cleaned, the deposit is very compact and finely crystalline, and adheres very firmly to the surface of the plate. When the strength of the solution is increased the size of the plate can be slightly diminished, but not by any means in the proportion of the increased quantity of silver in the solution; the deposit is then more roughly crystalline, will not bear lengthened application of the current, and adheres much less firmly to the plate. So far as these experiments have gone, it seems a mistake to use a solution containing more than, or even as much as, ten per cent. of silver nitrate. Solutions containing from three to thirty per cent. have been repeatedly tried; but the best results have always been obtained with solutions containing from four to ten per cent. There seems no reason for using strong solutions except a slight difference in the original cost of the plates; but as these may be of very thin metal, the cost is a small matter compared with the risk of either total or partial failure of the experiment. When the loss or anode-plates are to be used simply to supply silver to the solution, they need not be larger than the gain or cathode-plates; and there is some advantage in making them smaller, so as to increase the distance of the edges of the cathodes from those of the anodes. When the plates are small the surface becomes very soft and spongy, and if the density of the current exceed a certain moderate amount they will blacken, and the resistance of the cell is apt to become variable, due to the liberation of gases. It is better to make the anodes a good deal larger than the cathodes, because in of Silver and of Copper. 395 that case the surface of the plate remains bright and moder- ately hard, so that the plate can be washed and weighed if that be thought necessary. The density of the current at the anode should not exceed one ampere to 400 square centimetres of surface, and should be even less if the plates are to be weighed. The action of the current in causing the solution of the anode is somewhat curious, especially if the plates be made of rolled sheet. If the plates be simply washed and placed in the cell with the bright polished surface still on them, it is seldom that the outside skin will be dissolved. ‘The silver is taken from the interior of the plate, leaving a very thin skin lying loose on the surface, which is ready to fall off either when the plates are lifted out of the cell or when they are placed in water for the purpose of washing. Whether this is due to the mecha- nical state of the silver on the surface or to a peculiar difficulty in so cleaning the surface of the plate that it is properly wetted by the liquid, is not yet quite clear. A plate of silver when repeatedly used for an anode becomes soft and almost devoid of elasticity, due to the solvent action taking place deep into the interior of the plate ; and it is well to heat the plates to about a red heat in a spirit flame between different experiments, so as to keep it hard and prevent loss of silver. In the case of copper and copper sulphate, the size of the plate may be almost anything from twenty square centimetres: to the ampere upwards; but for experiments which are to be continued for two or three hours, the cathode-plate should present about fifty square centimetres of surface per ampere of current. With plates of from this size upwards the cur- rent may be allowed to flow for almost any length of time without introducing the least difficulty as to loss of copper in the subsequent washing. At the higher limit of current- density here mentioned (one fiftieth of an ampere per square centimetre), there is a slight tendency for the deposit to thicken at the edges of the plate and become rough, but as the current- density diminishes this becomes less and less marked. It ma be taken as certain that the aggregation of the deposit at the edges of the plate is due to the current-density becoming too great at the sharp edges, and thus causing the formation of sharp crystals, which in their turn intensify the action. If the plates are made large enough, this critical current-density is never reached, and the deposit is as smooth on the edges as. elsewhere. It is mainly in the ease with which a perfectly uniform and solid deposit can be obtained that the great advantage of copper over silver for ordinary use in electro- lytic measurements lies. The anodes in the copper cell behave in a similar manner to: 896 Mr. T. Gray on the Electrolysis that above described for the silver anodes, with the exception that they show no tendency to become soft and inelastic. If the current-density at the anode exceed the fortieth of an ampere per square centimetre, the current is apt to be variable and may almost stop completely, even when an electromotive force of 25 volts is used to produce it*. This is due to excessive resistance at the surface of the anode. In some cases the current will, after a few minutes, resume nearly its former strength, and gases are then freely given off at the anode. When the current-densityis small, from the two-hundredth of an ampere per square centimetre downwards, the plate can generally be washed, and used to show the loss of copper during the passage of the current. This loss does not in any case agree with the gain on the cathode, in consequence of the solution taking up copper during the experiment ; some examples of the results which have been obtained, and some further remarks on this subject are given below. In almost all cases there remains on the surface of the anode more or less of a very fine brown powder, which increases in amount as the current-density is increased, and which also seems to depend somewhat on the nature of the plate. If, for example, a plate of electrotype copper be used, the surface will be found to have become a dark red when the plate is removed from the cell, but no loose copper will be found. ‘That is certainly the case at least so long as the current-density does not exceed the one hundred and fiftieth of an ampere per square centimetre. Currents of greater density were not tried with electrotype copper. This result seems to point either to the mechanical state of the copper in the rolled sheet, or the presence of a somewhat in- soluble oxide, as the cause of the loose powder being left on the plate. Local action, due to unequal quality of the plate itself, may have something to do with it; itis certainly greatly increased by frequent reversals of the current in the cell. The effect of varying the density of the solution of copper sulphate is not great until the density falls to about 1:05, when the deposit begins to be less adherent. There is, however, greater danger of error due to oxidation of the deposit in weak than in strong solutions, when no acid is added, owing to the fact that the solution becomes more quickly saturated with copper. Any density between 1 and 1:18 will be found to answer perfectly, but it is not advisable to use a saturated solution because there is then a risk of crystals forming on the plates. Preparation of Plates.—In all experiments on electrolysis the proper treatment of the plates previous to their immersion in the liquid is a most important consideration. The treat- * See note on page 413. of Silver and of Copper. 397 ment which I have found to give the best results with silver is extremely simple. Suppose a new sheet of silver is to be used either for a cathode or anode—first make the corners and the edges round and smooth, and then polish the surface thoroughly with a soft clean pad, water, and fine silver sand, so as to remove the skin which has been in contact with the rolls in the manufacture ; rinse the plate in clean water, or by holding it in a rapid stream of clean water from a water- tap, so as to remove the sand, and then wash it first with clean soap and water and afterwards with clean water; next place it for a few minutes in a boiling solution of cyanide of potas- sium, and after that wash it thoroughly in a stream of clean water, taking care not to touch the part of the surface which is to receive the deposit with anything. If the surface of the plate be touched with the fingers, even when they appear to be clean, the markings of the skin will be reproduced by the deposit leaving the parts of the plate bare which were in contact with it. The plate may be dried either in a current of dry air in front of a bright fire, or in any other convenient manner which will insure that the surface of the plate will remain clean, and then accurately weighed. One point about weighing should be mentioned, because it is apt to be over- looked, that is, that the plate must be allowed to assume the temperature of the air before it is weighed, as even a very slight difference of temperature between the plate and the air inside the case of the balance is sufficient to produce quite a sensible error in the weight. For the preparation of copper-plates a very good plan, especially with large plates and powerful currents, where it is necessary to arrange the resistance of the circuit carefully when the electrolytic cell is included, is, after the edges and corners of the plates have been well smoothed and rounded, to proceed as follows :—Polish the plate thoroughly with silver sand, wash the sand off by holding the plate in a rapid stream of water and rubbing it with a brush or a piece of clean cloth ; place the plate in the cell, as a trial plate, and deposit a thin coating of copper over it, at the same time adjusting the current to the proper strength ; then remove the plate, wash it in clean water, and dry it, first in a clean blotting-pad, and then before the fire, taking care not to heat the plate sensibly. If the plate has not been properly cleaned, the deposit of copper will show any defect; and if it be found to be perfect, the plate may be weighed and the electrolysis continued. For small currents it is more convenient to use a battery of moderately high potential, say twenty or thirty volts, and to keep a resistance in circuit with the cells; and in 398 Mr. T. Gray on the Electrolysis this case it is not necessary, after the first trial at least, to adjust the current with the plates in position, as the resistance of the cells can be nearly enough allowed for. In these cases the plates were simply polished either with silver sand and water or with clean sand-paper, then washed in clean water - to remove the sand, placed in water slightly acidulated with sulphuric acid for a short time so as to remove any trace of oxide, then rinsed in clean water, and dried first in clean blotting-paper and afterwards in front of the fire or over a spirit-flame. If the plate be thickly oxidized on the surface but otherwise clean, as is often the case with new sheet copper, the quickest mode of cleaning is to wet the plates and then place them for a few seconds in strong nitric acid, rinsing them immediately afterwards in clean water, and putting them into water containing a few drops of sulphuric acid to prevent oxidation. The plates may then be lifted one by one out of the acidulated water, rinsed in clean water, and dried in the manner above described. The method of drying here recommended is important, as the blotting-pad removes almost at once nearly the whole of the water, and hence the drying can be completed in a few seconds, either in front of a fire or over a spirit-flame, with- out the slightest oxidation taking place. A plate may be washed and dried in this manner over and over again without producing as much as the tenth of a milligramme of difference in its weight. Washing the Deposit.—The operation of washing the de- posit at the conclusion of the electrolysis is one on the im- portance of care in which Lord Rayleigh has laid great stress, The method adopted for silver in these experiments was to lift the bar b (fig. 1) with the plates attached, and either to dip them several times in clean distilled water contained in a clean glass vessel before removing them from the clips, or to remove them as quickly as possible from the clips, and dip them several times one by one into the water; in either case observing carefully whether any small crystals were removed in the operation. If there is danger of loss of silver from the anodes, it is better to remove the plates from the clips before they are dipped in the water. After this preliminary rinsing to wash off the greater part of the silver nitrate solution, the plates were laid in the bottom of a shallow glass tray con- taining distilled water, and the water made to flow backwards and forwards over them for a minute or two by raising and lowering one edge of the tray. The plates were then re- moved and placed in another similar tray containing clean, but not necessarily distilled, water, washed in a similar of Silver and of Copper. 399 manner, and allowed to soak for about fifteen minutes before being dried. It is important to bear in mind that the water used for the first rinsing and washing must be clean water which has been recently distilled, and which has not had an opportunity of absorbing any impurity, such, for example, as a minute quantity of common salt, by coming in contact with anything which has not been recently and thoroughly cleaned. The water supplied by the Glasgow Corporation for domestic purposes, although very nearly pure, clouds imme- diately on contact with silver nitrate, and throws down a con- siderable precipitate of silver chloride, no doubt due to the fact that the water contains an appreciable percentage of chlorides in solution. ;SOULUIBIS ae : : fs nbs 19d SsoyT a 5 =— : 1230, = A : = ai = 5 zed Sout ates re : ~~ ee ead U aes ae on) eS AISS900 ee 3 E28 =| : ns Surinq Heise 7 =ane ® a TIP SSO] Sue Bolas = 28 © “UOT "<0I = 8 josey | | | | hee: ee 1yW99 Ore a 5 : : aE nbs 10d me daa i : : 3 SSOT [PIO a i = ‘ ABB OSE Chet ~ qed samun *wTyue = ink ° oO for) 9 mudd U aoa a a AISS900 pane . z ns Surmp a ee aie ' o ab AIL ae [JO 990 Smee uw T= UruL i , : a on tue e ee : ® orenb ae : — 4 | : § 19d ssOoT a esa aa S S Ss ae 2 ae a e a : zed sour eee Re = | : . OL ai a d oe me IES Ch aS) = oO OAIS8O cama | a = Issa00ns SurAn} eee | cam - 1INp S80 : = FS : a4 ur ‘cOI + ae a : sen sO soUIUIeAS UL seer : ss : ! dod ssoy [230 SAAB ce qi Bee ae a ——— i ‘ ‘gOlea 17909 a1enb —— ae : = or) . ATsse00n aoe ate saat ? § SUT aes a 2 : rn wal ‘a ; Th co pssorjo ove | | aoe Sy ee ‘cOL — sourur : : Hii i oR wa1}m90 nibaraa se re ; : IEE : i : pete SRA SSAS 1% ae oD oD 3 oo 6 54! =) rod dann eae : i : : as oarenb = i = # a} AIsse0on ie = ao a peu ee . PD we a "cOl = oa) a ee : T1}99 oo oenon i me : i - iad ssoy [e930 ES a a ° = SS zed 01 ae : ; - - ; : el ae Wm90 orenb — : : : aE Q,% : AIS8909 Le 238 NA ns Surmyq noone Bue 2988 a8 TR eS P S80] Jo 78 miso ls = z og Se ea I 3 hah : 14u99 atenbs pee i os aoe ty DO iS . : ee : : ; si 101, aN st 29 ° E 4 iad sourur anaes : = : = ~ a. lay! =H SATSS900NS ibe ee on e sae eo oo urIMp sso "Oude TEE #88 4 [Jo 998N ret ek ac) “un *e0T = | oy re an ee souUIRI | a E me renbs Taek wie ce é zed ssoq [24 sy “a x am AAA A os Gm = zod ‘cOL + “Un - = Hi ; 89 a jo) o> ie) oak Cates Ul cf canbe | a : : | : : ie 5 isse00ns Sump He Bee | Soecee i : 1 a 3 ! . e e = 15 (e) om en +e SS NAN elie} 4 eo eet aces 7 coca Cy . iS Ree ad S80] [240 A 5% is 23% fo) =) 19 zed SO an xe = “ : sou rast = ae | : i x us axenbs | a s © La en 1sse00ns 3 ais Se 2588 nt Hae eo Qk of) per cent. of sulphuric} . acid added. 7. ( ‘2359 |Notweighed)|Ordinary pure sul- | phate, y>5 per cent. 2361 a acid added. Accidentally ‘2360 ‘3 Burgoyne Burbidges lost. as above, } per cent. acid added. ‘2374 * Ditto, stood for four} ' days with a copper plate in it and no acid added. "3000 ‘5 Ordinary pure sul- *B549 ” Ditto, ditto, with vo 8549 55 per cent. acid added. "3549 ‘: Burgoyne Burbidges with § per cent. acid added. 3060 ie Ditto, with no acid added. 2368 of last, with the ex- 2367 e ception of the third, "2369 ¥ which had 5 per 9. ‘2368 ij Solutions the same as "2371 45 cent. of acid added. of Silver and of Copper. 407 mercial copper sulphate as from the various specimens of pure sulphate experimented on; and for the first few hours no difference was found, whether acid had or had not been added to the solution. If, however, the same solution is to be used for several successive experiments, acid should be added, as will be evident from the results given in Table IV. The first column of this table gives the numbers of the experiments in the order in which they were made ; the second column the ratio of the electrochemical equivalent of copper to that of silver, obtained from the experiments when the current-density -at the cathode was one fiftieth of an ampere per square “centimetre ; the third column the same ratio when the current- density at the cathode was one two hundred and fortieth of an ampere per square centimetre. ‘Two or more silver cells were always in circuit with the copper cells, but as the differences between them, except in one or two cases where one of them was known not to be reliable because of silver lost previous to weighing, were never so great as to enter into the figures here given, it does not seem necessary to record the numbers in detail. Tasie LY. Ratio of the Electro- | chemical equivalent of Cop- N ee per to that of Silver. ie Roar ts AC AL aa a Remarks. ae Area of plate | Area of plate * | 50 sq. cms, | 240 sq. cms. per ampere. | per ampere. t "2939 "2930 Fresh solutions. 2. "2944 "2929 Solutions interchanged. 3. “2941 ‘2935 Solutions as in 2. 4, 2942 "2939 ri x 5. "2944 "2942 : = 6. 2947 22932 Solutions again interchanged. .{ *2940 °° N lutj : v7, 9940 ‘2934 ew solutions once previously 2940 used. — 8 | ae 2932 | Fresh solutions containing a : | -2940 9929 | little acid. In all these experiments, with the exception of the 8th, no acid was added to the solution. The quantity of liquid con- tained in the cells for the experiments 1 to 6 was about 100 cubic centimetres and was the same for the large as for the small plates. The total area of copper plate, including both gain and loss plates, was about 30 square centimetres for the small plates and 60 square ear for the large plates. 212 408 Mr. T. Gray on the Electrolysis These experiments show clearly a gradual increase in the ratio with successive experiments, and that in a more marked de- gree with the small current-density than with the large. Omitting the results of experiments 2 to 6, the average ratio for the two current-densities here used are ‘2940 and °2931, and the ratio of these two numbers will be found to be in good agreement with the results of special experiments on the effect of size of plate given in Table V. and illustrated in curve 13. 3500 _ S 20) 400 600 800 Abscisse give area of cathode in square centims. per ampere. Liffect of Size of Plate or Current-Density.—The two ex- periments on the effect of the density of the current at the cathode, the results of which are given in Table V. and illus- trated in curve 13, were made on the 25th and 28th of June respectively; the current, as measured by one of Sir William Thomson’s galvanometers, was about ‘0997 ampere and was continued for three hours. The first column gives the area of the deposit, the second and third the amount of copper deposited. 3 TABLE V. Ayea of plate | “eight of deposit| Weight of deposit pi Hes in grammes in grammes aah : (first experiment).| (second expt.). 3 3034 3533 5 3030 3529 11 3528 3530 18°5 3526 3527 36 3524 3521 73 3003 3502 The results given in Table IV. indicate a slightly greater difference between the results obtained with a current-density of a fiftieth of an ampere and those obtained with a two hundred and fortieth of an ampere. per square centimetre of of Silver and of Copper. 409 the cathode; but both sets agree in indicating rather less than a tenth per cent. as the correction which has to be added to the results obtained with the denser of these currents, in order to arrive at the true value of the electro-chemical equivalent of copper. The results of Table IV. give for the amount of copper deposited by a coulomb of electricity, when the cathodes ex- pose a surface of fifty square centimetres to the ampere, - 0003287 gramme, if we assume :001118 as the correspond- ing number for silver. They would thus indicate a value not differing much from ‘0003290 as the true value of this con- stant for copper. In the use of copper for the measurement of currents by electrolysis the absolute value of this latter number is not of much importance; what is wanted is the proper number to use for a certain current-density, and at ordinary temperature this number will not differ much from the number obtained by using ‘0003287 for cathodes of fifty square centimetres per ampere, and correcting for other sizes by the dotted line in curve 13. : A few experiments were made with very weak current- densities, the circuit being kept closed for about a week in each case. The results show that when the current-density is very small the rate of loss by direct action of the liquid is much the same as if no current were flowing. The deposit was patchy and did not usually cover all the surface of the plate, a result which was perhaps due to inequalities in the plate itself. The difference between the gain and loss of weight by the plates in the electrolytic cell is usually very much greater than can be accounted for by the loss of similar plates placed in the same liquid when no current is flowing, a result which appears to be largely contributed to by the anode-plates losing very much more when the current is flowing than when it is not. If the difference between the gain and loss be divided by the sum of the areas of the anodes and cathodes, and the quotient, multiplied by the area of the cathode, be added to the gain, a result is obtained which is always too high to give the true electro-chemical equivalent, but which is very nearly constant for different cells, even when they begin to give uncertain results from the gain of weight taken by itself. Arrangement of the Circuit.—For experiments with weak currents, such as those the results of which are given in the Tables II. to V., the circuit generally included a battery of twenty-four tray cells, the electrolytic cells, one of Sir William Thomson’s improved rheostats, and a galvanometer. The galvanometer was in some of the earlier experiments one of Thomson’s lever voltmeters, but in the later experiments one 410 Mr. T. Gray on the Electrolysis of his “ school galvanometers” arranged so as to have a ‘‘false zero,” and having its needle suspended in a strong field produced by per manent ma gnets, was used. An ordinary index galvanometer whenarranged in this way, with its zero well off the scale, gives ample sensibility, and is very compact and convenient. The current was adjusted by the rheostat until the index was exactly over a division of the scale, and it was kept there by turning the rheostat, if the current varied, all through the experiment. Standardizing Arrangements.— The arrangement of the apparatus for standardizing instruments intended to measure strong currents such as deca and hecto ampere-metres is shown in figure 7, which may be taken as a plan of the arrangements on the top of the standardizing table, with the exception that the galvanometer G is, for convenience in the figure, shown much too close to the conductors. On one end of the table six of the Electric Power Storage Company’s accumulators are arranged, and connected by means of thick copper rods to a mercury-cup commutator asshown. ‘The mercury cups m are shown joined across by bridges which are of thick copper in such a way that the cells are in series, but for most purposes these are removed and the cells joined in parallel by means of two rods of copper, provided with teeth at the proper distances apart to fit into the cups m, and thus join all in each row together. The battery, when fully charged and joined in this way, is capable of maintaining a current of two hundred amperes for ten hours. The commutator is joined by means of two copper rods 7, 7, — to a distributing board A, by means of which one or more instruments can be put in the same circuit. In the arrange- ment shown, a set of conductivity-bridges D and a rheostat R are introduced between the cups 1 and 2; a galvanometer G between the cups 4 and 5; a pair of large electrolytic cells, joined together by means of a moveable cup M, between the cups 6 and 7; and an electric-current balance between 11 and 12. The arrangement here shown, together with details of the conductivities D and the rheostat R, will be found described by Sir William Thomson in a recent patent specifi- cation. The following description will give a general idea of the apparatus and the mode of using it. A perspective sketch of two of the conductivity-bridges D is given in figure 8, where one that is in the circuit is shown standing in the mercury troughs ¢, ¢, and another insulated. These conductivity-bridges consist of ‘U-shaped pieces of plati- noid rod or wire accor ding to the conductivity required. The thick rods are bent into shape and the two limbs held at the proper distance apart by wooden blocks, while the thin wires are of Silver and of Copper. 411 supported all along their length by astrip of wood. The length of rod in each U must of course be sufficient for the metal not to become much heated by the potential of cell, one or two volts; the length actually used in the apparatus is four metres. The troughs ¢ are partially filled with mercury and have thick copper bottoms, against which the ends of the plantinoid rods press by their own weight. The thin wires have thick terminal pieces fixed to the wooden strips for this purpose. The con- ductivities of the different bridges, beginning at the end 4, ¢, fig. 7, are graduated so as to be nearly in the ratio 1, 1, 2, 4, 8, 16, &e.; so that any conductivity between the highest and the lowest can be put in by steps equal to the lowest conduc- tivity in the set. To bridge over these steps and thus make the variation from the highest to the lowest perfectly con- tinuous, the rheostat R, which has a minimum conductivity somewhat less than the lowest of the bridges, is introduced and forms in fact another bridge which can be readily varied. The wire of this rheostat is formed of a strand of fine copper wires, and is capable of carrying a current of ten amperes. The galvanometer G and the current-balance B may be of any form which it is desired to standardize ; they are simply put into the figure by way of illustration. The electrolytic cells E are rectangular earthenware vessels fitted in the manner illustrated in figs. 4-6, and described on page 392, above. The larger of these cells is fitted to receive plates the aggregate surface of which is sufficient for two hundred amperes, but when the current to be measured is smaller the plates are not all introduced. In this cell the plates are so large that there is considerable danger of their touching each other when they are left freely suspended in the liquid, and hence two U’s of thin glass rod are hung over each of the anode plates, so as to keep the cathodes from touching them. The small cell is convenient for currents of from 5 to 25 amperes. All these operations connected with the treatment of the plates for these cells;have been already described, and hence it only remains to indicate how the apparatus is generally used for standardizing instruments. Suppose, for example, a eurrent-balance is to be standardized, and that it is known, by preliminary trials or by comparison with another instrument, to require about fifty amperes to bring its index to the zero mark when a certain weight is put on the beam. Arrange the electrolytic cell so that the cathode plates expose a surface of about twenty-five hundred square centimetres, and join up as shown in the diagram, fig. 7, putting a bridge across from 4to5. Introduce conductivity into D until the index of the 412 Mr. T. Gray on the Electrolysis “s balance comes nearly to zero, and then adjust by the rheostat. Leaving everything in position, break the circuit and take out the cathode plates, wash, dry, and weigh them, in the manner already described. Again put the plates in the cell, and observing the time carefully on an accurate time-keeper, close the cireuit. The current will at once assume almost the exact strength, and what little deviation there may be can be adjusted in a second or two by the rheostat. From five to ten seconds usually suffices to bring the current accurately to the proper strength, and the electrolysis is as a rule con- tinued for an hour. Now suppose the average error during ten seconds amounts to as much as five per cent., the error on the total amount will only be about one-seventieth per cent., which may be neglected. The current is kept steady by means of the rheostat during the whole hour, and with most instruments this can be done with sufficient accuracy by observing the index of the instrument itself, but if the instrument is not sufficiently sensitive, or if the constancy of its indication is to be tested, a second instrument possessing the requisite sensibility and constancy is introduced into the circuit in the manner indicated at G, and the current kept steady by observing the index of it. The constant of this second instrument does not require to be known ; it should be of such a kind that its constant can be readily changed so as bring its deflection to the proper amount, and the index exactly to a scale-division. The position of the index of the instrument being standardized may then be observed from time to time and the variations, if any, noted. When instruments which are designed for the measurement either of very weak or very strong currents, as, for example, milliampere- or hectoampere-meters, are to be standardized, it is generally more convenient to send a stronger current in the former case, and a weaker current in the latter case, through the electrolytic cell. Several methods involving the use of auxiliary galvanometers have been used for this purpose ; but the method illustrated in fig. 9 is sufficient to indicate the general principle of divided circuits on which such methods are based. Referring to the figure, 7, 7, is a set of resistances, supposed in this case to consist of eleven straight wires each of exactly the same resistance and as nearly as possible of the same length and thickness. These wires are soldered to thick bars of copper, 06, b1, bs, the resistances of which are negligible in comparison with that of the wires. In the case here considered the resistances are supposed to be arranged in two groups of ten and one respectively, but for general purposes it is more of Silver and of Copper. 413 convenient to solder separate terminal pieces to one end of each wire, so that the number in each group may be varied by putting the terminals in one or other of two mercury troughs. Between the bars 0; and b, a zero galvanometer g is connected, and is used for the purpose of indicating when b, and b, are at the same potential. When this is the case, the current from 6 to 0, is to the current between 6 and 0, in the ratio of the resistance between 6 and 0, to the resistance between bandb,. Two sets of resistances, or two rheostats, R and R,, are introduced into the branches for the purpose of ad- justing the currents to the required strength. The electrolytic cells E and H, are placed in one or other of the branch circuits accordingas theinstrument to be graduated is designed for weak or for strong currents, and the instrument is included in the other branch. The circuit is closed through a suitable battery, which must be of such a kind as will maintain a nearly uniform electromotive force. The current through the instrument Gy, is kept constant by means of the rheostat Ry, and the difference of potential between b, and 0, is kept at zero by means of the rheostat R. When the instrument G, is to be standardized by means of astandard galvanometer or astandard current-balance, the arrangement and mode of operation is precisely similar, the standard instrument taking the place of the electrolytic cells. Note on the Hiffect of Excessiwe Current-Density at the Anode in the Electrolysis of Copper Sulphate.-—On page 396 above, the effect of excessive current-density at the anode of a copper- copper-sulphate electrolytic cell is referred to, and it is there stated that, unless the anodes presenta surface greater than 40 square centimetres per ampere, the current is apt to diminish greatly in strength after the first few minutes. As the result of some special experiments on this subject I find that the requisite size of anode depends greatly on the degree of saturation of the solution. When the solution is nearly saturated, say above 1°18 in density, the current is apt to be almost entirely stopped on account of the anode plates becoming completely covered by a finely crystalline deposit of copper sulphate which dissolves in the liquid very slowly. A current of one-tenth of an ampere derived from a battery of 24 tray Daniells was passed through an electrolytic cell charged with a solution of sulphate of copper of density 1°18, and having two anode-plates made of round copper wires presenting a total surface of 3°8 square centimetres. The current remained nearly constant for 12 minutes, when it gradually diminished, became practically zero in a few seconds, and remained so for 20 minutes. At the end of this time the 414 Mr. H. Tomlinson on certain Sources of Error in circuit was broken, and the plates taken out of the liquid. The cathode was found to be covered with a good deposit of copper, but the anode was completely enclosed in a glacial sheath of copper sulphate. Another cell, in which the anode had a surface of 4°4 square centimetres, was treated in the same way, and was found to carry a current of one-tenth of an ampere for 14 minutes. The plates were then found to be covered by a similar coating of copper sulphate. A similar cell in which the anode had a surface of 7 square centimetres was found to carry one-tenth of an ampere for 50 minutes, when the current fell off as before from the same cause. Hixperiments were then made in a similar manner with cells containing a solution the density of which was 1°11. When the current density at the anode exceeded one ampere per 20 square centimetres, the plate became covered with black oxide and the current diminished greatly in strength after a few minutes (about 7 minutes for a current-density of one ampere to 7 square centimetres, and 3 minutes for double that current-density). The current does not entirely cease, and will after a few minutes, if the battery be of sufficiently high potential, again assume nearly its former strength. The oxide then falls off and gases are liberated at the surface of the anode, forming a descending stream close to the plate and an ascending stream two or three millimetres further out. As the current-density is diminished, less and less oxidation takes place and it becomes a lighter brown in colour. With a current-density of one ampere to 30 square centimetres the anode became covered with brown oxide, which made the re- sistance of the cell high and variable, but little or no gas was generated. No gas was, so far as could be observed, given off at the cathode during any of these experiments. XULIX. On certain Sources of Error in Connection with Hape- riments on Torsional Vibrations. By Hrrerrt Tomuin- son, 6.A.* Introduction. (bs ee G a long series of researches on the torsional elas- ticity and internal friction of metals, I have come across certain sources of error in connection with torsional vibrations * Communicated by the Physical Society: read June 26, 1886. Connection with Experiments on Torsional Vibrations. 415 generally, which I venture to bring before the notice of the Physical Society in the hope that by so doing I may perchance save others who may pursue similar investigations from the pitfalls into which I have fallen, and from which I have managed to extricate myself only after a considerable expenditure of time and labour. In my earlier experiments a wire about 600 centims. in length and 1 millim. in diameter was suspended vertically, having its upper extremity clamped to a rigid support and its lower one clamped or soldered to the centre of a horizontal bar of brass, from which were suspended, by threads or fine wires, two cylinders of equal dimensions and mass, and placed at equal distances from the wire. ‘The torsional oscillations of the wire were observed by means of the usual arrangement of mirror, scale, and lamp, the main object of the inquiry being to determine as accurately as possible the logarithmic decrement of the amplitude of swing and the vibration-period. After one set of observations had been completed, the moment of inertia was altered by sliding the cylinders along the bar further away from or nearer to the wire, so that the latter might now vibrate in a different period. Ift be the period of vibration, & the moment of inertia, and / the torsional couple, wk The moment of inertia of the bar together with that of the suspended cylinders could be calculated with very fair accuracy; the value of ¢ was also calculated to a nicety; so that it was reasonable to expect that the values of / obtained with different moments of inertia should be very fairly in accordance with each other. For some time this proved to be the case, and several wires of different metals had already been examined when a curious phenomenon presented itself. A change of moment of inertia, had just been made, and the wire was then set in torsional oscillation; but instead of the amplitude of swing diminishing by slow degrees as in the previous experiments, the spot of light was seen to make five or six oscillations of very rapidly diminishing amplitude, and finally come nearly to rest. Soon, however, the amplitude began to increase until in a few vibrations it extended over some two hundred divisions of the scale, when once more diminution and, finally, rest nearly ensued. I will not trouble the Society with my conjectures at the time as to the cause of the phenomenon ; suilice it that at length I discovered, what perhaps I ought to have discovered at once, that the, at first sight, startling apparition was due to a very natural cause, 416 Mr. H. Tomlinson on certain Sources of Error in namely, the rotation to and fro of the cylinders about their axes, and that the rapid absorption of energy was owing to the fact that the torsional vibration-period of the wire’ nearly synchronized with the vibration-period of the cylinders about their axes. I need hardly say that after this discovery I con- sidered it necessary to re-try all my previous experiments with improved arrangements. With more perfect apparatus, in which the cylinders were clamped to the bar so as to be in- capable of motion independently of the latter, I was glad to find the main conclusions which I had previously drawn re- specting internal friction confirmed. On one occasion, how- ever, by what afterwards turned out to be the merest chance in the world, the old phenomenon appeared with all its curious concomitants, and beats were very plainly discernible between two sets of vibrations which nearly synchronized. These two sets of vibrations I set to work to disentangle in a manner kindly suggested to me by Prof. G. G. Stokes, with whom I had some correspondence on the subject. Suppose that in one or more beat-intervals the slower vibration is dominant”*, and afterwards the quicker. Ifthe amplitudes cross during a beat-interval, that one is to be deemed dominant which is dominant about the time of lull. Suppose there are m beat- intervals in which the slower is dominant, followed by m in which the quicker is dominant, and let N be the total number of vibrations countedt. Then the number of vibrations of the two component kinds will be :— For the slower, N—2n, For the quicker, N +2m. In this way I managed without any difficulty to find that one of the two vibration-periods was nearly a constant what- ever the moment of inertia; whilst the other, that due to the torsional elasticity of the wire, increased as the square root of the moment of inertia. At first I felt very strongly inclined to believe that the former of the two vibration-periods per- tained to some molecular state of the wire; and I was espe- cially deceived by the fact that always after rest the pheno- menon became less marked, and, finally, almost vanished, easily, however, to be reproduced in all its former intensity by the slightest shock given to the wire, or by raising or lowering the temperature slightly. The wire was of iron, and my pre- vious experience of the effect of rest on the molecular dispo- sition of this metal led me to draw the above-mentioned conclu- sion. Had I not been thus deceived, I might more quickly * By dominant vibration is meant the vibration of greater amplitude. 1 For accuracy, the counting should begin and end with a maximum. Connection with Experiments on Torsional Vibrations. 417 have arrived at the solution of the problem; but as it was four weeks were spent in endeavouring to find the effect of change of temperature, change of load, and change of mode of attaching the cylinders, until I discovered that the pheno- menon was simply caused by approach to synchronism between the periods of torsional and pendulous vibrations of the wire. Of course, if the axis of the wire passed accurately through the centre of mass of the vibrator, the phenomenon would not occur, but it is impossible quite to secure this; and as a con- sequence so-called centrifugal force on the one hand, and the force of gravity on the other, set up pendulous vibrations which may seriously interfere with the vibration-period due to the torsional elasticity, and still more so with the loga- rithmic decrement of are. Having thus discovered two enemies in ambush, I thought that the ground was now clear, at least as far as any danger from synchronism was concerned; but in this I was mistaken: for happening to pursue my investigations still further with comparatively light loads and small moments of inertia, I at length became acquainted with a third source of error arising from approach to synchronism between the torsional and transverse vibrations of the wire. It is, I believe, quite im- possible to excite torsional vibrations without at the same time exciting transverse ones; and should synchronism nearly occur between the periods of these vibrations, we may have beats quite as strongly marked as in the two cases before mentioned. There is yet another source of error to which it is desirable to draw attention. Whena wire has been recently suspended, the torsional vibration-period will always be found slightly greater than when it has been suspended for some time and frequently set in vibration. The length of time which must elapse before a wire has reached its state of maximum torsional elasticity depends upon the nature of the metal ; with copper, for instance, a few hours will suffice, whilst iron may require at least a couple of days. Now suppose that this state of maxi- mum elasticity has been reached: if we wish to preserve it, care must be taken not to jar the wire or to subject it to a change of temperature of even two or three degrees Centi- grade; for in either case the elasticity will be slightly di- minished, and a few hundred vibrations will be required to restore it to its former condition. Since it is very difficult to avoid such slight changes of temperature as those mentioned above, it is always advisable after any rest to cause the wire to vibrate a few hundred times before beginning fresh observations. 418 Error in Experiments on Torsional Vibrations. Determinations of Torsional Electricity. It is evident, from what has gone before, that when the cylinders are suspended from the bar by threads, the torsional elasticity, as inferred from the vibration-period, may be seriously affected should the torsional vibration-period of the wire approach to synchronism with the vibration-period of the cylinders about their axes; and to such an extent did I find this to be the case, that the values of the torsional couple for the same wire, as calculated from the previously given formula, differed from each other by 10 per cent. or more when the moment of inertia was varied. The effect of synchronism between the torsional vibrations of the wire and the pendulous vibrations is not likely, if proper care be taken in the construction and disposition of the bar and its appendages, to be great; but it may nevertheless be - very sensible. : Synchronism between the torsional and transverse vibra- tion-periods is not likely to occur except in extreme cases ; but it must be remembered that the torsional vibration-periods may be affected by synchronism between them and multiples of the transverse vibration-period, though of course not to the same extent as when the fundamental transverse vibration- period equals the torsional vibration-period. A similar remark to the above would also apply to the other two cases of synchronism. Determinations of Magnetic Moments. It is evident that determinations of magnetic moments by the method of oscillations may be similarly affected by the same causes as we have seen would affect determinations of the torsional elasticity. Damping of Magnets. Still greater error may be introduced from the above causes in experiments on the damping of vibrating magnets ; for the logarithmic decrement is much more seriously affected than the vibration-period. Determinations of Moments of Inertia. When it is impossible to determine the moment of inertia of a body by measurement of its dimensions and mass, it is usual to find the time of oscillation, and afterwards to redeter- mine this time when the moment of inertia has been altered to a known extent. | If t and ¢' be the two times of oscillation, K the required On the Self-induction of Wires. 419 moment of inertia, and & the known change of moment of inertia, 2 Now a small error in the determination of either t or ¢’ pro- duces a comparatively large error in the value of K; and itis therefore of considerable importance here that there should be no approach to the previously mentioned synchronisms. It is important also here to notice the fourth source of error which has been mentioned; for it is very difficult to change the moment of inertia without imparting some shock to the wire. L. On the Self-induction of Wires.—Part IV. By OLIVER HEAVISIDE*. ~* mentioned at the close of Part III., it would appear that the only practicable way of making a workable system, which will allow us to introduce the terminal con- ditions that always occur in practice, in the form of linear differential equations connecting C and V, the current and potential difference at the terminals, is to abolish the very small radial component of current in the conductors. This does not involve the abolition of the radial dielectric current which produces the electric displacement, or alter the equation of continuity to which the total current in the wires is subject. The dielectric current, which is SV per unit length of line, and which must be physically continuous with the radial cur- rent in the conductors at their boundaries, may, when the latter is abolished, be imagined to be joined on to that part of the longitudinal current in the conductors that goes out of existence by some secret method with which we are not con- cerned. We assume, therefore, that the propagation of magnetic induction and electric current into the conductors takes place, at any part of the line, as if it were taking place in the same manner at the same moment at all parts (as when the dielec- tric displacement is ignored, making it only a question of inertia and resistance), instead of its being in different stages of progress at the same moment in different parts of the line. This requires that a small fraction of its length, along which the change in C is insensible, shall be a large multiple of the radius of the wire. The current may be widely different in strength at places distant, say, a mile, and yet the variation in a few yards be so small that this section, so far as the propa- * Communicated by the Author. 420 Mr. O. Heaviside on the gation of magnetic induction into it is concerned, may be regarded.as independent of the rest of the line ; the variation of the boundary magnetic force or of C fully determining the internal state of the conductors, exactly as it would do were there no electrostatic induction. In a copper wire, in which w=1, and k=1/1700, the value of the quantity 4apkp is p/135. On the other hand, the quantity m in —s’=4apkp+m’ has values 0, w/l, 2a/l, &c., or a similar series, in which / is the length of the line in cen- timetres, so that jz / lis a minute fraction, unless 7 be exces- sively large. But then it would correspond to an utterly insignificant normal system. We may therefore take —s’=Arpkp. It will be as well to repeat the system that results, from Part II. The line-integral of the radial electric force across the dielectric being V, from the inner to the outer conductor (concentric tubes), and the line-integral of the magnetic force round the inner conductor being 47rC, so that C is the total current in it, accompanied by an oppositely direct current of equal strength in the outer conductor, V and C are connected by two equations, one of continuity of C, the other the equa- tion of electric force, thus :— | —~# =sv, eT aL,0+R0+Ry. . (41) Here e is impressed force, 8 the electrostatic capacity, and L, the electromagnetic capacity, or the inductance, of the dielec- tric, all per unit length of line; and R,” and R," are certain functions of d/dt and constants such that R,’C and —R,!’/C are the longitudinal electric forces of the field at the inner and outer boundaries of the dielectric, which, when only the first differential coefficient dC/dt is counted, become d 1 di? respectively, where R,, L,, and R,, L, are the steady-flow resistance and inductance of the two conductors. The forms of R," and R,! are known when the conductors are concentric circular tubes, of which the inner may be solid, making it an ordinary round wire. Now if the return con- ductor be a parallel wire or tube externally placed, it is clear that we may regard R,” and R," as known in the same manner, provided their distance apart be sufficiently great to make the departure of the distribution of current in them from symmetry insensible. We have merely to remember that it is now the inner boundary of the return tube that corresponds to the R'=R, +L R'=R, +1," Self-induction of Wires. 421 former outer boundary, 7. e. when it surrounded the inner wire concentrically. The quantity V will still be the line-integral of the electric force across the dielectric by any path that keeps in one plane perpendicular to the axes of the conductors, in which plane lie the lines of magnetic force. Also, the product VC will still represent the total longitudinal transfer of energy per | second in the dielectric at that plane, or, in short, the energy- current. As regards the modified forms of S and L,, there is, in strictness, some little difficulty, on account of the dielectric being necessarily bounded by other conductors than the pair under consideration, in which others energy is wasted, to a certain extent. This can only be allowed for by the equations of mutual induction of the various conductors, which are not now in question. But if our pair, for instance, be suspended alone at a uniform height above the ground, so that only the very small dissipation of energy in the earth interferes, it would seem, so far as the wire current is concerned, to be an unnecessary refinement to take the earth into consideration. There are, then, two or three practical courses open to us; as to suppose the earth to be a perfect nonconductor and behave as if it were replaced by air, or to treat it as a perfect con- ductor. In neither case will there be dissipation of energy except in our looped wires, which have no connection with the earth, but there will be a different estimation of the quan- tities L, and 8 required. For when we suppose the earth is perfectly conducting, we shut it out from the magnetic field as well as from the electric field. The electrostatic capacity S is that of the condenser formed by the two wires and inter- mediate dielectric, as modified by the presence of the earth (the method of images gives the formula at once), and the value of L, is such that L,S=pc=v—, where v is the velocity of undissipated waves through the dielectric; that is, as before, L, is simply the inductance of the dielectric, per unit length of line. On the ground there will be both electrifica- tion and electric current, due to the discontinuity in the electric displacement and the magnetic force respectively ; but with these we have no concern. In the other case, with extension of the magnetic and electric fields, the product L,8 still equals v-*. Neither course is quite satisfactory ; perhaps it would be best to sacrifice consistency and let the magnetic field extend unimpeded into the earth, considered as nonconducting, with consequently no electric current and waste of energy, whilst, as regards the external electric field, we treat it as a conductor. We must compromise in some way, unless we take the earth into account fully as an ordi- Phil. Mag. 8. 5. Vol. 22. No. 138. Nov. 1886. 2G 429 Mr. O. Heaviside on the nary conductor. Similarly, if the line consist of a single wire whose circuit is completed through the earth, by regarding it as infinitely conducting we replace the true variably distri- buted return-current by a surface-current, and, terminating the magnetic field there, have L,S=v-~?; but if we allow the magnetic field to extend into it, though with insignificant loss of energy by electric current, we shall no longer have this property. The property is intimately connected with the influence of perfect conductivity on the state of the dielectric. For per- fect conductivity will make the lines of electric force normal to the conducting boundaries, will make them cut perpendi- cularly the magnetic-force lines, which lie in the planes z = const. and are tangential at the boundaries, and will make L,S=v-?, irrespective of the shape of section of the conduc- tors. Now, at the first moment of putting on an impressed force, wires always behave as if they were infinitely conducting, so that, by the above, the initial effect is simply a dielectric disturbance, travelling along the dielectric, guided by the conductors, with velocity v, irrespective of the form of sec- tion. Of course dissipation of energy in the conductors immediately begins, and finally completely alters the state of things, which would be, in the absence of dissipation, the to-and-fro passage of a wave through the dielectric for ever. Hixcept the extension to other than round conductors, this does not add to the knowledge already derived from their study. The effect of alternating currents in tending to become mere surface-currents as the frequency is raised (Part I.) may be derived from, or furnish itself a proof of, the property above mentioned—that at the first moment there is merely a dielectric disturbance. For in rapid alternations of impressed force, we are continually stopping the establishment of the steady state at its very commencement and substituting the establishment of a steady state of the opposite kind, to be itself immediately stopped, and so on. When the dielectric is unbounded, not enclosed within conductors, there is also the outward propagation of disturb- ances to be considered; but it would appear, by general reasoning, that this is, relatively to the main effect, or propa-= gation parallel to the wires, a secondary phenomenon. It is clear that the same principles apply to conductors having other forms of section than circular, when V and C are made the variables, provided the functions R,” and R,! can be properly determined. The quantity VC being in all cases the energy-current, its rate of decrease as we pass along the line is accounted for (as in Part III.), thus, by making Self-induction of Wires. 423 use of (141), with e=0, d — 4 (vo) = 5 (G8V? + 41,0) + CRy"C+CR,"C 5 (142) that is, in increasing the electric and magnetic energies in the dielectric, and in transfer of energy into the conductors, to the amounts CR,"C and CR,"C per second respectively, which are, in their turn, accounted for by the rate of increase of the magnetic energy, and the dissipativity, or Joule heat per second in the two conductors ; or O/C 0-1 On CaO,2T, . «1 (43) Q being the dissipativity and T the magnetic energy per unit length of conductor. These equations (143) must therefore contain the enlarged definition of the meaning of the functions R," and R,". For it is no longer true that R,’C is, as it was in the tubular case, the longitudinal electric force at the boundary of the conductor to which R," belongs. It is a sort of mean value of the lon- gitudinal electric force. Thus, we must have WH oe ds CHC, 0... 2: C48) if KH be the longitudinal electric force and H the component of the magnetic force along the line of integration, which is the closed curve boundary of the section of the conductor perpendicular to its length. Butno extension of the meaning of V is required from that last stated. Let us, then, assume that R,” and R,! can be found, their actual discovery being the subject of independent investiga- tion. We can always fall back upon round wires or tubes if required. They are functions of d/dt and constants, if the line is homogeneous. But, as we have got rid of the radial component of current in the conductors, and its difficulties, the constancy of the constants in R,” and R, (as the conduc- tivity and the inductivity, or the steady-flow resistance, or the diameter) need no longer be preserved. Provided the con- ductors may be regarded as homogeneous along any few yards of length, they may be of widely different resistances &c. at places miles apart.. Then R,", R,!’ become functions of z as well as of d/dt, and S a function of z. Let our system be ~Hagyv, e—Lapie, ©. . (045) ep ~~ where both R” and 8” are functions of d/dt and z. As re- gards 8", it is simply S(d/dt) when the dielectric is quite non- conducting. But when leakage is allowed for, it becomes 2G 2 424 ‘Mr. O. Heaviside on the K+8(d/dt), where K is the conductance, or reciprocal of the resistance, of the dielectric across from one conductor to the other. Then both K and § are functions of z. The conduc- tion current is KV, and the displacement current SV, whilst their sum, or S’V, is the true current across the dielectric per unit length of line. We have now, by (145), with e=0, — © (VC) = V8/V + ORC = KV?4 Sasv+OR"C. . . . (146) The additional quantity KV? is the dissipativity in the dielectric per unit length, whilst now CR"C includes the whole mag- netic energy increase, and the dissipativity (rate of dissipation of energy) in the conductors. Let V,, C,, and V2, C, be two systems satisfying (145) with e=0. ‘Then , ~ £V,G)=Vi8"V24 ORG, —£ V:0,=V.8/Vi + OBC from which we see that if the systems be normal, d/dt be- coming p, and py respectively, we shall have d dz (V,0C,—V2C)) =CiC, =(p,— SV, Vo— (P p)4 ‘aut dagh, Lips R,’ and R,! being what R"” becomes with p, and _p, for d/dé. As the quantity in the $ \ is the U,,—T), of Part 1a: the first term is U,., we see that the mutual magnetic energy is T)=C,Co(Bi"—B")+(pi—p.). » « + G48 The division by p,—2 can be effected, and the right mem- ber of (148) put in the form C,f(p1) x C272). When this is done, we can find the mutual magnetic energy of any magnetic field (proper to our system) and a normal field, in terms of the total current in the wire and its differ- ential coefficients with respect to ¢; so that, in the expansion of an arbitrary initial state, C, GO, G, &e., may be the data of the magnetic energy, instead of the magnetic field itself. "pi Ry Rt iP (147) Self-induction of Wires. 425 We see also, from (148), that if T be the magnetic energy of any normal system per unit length of line, then dR! _ (72 A 2T=C cores Cae Stare (149) and therefore, if Q be the dissipativity in the conductors, dR" =) (150) Now consider the connection of the two solutions for the normal functions. Since the equation of C in general is, by (145), Q=R'C?—-T= C2 (R"=p d(1 aC ogi G5) =B'O-e Pat eee 45) the normal C function, say w, is to be got from a(1aw dz\$" dz with d/dt=p in R" and 8", making them functions of z and p. Let X and Y be the two solutions, making Wo MPG ge ws eS ES where g is a constant. The normal V function, say w, is got from w by the first of (145), giving Dee SE CNY oy LS ims, Shas 8" if Me iat Va Y fda. In X and Y, which together make up the w in (153), p has the same value. Therefore, in (147), supposing C, to be X and C, to be Y, we have disappearance of the right member, making : 2 rade gem ap as ~ (V,C, —V.Cj) = 0, or V,C, = V,C, = constant, or XY'—YX'=S8" x constant=A8", say, . . (155) leading to the well-known equation v=x | i dz, connecting the two solutions of the class of equations (152) ; which we see expresses the reciprocity of the mutual activities 426 Mr. O. Heaviside on the of the two parts into which we may divide the electromag- netic state represented by a single normal solution. Also, by (147), integrating with respect to.z from 0 to J, r iB Pu aaa u i) Saute (oe ee =A . (156) 0 0 Pi7Pe Pi-Pa either member of which represents the complete U;,—Ty, of the line: The negative of this quantity, as in Part III, is the corresponding U,,—Ty. in the terminal arrangements; so that the value of 2(U—T>) in a complete normal system, including the apparatus, is : ‘dR" dZ aZ 2(U—T) =| Sutdz—) —w'dz—w? = ean (U1) =f 'seae—| rds O ip) if V/C=Z, and Z) at z=1 and 0, these being functions of p and constants, and wy, wp are the values of w at z=l and 0. Or, which is the same, aU-T)= [wf (-z)] (158) Te 2h ee as before used. There is naturally some difficulty in expressing the state at time t, thus :— V= LAue™, C=>Awe? a due to an arbitrary initial state, on account of the difficulty connected with (Ry”— Re”) = (pi—Po), and the unstated form of R”. But when the initial state is such as can be set up by any steadily-acting distribution of longitudinal impressed force (e an arbitrary function of z), so that whilst V is arbitrary, C is only in a very limited sense 2 + Wo arbitrary, and C, OC, &c. are initially zero, and certain definite distributions of electric and magnetic energy in the terminal apparatus are also necessarily involved; in this case we may readily find the full solutions, and therefore also determine the effect of any distribution of e varying anyhow with the time. In fact, by the condenser method of Part JII., we shall arrive at the solution (135); we have merely to employ the present w and w, and let M be the value of the right member of (158). The following establishment, however, is quite direct, and less mixed up with physical considerations. To determine how V and C rise from zero everywhere to the final state due to a steadily-acting arbitrary distribution of e put on at the time ¢=0. Start with e at z=z, and Self-induction of Wires. 427 none elsewhere, and let (X+q)Y)A, and (X+4q,Y)A, be the currents on the left (nearest z=0) and right sides of the seat of impressed force. We have to find q, q,, Ao, and Aj. The condition V=Z,C at z=0 gives us, by (153), (154), —(X,y' + qo) +S," = Z(X + 9Y,)3 therefore Beis Go —(Xq +80"Zy Xo) + (¥9 +8 0"Zo Yo). - . (159) Similarly, V=Z,C at z=1, gives us | Qy= — (Ky + 8,"Z,X,) = (V1 +8:"Z, Y3). . . (160) Here the numbers 9 and ; mean that the values of X, &e. and S” at z=0 and at z= are to be taken. Now, at the place z=z, the current is continuous, whilst the V rises by the amount e suddenly in passing through it. These two conditions give us (X,+ q0¥2)Ao=(Xe+9,¥9) Ai, — B22 + (Xo' +g Yo') Ay =(Xe' + 91 Y2) Ay, where the , means that the values at z=z,. are to be taken. These determine Ay and A, to be Mi AG ey ou Oita h la. A foe SoM) 2 AOS eT Chel Se (So!) "(Xe Yo — ¥2X9')(Go— 1) aie Now use (155), making the denominator in (161) to be h(go—M)- We have then, if Cy and C, are the currents on the left and right sides of the seat of impressed force, _ (X+q¥)(X2+ ns) 5 h(qo—-) _(X + Y) (X+ Xs), h(qo- M1) These are, when the p is throughout treated as d/dt, the ordinary differential equations of Cy and C, arising out of the partial differential equation of C by subjecting it to the ter- minal conditions and to the impressed force discontinuity, Now make use of the algebraical expansion I (Po) _ I(p) coy ag «(88) (162) 428 Mr. O. Heaviside on the the summation being with respect to the p’s which are the roots of @(p)=0, without inquiring too curiously into its strict applicability, or bothering about equal roots. Here has to be d/d¢ and the p’s the roots of P=hMG—H)=05 so that (162) expands to Caen aaa —- , 4 es Le (Jo- 41) rere where the single g takes the place of the previous gp or q, which have now equal values, and C has the same ex- pression on both sides of the seat of impressed force. But é, is constant with respect to ¢, whilst C is initially zero; hence es m eo(1 — ¢Pt) djdt—p —p ’ which brings (164) to X+qY)(X2+gY Gay SELES) (1 i aay PF p which is the complete solution. By integration with respect to z we find the effect due to a steady arbitrary distribution - of e put on at ¢=0; thus } w( ewdz 0 Dae es eer (12), 72). 56 Soe where $’=dd/dp, and w is the normal current-function X+gY. To express the V solution, turn the first w into wu. The extension to e variable with ¢, as in Part III., is obvious. But as the only practical case of e variable with ¢ is the case of periodic e, whose solution can be got immediately from the equations (162) by putting p’?=—n’, constant, the extension is useless. Note that g) and g, are not equal in (162), and therefore in the periodic solution obtained from (162) direct they must be both used. The quantity —q¢' which occurs here is identical with the former complete 2(U—T) of the line and terminal apparatus of (157) or (158). Self-induction of Wires. 429 Let Cy be the finally reached steady current. By (166) it is gee | (cua 167) ‘= (— 5) (ew EcuiLe cane Neen Nias | To this apply (163), with p=0. Then a finite expression for Co is | 1 Co= (o/h) ae tera? Se On where wy and dy are what w and ¢ become when p=0 in them. Or, rather, it would be so if go and g; taken as identical could be consistent with py=0. But this is not generally true, so that (168) is wrong. To suit our present purpose, we must write, by (162), i. Zz L iS Sp S+a¥) | eX +aVder K+ ye) [eK +ayVer} te 3-28) { 1 (eyde +n (lesa) s. my eo 8) the g) being used in wo, the g; in w,. Now we can take p=0, and get the correct formula to replace (168), viz. z L = eA wr {evn de + wy | ewyode } 5 « (L70} 0 z | the second , meaning that p=0 in w, and 2. If there is no leakage (K=0 in 8”), Cy becomes a constant, given by 1 1 C= { ei | Rdz+R)+R, t,. Gay) 0 0 where the numerator is the total impressed force, and the denominator the total steady-flow resistance ; R, R,, and R, being what R’, —Z,, and Z, become when p=0 in them. But when there is leakage (170) must be used ; it would require a very special distribution of impressed force to make C, the same everywhere. To find the corresponding distri- bution of V, say V,, in the steady state, we have then —d0,/dz=KV, so that a single differentiation applied to (170) finds We Knowing thus C, finitely, we may write (166) thus,» 430 Mr. O. Heaviside on the C=0,—3(—ulp9 [evde er oi, ao aes 0 where C, is given in (170). The summation here, with ¢=0, is therefore the expansion of C,. The internal state of the wire is to be got by multiplying the first w by such a function of 7, distance from the axis, and of whatever other variables may be necessary, as satisfies the conditions relating to inward propagation of magnetic force, and whose value at the boundary is unity. In the simple case of a round solid wire, (172) becomes, by (87), Part IL., r J1(syr) w | ewde aI 1(s1%1) (—p¢') This gives C, the current through the circle of radius 7, less than a, the radius of the wire, Co, being the final value. The value of s, is(—4744,p)?. Here of course we give to 1, &;,and a, their proper values for the particular value of z. As before remarked, they must only vary slowly along z. In the case of a wire of elliptical section it is naturally suggested that the closed curves taking the place of the concentric circles defined by r=constant in (173) are also ellipses ; and that in a wire of square section they vary between the square at the boundary and the circle at the axis. The propagation of current into a wire of rectangular section, to be considered later, may easily be investigated by means of Fourier series, at least when the return current closely envelops it. oo = Cor— > et, ee (173) As an explicit example of the previous, let us, to avoid introducing new functions, choose the electrical data so that the current-functions X and Y are the J, and K, functions. This can be done by letting R” be proportional and §! inversely proportional to the distance from one end of the line. Let there be no leakage, and i hz, Se cas where §, is a constant, and Ry" a function of d/dt, but not of z. The electromagnetic and electrostatic time-constants do not vary from one part of the line to another. The equation of the current-function is 1d z dz 22) =Re'Sypu 5 Lt, Oa. Self-induction of Wires. 431 from which we see that X=J)(fz), Y=Ko(/2), S= (—Ro'Sop)?. But, owing to the infinite conductivity at the z=0 end of the line, making K,(fz)=© there, we shall only be concerned with the Jy function, that is, on the left side of the impressed force, in the first place. Since V is made permanently zero at z=0, the terminal condition there is nugatory. So =e); and w=J,(f2) +uK,(f) ; u=(f/Sp)JiC(fz), and u=(f/Sp){JiC fe) +n Kul fz)} s on the left and right sides of an impressed force, say at z= 2p. The value of g;, got from the V=Z,C condition at z=1, is (FUSop)IV FD ~ZSo( FD) (1602) ~ Z.Ko(f)— (flop) Kul fy We have also XY'—Yx! AN Ss =r ace TRA ag (1554) and the © solution (166) becomes 1 C=¥(—po ILL) | eI Hd. A—e), (1660) 0 where 6=—4q,/Sop, and g, is given by (160a). If we short-circuit at z=l, making Z,=0, we introduce peculiarities connected with the presence of the series of p’s belonging to f=0. The expression of g; is then, by (160a), © n=—J;(fl)/Ky(f)). It seems rather singular that we should have anything to do with the K, function, seeing that C and V are expanded in series of the J, and J, functions. But on performing the differentiation of @ with respect to p it turns out to be all right, the CRC in (166a) becoming —pp=— BSF De 7, (Ro?) in general ; whilst in the f=0 case, ae makes 6=4R,'/, we have where —p¢'=— ppl —— The value of @ when p=O in it is, by inspection of the expansions of J, and Ky, simply $R,/?, the steady-flow resist- 432 Mr. O. Heaviside on the ance of the line; R, being the constant that Ry!’ becomes with p=0. We may therefore write (166a) thus :-— Uy Uy dR, C= f edz-4R,P—S edz .(—p¢')7 ul ewdz.e! . . . (176A) 0 The value of ¢' here, got by differentiation with respect to p, may be written in many ways, of which one of the most useful, for expansions in Fourier series, is the following. Let w=(1+ 9’)? cos(mz+ @); then OO om t m Zi— | dp 8" cos’0 = a (g (m/S")? + oy Z corn) sit Yi oR { costml 5 ¢ a: —1 . ~ 28" cos’? dp d(mi) \8" mas" “a a8 PE L,) W Corresponding to this, m Z,—Z, S” (m/8")? + Z,Z, finds the angles ml; it is got by the union of tan9=8"Z,/m, tan(ml+6)=S"Z,/m, . . (179) which are equivalent to (160 0). For example, if we take RK’ =R, constant, thus abolishing inertia, and S”=Sp, no leakage, and 8 constant (R and 8 not containing p, that is to say), the expansion of V, an 1 ane function of z is (178) tan ml= iE Vo arr + @)dz V j= sin (mz +9) m Ly — | 1—cos'ml 7 ao Sp agEse Tl, } no] ™ (177) (180) Self-induction of Wires: 435 subject to (178). Here p= —m’/RS, so that the state of the line at time ¢ after it was V,, when left to itself, is got by multiplying each term in the expansion by e-m%/83 "The corresponding current is given by RO=—dV/dz. But the solution thus got will usually only be correct, although (180) is correct, when there is, initially, no energy in the terminal apparatus. If there be, additional terms in the numerator of (180) are required, to be found by the energy-difference method of Part III. They will not alter the value of the right member of (180) at all; they only come into effect after the subsidence has commenced. Similar remarks apply what- ever be the nature of the line. It is, however, easy to arrange matters so that the energy in the terminal apparatus shall produce no effect in the line. For example, join the two conductors at one end of the line through two equal coils in parallel ; if the currents in these coils be equal and similarly directed in the circuit they form by themselves, they will not, in subsiding, affect the line at all. Returning to (177), or other equivalent expression, it is to be observed that particular attention must be paid to the roots m/=0, which may occur, or to the series of roots p belonging to the m=O case, when we are working down from the general to the special, and happen to bring in m=0. Take Z,=0 for instance, making, by (175) and (1605), m j= Bi Sr tan ml, where m’=—SpR". Then dd dZ, tanml/dR" RR’ ee l — dp i 2m \ dp 2) 2 a dp Now, as long as Z, is finite, m cannot vanish; but when Z, is zero, giving ml= any integral multiple of 7, m=0 is one case. we have, when m is ei db _1/dR" RY dp’ : apa : ap page > and Pga 5 5 eB ys 5 Ghee but when m is zero the middle term on the right of the pre- ceding equation becomes finite, making dg /dp=1(aR" /dp). The result is that the current solution contains a term, or infinite series, apparently following a different law to the rest, with no corresponding terms in the V solution. This merely means that the mean current subsides without causing any = =) (181) 436 Mr, O. Heaviside on the electric displacement across the dielectric, when the ends are short-circuited (Z=0); so that if, in the first place, the cur- rent had been steady, and. there had been no displacement, there would have been none during the subsidence. The transition from the combined inertia and elasticity solutions to elasticity alone is very curious. Thus, let Z=0 at both ends, and R’=R-+ lip, where R and L are constants not containing p. The rise of current due to e is shown by 1 1 ( edz 9 _ Cosme {e cos mz dz VOB we eo e~ Re/L 0 pt cpr as > R+42Lp ert 1839 the m’s in the summation being m/l, 27/l, &e.; and each having two p’s, given by O=m? + RSp + LSp”. The m=0 part is exhibited separately, and is what the solu- tion would be if e were a constant (owing to the constancy of R). But, whatever e be, as a function of z, the summation comes to nothing initially, on account of the doubleness of the p’s, just as in (172 a) the part in the second line vanishes by reason of every p summation vanishing when t=0. Now, in (183), let L be exceedingly small. The two p’s approximate to —m’/RS, the electrostatic one, andto —R/L, the electromagnetic one, which goes up to «©, the storehouse for roots. The current then rises thus : 1 ( edz 1 ae ry (1 —e-Bh) 4 mi > Cos me { ecos mz dz.(1 — Ril) 0) O= ih Li og mPef88) (184 - > cos ms \ecos mz dz. (1—e san But the first line on the right side is equivalent to (e/R) (1—e-®#%), and here the exponential term vanishes instantly, on L being made ee zero, so that (184) becomes 1 Ce R- — a3 COs me ecos mz dz. Cl _ enum (185) except at the very first moment, when it gives C=e/R, which is quite wrong, although the pr receding for mula, giving C=0 at the first moment, is correct. Or, (185) is equivalent to arta Self-induction of Wires. 437 from which inertia has disappeared. Here V is given by (188) below. The process amounts to taking one half the terms of the summation in (183), and joining them on to the preceding term to make up e/R, whichis quite arbitrary. An alternative form of (185) is ez C=~2 1 pape - > cos mz {\- cosume da .ec™ Ee...) (186) On the other hand, there is no such peculiarity connected with the V solution in the act of abolishing inertia. The m=() term is ails. sin mz ( edz )=0, because m is zero and p finite. Therefore V rises thus, l m sin me e cos mz dz ee DS SES A ig Pe : Y i mane ay pane es before abolition of inertia. But as L is made zero, the deno- minator becomes m? for the electrostatic p, and for the other ; thus one half the terms vanish, leaving (187) L v=35 sae { ecos mzdz(1—e-”"4B8), , (188) 0 where L=0, without any of the curious manipulation to which the current formula was subjected. Next let us consider the transition from the combined elasticity and inertia solution to inertia alone (of course with resistance in both cases, as in the preceding transition). It is usual to wholly ignore electrostatic induction in investiga- tions relating to linear circuits. This is equivalent to taking S=0, stopping elastic displacement, and compelling the cur- rent to keep in the wires always, ¢. e. when the insulation is perfect, as will be here assumed. We then have, by (145), Be eT = BV. ers 1) By integrating the second of these with respect to z we get rid of V, and obtain the differential equation of C, ‘ edz= {{" Rldz + Z,—Z, \0=9,0, Baye a (£90) whence follows this manner of rise of the current, when e is Plul. Mag. 8. 5. Vol. 22. No. 188. Nov. 1886. 2H 438 Mr. O. Heaviside on the steady, and put on everywhere at the time ¢=0, reaching the final value Co, C=C)— 3 (—p) (. ev ae (191) ¢:=0 finding the p’s. We can find V at distance z by inte- grating the second of (189) with respect to z from 0 to z thus v={ ete { "Rlde\C, (199) wherein C is to be the right member of (191). This finds V by differentiations with respect to ¢ performed on C. In the final state put R,” for R"’ and —R, for Z,, steady-flow resist- ances. V will usually vary with ‘the time until the steady state is reached ; but if the line is homogeneous, with only the two sane R and L, and if also Z, and Z, are zero, V will be independent of. t, and instantly assume its final distribution. Thus, on these assumptions, we shall have cl i ‘edz/Rl) (1—e- 2), v— (eae— (2/ 0 as showing the current to rise independently of the distribution of e, and V to have its final distribution from the first moment, which, when the impressed force is wholly at z=0, of amount € 18 €&(1— i eee as the equation of the free surface, the known solution for simple harmonic waves in water of depth D gives, m(D—y) —m(D—y) = uf i+ mh — sine}, e7mD ie —e-~mD-y) — Uinh — — cos me, - (2a De where g ee ene \. U= Ae m “emD 4 g—ml + e—~mD Hence, where 2=0, as in the nodal section §§ P B, em(D—y) _ ¢—m(D—y) emD —_ _—mD iia (22); u=U, and v= Umh also 2mD __ e—2mD AS AmD D { ody = 4 Umi? omy e ° ° ° (23), 4mD =igh? 41— ees - +. (24). Now going back to (19) we see that when U approaches D,’ “ gD 1D, + Dy’ , the first term might become important, even though the corrugations at a great distance down-stream from the inequalities were infinitesimal. Reserving consideration of this case, and supposing for the present U to be considerably smaller than the critical value, we may neglect the first term in comparison with the second, remembering that in fact quantities comparable with the first the critical velocity Waves in Flowing Water. 451 term are neglected in the approximation (24) to the value of the second ; and we have, as our final approximate result, 4mD There is no difficulty in understanding the permanent steadiness of the motion which we have now been considering: to any finite distance, however great, on either the up-stream or down-stream side of the inequalities, if the water in the finite space considered is given in this state of motion, and if water is admitted on the one side and carried away on the the other side conformably. But it is very interesting and instructive to consider the initiation of such a state of things from an antecedent condition of uniform flow over a plane bottom. Suppose, as the primary condition, an inequality, whether elevation or depression, to exist in the bottom, but to be carried along with the water, so that the flow of the water is everywhere uniform and in parallel lines. If the inequality is an elevation above the bottom, our supposition is that the whole projecting piece, moving with the water, slips along the bottom. If the inequality be a depression in the bottom, the more awkward supposition must be made of a plasticity of the bottom, and the form of the inequality carried along, while the bottom is kept rigidly plane before and after this depression. Suppose, now, the inequality is gradually or suddenly brought to rest, what will be the resulting motion of the water? The question is identical with that of finding the motion of water in a canal, when by an external force, such as that of a towing-rope, a boat is gradually or suddenly set in motion through it; or, rather, it would be identical if the boat were a beam filling the whole breadth across the canal, so that the motion of the water shall be purely two-dimen- sional. I hope in a later article (Part III. or Part IV. of the present series) to investigate the formation of the proces- sion of standing waves in the wake of the obstacle, and its radual extension farther and farther down-stream from the obstacle, the motion having become sensibly steady in the its neighbourhood, and becoming so to greater and greater distances down-stream by the completion of the growth of fresh waves. ‘The disturbance sent up-stream from the initiating irregularity must also be considered. Equation (15) shows that whether the irregularity be an elevation, as in our first diagram (fig. 1), or a depression, as in fig. 2, a rising of level must travel up-stream, at a velocity relatively to the water which we know must be WgD,!, where D,! is inter- 452 On Stationary Waves in Flowing Water. mediate between Dy and the smaller depth, which we shall call D’, in the undisturbed stream above. But however gra- dually the initiating irregularity may have been instituted, this travelling of an elevation up-stream must develop a bore; because the velocity of propagation is, as it were, different in different parts of the slope, being gD! at the commence- ment of the slope, and ranging from this, through /gD,/, to a/ gDo as the depth rises from D’ to Dy; so that, as it were, the brow of the plateau in its advance up-stream overtakes the talus, till the slope becomes too steep for our approximation. The inevitable bore and “‘ broken” water (inevitable without viscidity of the water, or some surface-action preventing the excessive steepness) would modify affairs down-stream ina manner which it is difficult to imagine. It becomes, there- fore, interesting to see how it may be avoided, whether by surface-action, or by giving some viscosity to the water. It is more interesting to do this by surface-action, and to allow the water to be perfectly inviscid, so that our standing waves down-stream may be perfectly unimpaired. And we may do it very simply by covering the free surface all over (up-stream and down-stream) with an infinitely thin viscously elastic flexible membrane, stiffened transversely (after the manner of the sail of a Chinese junk) by rigid massless bars with ends travelling up anddownin vertical guides on the sides of the canal. If we suppose the motion of these ends to be resisted by forces proportional to their velocities, and the membrane to exercise (positive or negative) contractile tensional force in simple proportion to the velocity of the change of its length in each infinitely small part ; we have a mechanical arrangement by which is realized the mathematical condition of a surface normal pressure varying according to normal component velocity of the otherwise free surface, and in simple propor- tion to this normal velocity when the slope is infinitesimal. By making the viscous forces sufficiently great, we may make the progress of the rise of level up-stream as gradual as we please, and perfectly avoid the bore. We may also make the progress of the procession of stationary waves down-stream as slow as we please. The form of the water-surface over the inequality or inequalities, and to any distance from them, both up-stream and down-stream, is not ultimately affected at all by the viscous covering ; and it becomes, as time advances, more and more nearly that of the mathematical solution for steady motion, which I hope to give, with graphic illustrations drawn according to calculation from the solution, in Part III. pode LIII. New Geometrical Representation of Moments and Pro- ducts of Inertia in a Plane Section; and also of the Re- lations between Stresses and Strains in two Dimensions. By AuFRED LopGe, M.A., Coopers Hill, Staines*. i object of the first part of this paper is to give two methods by which the connection between the moments and products of inertia about pairs of rectangular axes through a point may be represented by means of a circle, without the necessity of drawing the ellipse of inertia. If a, 6 are the radii of gyration about the principal axes OA, OB at the given point O, and &, / those about any other pair of rectangular axes through the same point, h being the product-coefficient about the same pair (t.e. the product of inertia divided by the area of the section), we have k= Chal pb. sm7 a oh. (6. ie) Fa sin Go eds ae a ter & pe, fe fe (= @7 psi 6 Cos. Ors 8 ates CBD where @ is the angle KOA, considered positive when measured from +OA towards +OB, and when the right angle from +OK to + OL is measured in this positive direction. First Method of Geometrically representing the above felations, With diameter equal to Fig. 1. a+b describe a_ circle passing through O, but otherwise in any position whatever in the plane of the section, cutting the principal axes OA, OB in A,B respectively. This may becailed the gyration- circle at O. - On the diameter AB of the circle take a point P, such that PA=a, and bb. Then, if OK, OL are a pair of rectangular axes, cutting the circle in K, L respectively, PK is the radius of gyration about OK ; PL is the radius of gyration about OL ; - * Communicated by the Author. Phil. Mag. 8. 5. Vol. 22. No. 138. Nov. 1886. 21 454 Mr. A. Lodge on a New Geometrical Representation of and twice the triangle KPL is the product-coefficient about the pair OK, OL. For, draw PM perpendicular to AK ; then PM=a cos 8, MK=6O sin @ ;s hence PK?=a? cos? +0? sin? 0=k’. Similarly, ¢¢ Pir =i. Also LK (the base of the triangle KPL)=a+6; and, if 8 is the centre of the circle, SP =4(a—6), and the angle KSA= 28, therefore the height of the triangle =43(a—b) sin 20=(a—b) sin 0 cos 8 ; ... twice area of triangle KPL=(a?—2?) sin 0 cos 0=h. Q. H. The sign of the product 4 is positive if the positive direc- tions of OK, OL include between them the axis of minimum moment (as in the figure), for in that case @ is positive and less than a right angle, and a?—0? is positive ; or @ is nega- tive and less tlian a right angle, and a?—b” is negative. This condition is the same as the following :—If the centre S is in the positive quadrant KOL, the product of inertia is positive if O, P are on opposite sides of KSL. If 8 is taken on one of the axes OK, OL, when the product is positive P is in the positive quadrant. If Q be taken on AB so that SQ=SP, the figure PKQL is a parallelogram with sides equal to the radii of gyration about OK, OL, whose area equals the product-coefficient about OK, OL, and one of whose diagonals is the sum of the principal radii of gyration, the other being the difference ; P, Q being the ends of the shorter diagonal. Hence, if the radii PK, PL, and the product-coefficient about OK, OL are given, it is easy to find the radius SK of the gyration-circle at O, and to construct for the principal axes. It is worth noticing that if PK is drawn perpendicular to OK, LQ lies along LO, and P is on the positive or negative side of LO, according as the product of inertia is positive or negative. It is not difficult, being given the position of the centre of area G, to construct for the central principal axes. For, suppose G lies on OK ; draw GL’ parallel to OL. Then the radius about GK is known, the product of inertia about GK, GL’ is equal to that about OK, OL, and they are both there- fore represented by parallelograms of the same height, and with one pair of sides of the same length. The other sides of the new parallelogram are of length equal to VW PL?—OG?, Moments and Products of Inertia in a Plane Section. 455 and are therefore easily constructed. This completes the parallelogram, giving all data for the gyration-circle belonging to G. Second Method of representing the Relations (1), (2), and (3). Let UA=the mo- Fig. 2. ment of inertia about eee and Ub=AV= the moment about OB, OA and OB being, as before, the principal axes at O: so that UV is the swm of the prin- cipal moments, and AB their difference. Describe the circle AOB with centre T. Let any axis OK cut the circle in K, and from K draw KM perpendicular to UV. Then UM is the moment of inertia about OK, VM is the moment about OL (perpendicular to OK), and KM is the product of inertia about OK, OL. For, taking the area of the section as unity, which does not affect the relations between the quantities, UM=UT+TM=3(¢ +0) +4(a’—6”) cos 26, =a’ cos? +0? sin"d=K’ ;sx VM=VT—TM =a’ sin’0 + 0? cos? @=/ ;sx KM=TK sin20=3 (a’— 0’) sin 20 =h. These relations hold in whatever position the circle OAB is, and therefore O may be in the same straight line with MK, in which case OM=MK. Hence, if the moments and product about OK, OL are given, we have the following construction for the principal axes :— On OK take OM=the given product of inertia, measuring OM in the positive or negative direction according as the product is positive or negative. From M draw MU parallel to OL in the positive direction equal to the moment about OK, and in the opposite direction draw MV equal to the moment about OL, so that UV is the sum of the moments. Bisect UV in T, and with centre T, radius TO, describe a circle cutting UV in A and B. Then OA, OB are the principal axes, and UA, UB the moments about them. 212 456 Mr. A. Lodge on the Relations between The proof is the converse of the Fig 3. preceding proof. The only thing requiring special consideration is whether the axis of minimum moment will fall in the proper quadrant KOL, 7. e. in the positive quadrant if OM is positive, and out of it if OM is negative. The axis of minimum moment will evidently be in the same quad- rant as the point U from which the moments are measured, and by the construction U will be in the first quadrant if OM is posi- tive and in the second if OM is negative ; which completes the proof. _The expressions for the principal moments in terms of the given moments and product are easily deduced from the figure, V1Z.:— a, P=UT+TA=UT+ VTM?+ MEK? =F +P) +4 / (PP —PP EH, 2. ee and the angle AOK (8) is given by the equation tan 26=tan KTA=7 + ee These equations are of course deducible from the relations (1), (2), and (8). Stresses and Strains. The above construction is also very useful for the graphic determination of principal stresses and principal strains from given two-dimensional stress or strain conditions in any two rectangular directions, as the equations are exactly of the same form as (4) and (5). Thus, let OX, OY be two given rectangular directions, and let p, be the normal component and q the tangential compo- nent of a given stress on planes perpendicular to OY, and P2,q the normal and tangential components of a given stress on planes perpendicular to OX, the whole action being restricted to the plane XOY; and let 1, p, be considered positive when tensile, and q positive when it tends to produce positive sliding, i.e. when it tends to diminish the angle between +OX and + OY. Stresses and Strains in two Dimensions. 457 On OX take OM=g in magnitude and direction. From M draw MU=p, in magnitude and direction (7. e. upwards if tensile, downwards if compressive), and on UM take MV=p, Fig, 4. in such way that UV=p,+ 2. [In the figure p; and pe are both tensile, and ¢ is positive. | Bisect UV in T, and with centre T and radius TO describe a circle cutting UV in A, B. Then OA, OB are the directions of the principal planes, and UA, UB the corresponding principal stresses (p, Pp’): 2 For, if AOK=@, tan 20= 7, and ja aes a UA, UB=3(pi+p2) +3 V(Pi-P2) +40, which are the equations for the directions and magnitudes of the principal stresses. The principal strains 7, 7’ can also be shown by an extension of the same figure; for if E is Young’s modulus for the ma- terial, and 7 the ratio of lateral compression to longitudinal strain for a single stress, the following relations hold :— B(i+7/)=(1—) (pt+p); EG—1’)=(1+7) (p—p’). 458 Prof. M. A. Cornu on the Distinction between Hence, if on the diameter OTD a point T’ be taken such that OT’=(1+7) OT, and if through I’ a parallel to AB is drawn cutting OA, OB in A’, B’ respectively and DU, DV in U’, V’ respectively, we have U'V’=2U’T’=(1—n)(p+yp’), and A'B’/=2A/T’=(1+)(p—p’); so that U/A’=Mi, and Wes — Ei: Also, if with T’ as centre and T’O as radius a circle be drawn, it will pass through A’, B’, and will give the strains in any directions, in the same way as the circle OAB gives the stresses. For example, the stress on the plane OHH’ has the _ normal component UF, and the tangential component FE ; while the normal strain is proportional to UF” and the sliding (g) to 2E’H’. The figure also shows the relation between Hi and the coefficient (G) of elasticity of sliding; for FE = Gg, and 21’ Ki’ = Kg, therefore H:G=2P'H’ : FH=2(1+7). Coopers Hill, October 11, 1886. LIV. On the Distinction between Spectral Lines of Solar and Terrestrial Origin. By Prof. M. A. Cornu*. [Plate VIII.] ‘LA\RAUNHOFER, when he discovered the dark lines with which the solar spectrum is crossed, gave them names in order to facilitate description ; the principal lines were de- signated by the letters A, B,...H in such a way as to separate approximately the seven principal colours of the spectrum. The subsequent observations of Brewster, Dr. Gladstone, and M. Janssen proved that, notwithstanding the symmetry of denomination and the identity of appearance, these dark lines belonged to two distinct classes. Indeed, the one preserves always the same aspect, while the other becomes broader and darker as the sun approaches the horizon. The first, most of which have been identified with the bright lines due to metallic vapours (iron, magnesium, calcium, nickel, &c.), have been attributed, since the researches of Prof. Kirchhoff, to the absorption produced by metallic sub- stances in a state of vapour on the surface of the sun. ‘The other, in consequence of their intensity varying with the thickness of the atmosphere traversed by the sun’s rays, are explained by the selective absorption due to the cold gases or * Communicated by the Physical Society : read June 12, 1886. Spectral Lines of Solar and Terrestrial Origin. 459 vapours of the earth’s atmosphere. We ought then to distin- guish the solar from the telluric lines. Thus, of the eight principal Fraunhofer-lines, six are cha- racteristic of metallic elements and are of solar origin (C and F, hydrogen; D, sodium; H, G, iron; H, calcium) ; the other two (A and B) are telluric. Fraunhofer had besides distinguished two complex groups, namely a band a, very broad, in the extreme red, and a well- marked triple line, b, in the green; 0 is solar (magnesium), and a of terrestrial origin (fig. 1). Brewster, on discovering Fig. 1. Solar spectrum, with principal lines marked. new bands of variable intensity in the spectrum, added new designations; it is sufficient here to mention the band a situated in the orange, and the ,band 6 inthe yellow. These symbols have been adopted by Angstrom (Spectre normal du Soleil). Up to the present time it has been considered a difficult, and in any case a troublesome, matter to distinguish between these two kinds of lines. It was necessary, in fact, to observe the solar spectrum at two very different altitudes of the sun, under various meteorological conditions, to be able to affirm that the spectrum-lines do or do not change in intensity with the thickness of the atmosphere or the quantity of water- vapour traversed by the solar rays. The improvement of spectroscopes, in respect of the sharp- ness and especially of the dispersion of the lines, has allowed me to arrive ata method which renders the distinction between the two kinds of rays in a certain sense intuitive. This method is founded on a principle due to M. Fizeau*— the principle of the displacement of the spectral lines of the light emitted by a source which is in absolute or relative motion. We easily obtain the expression for the apparent wave-length X’ of a radiation from a point in motion with a * Bulletin de la Société Philomathique, décembre 1848; and Ann. de Chim. et de Phys. 4 série, t. xix. p. 211. 460 Prof. M. A. Cornu on the Distinction between relative velocity v, the true wave-length being A, viz. :— v=a(1- 3) This principle may be applied directly to the light emitted by the solar disk at the two extremities of an equatorial diameter. The absolute velocity v of a point on the solar equator is very sensibly 2 kilometres per second, that of the velocity of light, V, 300,000 kilometres per second. Hence we shall have a variation of wave-length equal to Di whet r | 300,000 + 150,000” + or — according as we take the same radiation at the eastern or western end of the solar equator. . If we wish to know numerically the magnitude of the dis- placement of a spectral line corresponding to this variation of wave-length, it is sufficient to substitute for X the numerical value which defines the region which we wish to observe. Consider, for example, the two D lines (A; =588°40, A. = 588'89). The displacement of a line having X=589 will be 589 An— it T5p UO If we wish to compare this displacement with the distance between the two D lines (i. e. with A,—A,=0°49), we shall have as the relative displacement, An 589 1 Mye—my 49x 150,000 ~ 1248 AN= +2 X + The double displacement will therefore be vee or ee of the distance between the two D lines. 1248 62-4 This total displacement, small as it is, is perfectly appre- ciable with spectroscopes of very high dispersion, such as the prism-spectroscope of M. Thollon, and the instruments with diffraction-gratings constructed by Prof. Rowland of the University of Baltimore. With the magnificent grating presented by Prof. Rowland to the Hcole Polytechnique, the double displacement may become sensible almost with all the points of the solar contour ; that is to say, even with those which are far from giving the maximum separation. The experimental method consists in causing the images of the two extremities of the solar equator to fall alternately on the slit of the spectroscope. Tor this purpose the solar beam is received on a condensing-lens, which produces in the plane of the slit a sharp image of the solar disk. The substitution Spectral Lines of Solar and Terrestrial Origin. 461 of one of the extremities for the other produces the double displacement of the solar radiations, while it has no influence upon the position of the absorption-lines of the atmosphere. The analysis of the optical conditions shows that perfect sharpness of the displacement can only be obtained when three experimental conditions are fulfilled :— 1. The spectral images at the focus of the telescope of the spectroscope must be aplanatic ; that is, such that the vertical lines (spectral lines) and the horizontal lines (due to the im- perfections of the slit) are equally sharp in the same plane. 2. This plane must coincide accurately with that of the cross wires. 3. The focal image of the solar disk must be exactly in the plane of the slit of the collimator. The omission of any one of these adjustments would involve an abnormal displacement of the lines consequent upon a dis- placement of the collecting-lens. On the other hand, when all these adjustments have been made the dark lines of terres- trial origin are perfectly fixed, while those of solar origin move with extreme sharpness. If we take as a fixed mark a dust particle on one of the cross wires, such as are always to be found on wires, we are able to distinguish immediately, at a glance, solar from telluric rays. Finally, I may add that the displacement becomes still more sensible when the substitution of one solar edge for the other is effected rhythmically. This is done by causing the collecting-lens to oscillate two or three times per second *. * The following (see fig. 2) is the arrangement employed :—The col- lecting-lens C is carried by a socket S S’ which rests on the table U U’ of Fig, 2. J : ” /P iD the spectroscope by two points P (one lies behind the other in the figure), about which it can turn. The optic centre C of the collecting-lens describes a small horizontal element, and transmits the same motion to the image of the solar disk. The observer produces this motion by the aid of the lever L, The pinion H serves to put the collecting-lens exactly in focus on the slit. 462 On Spectral Lines of Solar and Terrestrial Origin. The eye then becomes sensitive to the least oscillation of the lines. The distinguishing of the lines becomes then entirely intuitive. We observe each spectral line, and the result is such as would be produced by shaking it. If it remains im- movable, it is a telluric line ; if it oscillates, it is a solar line. The study of the telluric lines of the solar spectrum becomes therefore infinitely easier than heretofore. I have devoted to this study the fine days of the last few years, and the results have been very fruitful. Amongst the most interesting of the observations I may mention :— (1) the anatomy of Angstrém’s group « (see Plate VIII.), in which I have succeeded in detecting a group possessing the same constitution as the bands A and B, according to the beautiful observation of Prof. Langley ; (2) the telluric nature of a certain number of lines beyond the band 6; and (3) the solar origin of the line 1474 of Kirchhoff: this line is double under strong dis- persion, and as it oscillates we may conclude that the vapour which absorbs the radiations of which it takes the place is carried round by the rotation of the sun *. * The oscillation of the lines is not the only means of distinguishing the two kinds of lines. If (by means of a Wollaston’s double refracting- prism) we obtain two images of the solar disk in such a way that the two opposite extremities of the equatorial diameter are tangent and normal to the slit (fig. 3), the telluric lines T T’ of the spectra of the two images Fig. 3. S x Ss’ Fig. 5. as T’ s’ are upon the same line, while the solar lines SS’ are dislocated. This dislocation is very sharp if the adjustments enumerated above have been properly carried out; otherwise we obtain the confused appearance of fig. 4. If we employ a collecting-lens of very short focus, giving a very small image of the solar disk, it is easy to show that the appearance of Notices respecting New Books. 463 To sum up :—Physicists and astronomers have now in their hands an extremely useful method for studying the constitu- tion of the solar spectrum, a method which enables us to dis- tinguish at a glance between lines of solar and terrestrial origin. LV. Notices respecting New Books. The Volcanoes of Japan. By Joun Mitne. Transactions of the Seismological Society of Japan, Vol. IX. Part Il. 8vo. Yoko- hama, 1886: pp. 184, with a map and numerous sketches. N R. JOHN MILNE, F.G.S., occupies the whole of this Part of the Transactions with another of his important papers on the Volcanic Phenomena of Japan. Richas Japan is in the possession of Volcanoes, there does not appear to have been any native author who has endeavoured to describe in full these conspicuous land- marks. From the limited bibliography of native works on the subject (avowedly imperfect, and no doubt difficult to collect) given in this Memoir, we can only note one which, from its title, would appear to bea definite treatise, namely no. 26, Kyushi Kwazan Ron, by Aoe Shin, giving an account of the Volcanoes in Kyushu. Of European writers, Naumann, Von Drasche, and D. H. Marshall have contributed to our scanty knowledge; but it has been left for John Milne, in addition to his many notes on the subject, to supply us with this systematic and exhaustive work on the “ more important Volcanoes of Japan.” After the bibliography, which consists of 42 references, and which would have been more valuable had the dates been given, Mr. Milne proceeds at once to take the Volcanoes individually, on a plan of his own, not easy to follow and apparently with no definite arrangement. A minute account of the date and extent of each recorded eruption is given, together with many curious and inter- esting particulars as to the religious signification attached thereto, and the folk-lore concerning the mountain and its outbursts. This is followed by a geological description, and accompanied by an outline-sketch of the volcano, generally from a photograph. He finishes this part of the Memoir with a table of all the known eruptions, showing a comparison of the activity in summer and winter months. The Japanese measurements, often used here, should have been translated or reduced to their English equivalents, in a table, or throughout, instead of in the rare and scattered instances met with. One hundred and sixty-two pages are devoted to these im- portant points; and Mr. Milne then goes on to give his conclu- sions, from personal observation, that the two islands, [turup and Kunashiri, are older than any other members of the Japanese the figure (3) is modified. The telluric lines remain vertical, while the solar lines are not only dislocated, but become oblique (see fig. 5). We have here a very delicate test which lends itself to photographic obser- vations. 464 Notices respecting New Books. group, and that they “form the first of a series of stepping- stones which connect Japan, by means of Kamschatka, with the remainder of Asia.” A few observations on the slope of the cones follow; and Mr. Milne records one instance, that of the “small cone rising from the upper crater of Cha-cha-Nobori,” in which the inclination is 37°. A reference to another cone—Atatsu-Nobori, at p. 147,—‘‘one side having a slope of 50°, and the other of 49°”—might have been repeated here. Mr. Milne says that Atatsu-Nobori is ‘the steepest volcanic cone I have yet seen.” Of the Kuriles, Mr. Milne writes, ‘“‘They are, so to speak, amongst the last of the links which together build up the Volcanic chain which bounds the shores of the West Pacific,” and “ they are probably contemporaneous with the younger volcanoes of Kam- schatka and Japan.” The author lays great stress on the importance of the Japanese volcanoes as land-builders, and refers the low altitude of the Kuriles, as compared with the Volcanic Peaks of Klutchewsk (16,500 ft.) in Kamschatka, and of Fuji-yama (12,450 ft.), to the fact that they were “probably built up from the bottom of an ocean which is perhaps the deepest in the world.” The rocks that he and his assistants collected appear to be augitic andesites. The absence of lava-streams is noticed, and pointed out as “ sug- gestive of the way in which these islands have been built up ”— by cindery accumulations or ash-beds. One of the most im- portant parts of the Memoir is the map of the Volcanoes of Japan, in which the exact positions of one hundred and twenty-nine active or extinct craters are indicated. In a table accompanying it, a note is given of the height, and the nature of rock, with general remarks concerning each volcano. Of the 129, fifty-one are still active. Thirty-nine have symmetrically formed cones. The paper concludes with well-considered remarks on the Relative Age of all of them, the characters of their Lavas and other Rocks, the intensity of Eruptions, and the general form of the Volcanoes. To assist the student in followmg the author on the last-mentioned subject he has reproduced the diagrammatic plate from the ‘Geological Magazine’ for 1878, and added a plate of various profiles of Fuji-san (“called by foreigners Fuji-yama”) from photographs and surveys of the mountain ; and he quotes an inter- esting series of causes which help to modify the natural curvature of a volcano. Some important notes also are given as to the height of Fuji-san; and the conclusion Mr. Milne arrives at is that the proper height should be taken as between 12,400 and 12,450 feet. The memoir is profusely illustrated with lithograph sketches, chiefly exact outlines of the various cones referred to. Discussions on Climate and Cosmology. By JamuEs Crouu, LL.D., F.RS. Edinburgh: Adam and Charles Black. 1885. Ir is now nearly a quarter of a century since Dr. Croll first enun- ciated the outline of the theory of the cause of glacial phenomena, Notices respecting New Books. 465 a problem which previous to that time had engaged and perplexed many minds. ‘Till Dr. Croll’s famous theory appeared in the pages of this Magazine, no hypothesis bad been propounded which satisfied the conditions of the problem even as known at the time; while the various tentative explanations for the most part demanded the calling into existence of agencies and of terrestrial changes which would themselves be more difficult to account for than the phe- nomena they were meant to explain. It was from the first seen and conceded that the varying eccentricity of the ecliptic, upon which Dr. Croll bases his theory, has a real existence in nature ; and the only points then, and now, in dispute have relation to the manner and extent to which these cosmical causes affect terrestrial climate and conditions. arly in the century, Sir John Herschel, Arago, and others had given attention to the question of the rela- tion of eccentricity to climate; and they came to the conclusion that it neither affects the amount of heat received from the earth nor, to any appreciable extent, its distribution. Dr. Croll thus brought out his theory in the face of, and against, the confidently expressed opinion of some of the greatest authorities in physical science. The door which he sought to open had been officially barred for a long generation, and when he pushed through it he found no beaten path on the other side. Notwithstanding all that had been written about the Glacial Epoch, the facts and phe- nomena of the period were but imperfectly stated and understood, and the science of Climatology was either non-existent or in the most chaotic condition. Gradually and laboriously he built up his theory ; he accumulated facts and observations with the most pains- taking industry ; and these he marshalled and arranged with con- summate skill, till he built up and solidly buttressed one of the most important and far-reaching doctrines which has been enun- ciated in the whole range of geological science. In dealing with the climatological and other physical facts and bearings of his sub- ject, Dr. Croll had little help from the investigations of those who had gone before him. He had, indeed, to combat many erroneous notions which had become generally entertained from the works of popular writers. Of the great physicists, some had given opinions directly opposed to Dr. Croll’s contentions; others had given no consideration to the climatological relations of terrestrial and cos- mical phenomena; and many were by no means agreed as to the effects of such phenomena. Thus there was scarcely any ready- made help available for incorporation with Dr. Croll’s work; he had few witnesses to call in his favour; he had many stumbling- blocks to remove, much to explain away, and much to argue against. With indomitable patience and perseverance he set himself to his task ; with calmness and temperance, and yet with marvellous in- tellectual alertness, he met the arguments of opponents, adhering to his position, and maintaining his view, with modest tenacity and resolution, which commanded the respect and esteem of his most powerful opponents. | The greater part of what Dr. Croll has written on the subject of 466 Notices respecting New Books. geological climate, and the principal controversies which have arisen in connection with the theory, have appeared in the pages of this Magazine, and may be presumed to be familiar toits readers. His own contributions to the controversy during the last ten years Dr. Croll has now gathered together, and presented to the public in the volume which forms the occasion of this notice. The work is largely controversial in tone, more taken up with maintaining the position of the theory as enunciated in ‘ Climate and ‘Time’ than in broadening its foundations by new views, or even in resolving difficulties and clearing the logical path along which Dr. Croll leads his adherents. Dr. Croll is not prepared to accept any of the modi- fications of his theory which have been proposed; still less is he inclined to coincide in the arguments which would be subversive of his position. In the preface to this volume he intimates that he has now spoken his last word in defence of his theory. ‘“ There are many of the topics discussed,” he says, ‘‘ which I could have wished to consider more at length; but advancing years and declining health constrain me to husband my remaining energies for work in a wholly different field of inquiry—work which has never lost for me its fascination, but which has been laid aside for upwards of a quarter of a century.” Such a decided intimation of withdrawal from a controversy in which he has so long been the principal figure surely forms a fitting occasion for an acknowledgment of the vast and varied services Dr. Croll has rendered to an obscure branch of science, of the amount of intellectual activity and industry to which his writings have given rise, of the large measure of light he has thrown on what appeared to be the most complex and puzzling phenomenon which meets the student at the very gateway of geo- logical investigation, and to the stimulating and suggestive models he has afforded for dealing with the ravelled skein of geological history in many of its departments. Dr. Croll’s theory, all opposition notwithstanding, holds the field. Many have taken objection to the entire fabric, and still more have been opposed to various propositions by which the theory is sup- ported; but no opponent has been able to suggest that most powerful of all arguments—a counter-theory. It is almost a me- lancholy sight to see the learned President of the British Association falling back upon the antiquated Lyellian doctrine. Whatever may satisfy the conditions of the problem of Glacial periods, it is now well known that Sir Charles Lyell’s theory of polar continents and tropical oceans will not supply the key; but Principal Dawson can discover, or has at his service, no other. But although no effective counter-theory has been started, even as a stalking-horse, against the Croll doctrine, it is not to be concluded that the con- troversy is at an end, and that henceforth the theory that periodic changes of climate are primarily caused by Eccentricity is to be accepted as an article of faith in the schools of geological and phy- sical orthodoxy. It would indeed be a misfortune were the con- troversy, which has gone on so briskly for about twenty-five years, to be allowed to subside. Authorities who are prepared to accept Intelligence and Miscellaneous Articles. 467 the cardinal points of the doctrine are by no means agreed as to many of the assumptions on which it is based, nor as to the issues of the argument for the case. The whole subject is compa- ratively new, and the men who are entitled to express an opinion on it are not numerous. LHre a general concensus of opinion can be arrived at much investigation and debate are necessary ; foot by foot solid ground must be conquered, and it will be found, as truth emerges, that it willillumine, not only much that is yet obscure in the theory of climatic perturbations, but that, like all truth, it will become the fertile mother of a long line of exact knowledge. LVI. Intelligence and Miscellaneous Articles. ON THE MAGNETIC ROTATION OF MIXTURES OF WATER WITH SOME OF THE ACIDS OF THE FATTY SERIES, WITH ALCOHOL, AND WITH SULPHURIC ACID; AND OBSERVATIONS ON WATER OF CRYSTALLIZATION *. BY W. H. PERKIN, PH.D., F.R.S. — the Author’s previous work on the magnetic rotation of compounds it was found that the molecular magnetic rotation of water, which is taken as unity, is not the same as the sum of the values of oxygen and two of hydrogen, as deduced from the molecular magnetic rotation of other bodies. Thus hydrogen is found to be 0°254, whilst hydrogen in hydroxy] varies from 0:194 in ordinary alcohol to 0-137 in monobasic acids, and is 0-261 in carbonyl; so that, taking the lowest number, it gives H,+O= 0-645, and taking the highest it is 0°769. From these facts it appeared that the determination of the mag- netic relation of hydrated bodies might give numbers which would show whether they contained water or whether the substances with which it was mixed had combined so as to form new compounds. If the former were the case the molecular rotation should represent that of the compound + that of water ; if the latter, it should be lower than this. For example, if formic acid were mixed with water, molecular proportions being used, either H.COOH+H,0 or H.C(HO,) might be produced. In the first case the rotation should be WOLMIG ACIG! oars - Se aI: 1671 ANG aera eller Beagle Stee Na er 1:000 2-671 In the second, taking the highest value of H,+ 0, it would be GRIME eI. cece os cS sks 671 Nia ae he ey eh ce ces 0°769 2-340 which is considerably lower. * Discussion on the “ Nature of Solution,” British Association, Birm- ingham Meeting, Section B. 468 Intelligence and Miscellaneous Articles. As some of the fatty acids have been believed to unite with water, to form trihydric alcohols, they were elected for examination ; and at the same time hydrated alcohol was also examined, because it could not form a compound with water, and would therefore act as a check upon the results. The results of the examination show that formic, acetic, and propionic acids, when mixed with water, do not form new com- pounds, but that the products simply consist of these bodies and water. Sulphuric acid in the pure and hydrated conditions was next examined, viz. H,SO,, H,SO,+H,0, H,SO,+2H,0, and H,SO,+ 3H,O. The numbers obtained in this case show that combination takes place chiefly when one molecule of water has been added, only to a small extent in the case of the second addition, and scarcely at all when the third is added; and the author gives reasons for considering that sulphuric acid combines with one molecule of water only, forming the compound (HO),SO. Whilst studying the nature of the hydrated products, the author’s attention was drawn to the subject of water of crystallization ; and from the inconsistency as to the presence or absence of water of crystallization in compounds of the same class—as, for example, in those of silver, potassium, and sodium, and also methyl bromide, as compared with analogous compounds—he considers it impossible to believe that water of crystallization has any relationship to chemical combination ; that of course refers to water when it exists as such, and not to hydrogen and oxygen present in the proportions found in water but otherwise combined. If this be so, it is thought that its association with chemical compounds is most likely con- nected with the building-up of the crystalline form, it being diffi- cult to see what other part it can play ; and the reason why some compounds crystallize without and some with water of crystalliza- tion is probably determined by the tendency to produce that form which can be the most readily built up: if that form can result from the anhydrous salt, anhydrous crystals are formed ; if with the salt and water, then the crystals will contain water of crystallization ; and it is well known how change of conditions will cause variation in the proportions of water of crystallization, and also form of crystal. These observations would also apply to compounds crystallizing with alcohol, acetic acid, benzene, &c., and to some double salts, where one or more of the constituents would act like water of crystallization. Attention is drawn to the fact that if the above view of water of crystallization be correct, it is evident that a salt containing water of crystallization will be resolved into water and the salt on breaking up of the crystalline form by solution, which is believed to be the case by many who have studied the subject of solution.— The Chemical News, Oct. 22, 1886. THE LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIES. ] DECEMBER 1886. LVII. Notes on Electricity and Magnetism.—ll. The Selj- induction and Resistance of Compound Conductors. By Lord RayueieH, Sec. £.S., D.C.L.* _ alee entering upon the proper subject of this com- munication, I take the earliest opportunity of correcting a mistake of some importance into which I fell in connection with induction-coils. From the fact that a closed magnetic circuit acts very ill as a store of energy, I argued that closed electromagnets were to be condemned as secondary generators (Phil. Mag. August 1886, p.179). Dr. Hopkinson and Prof. Ewing have independently drawn my attention to the fact that the success of induction-coils does not depend upon storage of energy, and that they might work with high effi- ciency even though the whole of the energy put into the iron were wasted. Prof. Hwing’s remark, to which I took excep- tion, is thus perfectly correct. In his inaugural address to the Society of Telegraph Engi- neers, and in a subsequent communication to the Royal Societyt, Prof. Hughes has described a series of interesting experiments, which have attracted a good deal of attention in consequence both of the official position and known expe- rimental skill of the author. Some of the conclusions which he advances can hardly be sustained, and have met with severe criticism at the hands of Weber, Heaviside, and others. There * Communicated by the Author, + Journ. Tel. Eng. vol. xv. (1866) p. 1. t Proc. Roy. Soc. vol. xl. (1886) p. 451. Phil. Mag. 8. 5. Vol. 22. No. 189. Dec. 1886. 2K 470 Lord Rayleigh on the Self-induction and are certain other points raised by him, or suggested by his work, which seem worthy of consideration ; and I propose in the present paper to give an account of some investigations, mainly experimental, carried on during the summer months, which may, I hope, tend to settle some controverted questions. Prof. Hughes’s first apparatus consists of a Wheatstone’s | quadrilateral, with a telephone in the bridge, one of the sides of the quadrilateral being the wire or coil under examination, and the other three being the parts into which a single German-silver wire is divided by two sliding contacts. If the battery-branch be closed, and a suitable interrupter be intro- duced into the telephone-branch, balance may be obtained by shifting the contacts. Provided that the interrupter introduces no electromotive force of its own™, the balance indicates the proportionality of the four resistances. If P be the unknown resistance of the conductor under test, Q, R the resistances of the adjacent parts of the divided wire, 8 that of the opposite part (between the sliding contacts), then, by the ordinary rule, PS=QR; while Q, R, 8S are subject to the relation Q+R+S=W, W being a constant. If now the interrupter be transferred from the telephone to the battery-branch, the balance is usu- ally disturbed on account of induction, and cannot be restored by any mere shifting of the contacts. In order to compensate the induction, another influence of the same kind must be introduced. It is here that the peculiarity of the apparatus lies. A coil is inserted in the battery and another in the telephone-branch, which act inductively upon one another, and are so mounted that the effect may be readily varied. The two coils may be concentric and relatively movable about the common diameter. In this case the action vanishes when the planes are perpendicular. If one coil be very much smaller than the other, the coefficient of mutual induction M is proportional to the cosine of the angle between the planes. By means of the two adjustments, the sliding contact and the rotating coil, itis usually possible to obtain a fair silence. In his address Prof. Hughes interpreted his observations on the basis of an assumption that the self-induction of P was represented by M, irrespective of resistance, and that the resistance to variable currents could (as in the case of steady currents) be equated to QR/S. In the discussion which fol- lowed I pointed out that this was by no means generally true, * This condition is not always satisfied: With the reed-interrupter (see below) a loud sound may sometimes be heard, although the batterv- branch be open. Resistance of Compound Conductors. 471 and [ gave the following formulz* as applicable to the case in which the only sensible induction among the sides of the quadrilateral is the self-induction L of the conductor P:— QOR—SP=pMb,... «+ (Cd) Per + @-PR+ S)=Sh., yo sO) The electrical vibrations are here supposed to follow the har- monic law, with frequency p/27. “It will be seen that the ordinary resistance balance (SP=QR) is departed from. The change here considered is peculiar to the apparatus, and, so far as its influence is concerned, it does not indicate a rea alteration of resistance in the wire. Moreover, since p is involved, the disturbance depends upon the rapidity of vibra- tion, so that in the case of ordinary mixed sounds silence can be attained only approximately. Again, from the second equation we see that M is not in general a correct measure of the value of L. If, however, P be very small, the desired condition of things is approached ; since, by the construction of the apparatus, Q+R+8 is constant (say W), and if P be small enough 8 does not differ much from W, 7. e. most of the wire forming the three sides of the combination is devoted to the member opposite to P.”’ The formule are easily proved. Since there is no current through the bridge, there must be the same current (#) in P and one of the adjacent sides (say) R, and for a like reason the same current y in the sides Q and 8. The difference of potentials at time ¢ between the junction of P and Rand the junction of Q and S may be expressed by each of the three following equated quantities :— dx A(e+y) _ ; Qy—Px LT= M e =Re—S8y; from which the required results are obtained by elimination of the ratio w: y, and introduction of the supposition that all the quantities vary harmonically with frequency p/27. The inadequacy of Prof. Hughes’s original interpretation has been remarked upon also by Prof. Webert and by Mr. O. Heavisidet, who have obtained the corrected formule. I give them here because I agree with Prof. Weber that ‘this form of apparatus possesses distinct advantages. As he points out, if P be known, the application of (2) really presents no difficulty, and allows of L being readily found in terms of M. * Journ. Tel. Eng. vol. xv. p. 54. “ + El. Rev., April 9, 1886; July 9, 1886, t Phil. Mag. August 1886. 2K2 472 Lord Rayleigh on the Self-induction and There are many cases in which we may be sure beforehand that P (the effective resistance of the conductor, or combina- tion of conductors, to the variable currents) is the same as if the currents were steady, and then P may be regarded as known. There are -other cases, however,—some of them will be treated below,—in which this assumption cannot be made ; and it is impossible to determine the unknown quantities L and P from (2) alone. We may now fall back upon (1). By means of the two equations, P and I. can always be found in terms of the other quantities. But among these is included the frequency of vibration ; so that the method is only prac- tically applicable when the interrupter is such as to give an absolute periodicity. A scraping contact, otherwise very convenient, is thus excluded*; and this is undoubtedly an objection to the method. My own experiments have been made with three different forms of apparatus. The first was constructed upon the model of that originally described by Hughes, and still to be pre- ferred for some purposes. The others will be described in due course ; but it will be convenient to consider first those parts which are common—the interrupters and the induction- compensators. The Interrupters. When regular vibrations are not required, a scraping con- tact interrupter is the least troublesome. Mine is of the roughest possible construction. It is driven by a small jet of water issuing from a glass nozzle in communication with a tap, and impinging upon blades bent in a piece of tin plate and revolving about a vertical axis. The upper part of the axis carries a small cylinder of roughened iron, against which a brass spring lightly presses. As in Hughes’s apparatus, the scraping contact is periodically broken altogether by a pro- jecting finger, which during part of the revolution pushes back the brass spring. This is a point of some importance, for a faint scraping sound is far better heard and identified when thus rendered intermittent. The apparatus stands in the sink, so that the water scattered from the revolving blades runs away without giving trouble. The pressure exercised by the contact-spring requires readjustment from day to day if the loudest sound is wanted. But for many of the most interesting experiments a scraping contact is unsuitable. Prof. Hughes has found, indeed, that * A toothed-wheel interrupter, as usually employed, does not give a regular vibration of the period corresponding to the passage of a tooth, Resistance of Compound Conductors. AT3 in some cases the natural pitch of the telephone-plate is pre- dominant; so that the vibration, as it reaches the ear, is not quite so mixed as might have been expected from its origin. When, however, the induction and resistance under observa- tion are rapidly varying functions of the frequency of vibra- tion, it is evident that no sharp results can be obtained without an interrupter giving a perfectly regular electrical vibration. With proper appliances an absolute silence, or at least one disturbed only by a slight sensation of the octave of the principal tone, can be obtained under circumstances where a scraping contact would admit of no approach to a balance at all. A thoroughly satisfactory interrupter of this kind has not, to my knowledge, been constructed. Tuning-forks, driven electromagnetically with liquid or solid contacts, answer well so long as the frequency required does not exceed 128 or 256 per second; but here we desire frequencies of from 500 to 2000. My experiments have been made with harmonium- reeds as interrupters, the vibrating tongue making contact once during each period with the slightly rounded end of a brass or iron wire, which can be advanced exactly to the required position by means of a screw cut upon it. Blown with a well regulated wind, such reeds have given good results even up to 2000 (complete) vibrations per second ; but they are often capricious and demand frequent readjust- ment. The reed which I have usually employed makes about 1050 vibrations per second, and answered its purpose fairly well. Hitherto I have not been able to satisfy myself as to the cause of the falling off in efficiency, which often sets in suddenly, and persists until cured by a readjustment. Another objection to this interrupter is the simultaneous pro- duction of loud aerial sounds, which must be prevented from reaching the ear of the observer at the telephone by several interposed doors. | The Induction- Compensators. Two instruments, similar in all respects, were made by my assistant Mr. Gordon, much after the pattern employed by Prof. Hughes. In each there is a small coil mounted so that one diameter coincides with a diameter of a larger coil, and movable about that diameter. The mutual induction M between the two circuits depends upon the position given to the smaller coil, which is read off by a pointer attached to it and moving over a graduated circle. The circles are so divided that the reading (@) would * be zero when the aves * The position is mechanically unattainable, ATA Lord Rayleigh on the Self-induction and of the coils were coincident, or the planes parallel. In this position M is arithmetically a maximum (M,); and we con- sider its algebraic sign to be positive. At 90°, when the axes are at right angles, M=0. At 180° M would be nega- tive, and of the same arithmetic value (M,) as at O°. The coils are wound upon boxwood rings, and in each there are 45 convolutions. The mean diameters are about 3 inches and 14 inch. Some of the earlier experiments were interpreted by a theory of the compensator, which I knew at the time to be very rough. If the small coil be treated as infinitely small, then M=M, cos 0. On the same supposition we have, from the roughly measured dimensions, M,= 60,000 centim. The law of the simple cosine was found to lead to consi- derable anomalies; and when at a later date (August 19) I carried out my intended calibrations, some very curious results revealed themselves. The best arrangement for calibration and for determination of the constant of the instruments is to institute distinct pri- mary and secondary circuits. ‘The former included a battery, a scraping contact (p. 472), and the two outer (larger) coils of the compensators. The latter included the two inner coils and a telephone. The precise procedure will depend upon whether we can assume the exact equality of the two com- pensators. In that case we may introduce, and retain during the observations, another pair of induction-coils, one of course in the primary and the other in the secondary circuit, and of such power as to produce a displacement of about 30° of the compensator. Thus, while in the absence of the additional coils, balance would be obtained when both compensators stand at 90°, their introduction would lead to such readings as 90°, 60°; 100°, 70°; 110°, 80°; &c. By this means various parts of the scale of one compensator can be compared with non- corresponding parts of the other; and this is sufficient if the two are similar. This method was used ; butit is perhaps better to arrange so that each compensator is calibrated independently of the other. In this case alternate readings are taken with and without the cooperation of the additional coils; and the equivalent induc- tion is found for each compensator at various parts of its scale. The following set of readings will give an idea of the modus operandi. Resistance of Compound Conductors. 475 Additional coils in. Additional coils out. Reading of I, | Reading of IT. | Reading of I. | Reading of II. 92 30 92 432 1023 423 1023 58 113 53 113 64 1243 64 1242 754 1354 73 1353 863 1483 863 1483 98 Tt will be seen that the adjustment is made alternately on the two compensators. Thus in the second compensator the steps 80°-421°, 423°-53°, 53°- 64°, &e. have all the same value, whatever may be the construction of the first compen- sator, which indeed need not be graduated at all. In like manner, the steps from 92°—-1024°, 1023°- 113°, &e. on the first compensator have an equal value. An examination of these and other results, not here re- corded, leads to the unexpected conclusion that from 40° to 140°, 7. e. through a range of 100° about the perpendi- cular position, the scale of induction does not differ appre- ciably from the scale of degrees. From 30° to 40°, or from 140° to 150°, the induction is something like a tenth part. less than that corresponding to 10° in the neighbourhood of 90°. Within the whole mechanical range of the instruments, from 30° to 150°, there could scarcely be an error of 2 per cent. in assuming M proportional to the angle measured from perpendicularity, 2. e. (4a4—8@), or, say, &. The general explanation of this very convenient property is not difficult to understand ; since for high values of 6’ the approximation over the whole circumference of coils of not very unequal diameters must lead to a more rapid increase of M than if the smaller coil were very small; and it is con- ceivable that for some particular ratio of diameters the increase may just so much exceed that represented by sin 6’, as to correspond nearly to @’. I was desirous, however, of explaining this very peculiar relation more completely, and have therefore developed the theory for the case of a ratio of 2:1 (nearly that of my apparatus) on the basis of formule given by Maxwell. The details of the calculation are given in the form of an appendix (p. 498). 476 Lord Rayleigh on the Self-induction and It may suffice here to say that the experimental result is abundantly confirmed ; and that reason is found for the con- clusion that the proportionality of induction to angle would be even better maintained if the diameter of the smaller coil . were increased from ‘50 to ‘55 of that of the larger. The non-mathematical reader may be content to accept this pro- portionality over most of the range of the actual instruments upon the experimental evidence. The absolute value of the induction-coefficient corresponding to each degree of the compensators was determined at the time of the calibration by comparison with the calculable induction- coefficient between two coils wound in measured grooves cut on the surface of a wooden cylinder. These coils contained respectively 21 and 22 convolutions; and the induction- coefficient for the mean windings is found to be 277°3 centim. by a calculation of which it is not necessary 2 record the details. Hence, for the actual coils, M=21 x 22 x 277°3=1°281 x 10° centim. The obvious procedure for the comparison would be to com- bine the compensators without additional coils, so as to obtain a balance at the telephone when both stand at 90°, and then to observe the displacement or displacements necessary when the standard cylinder coils are introduced. In my case, however, the range of induction provided by the compensators was insufficient to balance the standard, if used in this way, even when displacements were made in such (opposite) directions as to cooperate in changing the total induction: An addi- tional pair of coils was therefore introduced, for the purpose, as it were, of shifting the zero, of which nothing required to be known, since they remained connected whether or not the standard coils were in operation. With this modification, balance could be attained in both cases. With standard coils in, one pair of readings was 130°, 434°; and with standard baa out, 50°,1274°. The commechione were such that when there was no external change, the cor- responding readings would move in the same direction (p.475) ; and thus the number of degrees of one compensator equivalent to the standard is 130 —504+1274—421, viz. 165. Accordingly, every degree of either compensator, not too far removed from the middle of the range, represents 776°3 centim. of induction-coeflicient. The maximum coefficient, when 9=0, according to Table IT. (Appendix), would be about 56100 centim, Resistance of Compound Conductors. AT7 In view of the statement * “that the coefficient of mutual induction is less in iron than in copper wires,’ I may mention an experiment made with the aid of one of the compensators, in which the effect of the substitution of iron for copper is directly examined. The mutual induction measured is that between two circuits, one of which was composed of the two copper coils of 21 and 22 convolutions spoken of above connected in series; and the other of a single turn of wire situated midway between, and lying in a shallow scratch or groove on the wooden cylinder, by which its position was accurately defined. The arrangements being the same as in the determination of the constant of the compensator, the value of the double induction (obtained by reversal) between the circuit of a single turn of copper wire and the circuit of 43 turns was40°°7. The single turn of copper wire was now replaced by a turn of iron wire of equal diameter and bedded in the same scratch, with the result that the double induction was 40°°6, the same value being obtained whether the iron wire were inciuded in the primary circuit with the battery and interrupter, or in the secondary with the telephone. Care had, of course, to be exercised in the disposition of the leads, in consequence of the use of a single turn only for one of the circuits. So far as the experiment could show, the induction is absolutely the same, whether the single turn be of iron or of copper. To return now to the bridge arrangement, the following are a few examples of the use of the original form of appa- ratus. The scale of the wire readings was in 3), inch, the whole length (Q+R+8) being 1960. In ohms the resist- ance of the whole wire is 4:00. The interrupter was the “reed,” making about 1050 (complete) vibrations per second. Thus p=2r x 1050. The first case is that of a helix of insulated copper wire, without core of any kind. To get a balance the compensator had to be placed at 954°, so that M=386°, each degree re- presenting 776 centim. ‘The resistances also necessary were Q=610, R=190; therefore S=1160. They are expressed in scale-divisions, the value of each of which is 4:00 x 10° , centim. eee ei eee If, as we are almost entitled to do, we assume that the resist- * Proc. Roy. Soc. vol. xl. p. 468, 478 Lord Rayleigh on the Self-induction and ance P to variable currents is the same as that readily found with use of steady currents, viz. 37°3 scale-divisions, we may at once deduce P+Q+R+8 386x776 x 2833 S ia 1160 We will, however, dispense with this assumption. Hlimina- ting L between the two equations, we get for the determina- tion of P, pie Fe ao Af ee Oe}. L=M = 68200 centim. ata In the fractions containing M? the resistances must be expressed in absolute measure. We find pPM? _ 4a? x 1050? x 36? x 776? See TCO 1020220 pM?(Q+R+8)_ 47’ x 1050? x 36? x 776? x 1960 S.Q.RiY)~ 1160 x610'« 190 x 1072-0 ='0061 ; =o so that — P=-876 Ee. differing some 12 per cent. from the value (QR/S) given by the usual formula. Inserting the values of Q, R, 8, we have P=87'5 scale-divisions. This is the effective resistance to variable currents of the frequency in question. With steady currents the readings were Oot, ke 190) Si lr so that Dat KeLIO ee | Di = 87:3. The resistance to variable currents, calculated by the correct formulee with knowledge of the frequency of vibration, is thus almost identical with the value found with steady currents; whereas if we were to ignore the disturbance of the ordinary resistance rule by induction, we should erroneously conclude that the resistance to variable currents was some 12 per cent. higher than to steady currents. _ Of other experiments made with this coil I will only men- tion one. When a stout copper rod was inserted, the circum- Resistance of Compound Conductors. ATY ferential secondary currents induced in it altered the readings with variable currents to = 660, jR—190;.. M292". The effective self-induction is evidently diminished, and the effective resistance increased in accordance with the universal rule*, The precise values may be obtained from our two fundamental equations, in the manner exemplified above. If the foregoing experiment (with the copper core) be attempted with the scraping-contact interrupter, giving a mixed sound, no definite balance is obtainable. The second example that I shall give is of a wire of soft iron about 14 metre long and 3:3 millim. diameter. Here with variable currents from the reed-interrupter, of the same period as before, Q=178, R=190, S=1592; M=8x 776 centim. ; from which we find p=-935 2-7 = 20-93 scale divisions In the present case the ordinary simple rule (QR/S) would lead to an error of 14 per cent. only. The resistance to steady currents is given by 10 £90) 22! Po= apap = 11:38. We may conclude that the effective resistance to variable currents of this frequency (1050) is 1°84 times the resistance to steady currents. A long length of wire from the same hank was examined later by another method (p. 488), and gave for the ratio in question 1°89. In some of his experiments | Prof. Hughes found that it made but little difference to the self-induction of an iron wire, whether it was arranged as a compact coil of several turns, or asa single wide loop. The question is readily examined with the present form of apparatus; for, since the resistance is not altered, the compensator readings give an accurate relative measure of the self-induction. A hank of nineteen convolutions of insulated soft iron wire required for balance 25°-8; but when opened out into a single (approximately cir- cular) loop, the reading was only 11°2,a much greater differ- ence than that mentioned by Hughes. * See equations (8), (10), (11), (12), Phil. Mag. May 1886, pp. 373- 375. + Proc. Roy. Soe, vol. xl. p. 457. 480 Lord Rayleigh on the Self-induction and A better experiment may be made with a coil of doubled .wire. The length previously used was divided into halves, which were tied together closely with cotton thread and bent into a compact coil of 9 convolutions and about 44 inches dia- meter. When the two wires were connected in series, in such a manner that the direction of electric circulation was the same in both, the self-induction was represented by 242 ; but when one wire was reversed, the self-induction fell to 9°°2, the large difference depending entirely upon the mutual induction between the two iron wires. I had intended to apply this apparatus to investigate the self-induction of wires of various materials and diameters, formed into single circular loops, but the subject has been so ably treated by Prof. Weber as to render further work un- necessary, at least as regards the non-magnetic metals. That the circle is the proper standard form for accurate measurement cannot be doubted ; but the effect of magnetic quality is shown most markedly when the wires compared are of given length and diameter, and doubled so as to form single close loops. For a total length 2/ of copper wire, the self- induction is smallest when the wires are just in contact, and then * L=3°772 1. In practice some interval is required for insulation, so that the coefficient of 1 may perhaps be taken to be 4. To iron wires the theory is not strictly applicable t, but we may pro- bably assume without serious error L=/(4 log 2+ w)=1(2'7724 pw), p being the permeability. Prof. Hughes finds for the ratio of iron to copper under these circumstances 440 : 18 {; accord- ing to which we should have d+m 40 SEA nS” or #=95, in approximate agreement with values found by other methods. Although the original apparatus of Hughes is capable of very good results, and is especially suitable when the wires under test are in but short lengths, the fact that induction and resistance are mixed up in the measurements is a decided drawback, if it be only because the readings require for their interpretation calculations not readily made upon the spot. * Maxwell’s ‘ Electricity and Magnetism,’ §§ 686, 688. } Phil. Mag. May 1886, p. 383. $ Loe. cit. p. 457. Resistance of Compound Conductors. 481 The more obvious arrangement is one in which both the in- duction and resistance of the branch containing the subject under examination are in every case brought up to the given totals necessary for a balance. To carry this out conveniently we require to be able to add self-induction without altering resistance, and resistance without altering self-induction, and both in a measurable degree. The first demand is easily met. If we include in the circuit the éwo coils of an induction-com- pensator, connected in series, the self-induction of the whole can be varied ina known manner by rotating the smaller coil. For the self-induction of the instrument, used in this manner, may be regarded as made up of the constant self- inductions of the component coils taken separately, and of twice the positive or negative mutual induction between them. The first part, in consequence of its constancy, need not be regarded; and thus every degree (within the admissible range) may be taken as representing 2x 776°3, or 1552°6 em., of self-induction. The introduction, or removal, of resistance without altera- tion of self-induction cannot well be carried out with rigour. But in most cases the object can be sufficiently attained with the aid of a resistance-slide of thin German-silver wire. It may be in the form of a nearly close loop, the parallel out- going and return parts being separated by a thin lath of wood. A spring of stout brass wire making contact with both parts short-circuits a greater or less length of the bight. In the Wheatstone’s quadrilateral, as arranged for these experiments, the adjacent sides R, 8S are made of similar wires of German silver of equal resistance ($4 ohm). Being doubled they give rise to little induction, but the accu- racy of the method is independent of this circumstance. The side P includes the conductor, or combination of conductors, under examination, an induction-compensator, and the resis- tance-slide. The other side, Q, must possess resistance and self-induction greater than any of the conductors to be com- pared, but need not be susceptible of ready and measurable variations. But, asa matter of fact, the second induction- compensator was used in this branch, and gave certain advan- tages in respect of convenience. Sometimes also a rheostat was included ; but during a set of comparisons the condition of this branch was usually maintained constant, the necessary variations being made in P. In order to avoid mutual in- duction between the branches, P and Q were placed at some distance away, being connected with the rest of the apparatus by leads of doubled wire. It will be evident that when the interrupter acts in the 482 Lord Rayleigh on the Self~induction and battery branch, balance can be obtained at the telephone in. the bridge only under the conditions that both the aggregate self-induction and resistance in P are equal to the correspond- ing quantities in @. Hence when one conductor is substi- tuded for another in P, the alterations demanded at the com- pensator and in the slide give respectively the changes of self-induction and of resistance. | In this arrangement the induction and resistance are well separated, so that the results can be interpreted without cal- culation. During the month of July a large number of observations on various combinations of conductors were effected, but the results were not wholly satisfactory. There seemed to be some uncertainty in the determination of resis- tance, due to the inclusion of the two movable contacts of the resistance-slide in one of the sides (P) of the quadrilateral *. I therefore pass on to describe a slight modification by means of which much sharper measurements were attainable. In order to get rid of the objectionable movable contacts, some sacrifice of theoretical simplicity seems unavoidable. We can no longer keep Q (and therefore P when a balance is attained) constant ; but by reverting to the arrangement adopted ina well-known form of Wheatstone’s bridge, we cause the resistances taken from P to be added to Q, and vice versa. The transferable resistance is that of a straight wire of German silver, with which one telephone terminal makes contact at a point whose position is read off on a divided scale. Any uncertainty in the resistance of this contact does not influence the measurements. Fig. 1. The diagram shows the connection of the parts. One of the telephone terminals goes to the junction of the ($ ohm) resistances R and 8, the other to a point upon the divided wire. The branch P includes one compensator (with coils connected in series), the subject of examination, and part of the divided wire. The branch Q includes the second com- pensator (replaceable by a simple coil possessing suitable * Prof. Hughes appears also to have met with this difficulty in his second apparatus. Resistance of Compound Conductors. 483 self-induction), a rheostat, or any resistance roughly adjust- able from time to time, and the remainder of the divided wire. The battery branch, in which may also be included the in- terrupter, has its terminals connected, one to the junction of P and R, the other to the junction of Q and 8. When itis desired to use steady currents, the telephone can of course be replaced by a galvanometer. In this arrangement, as in the other, balance requires that the branches P and Q be similar in respect both of self-in- duction and of resistance. The changes in induction due to a shift in the movable contact may usually be disregarded, and thus any alteration in the subject (included in P) is measured by the rotation necessitated at the compensator. As for the resistance, it is evident that (R and S being equal) the value of any additional conductor interposed in P is measured by twice the displacement of the sliding contact necessary to regain the balance. The position of the contact was read to tenths of an inch ; and, since the actual resistance per inch was *0246 ohm, a displacement of that amount represents ‘0492 ohm. ‘To save unnecessary reductions, the resistance of any conductor will usually be expressed. in terms of the contact displacement caused by its introduction, just as the self-induction is ex- pressed in degrees of the compensator. In order to compare the behaviour of iron and copper, two double coils were prepared as nearly similar as con- veniently could be. The iron coil was that already spoken of (p-479). The resistance of each wire was ‘9 ohm, and the diameter ‘032 inch. In the double copper coil the resistance of each wire was ‘1 ohm, and the diameter ‘037 inch. Each coil consisted of 9 (double) convolutions, of diameter about A+ inches. The two iron wires being connected in series, the large self- induction (when the current circulated the same way in both wires) was found to be 65°11 ; the small self-induction (when the directions of circulation were different) was 23°1. On the other hand, with the copper wires the large self-induction was 45°), and the smali only 1°0. Thus, although the manner of connection makes far more relative difference in the case of copper than in the case of iron, the absolute difference, which represents four times the mutual induction of the two wires, is nearly the same, viz. 44°°0 for copper and 42° 0 for iron. There is here no evidence of any distinction in the mutual induction of iron and copper, the slight want of agreement being easily attributable to different degrees of A84 Lord Rayleigh on the Self-induction and closeness of approach in the two cases. The readings for re- - sistance were sensibly the same, whether the currents were steady (balance being tested by a galvanometer), or were variable with a frequency of 1050 per second. They were also unaffected by the reversal of one of the wires. With the same pair of double coils an interesting experiment may be made by observing the effect of closing the second wire upon the apparent resistance and self-induction of the first. To steady currents the resistance of one of the copper wires was 1°75, unaltered by closing the circuit of the other wire. With secondary open, the same resistance was found to apply to periodic currents of frequency 1050; but when the secondary was closed the resistance rose to 2°67. On ‘the other hand, the closing of the secondary reduced the self- induction from 11°:2 to 4°7. It will be instructive to com- pare these results with Maxwell’s formule :— Alt grea = ® ° e ° (3) "M?N L!=L 5a wate ° e ° ° (4) which we may do by means of a value of M (the mutuai in- duction) deduced from the previous experiment, in which the wires were connected in series. ‘Thus M=11°'0=11-0 x 1553 centim. From the present experiment, | 3 9 centim. sec. L=N=11°2 of compensator= 11-2 x 1553 centim., p=2nr x 1050 ; R= S=1°75 inches of slide=1'75 x 0492 x 10 so that fis Thus, according to the formule, the resistance R! to the periodic currents should be R’/=R+'60R = 2°80 inches of slide. This compares with the 2°7 actually found. In like manner L’=L—-60L = 4° 5 of compensator, serene as well as could be expected with the observed value Similar experiments were made on the double coil of iron wire. With secondary open, the resistance of one wire to Resistance of Compound Conductors. 485 steady currents was 16°79. To periodic currents of fre- quency 1050 the resistance was just perceptibly greater, viz. 16°85, which increased a little further (17°15) when the secondary was closed. The closing of the secondary left the self-induction sensibly unaffected at 21°°2. The much slighter influence of the secondary here observed is due mainly to the higher resistance of the iron as compared with copper. A calculation carried out as before gives pw? R? + pL? agreeing pretty well with the proportional change observed in the resistance. The corresponding change in self-induc- tion would be barely sensible. In the case where the primary and secondary circuits are similar (S=R, N=L), Maxwell’s general formule may be written in the form R—-R_L-L! pM? 6) Mabrengs Tin id Ree Let bth ¢ and we may note two extreme cases. When p is small, or, more fully expressed, when the period of vibration is long in comparison with the time-constant of either circuit, viz. L/R, the reaction of the secondary currents is of small importance. On the other hand, when p is large, the right-hand member of (5) approaches to the torm M’/L’; and this again does not differ much from unity when the two circuits consist of a double coil of non-magnetic wire. Under such circumstances the reaction of the secondary tends to destroy the self-induction _ and to double the resistance of the primary. Being desirous of investigating an example approximating to the second extreme, I prepared a double coil of stouter wire than the preceding. The diameter was about ‘08 inch, and the length of each wire was 318inches. There were 20 (double) turns, so that the mean diameter of the coil, wound as compactly as possible, was about 5 inches. The resistance of each wire was about ‘05 ohm. The coefficient of mutual induction was determined by comparison of the self-induction (L) of one wire with that of the two wires connected oppositely in series, viz. (2L—2M), In this way. it appeared that M=43°1 =43°1 x 1553 centim. The interrupter was the reed, of frequency 1050. Observation showed that the closing of the secondary diminished the self-induction from 44°4 to 3°-4. The re- Phil. Mag. 8. 5. Vol. 22. No. 1389. Dec. 1886. 2L 016, 486 Lord Rayleigh on the Self-induction and sistance to steady currents was ‘92inch. The resistance to the periodic currents was ‘97 with secondary open, and 1°74 with secondary closed. Taking then L=44-4 x 1558 contim., R=-97 x-0492 x 10.2, we get pe 10 x 1951 R?4+ pL? 10% x-0234107x 2-071 According to the formula, therefore, L’=:068 L=:068 x 44:4=38"0, which is to be compared with the observed 3°-4. The application of the formula to the calculation of R’ is somewhat embarrassed by the observed difference of resist- ances to steady and to periodic currents when the secondary was open, of which the theory takes no account. It is true that the difference is small, but it appeared to lie outside the limits of error of observation. It is not accounted for merely by the tendency of periodic currents to adhere to the outer parts of a conducting cylinder. If this observation stood alone, one would be inclined to attribute the discrepancy to some action, whether electro-magnetic or electro-static, of the neighbouring secondary, even though open ; but, as we shall have occasion to notice, a similar tendency of the resistance to increase when periodic currents are substituted for steady ones is to be observed in cases where no such explanation is available. The effect was, however, too small to be investi- gated further without some modification in the apparatus, or in the nature of the conductors submitted to examination. If we take, as found for the periodic currents, R=‘97, we get 932. R/=1-93 x 97=1:87, instead of 1:74 as observed. On the other hand, if we take R="92, we get R= 193 x-o2= 0°77. The next experiment was contrived to illustrate the beha- viour under periodic currents of a system composed of two conductors connected in parallel. The general theoretical formule are given in a former paper*; but the more special case selected for experiment was one in which the mutual induction of the two conductors (M) and the self-induction of * Phil. Mag. May 1886, formule (13), (14), p 3877. Resistance of Compound Conductors. 487 one of them (N) can be neglected. The formule for the re- sistance and self-induction of the combination then reduce to ae pL? = EG8 [2 +F asap) ern VL L/= (R+8)4+p2L” e e r e e ° e e (7) in which SR(R+8) represents the resistance to steady cur- rents (p=0). The peculiar features of the arrangement are brought out most strongly by taking a case in which 8S (the resistance of the induction-less component) is great compared with R. It is then obvious that steady, or slowly alternating, currents flow mainly through R, and accordingly that the re- sistance and self-induction of the combination approximate to. Rand L respectively. Rapidly alternating currents, on the other hand, flow mainly through 8, so that the resistance of the combination approximates to S, and the self-induction to zero. These common-sense conclusions are of course embodied in the formule. The conductors combined in parallel were (1) the coil of stout copper (p. 483) with its two wires permanently con- nected in parallel so as to give maximum self-induction (L), and (2) a moderate length of somewhat fine brass wire. With steady currents the resistances were Ha45, AS = 2°29 i yg ="). It had been expected that the resistances R, 8, of the sepa- rate conductors would have been sensibly the same whether tested by steady or by periodic currents ; but the resistances in the latter case tended always to appear higher. Thus with the same reed as interrupter, R='52, S=2:33, L=43°7, N=-3°; and for the combination, ih’ = 2°04, Li=5""0), These results of observation illustrate satisfactorily the general behaviour of the combination to periodic currents of high frequency, and they agree fairly well with the formule. According to these, if we take the values of Rand 8 as ob- served with periodic currents, we have R/=2-16, L/=2°61. The altered distribution of current under the influence of ‘induction, and consequent increase of resistance, exemplified 2L 2 488 Lord Rayleigh on the Self-induction and in the above examples, is an extreme case of what may happen to a sensible extent within a simple conducting cylinder, especially of iron, when the diameter is not very small in relation to the frequency of electrical vibration. In order to avoid magnetizing the material of the conductor, the current tends to confine itself to the outer strata, in violation of the condition of minimum resistance. Prof. Hughes has already given examples of this effect; but they are difficult to com- pare with theory in consequence of his employment of a vibration of indefinite pitch. The following observations were made with the usual reed interrupter, giving about 1050 vibrations per second. A somewhat hard Swedish-iron wire, 10:03 metres long, and 1°6 millim. diameter, was first examined. ‘The resistance to steady currents was 10:3, and to the variable currents given by the reed, 12:0. The wire was then softened in the flame of a spirit-lamp, after which the resistance to steady currents was 10°4, and to variable currents 12:1. Hxpressed in ohms, the resistance to steady currents is 7 10:4 x °0492=°51 ohm. From these data we may deduce an approximate value of the magnetic permeability (w) of the material for circumfer- ential magnetization. Jor if / be the length, R the resistance. to steady currents, p/27 the frequency of vibration, we have for the resistance (R’) to variable currents the approximate expression * On 1 pip 1 ptltpt BaR{1+ oe - aoe tf so that for the rough determination of « we may take in the present example, ail oa al oe 3 1 1 Warker” 104 The result is w= 108s A more accurate use of the formula would bring out a sensibly higher value ; but it is hardly worth while to pursue the matter, inasmuch as any deduction of uw from the small observed difference of resistance (1°7) is necessarily subject to considerable error. | In order to get better materials for a determination of « by this method, a stouter wire of Swedish iron was next tested, 18°34 metres in length and 3:3 millim. in diameter. The metal was rather hard. The resistance to steady currents was found to be 4:7, and to the variable currents from the * Phil. Mag. May 1886, equation (19) p. 387. Resistance of Compound Conductors. 489 reed 8:9. These are, as usual, in terms of the scale of the apparatus. The absolute resistance to steady currents centim. sec. R='230 x 10° In this example, the change of resistance (in the ratio 1°89 : 1) is so great that no use can be made of the approxi- mate formula quoted above, but we must revert to the original series. In the notation employed in the paper referred to, if (zx) denote the function *, a ie UFO teperga t+ ye hegacga et Ss auues the resistance to variable currents (R’), and the self-induction, L’, are given by Ra) os al $ (iplw/R) — SS = SSS SESE e . 8 Bee te oh aaa yd sr (°) so that the real part of the fraction ¢/¢’ gives the ratio R’/R. By calculation from the series I find $(0X5'2865) — —4:58934+7ix1:5171_ git ae $i (Gx 52305) ~ —1-0297 Fix 16662 18006 +x 19899, in which the first term on the right agrees sufficiently nearly with the observed value of R’/R. We may conclude that ply | R=5:2365, whence p= ooo: In order to give an idea of the degree of accuracy with which # is determined by the observed value of R’//R, it may be worth while to record another numerical result, viz. :— (tx 5°6815) gf’ (i x 5°6815) In these calculations it is assumed that the increase in R, observed when variable currents are substituted for steady ones, is due simply to a less favourable distribution of current over the section. If there were sensible hysteresis in the magnetic changes, R would be still further increased. I believe, however, that under such magnetizing forces as were at play in these experiments, there is no important hysteresis, and that « may be treated as sensibly constant. The increased self-induction and resistance of an iron wire, =1°9596 +7 x 16544. * The relation of d to Bessel’s function of order zero is expressed by (a) =J3 (2iv2). 490 Lord Rayleigh on the Self-induction and due to its magnetic quality, are doubtless disadvantages from a telephonic point of view. If found serious they may be mitigated, as Prof. Hughes has shown, by the use of a stranded wire, in which the circumferential magnetic circuits are interrupted. There has been some confusion, I think, in connection with the notion of “ retardation.” If we had the means of observing the passage of signals at various points of along cable, we should find them not merely retarded (which would be of no consequence) as we recede from the sending end, but also attenuated. The amplitude of a periodic, é. g. telephonic, current sent into a cable becomes less and less as the distance increases. Nothing of the kind can happen in a well-insulated iron wire of negligible electrostatic capacity. Its resistance and self-induction may oppose the entrance of a current, but whatever current there is at any moment at the sending end of the wire must exist unimpaired throughout its whole extent. I will now record a few experiments as to the effect of an iron core upon the apparent self-induction and resistance of an encompassing helix. The wire was wound in one layer upon a glass tube; the total number of turns is 205, occu- pying a length of 28°6 centim. The length of the wires forming the cores was 24:1 centim. The results given are the differences of the readings obtained with and without the cores, so that the resistance and self-induction of the helix itself are not included. The interrupter was the same reed as in previous experiments. _ A comparison was made of the effect of a solid iron wire 1-2 millim. in diameter and of two bundles of wires of similar iron (drawn from the same specimen) of equal aggregate section and weight. One bundle contained 7 wires, and another 17. The results were :— 1 wire. 7 wires. 17 wires. Resistance.........+.000. 1:3 0°3 0:2 Self-induction ......... 13° 18° 18° showing that when the wire was undivided the secondary currents developed in it increased the apparent resistance of the helix by 1-3, and diminished the apparent self-induction. A similar experiment was tried with a stouter wire, 3°3 millim. in diameter (from the same hank as the length of 18°34 metres treated as a conductor). In the hard condition. Resistance of Compound Conductors. 491 the self-induction due to this was 244°, and the resistance 3°8; numbers altered to 283° and 44 respectively by softening with a spirit-flame. The etfect of a bundle of thirty-five soft wires of the same iron and of equal aggregate section was 84° of self-induction and 1°6 of resistance *. There is nothing surprising in the conclusion, forced upon us by the observations, that the magnetic effects of iron rods 33 millim. in diameter are seriously complicated by the formation of induced internal eiectric currents. As I have shown on a former occasion f, the principal time-constant of a cylinder of radius a, specific resistance p, and per- meability w, is given by A Anrpwa? ~ (2°404)?p° This means that circumferential currents started and then left to themselves would occupy a time 7 in sinking to e-! of their initial magnitude. Whether the effects of such currents will be important or not depends upon the relative magnitudes of rand of the period of the magnetic changes actually in progress. In the present case, with U0, p— JOA... 20-— "a0, the value of 7 is about 5,459 of a second, that is, about half the period of the actual electrical vibration. The theory of an infinite conducting cylinder exposed to periodic longitudinal magnetic force (Le?’) has been given by Lamb {, who finds for the longitudinal magnetic induction at any distance r from the axis J (kr . om He ales Me abate ur mata py. where pe ee et tN When the changes are infinitely slow, ¢ reduces to wle?’, as should evidently be the case. A more complete solution was worked out a little later by * It may be worth while to remark that in these experiments no approach to a balance could be obtained when a scraping contact inter- rupter was used. With the reed there was complete silence, or at most a slight perception of the octave. The failure of the scraping contact is due, of course, to the mixed character of the vibration, and to the fact that the adjustments necessary for balance vary rapidly with pitch, + Brit. Assoc. Report, 1882, p. 446. t Math. Soc, Proc. Jan. 1884, vol. xv. p. 141, 492 Lord Rayleigh on the Self-induction and Oberbeck *, including what is required for our present pur- pose, viz. the value of 27 ir crdr. In terms of the func- tion @ previously used (p. 489), (8) becomes pt Pipe. 77"/p) t : OC pi wa lpy ee as whence is readily deduced a Wee 2 eid? ipt P (ype. 7a°/p) _ 0 ata am or dr Ta’. ple Alto ain (12) c=ple where Chea x ‘ The mathematical analogy between this problem and that of the variation of a longitudinal electrical current in a cylindrical conductor has been pointed out by Mr. Heaviside, who has also given the full solution of the latter. Maxwell’s investigation, somewhat further developed in my paper f, relates principally to that aspect of the question with which experiment is best able to deal, viz. the relation between the total current at any moment and the corresponding electro- motive force. . That the argument in ¢, ¢’ is the same in (12) as in (8) will be evident, when it is remembered that R in (8) denotes the resistance of unit length of the cylinder; so that b oma" Be Hence, if we may assume that the material is isotropic, the same numerical results are applicable to a given wire in both problems. But from this point the analogy fails us. What we require here to express is the ratio of the total magnetic induction to the external magnetizing force, and not the inverse relation, corresponding in the other problem to the expression of the electromotive force in terms of the total current. The experimental results are the reaction of the core upon the magnetizing circuit, expressed as alterations of apparent self-induction and resistance. Now if m be the * Wied. Ann. vol. xxi. (1884), p. 672. There seems to be some error in the way in which the magnetic constant appears in Oberbeck’s solution (47). According to it (as I understand) a copper core would be without effect. + Phil. Mag. August, 1886, p. 118. t Phil. Mag. May 1886, p. 386, Resistance of Compound Conductors. 493 number of turns per unit length in the magnetizing helix and C the current (proportional to ¢”*), we have en ee eae cos. a CLS) and for the electromotive force (E) due to the change of mag- netic induction in the core, reckoned per unit length, B=m4 {onl ‘rar } aA neem a pois IC Dei x) 00/018) jai LD) In order to interpret this, we must separate the real and imaginary parts of ¢’/¢. If we write =P-iQ, then the part of E which is in the same phase as dC /dt is Amn’a’y .ipC.P; and the part which is in the same phase as C is 4m?r’a’u.pC.Q. ‘The first manifests itself as an increase of self-induction, and the second as an increase of resistance. If p=oo, P=1, Q=0. What we require to know for our present purpose is the effect of introducing the core; and to obtain this we must subtract any part of E which remains when we put p=o, w=1. Calling this H,, we have E,=4m' n'a" .ipC, and E—H,=4m?r'a"{ipC(uP—1) +upC. Q}. Thus if 6L, dR be the apparent augmentations of self- induction and resistance in the helix due to the introduction of the core, reckoned per unit length, dL=4m?r’a?(wP—1), 1 (16) SR=4m' ra". pQ. From the calculation already made for the purposes of the other problem, we have Lite . CSE aaGR =(1:8906 +i x 15859), ='31047—7 x ‘26044; so that for the stout iron wire of 3°3 millim. diameter and eo, P='31047, Q=+26044. 494 Lord Rayleigh on the Self-induction and With these values the effects 6L, 6R of the core of 3:3 millim. diameter may be calculated ; but no very good agree- ment with observation is to be expected, since the conditions of infinite length, isotropy, &c. were but inadequately satisfied. Inserting in (16) m=205/28:6, a="165, w=99°5, we get SL=1650, S8R=10°x9-436. These are expressed in absolute measure, and reckoned per unit length of core. To obtain numbers comparable with the experimental readings, we must multiply by 24:1 (the length of the core), and reduce 6L by division by 1553, and OR by division by 10°x 0492. The result is (SL) =25°6, (8R)=4°6; which agree moderately well with the observed values, viz. (8L)=241°, (8R)=3'8, __ If the material composing the core were non-conducting, P=), and SL, =4m? n'a? (w—1). The ratio of the actual effect to that which would be got from the same aggregate section of a bundle of wires, infinitely thin and insulated from one another, is thus 0 eae Obs 2 er of which the numerical value in the present example is *303. The corresponding ratio of observed effects for the solid wire (softened), and for the bundle of 35 wires of the same aggre- gate section was 282 / $4=+339. The general result of these experiments is to support the conclusion arrived at by Oberbeck that the action of iron cores, submitting to periodic magnetizing forces of feeble intensity, can be calculated from the usual simple theory,, provided we do not leave out of account the induced internal currents which often play a very important part. Oberbeck’s observations were made with the electrodynamometer, and with rather low frequencies of vibration—about one tenth of that used in most of the observations here recorded. We have seen in several examples that the self-induction of a combination of conductors, being a function of the pitch, admits of an indefinite series of values; and the question suggests itself to which (if any) of these corresponds the Resistance of Compound Conductors. 495 value obtained by galvanometric observation of the transition from a state of things in which all the currents are zero to one in which they have steady values under the action ofa constant electromotive force. In the ordinary theory of Max- well’s method for determining self-induction from the throw of the galvanometer-needle in a Wheatstone’s bridge (a resist- ance-balance having been already secured), the conductor under test is supposed to be simple. The general case of an arbitrary combination of conductors can only be treated by a general method. An. investigation founded upon the equa- tions of my former paper* shows that the result which would be obtained by Maxwell’s method corresponds to the self- induction of the combination for infinitely slow vibrations. We have supposed that the behaviour of the compound conductor is not influenced by electrostatic phenomena ; other- wise the representation of the part of the electromotive force in the same phase as dC / dé as due merely to self-induction would be unnatural. So far as experiment is concerned, we have no means of distinguishing between an effect dependent upon dC /dt and one dependent upon Ke dt, for the phase of both is the same. We may contrast two extreme cases—(1) a simple conductor with resistance and self-induction, (2) a simple condenser with resisting leads. In the first case the electromotive force at the terminals is written | L.ipC+R.C; in the second —p’ .ip 3C+R.C, where yw’ represents the “ stiffness”’ of the condenser. If we persisted in regarding the imaginary part in the second case as due to (negative) self-induction, we should have to face the fact that the coefficient becomes infinite as p diminishes with- out limit. A number of combinations in which the induction of coils is balanced by condensers are considered by Chrystal in his valuable memoir on the differential telephonef. In a papert already referred to I have shown that when two conductors in parallel exercise a powerful reciprocal * Phil. Mag., May 1886, p. 372. The analysis may be simplified by choosing the first type so as to correspond to steady flow. The coeffi- cients 0,,, 0,,..., a8 Well as the final values of V,»W3--. are then zero, and the result may be expressed, \*, di=a,, hitb) dt. + Edinburgh Transactions, 1879. { Phil. Mag. May 1886, p. 378. 496 Lord Rayleigh on the Self-induction and induction, very curious results may follow the application of a periodic electromotive force. I have lately submitted the matter to experimental test, by which theoretical anticipations have been fully confirmed. The two conductors in parallel were constructed out of the three wires of a heavy and compact triple coil of copper wire* mounted in a mahogany ring, which has been in my possession for many years. Of these wires two are combined in series (with maximum self-induction) to constitute one of the branches in parallel. The other branch is the third wire of the triple coil, so connected that steady currents would circu- late the same way round them all. The variable currents were obtained from a battery and scraping-contact apparatus (p. 472), connected directly. Under these conditions, if the intermittence be rapid enough, the currents distribute them- selves in the two branches so as nearly to neutralize one another’s magnetizing-power ; and this requires that the cur- rent in the single wire should be of about twice the magnitude of the current in the double wire, and in the opposite direction. If we call these currents 2 and —1, the current in the mains must be +1. As may be seen from formula (13){, such a state of things leads to a high equivalent resistance for the system ; and the question might be investigated on this basis with the apparatus already described. I preferred, however, to examine directly whether it were true that the current in one of the branches exceeded that in the main ; and this could be readily done by “tapping”? with the telephone. For this purpose the two Fig. 2. Lorem iy Eengest us one wire. two wires. i AGlik, BAe ai Wea ae we main. eRek branches and the main were led through short lengths of similar German-silver wire to the junction, composed of a copper plate to which the wires were soldered (fig. 2). One telephone terminal was soldered to the plate ; the other was brought into contact with some point of the German-silver * The three wires were wound on together. t Loe. cit. p. 377 Resistance of Compound Conductors. 497 wire carrying the current to be observed. It is evident that if the three alternating currents were of equal magnitude, sounds of like loudness would be heard at equal distances from the copper plate, whichever of the wires was touched ; and, further, that the distances required to produce equal sounds are inversely as the magnitudes of the corresponding currents. A moment’s observation proved that the currents in A and B were about equal, and that in C much greater. Numerical estimates are best made with the aid of a second observer, who does not see what contacts are being tried. My assistant considered that about 64 inches of B and about 84 inches of C were required to give the same loudness as 6 inches of A. This agrees with the approximate theory as well as could be expected. If the single wire be reversed, then, according to theory (resistances of German-silver wires neglected), the distribution should be much the same as of steady currents under the sole influence of resistance; that is, the currents in the branches should be as +2 to +1, so that on the same scale the main current would be +38. According to this the equivalent lengths of the German-silver wires would be 6, 9,18. The numbers actually found by experiment were 6, 8, 174. In the first part of this experiment the current in one of the branches is greater than in the main; but I wished to examine a case where both parts of the divided current exceeded the whole. This could be done with a fivefold coil, as described in the previous paper ; but such was not ready to hand. In default thereof a common double coil, belonging to a large electro-magnet, was enveloped with a single layer of extra wire, which was combined in series with one of the original wires. ‘This arrangement is less favour- able than one in which the two branches are in close juxta- position throughout; but I thought that with the aid of an iron core it could be made to answer the purpose. Such a core was provided in the form of a bundle of fine wires, solid iron being obviously inappropriate. The two wires were now connected in parallel and replaced the triple coil, the arrangements in other respects remaining unchanged. The currents in the shorter branch (composed of one original coil simply), in the longer branch (composed of the other original and of the additional coil), and in the main were now found to be inversely as the measured distances ‘9, 1:3, 2°3, no regard being paid to sign, viz. as 1°11, ‘77, ‘43. These numbers cannot be quite correct as they stand, for the third should be equal to the difference between the 498 Lord Rayleigh on the Self/-induction and first and second. If we suppose the second and third to be correct, the first would have to be 1:20 instead of 1:11. Such an error as this may easily occur in estimating the equality of sounds heard successively ; and there can be no doubt that the smaller branch current largely exceeded the mai current*. ApprnpDix.— The Induction-Compensators (p. 473). For the mutual induction-coefficient between two circular circuits, subtending angles @,, a, at the point of intersection of their axes (lines through their centres and perpendicular to their planes), and distant c¢,, c, from that point, Maxwell gives M=47’ sina, sin? { 4 . Q' 1641) Qs (a2) Q, (8) +eeT ch ()2/@9(0)+ ... ban) the angle between the axes being denoted by 0. Q,.. - denote Legendre’s coefficients (more usually represented by P,), and the dash indicates differentiation with respect to p. In our present application the circuits are concentric, so that a;=a,=4, and ¢, c, are equal to their radii. Moreover {Q/(47)}? vanishes if 7 be even; while if ¢ be odd (2n+1) we have ae ae Be Te ae (2a lee (Qantn) P= pr ge 5 heey so that C Ose M gts =1Q0)+ gz mp (3) OO) 1 Poe D7 A aren O) vive Soin 1 Be Ged 1 ZN * QntL)Q@n+2) 2.2.6. . Bn)” & Qert1(8), 9) which is what we have to calculate for various values of @ on the supposition that c,=4c. | The following are the values of Qon41(@) at intervals of 10°. It is unnecessary for our purpose to go further than Q,. _* These experiments were described before the British Association at Birmingham, September 3. t ‘Electricity and Magnetism,’ § 697. Resistance of Compound Conductors. 499 a TAPE l. e. | a0) | @(6). | Q,(0). | ,(6). 90° ‘00000 00000 0000 “0000 80 | + 17365 |— -24738 | + -2810| — -2834 70 | + -34202 |.— -41301 | + -3281 | — +1486 60 | + °50000 | — -43750] + -0898 | + -2231 50 | + -64279 | — -30022 | — -2545| + -2854 40 |4+ -76604 | — 02523 | — -4197] — -1006 30 |-+ -86603 | + -32476 | — -2233| — 4102 290 | + -93969 | + -66488 | + -2715| — 1072 10 |+ -98481 | + 91057 | + -7840| + -6164 O | +1-00000 | +1:00000 | +1-0000 | +1:0000 | From these the values of (19) were computed. They are shown in Table IJ., together with the sines of 0’ and the differences for each step of 10°. Taste II. e&='25 ci. | 6. 6’. Induction. Diffs. Sin 6". Diffs. ; 90° 0° 0000 ‘00000 . 80 10 0769 0769 + ‘17365 1736 70 20 "1538 ‘0769 + +34202 "1684 60 380 2304 0766 + :50000 "1580 50 40 3058 0754 + 64279 "1428 | 40 50 3786 0728 + :76604 "1932 | 380 60 ‘4460 ‘0674 + 86603 ‘1000 20 70 5029 0569 + 93969 0737 10 80 5416 0387 + -98481 0451 0 90 "D559 0143 +1:00000 _ 0152 The column headed Induction gives the value of 1 A ee | Tranoe 291(9) +aq gz 4 Q3(0) + 5% pq? qe (8) + hs EGS It will be seen that for moderate values of 0’ the differences are very nearly constant, far more so than the differences of sin 0’, which latter would apply to the induction on the sup- position of a very small interior coil. The results of the ex- perimental calibration are thus confirmed and explained. An inspection of the table suggests that the proportionality to 0’ might be improved yet further if the value of ¢,/c, were a little increased: The following numbers calculated for a twenty per cent. increase of ¢3/c}, viz. for ¢.='54772 ¢, con- firms the idea. Such a proportion, applicable to the mean radii of the coils, might well be designedly chosen. 500 Mr. R. H. M. Bosanquet on TABLE III. 2 2 oi 2 Cle 0. Induction. Diff. 90° ‘0000 80 0752 0752 70 °1508 0756 60 2268 ‘0760 50 *3025 0757 40 3769 0744 30 "4479 ‘0710 20 5099 "0620 10 D532 "0433 0 5695 "0163 The numbers in the column headed Induction are the values of : ii ae ica hese 3Qi(9) +37 5 (3)Q5(9) + 5% orgs (3)’Q,(0) +... The last two entries are liable to a small error from omission of Q, &e. LVIII. Permanent Magnets.—III. On Magnetic Decay (con- tinued). By R. H. M. Bosanquet, St. John’s College, Oxford. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, Le the Philosophical Magazine, ser. 5, xix. 1885, pp. 57-59, -. I gave an account of the decay of the magnetism of a pair of permanent magnets during some months after they were made. Observations for H have been coatinued at intervals with the same pair of magnets, both in the laboratory and the non- magnetic room ; incidentally a large number of determinations of the moments of these magnets have been obtained, and from these an interesting result appears to follow with considerable certainty. The decay of the magnetism proceeds much more rapidly in summer than in winter; and in all probability the rate of decay is quicker for higher temperatures and slower for lower temperatures. It will be remembered that Joule proved, by burying systems of magnets for several years, that the maintenance of a constant temperature did not secure the constancy of the magnetic intensity (Reprint, vol. i. page 592), on which Joule Permanent Magnets. 501 remarks, ‘so that the cause of the gradual decline of power has yet to be discovered.” Regarding heat as a species of molecular agitation, it seems reasonable to expect that this agitation should act in a manner analogous to mechanical disturbances, in facilitating the decay of permanent magnetism. In fact we know that, by exposing permanent magnets to moderately high temperatures, we can destroy their magnetism altogether. The following are the mean results of the sets of observa- tions that have been made. The changes in the value of M are shown in the figure. It will be seen that they are much greater in summer than in winter. &c [6 6) or Mar June Sept. = e oo oO for) Mar June Sept a aBANS : ® o io) = | eRe 1884.50 N.B. The temperatures were always noted, and the tempe- rature-coefficients of the magnets were determined by a careful discussion. They are not large enough to produce any differ- ence in the numbers which need be taken into account for the present purpose. Phil. Mag. 8. 5. Vol. 22. No. 189. Dec. 1886. 2M 502 Dr. W. W. J. Nicol on the Vapour-pressures Nonmagnetic Room. Mean date. No. of sets.) Temp. F. 1884. Rebruarypl Sigeasyeiel DO: Fey ign cae Mar els irene ccence et Oakes September 18 ......... NO alee Mewes 1885. 3 amuary 4.2... sees. tos 11 41°8 Mar 28455 5..050% see 17 61:4 October 24 .te-eneco-2 10 53°5 1886. April AO accel 10 50°7 September 24 ......... 22 60:4 Laboratory. Mean date. No. of sets.| Temp. F. 1884, March Dieu casce bes: 12 ac telat ss 2 Oa Rte is Sen ghe lao ene e ae ASS es (Pr esesae pe September 18 ......... | Se arated Beau hneee 1885. e CE Tai Ei Bre neponen aren 16 42-7 Miaysa seek eee 26 52°6 October 24 ites 9 60:1 1886. Agr cir eusiecce unas 21 47-5 Sepbember 2e. wearecis 30 580 M. 11,767 11,620 11,121 11,095 11,002 10,819 10,805 10,661 “LIX. On the Vapour-pressures of Water from Salt-Solutions. By W.W. J. Nicou, M.A., D.Se., FR.SE., Lecturer on Chemistry, Mason College, Birmingham*. i igs 1835 Legrand{ published a paper on the effect of various salts on the boiling-point of water in which they were dissolved in varying proportions. The results he obtained were exceedingly interesting, but incomplete; and the experi- ments of Wiillnert{t and others have done but little to extend our knowledge of this branch of the wide subject of solution. With the desire of arriving at some definite conclusion as * Communicated by the Author. + Ann. de Chim. et Phys. lix, p. 428. t Pogg. Ann. cx. p. 564, of Water from Salt-Solutions. 503 to the nature of the action of salts on the vapour-pressure of the water in which they are dissolved, I commenced in April 1884 a series of experiments on the subject, the first portion of which, relating to the boiling-points of saturated solutions, has been already published*; while the general conclusions arrived at up to the middle of 1885 are to be found in a report on Solution, presented to the British Association in that year. Since then I have repeated my experiments with improved apparatus, with the result that my previous conclusions are extended and confirmed, so that I feel myself justified in publishing this second part of the research. Before proceeding to an account of my own experiments, it will be advisable briefly to summarize the results obtained by Legrand (loc. cit.). His experiments were conducted as follows :— After he had determined the boiling-point of pure water by heating it in contact with some granulated zinc in a wide test-tube, in the neck of which was fastened a delicate ther- mometer the bulb of which was immersed in the water, he introduced a known quantity of the salt under examination and again observed the boiling-point. The tube was weighed, and thus the quantity of water present was ascertained. A further quantity of salt was added, and the operations repeated until a saturated solution was obtained, the amount of salt in which was determined in the ordinary way. From the results thus obtained curves were drawn, and thus the quantities of salt required to raise the boiling-point of water successive half-degrees were found. On an examination of Legrand’s figures, we find that the seventeen salts examined in the above manner are divided into four classes by their effect on the boiling-point of water. 1. Several salts agree in this, that more salt is required to raise the boiling-point the first half-degree than to raise it the second, and this more than the third, and so on: thus, 7°5 per cent. NaCl raise the boiling-point to 101° C.; 5:7 per cent. raise it from 101° to 102° C.; 4:9 per cent. raise it from 102° to 103° C.; &c. The salts which behave in this way are NaCl, KCl, Na,CO3, Na, HPO,, and BaCl,. 2. One salt only, KC1Qs, has its effect on the boiling-point of water represented by a straight line: n(14°64 per cent.) KC1O; raise it n° C. 3. Other salts behave in a manner the converse of those in class 1; 7. e. as the concentration increases so does the amount of salt required to raise the boiling-point each successive degree. NH,NO;3, NaNOs, and KNO; behave in this way. %* Phil. Mag. October 1884. 2M 2 504 Dr. W. W. J. Nicol on the Vapour-pressures 4. The most general case is a combination of the three foregoing classes in the order given. At first the quantity of salt diminishes rapidly with the concentration ; then more slowly till it becomes practically a constant for a few degrees; then it increases more and more rapidly up to the saturation- point. K,CO3; may be taken as an instance. From 100° 101° C. 13 per cent. of salt are required ; from 101°-102° C. 9°5 per cent. For each degree between 113° and 118° C. A-8 per cent. only are required ; and above this the amount increases till from 130°-1381° C. 5:4 per cent are required ; and with other salts the increase is still more marked. K,COs, NaC,H3,0,, K,C,H,0,, NH,Cl, CaCl, SrCl, Ca(NOs)2, KO,H;0,; that is, eight out of seventeen salts examined belong to this class alone. It appears probable, as indicated above, that there is but a single curve of the form indicated in the diagram below ; and that it is complete in the case of some salts, while others cor- respond to the first, second, or third parts of the curve. How far this hypothesis is borne out by my experiments will be shown later on. Percentuge Sact . Apart from the general inaccuracy attending Legrand’s method of experiment, there is the serious objection to it that, though the pressure was constant, the temperature was vari- able. It would be difficult, if not impossible, to convert his results into a form in which they would be available for the purposes of the present paper. As it is, I believe they can be used only qualitatively, and that even with hesitation. of Water from Salt-Solutions. 505 In my experiments I had two methods available ; either the barometer-tube method as modified by Konowalow, or the boiling-point method modified so that the temperature was constant and the pressure the variant. The latter is by far the more convenient of the two methods, and is the one that I employed; but it was only after repeated trials that I succeeded in overcoming the danger of superheating and the conse- quent error of experiment. A flask, about 200 cubic centim. capacity, communicates by a wide side-tube with an upright condenser ; this in its turn is connected with a three-way tap and an air-reservoir, which communicates with the gauge and the water-pump. The flask stands on a small burner, and is protected from the heat by a metal ring, except about an inch and a half at the very bottom. In the neck of the flask is fitted the thermometer. The course of experiment was as follows:—In the flask was placed 50 grammes of zine and 60 cubic centim. of water, then 35 of the molecular weight of the salt in grammes were added. This gives a solution of almost exact molecular strength, as ~89, =<). The thermometer was introduced, care being taken that the bulb was completely immersed ; the side-tube joined to the condenser, the pump set in action, and the burner lighted. When the solution boiled, the pres- sure was slowly increased till the thermometer rose to 70° C. The barometer and gauge were then read by two telescopes, the pressure was again diminished slightly, a tube full of mercury introduced into the reservoir of the barometer, and another reading made at 70°, the mean of the two being taken. The same operations were repeated at intervals of 5° C. up to 95° C., when the flask was disconnected, washed, and dried, and the whole set of observations repeated with a fresh solution of the same strength. Thus each determination given in the following Tables is the mean of four results, ob- tained with two separately prepared solutions. The granulated zinc is absolutely necessary in large quantity, otherwise super- heating takes place. In no case was the zinc dissolved by the solution ; nor was it appreciably attacked, except by the very strongest solutions of KNO; and NaNO, at the higher tempe- ratures. In these cases appreciable amounts of ammonia were evolved, a point I intend to investigate further ; but even after prolonged boiling the boiling-pressure was found unaltered ; thus the results are not affected by the small de- composition of the salt in these cases. The remainder of the method of experiment is the same as that described in my previous paper. It must not be forgotten that, though the temperatures are arbitrary, still, as the determinations were 506 Dr. W.W. J. Nicol on the Vapour-pressures all made with the same thermometer and are thus strictly comparable among themselves, and as the boiling-pressures of pure water were determined at the same time with the same thermometer, we have a natural standard by which the results may be corrected if need be. Nothing, however, would be gained by such a correction ; for, as the figures in Table I. show, the thermometer, which was by Geissler and divided into tenths, was very nearly exact, the results agreeing well with Regnault’s, when it is remembered that his figures are for steam and mine for water. With regard to the probable error for any single determination, that does not much exceed +°3 millim., and is of course proportionately less for the mean results of four determinations which are given in the Tables. TABLE I. Boiling-pressures of pure Water (Zinc). Regnault ...) 2333 | 2888 | 38549 | 43832 525°5 | 633°7 Found ...... 228°3 2832 | 3494 | 4283 | 521-4 631°0 Difference ... 50 56 5:5 4:9 4:1 2°7 At present I have examined only four salts, which may be regarded as typical of the class which crystallize without water; they are NaCl, KCl, NaNO;, and KNO3. I intend soon to examine the behaviour of some typical hydrated salts. The results thus obtained enable us to examine, not only the effect of varying amounts of salts at the same temperature, but also that of fixed amounts of salts at varying temperatures; that is, the Tables given may be read either vertically or horizontally. Hach Table consists of two halves. The first contains the value of »—p’, where p = the vapour-pressure of pure water at the given temperature as contained in Table I, p’ the pressure of water-vapour from the salt-solution of the composition m molecules of salt per 100 molecules of water. / : The second half of the Table contains 2 —f ; thatis, the effect on the vapour-pressure of pure water by the presence of one -salt-molecule in solutions of the stated composition. A glance at the Tables shows at once that the four salts range them- selves in two classes, when the effect of varying amounts of of Water from Salt-Solutions. 507 TABLE IT. p—p’ for nNaCl 100 H,0. n. 70°. f°. 80°. 85°. 90°. 95°. eee P=? for nNaCl 100 H,0. n 2. 4-25 5:40 6°85 8:20 10°20 12:30 ze 4°50 5°63 7-00 8:45 10-23 12°55 5. 4-52 5°70 7:04 8:60 10:50 12°66 6. 4:73 5-90 125 875 10°58 12:83 8. 4-88 6:08 749 9°10 11-03 13°39 10. 5:04 6°24 7°66 9°31 11-28 13°64 salt at the same temperature is considered. In the case of NaCl and KCl, the restraining effect of each molecule of salt increases with the concentration. The reverse holds good Taste III. p—p’ for nKCl 100 H,0. n 70° 75° Dae 85° 90° 95° 2 76 9°6 12:2 14:7 17-4 22'3 4 16°6 20°5 251 31:0 o7'4 46:0 6 25°3 31:2 38°9 476 578 70°6 Adee PUP for nKCl 100H,0. nr 2. 3°80 4:80 6°10 7:35 8°70 1¥-15 4. 4°15 5:13 6°28 7°75 9°35 11-50 6. 4-22 5°20 6°48 7:93 9°63 77 8. 431 5°34 6°65 S11 9°84 11-89 10. 4°43 5°47 6°75 8:22 9:97 12:07 508 Dr.W.W. J. Nicol on the Vapour-pressures Tasie IV. p—p' for nNaNO,; 100H,0. wae an eewccccacccs | comewemconencann.|*"oecceseccnccecs | sasnconsccsssece: |saccennecesencce= |. ceecenscneccnnn- | coucccecescecoc== 15 51-7 64:2 79°5 97°5 119°3 144-7 20 65°8 80:9 99°4 122-7 149°4 181-2 25 76°8 94:8 117-4 144°3 176°2 2129 egy: PP for nNaNO, 100H,0. n 2. 4:25 5°45 6°65 8:10 9°85 11-90 4, 4:03 4:90 6:05 7:48 9:00 11 03 D. 3°96 4°92 6:14 7:48 9:12 11:16 6. 3°82 4-77 6°02 7°35 8:98 10°92 8. 3°84 4:74 5°86 718 8:75 10-70 10. 3°67 4-58 5°67 697 8°49 10°83 15. 3°45 4:28 5:30 6°50 7:95 9°65 20. 3°29 4:05 4:97 6°14 TAT 9:06 25. 3°07 3°79 4°69 577 7:05 8°52 with NaNO; and KNO,; the more concentrated the solution the less the effect of each molecule of salt present, though the total result increases with the quantity of salt present. TABLE V. p—p’ for n KNO; 100 H,0. of Water from Salt-Solutions. 509 Table V. (continued). ey PUP for n KNO; 100 HO. nN 2. 10°. De. 80°. 85°. 90°. 95°. 1. 3°90 5:20 6°80 7:80 9-40 11-10 2. 3°75 4°85 6-10 7:50 9-10 10°60 3. 3°33 4:20 5°23 6°57 8:00 977 4, 3°35 4:18 5:20 6:50 7:93 9°75 5. 3°16 4-04 5-00 6°16 7:58 9°30 10. 2°73 3°46 4:30 5:37 6°62 $13 15. 2°43 3°08 3°86 4-79 5°91 7:25 20. 2°25 2°84 357 4-42 5-47 6-71 25. 2:05 2°63 3°30 4:08 501 6:16 The accuracy of the above results is undoubted, though it is at variance with the results obtained by Wiillner (Joc. cit.), who found that the effect of the above salts was in direct pro- portion to the quantity present; nor can the conclusions of Wiillner be explained by the fact that his solutions were of percentage composition, while the above are molecular, corre- sponding to parts per hundred of water; for though this might explain one class of salts, it fails if applied to the others. My results, on the contrary, are confirmed by those of Tammann™%, obtained by the barometer-tube method, as will be shown later. I reserve the discussion of the probable cause of this different behaviour of the salts to the end of the aper. * The behaviour of the salt-solutions of the same strength, but at different temperatures, has now to be examined, a point which is by no means so simple as the preceding. Per- haps the best method is a comparison of the values of the Uf / Pp _ 2 var i — £ in which the variation in vapour- fraction br by / pressure is eliminated. The values of (? Lx 10,000) are YL given in Tables VI., VII., and VIII. for NaCl, NaNOs, and KNO;; that for KCl being practically a constant for each solution and independent of the temperature. Shey, AeGT 8 ye 9(F P’ 10,000) =172-45, ae = =181+2. =a =185+2. = 8 =189+2. =10 =193+1. * Wiedemann’s Ann, xxiv. 510 Dr. W. W. J. Nicol on the Vapour-pressures TABLE VI. al Zs x 10,000 for n NaCl 100 H,0. n. ee: Wome 80°. 85°. 90°. 95°. 186 191 196 19] 196 195 2. 45 197 192 200 - 197 196 199 5. 198 201 202 201 202 201 6. 207 208 208 204 203 203 8. 214 215 214 214 212 212 10. 221 -| 220 219 217 216 216 TABLE VII. Ce P=P 10,000 for n NaNO, 100 H,0. ep nN. (Dee Pipa. 80°. 85°, 90°. 95°, —— 186 193 190 189 189 189 176 173 173 175 173 175 174 174 176 175 175 177 167 168 172 172 172 173 168 167 167 168 168 170 162 162 163 163 164 151 151 152 152 153 153 144 143 142 143 143 144 135 134 134 135 1385 135 dS bo SUS OS DS OUP bo — (=p) a Tape VIII. P-P ¥ 10,000 for n KNO; 100 H,0. mp Though not very much can be gathered from the individual results, still I think that they give reliable results when treated en masse. When we divide the whole of the figures for each of Water from Salt- Solutions. 511 salt into two groups—A, for temperatures 70°, 75°, and 80°; B, for 85°, 90°, and 95°—and add together all the figures in group A and group B, then we find that, in the case of NaCl, group A exceeds group B by 20, with KC] A=B, and with NaNO; and KNO; group B exceeds group A by 30 and 91 respectively; thus showing that the net result for all strengths of solutions of the salts is that rise of temperature diminishes the restraining effect of the salt in the case of NaCl, does not affect it in the case of KCl, but increases it in the case of NaNO; and KNOs, and more so with the latter than with the former. I am fully aware of the danger of drawing conclusions from differences so small as these; still the number of experiments in each of the groups (eighteen being the minimum) is probably sufficiently great to justify my doing so. In any case, the accuracy of the results is at least as great as those attainable by the barometric method; and I fail to see that more accurate results can be in any way obtained. Comparing now the weak solutions with the strong ones at the low and high temperatures, we find that, taking the figures in Tables IJ. to V.,and subdividing the groups A and Bat the dotted lines, and calling these subgroups A, and Ag, B, and B, respectively, then the ratio of — to B 1s :— For NaCl, 1:0°983; for KCl, 1:0:990; for NaNOs, 1:0°999; for KNO3, 1:1°:018; that is (in words), concentration in- creasing and temperature rising, the result is diminution of the restraining effect of the salt—considerable in the case of NaCl, appreciable with KCI, little or none with NaNO,, and, on the contrary, a considerable increase in the restraining _ effect of KNO3, Turning now to the results of Tammann (loc. cit.), we find general confirmation of the conclusions arrived at above. ‘Tables IX. and X. contain his results compared with mine for the four salts examined by me. Tammann made his determinations at no definite temperatures, but at irregular intervals on the temperature-scale, and he employed the vapour-pressure of water observed at the same time with the salt-solutions as his temperature-indicator ; nor did he use mo- lecular solutions. The quantity of salt present is expressed in parts per 100 of water; and he gives the observed values of T—'T’, i. e. (p—p’) and also what he terms “die relative Spann-kraftserniederungen,” or relative diminution of vapour- (T—T’) x 1000 T pressure ; this is 2 , where m=parts of salt per ae Dr. W.W. J. Nicol on the Vapour-pressures 3 oe 100 water. This is connected with the value #—? x 10,000, ay which I have used above in the following way :— n= where a=mol. wt. of salt, 18 Lily Sedelep “ (oS )x 100=2—F* x 10,000. ey ae mp Now as Tammann’s temperatures are variable, it is necessary to use this value in comparing his results; and, again, as his figures vary within rather wide limits, better results are Bagh obtained by taking the mean value of =a x 1000 for tem- peratures lying on either side of the desired temperature. Table IX. compares results for #?=7U° and 90°, and for various strengths of NaCl, and for various solutions of the TABLE IX. TAMMANN. NIcou. Salt 0°. n £2 n NaCl vacws WAZ, 4°55 190 70° 5 198 Da) ita G2 6°75 208 a 6 207 See aeaaes 712 16:97 219 S 10 221 NaCl »....2 90°8 4°55 190 90° 5 202 BAe crc Sal 90°8 6°75 201 PA 6 203 ie A 90°8 10-97 216 i 10 216 KCl .. 71:6 3°30 179 70° 4 18142 PT Sane one ss 6°45 183 - 6 185+2 NaNO, 72:0 322 170 70° 4 176 5, m3 5'58 164 os 6 167 - a L512 160 3 10 161 - <3 19°30 136 . 20 144 BNOZ. dais 69:8 2°26 151 70° 3 144 Fe ies eee (PATE 315 Jil 5 3 146 Bae Cae 69°8 Toe 95 a 5 138 ae ed 15°31 96 és 15 107 of Water from Salt-Solutions. . tai other salts at =70°. The agreement is, on the whole, satis- factory, and my general conclusions are completely supported. In Table X. I have taken the whole series of Tammann’s results, and divided it in two from the temperature point of view, and then taken the mean value in each half. Thus the values obtained are not quantitatively comparable with mine, but are qualitatively so, inasmuch as they show the effect of TABLE X. Salt n to. oon “© aan Wael: eo: 4:55 | 196 192 ~ SELF nib 6-75 | 208 203 = >20 ae Fe th 10-97 220 205 = ol 3°33 180 182 SO , eee 6°45 181 185 < a NaNO, 3-22 144 169 < Fe 558 152 167 < say if 11°12 152 158 < i. 19°30 139 142 = KNO,...... 226 150 156 =< ees 3°15 114 119 =< = ay ei 7-31 92 95 =< <91 sated Ht 15°31 98 103 =< een whe: 20°59 98 1¢9 a rise of temperature on the vapour-pressure with solutions of constant strength. Here also there is confirmation of my results, and we arrive at the following conclusions :— 1. Temperature constant.—The effect of increase of concen- tration is, in the case of NaCl, an increase of the restraining effect of each salt molecule. Such is also the case, though less marked, with KCl, while with NaNO; and KNO, precisely the reverse is the case, the four salts forming a series, as is shown diagrammatically above. 514 Dr. W. W. J. Nicol on the Vapour-pressures 2. Concentration con- stant.—The effect of rise of temperature is to di- minish the restraining effect of the salt in the case of NaCl, to leave KCl unaffected, and to increase the effect with NaNO; and KNO,. The four salts forming again a series in the same order as above. 3. When both temperature and concentration increase the salts form the same series—diminution of the restraining effect with NaCl and KCl, little or none with NaNO,, and a marked increase with KNQ,. 4. When the solubility as a function of the temperature is considered, the same series is observed. ‘The solubility of NaCl increases only slightly with rise of temperature, that of KCl rather more so. Still more marked is the case with NaNOs, while the solubility of KNO, increases enormously with the temperature. 5. There is clearly a connection between increase of solu- bility of the salts and the effect they have on the vapour- pressure of the water in which they are dissolved. Any attempt to explain the behaviour of the salts must of necessity, in our present state of knowledge of the nature of “solution, partake largely of the nature of hypothesis. Still I -cannot refrain from pointing out how completely the Theory of Solution I put forward some years ago explains these very varied phenomena. According to this theory, solution is an entirely physical process, and results from the action of purely physical forces; thus differing entirely from all the modifications of the Hydrate Theory which have hitherto found favour with expe- rimenters. Solution is the result of the tendency of three forces towards equilibrium. These three forces being the attraction of water for salt, that of salt for salt, and that of water for water. For the sake of brevity the attraction be- tween similar molecules may be spoken of as Cohesion, that between dissimilar molecules as Adhesion. Thus, where the salt is soluble, the adhesion of salt and water is greater than the cohesion of the salt plus the cohesion of the water. When the salt is insoluble, the reverse is the case. When the salt dissolves, the adhesion of water to the salt is more and more diminished by the presence of the dissolved salt till a point is of Water from Salt-Solutions. 515 ‘reached at which the three forces are in equilibrium ; this is the saturation-point for the temperature under consideration. At the saturation-point as many salt molecules meet. and unite to form solid salt in unit time as are dissolved by the action of the water on the still undissolved salt. The effect of temperature on each of these three forces is to diminish each of them. It may be each to the same ex- tent, or it may be by unequal amounts ; and thus a salt may have its solubility unaltered, increased, or diminished by rise of temperature. That is, according as the difference between the adhesion and the sum of the cohesions at the higher tem- perature is >, =, < the same difference at the lower tempera- ture, so is the solubility at the higher temperature >, =, < than the solubility at the lower temperature. But this is not all, for the magnitude of these differences depends on the various effects of temperature on the adhesion and the sum of the cohesions; and this may be very different with different salts, for though the cohesion of the water is affected by tempera- ture always to the same extent, still the cohesion of the salt and the adhesion are variables. With this preliminary ex- planation, I will proceed at once to the explanation of the effect of salts on the vapour-pressure of water according to the above theory. The first point to be noted is that all four salts have, in dilute solution, very nearly the same restraining effect :— n. NaCl. KCl. NaNO,. KNO,. See 4:25 3°80 4:25 3°75 i) 2 ee 5°04 4°43 3°67 2°73 but that in strong solutions it is very different. Now the restraining effect of a salt can only be due to the adhesion of the water and salt*; this, then, is very nearly a constant for all four salts, and is still more nearly a constant for salts of the same metal. Next, the heat of solution of the salts is as follows :— NaCl. KCl. NaNO,. KNO,. —1180 — 4400 —95200 — 8500, still the same series before observed. Now, as pointed out in another paper, if the restraining effect of salt be a constant, or nearly so, then the heat of solution of various salts is a * It may be noted, in passing, that unless a salt molecule were capable of disturbing the equilibrium of the whole of the water molecules present, it could have no effect on the vapour-pressure of the water. 516 On the Vapour-pressures of Water from Salt-Solutions. measure of the physical work done in effecting the change of state of the salt—that is, in overcoming the cohesion of the salt. If this is so, then the cohesion of the four salts under consideration increases from NaCl to KNO3. Now the effect of concentration is to markedly increase the restraining effect of the salt in the case of NaCl, to a less extent with KCl; and, on the contrary, to diminish it with NaNO, and still more so with KNO;. But the cohesion is small with NaCl and increases up to KNO;. Is it not reasonable to sup- pose that, in the case of a salt with small cohesion, concentra- tion has but little effect on the adhesion; and that, on the contrary, when the cohesion is very large, the effect of con- centration is to slowly diminish the adhesion in each case till the saturation-point is reached ; and salts with cohesion inter- mediate between the extremes above will have an intermediate effect, on the vapour-presure of the water ? We have next to consider the effect of rise of temperature on the restraining effect of the salt. This we have seen is a diminishing one with NaCl, nil with KCl, and an increase in the case of NaNO; and KNO;. This, I submit, is fully in agreement with the above, and also with the observed solubi- lities ; for though an increase of solubility implies an increase of the difference between adhesion and the sum of the cohe- sions, still that is in few cases due to an actual increase of the value of the adhesion, but to the decrease of the cohesions exceeding the decrease of the adhesion. Now ina solution of a salt of constant strength, rise of temperature is necessarily attended by a decrease in the value of all the three forces. If the cohesion be diminished a little more than the adhesion the salt will be a little more soluble, and the effect on the restrain- ing effect of the salt will be to diminish it; for though the comparative value of the adhesion be increased, the absolute value is diminished. On the other hand, if the cohesion be very large and be largely affected by rise of temperature, the result will be to increase the comparative value of the adhesion ; and in strong solution to actually increase the absolute value, as is shown by the increase in the restraining effect of KNOs, especially in strong solutions. 3 Thus it appears that this theory of solution is able to explain the very varied phenomena of vapour-pressures of water from salt-solutions, as it has explained other phenomena of solu- tion. It remains to be proved whether this explanation is correct. alls aa LX. On Stationary Waves in Flowing Water.—Part III. By Sir Witu1am Toomson, F.R.S. [Continued from p. 452. ] In No. 138 (November), p. 446, equation (6), for (gD +?) read (gD +97). m* promised in Part I., we may now consider the appli- cation of the principles developed in it and in Part II. to the question of towing in a canal, and we shall find almost surprisingly a theoretical anticipation, 493 years after date, of Scott Russell’s brilliant ‘‘ Experimental Researches into the Laws of certain Hydrodynamical Phenomena that accompany the Motion of Floating Bodies, and have not previously been reduced into Conformity with the known Laws of the Resist- ance of Fluids”’*, which had led to the Scottish system of “fly-boat,’’ carrying passengers on the Glasgow and Ardrossan Canal, and between Edinburgh and Glasgow on the Forth and Olyde Canal, at speeds of from 8 to 12 or 13 miles an hour} by a horse, or a pair of horses, galloping along the bank. The practical method originated from the accident of a spirited horse, whose duty it was to drag a boat along at a slow speed (I suppose a walking speed), taking fright and running off, drawing the boat after him, and so discovering that when the speed exceeded a, gD the resistance was less than at lower speeds. Mr. Scott Russell’s description of the incident, and of how Mr. Houston took advantage for his Company of his horse’s discovery, is so interesting that I quote it in extenso :—“‘ Canal navigation furnishes at once the most interesting illustrations of the interference of the wave, and most important opportunities for the application of its principles to an improved system of practice. It is to the diminished anterior section of displacement, produced by raising a vessel with a sudden impulse to the summit of the progressive wave, that a very great improvement recently introduced into canal transport owes its existence. As far as | am able to learn, the isolated fact was discovered acci- dentally on the Glasgow and Ardrossan Canal of small dimensions. A spirited horse in the boat of William Houston * By John Scott Russell, Esq.,M.A., F.R.S.E. Read before the Royal Society of Edinburgh on April 3, 1837, and published in the ‘ Transac- tions’ in 1840. + One mile an hour is English and American reckoning of velocity, which, when not at sea, signifies 1‘60953 Iilometres per hour, or -44704 metre per second. Phil. Mag. 8. 5. Vol. 22. No. 139. Dec. 1886. 2N 518 Sir William Thomson on Stationary Eisq., one of the proprietors of the works, took fright and ran off, dragging the boat with it, and it was then observed, to Mr. Houston’s astonishment, that the foaming stern surge which used to devastate the banks had ceased, and the vessel was carried on through water comparatively smooth, with a resistance very greatly diminished. Mr. Houston had the tact to perceive the mercantile value of this fact to the Canal Company with which he was connected, and devoted himself to introducing on that canal vessels moving with this high velocity. The result of this improvement was so valuable inamercantile point of view, as to bring, from the conveyance of passengers at a high velocity, a large increase of revenue to the Canal Proprietors. The passengers and luggage are con- veyed in light boats, about sixty feet long and six feet wide, made of thin sheet iron and drawn by a pair of horses. The boat starts at a slow velocity behind the wave, and at a given signal it is by a sudden jerk of the horses drawn up on the top of the wave, where it moves with diminished resistance, at the rate of 7, 8, or 9 miles an hour” ™*. The ‘diminished anterior section of displacement pro- duced by raising a vessel with a sudden impulse to the summit of the progressive wave’’ is no doubt a correct observation of an essential feature of the phenomenon; but it is the annulment of “the foaming stern surge which [at the lower speeds| used to devastate the banks” that gives the direct explanation of the diminished resistance. It is in fact easy to see that when the motion is steady, no waves can be left astern of a boat towed through a canal at a speed greater than »/gD, the velocity of an infinitely long waye in the canal ; and therefore (the water being supposed inviscid) the resistance to towage must be nil when the velo- city exceeds »/gD. This holds true also obviously for towage in an infinite expanse of open water of depth D over a plane bottom. The formula (25) of Part II. for the whole horizontal com- ponent force upon an inequality or succession of inequalities on the bottom allows us to calculate the resistance on a boat of any dimensions and any shape provided we know the height of the regular waves which follow it steadily at its own speed in the canal, at a sufficiently great distance behind it to be sensibly uniform across the breadth of the canal, according to the principle explained in the middle of p. 354 of Part I. The principles upon which the values of .§ [the h of formula (25), Part II.] may be calculated are partly given in the remainder * Trans, Roy. Soc. Edinb, vol. xiv. (1840) p. 79, Waves in Flowing Water. 519 of the present article, and will be more fully developed in Part IV. To find the steady motion of water flowing in a rectangular channel over a bottom with geometrically specified inequa- lities, it is convenient, after the manner of Fourier, to first solve the problem for the case in which the profile of the bottom is a curve of sines deviating infinitesimally from a horizontal plane. For convenience, take OX along the mean level of the bottom, positive in the direction of U the mean velocity of the stream; and OY vertical, positive upwards. Let EE COST Ye ae a ne GL) be the equation of the bottom ; and eA = NY COS MG yk 2 of al we, (2d be the equation of the free surface, ) being height above its mean level. Let ¢ be the velocity potential; w, v the velocity components ; and p the pressure at any point (2, y) of the water at time ¢: so that we have d d u= & and =F e (3), and p=C—gy—t We) sw... a Now the deviation from uniform horizontal velocity is infini- tesimal, and therefore v and u— U, are infinitely small. Hence (4) gives p=C—gy—3 U’+U(u—U) (5). : é to OO cae ¢ must be a solution of the equation of continuity Ie 5 iy =0, and the proper one for our present case clearly is @— Use sirma(Ke + K'e-) x) xr 2 (6), where, because the motion is steady, K and K’ are constants. This, in virtue of (3), gives u—U=mcos me (Ke™+K’e“™), . . (7) v=msin ma(Ke™’—K’e~“™) . . . (8). Hence, as the values of y at the bottom and at the surface are infinitely nearly 0 and D respectively, we find respectively for the vertical component velocity at the bottom and at the surface, msin maz (K—K’), and msin ma (Ke™? —K’e—™), Hence, to make the bottom-stream-lines and surface-stream- 2N2 520 Sir William Thomson on Stationary lines agree respectively with the assumed forms (1) and (2), we clearly have mK=K)=mHU «> We and m(Ke™?—K’e-™) =mHU . . 9) whence §—He-™? Rages (11). rede qeeecuan ed RAN yelper Now at the free surface the pressure is constant, and hence, by (5), we have —gy+U(u—U)=constant . . . . (12): from which, by (2), (7), and (11), we find 2 H(em? + e—™P) —2H 0=—gH+mU emD — ¢g—mD whence 9H H= a ae an ee 6) e@D 4 e—mD __ Ue (e™D —e—mD) which is the solution of our problem, for the case of the bottom a simple harmonic curve. Suppose now the equation of the bottom to be h= («cos mz +x? cos 2ma+«3 cos 8ma+&c. )mA/m . (14); the equation of the surface, found by superposition of solu- tions given by (13), allowable because the motion deviates infinitely little from horizontal uniform motion throughout the water, is y=D=p=>_. 2«' cosimae . mA/a (15). eimD et e—~imD _ ; a (eimD — e—imD) To interpret the equation (14) by which the bottom is defined, remark that, by the well-known summation of its second member, it is equivalent to | fie gmAl/m.(1—«’) _ m A/a .« (cos ma—«) ~ 1—2Kcosmat+nK 1—2k cosme+ xk? (16). The series (14) is convergent for all values of « less than unity. According to the method of Fourier, Cauchy, and Poisson, the extreme case of « infinitely little less than unity will be made the foundation of our practical solutions. By £mA/T= Waves in Flowing Water. 521 (14) we see that (@or=0 Pog Rae NOT) 18)! (PTY te —1/m and hence by the first of equations (16) we see that ar/m ah aoe { a. gmA/m.(1—#') _ 4 Nepean eS) nim L—2Kcosmz+ Kk” Now when « is infinitely little short of unity the factor of dz in the first member of (18) is zero for all values of 2 dif- fering finitely from zero or 2iz/m, (i being an integer); and it is infinitely great when «=O or 2i/m. Hence we infer from (17) and (18) that a vertical longitudinal section of the bottom presents a regular row of symmetrical elevations and depressions above and below its mean level; the elevations being confined to very small spaces on the two sides of each of the points z=0 and #=2i7/m, and the profile-area of each eleva- tion being A. The depths of the depressions below the average level in the intermediate spaces between the elevations, are of course extremely small because of the exceeding shortness of the spaces over which are the elevations. For our complete analytical solution, not only must A be infinitely small, but the steepness of the slope up to the summit of A must every- where be an infinitely small fraction of a radian ; and of course therefore the infinitesimal lowering of the bottom be- tween the ridges, which the adoption of a mean bottom-level for our datum line has necessarily introduced, may be left out of account in our dynamical problem. If the slope of the ridge is not an infinitely small fraction of a radian our solution will still hold, provided its height is very small in comparison with the depth of the water over it. But the effective potency of the ridge would then not be its profile-area A, but something much greater; of which the amount would be found by taking a stream-line over it, far enough above it to have nowhere more than an infinitesimal slope, and finding the profile-area of such a stream-line above its own average level considered as the virtual bottom. With these explanations we shall speak of a ridge for brevity instead ofan “ irregularity ”’ or “ obstacle,”’ and call its profile-area A, simply the “‘ magnitude of the ridge;”’ this being, as we see by (15), the measure of its potency in disturbing the surface. When instead of a ridge we have a hollow, A is negative ; and when convenient we may, of course, call a hollow a nega- tive ridge. It is clear that (15) converges, and does not depend for its convergence on « being less than unity ; so that in it we may take « absolutely equal to unity, and we shall do so accordingly. 522 Sir William Thomson on Stationary To find now the effect of a single ridge, remark that if 1 be the length from ridge to ridge, Vin=Oafl oo. a er After the manner of Fourier now suppose / infinitely large ; which makes m infinitely small; and put : im=q and m=dq... .... = ee then with e=1, (15) becomes Se . 2 ae ae’ 0 efD 4 e— 7D _ = (eg? —e—9) where bg ee lee se, Equation (21) will be shortened, and for some interpretations simplified, by making gD=c, when it becomes p= { “do 2A/Dz . cos (cz/D) een evt+e-o— i (e7—e-7) The definite integral (21) or (23) seemed rather intractable, and the quadratures required to evaluate it, for many and wide- spread enough values of x to show the shape of the surface for any one particular value of D/b, would be very laborious, But I had found a method of evaluating it from the periodic solution for an endless succession of equidistant equal ridges (15), wholly analogous to analytical deductions from corre- sponding solutions for cases of thermal conduction and of signalling through submarine cables, to be found in vol. ii. pp: 49 and 56 of my Collected Mathematical and Physical Papers ; and, towards applying this method to a particular case of the disturbance due to a single ridge, I had fully worked out the periodic solution for the case represented by the diagram of curves (fig. 3, p. 529), when I found a direct and complete analytical solution for the single-ridge problem in a form exceedingly convenient for arithmetical computation, except for the case of # equal to zero, or from zero to a quarter or a half of the depth. The previous method happily gives the solution for small values of x, and indeed for values up to two or three times the depth, by very rapidly converging series, and thus between the two methods we have a remark- ably satisfactory solution of the whole problem. Before explaining the curves and their relation to the pro- blem of the single ridge, I shall give the new direct solution Waves in Flowing Water. 523 of this problem. It is founded on a well-known analytical method of Cauchy’s, of which examples are given in the Highteenth note (p. 284) to his Memoir on the Theory of Waves*. First, bring the denominator of (23) to the form of the product of an infinite number of quadratic factors, as follows: — Let 1 -ct bo CW eo Rag o8) +e ~ (¢ € spas (24). ie Expanding in powers of o, we have 1 1 PDT, aio 1 LD , +rysal(i-5z)ets f ae Hence, when 6 is greater than D, W is positive for all real values of ¢. But when 0 has any positive value less than D, W (which is always positive for small values of o*) is nega- tive for lar ge values of o?; and therefore at least one positive value of c? makes W zero. We shall see presently that only one positive value of o? does so. We shall see that all the zeros of W when 0 is less than D, and all but one when 6 is greater than D, correspond to Teal negative values of o%, This indeed is ae if for o? we put —6?, which gives W= = (coso—7 5) - = (26); D Laieis and which shows that the zeros of W are given by the roots of the well-known transcendental equation tan@ b @ ca Bi - - * ~ 5 “ (27). When 3 is greater than D this equation has all its roots real, and in the first, third, fifth, &c. quadrants. When 0 is less than D the root in the first quadrant i is lost, and in its stead we clearly have a pure imaginary ; while the roots in the third, fifth, &c. quadrants remain real. Let 6, 02, 03, &. be the roots of the first, third, fifth, &c. quadrants. As the first term of equation (25) is unity, we have * Mémoires de Académie Royale de Institut de France, savans étrangers, tome i. (1827). 524 Sir William Thomson on Stationary w=(1- 73) (1-55) es 7) be a o [285 W= (1+ 2) (1+ 52) (1+ 53) &e. where 0,?, 037, &c. are real positive numerics, while 0,? is real positive or real negative according as 6 is greater than D or less than D. Resolving now the reciprocal of W into partial fractions, we find == ae ~- ae + tbe +&e . . (29); 1+ 62 1+ 02 1+ 02 where —2 — (1—D/d) cos 6; cea ~~, (dW\ ~~ Dib— cos @, Lae, a ag ; —_ (1—D/b) sin 6; ~ 6,(1—b/D. cos? 6;) * For i=1 and D>8, 6; is, as we have seen, imaginary (its square real negative), and for this case the formula (30) may be conveniently mannTen N,=-< (30). 4(D/b—1) (e% +e7%) Dibes dor 2c areau) ie and the equation for finding o, is eben — Fo (ener) =0 mo (2) | (31) ; an equation which has one, and only one, real root when D>, and no real root when D<0. When b/D is given, it is easy to find, as the case may be, o, of (32) or O, the first-quadrant root of (27), by arithmetical trial and error; and the successive roots O>, é;, &c. more and more easily, by the solution of (27). It is to be remarked that, whatever be the value of b/D, these roots approach more and more nearly to the superior limits of the quadrants in which they lie: thus if we put 6; = (i— 4) 7— 4; ° ° e i ° (33), Na=(—Dh idea] (1—D/b) cos a; 1—b/D sin? «; we have =(—1)*#! (34); Waves in Flowing Water. 525 and sin @;[(t—4)r—a;]=D/b.cosa, . . . (35); or, as ig convenient for approximation, when z is very large, Se he Dh a OL, tan a; which shows that as 7 increased to infinity, the value of a; approaches asymptotically to D/b(i—43)m. Hence when 7 1s very large, the second member of (36) becomes approximately D/b . (1— 4.4?); and the equation becomes (1—4D/b)a2—(i-3)ra=—D/b . . (87); a quadratic, of which the smaller root when D is less than 36, and the positive root when D is greater than 30, is the required value of @;. Going back now to (23) and modifying it by (24) and (29), we have Hien _ A/Dr its CD: b= pp > ™ do se ® e e e (38); 0 + or, according to the well-known evaluation (attributed by Cauchy to Laplace) of the definite integral indicated, penile yonlety (39); Lom oo af ae ene ar : or with 6;, N; eliminated by (33) and (34), Fie (—1)*! cos a; a b= LA/D ° DS (1—2/D. sin? a;) é of ke (40), where 1, a, ... @; denote all the positive roots of (35). This series converges with exceeding rapidity when 2 is any thing greater than D, and with very convenient rapidity for calculation when wz is even as small as a tenth of D. When «=0, the convergence has the same order as that of 1—e+e?—&c., when e=1; and we find the sum by taking as remainder half the term after the last term included. The true value of the sum is intermediate between the values which we obtain by this rule for a certain number of terms, and then for one term more. When it is desired to obtain the result with considerable accuracy, a large number of terms would be required; and it will no doubt be preferable to use my first method as indicated above. It remains to deal with the first term for the case D>8, which makes it imaginary in the form (39), but real in the form (38) with —o,’ substituted for 0,7. For this case we 526 Sir William Thomson on Stationary have, by the well-known definite integral, first, I believe, evaluated by Cauchy, ; 4A/D oe get n= .o Ny sin 7, foe where o; arid N, are given by (32) and (31). It is to be remarked that, inasmuch as (38) has the same value for equal positive and negative values of x, the evalua- tions expressed in (39) and (41) are essentially discontinuous at v=0; and when wz is negative, —x must be substituted for « in the second member of the formulas. I hope in Part IV. to give numerical illustrations ; but with or without numerical illustrations, the analytical formula (39), with (41) for its first term and the sign of # changed throughout when x 1s negative, is particularly interesting as a discontinuous expression for a curve passing continuously from one to the other of the two curves _ 3A/D See: and, A/D y=— ae sin 45 for large negative values of « | oN, sin ca for large positive values of 4 (42). For the case of b > D every term of (39) is real, and (re- membering that the sign of # is changed when ~ is negative) we see that it makes equal for equal positive and negative values of x, and diminish asymptotically to zero as w becomes greater and greater in either direction. It expresses unam- biguously the solution (clearly unique when b> D) of the problem of steady motion of water in a uniform rectangular canal interrupted only by a single ridge of magnitude A across the bottom. ‘This is the case of velocity of flow greater than that acquired by a body in falling through a height equal to half the depth. It is otherwise in respect to uniqueness of the solution when the velocity of flow is less than that acquired by a body in falling through a height equal to half thedepth (b YgD. Putting now in (43) U?=gb, and comparing with (32), we see that mD=c,; and going back to equation (2) above we see that o\(“#—a feos eS a ane where § and a are arbitrary constants, is the addition which we must make to (39) to give the general solution for the case b == cae 2 0;N;e D when w is negative BC cosy +(C— where C and C’ denote arbitrary constants. 528 Sir William Thomson on Sationary The motion represented by this solution, with any values of 0 and O!, is steady and stable throughout any finite length of the canal on each side of the ridge, provided the water is introduced at one end of the portion considered and taken away at the other conformably. If the canal extends to infinity in both directions, and if the water throughout be given in the state of motion corresponding to the solution (46); the motion through- out any finite distance on each side of the ridge will continue for an infinite time conformable to (46). The water, if given at rest, might be started into this state of motion in the follow- ing manner :—First displace its surface to the shape repre- sented by equation (46), and apply a rigid corrugated lid to keep it exactly in this shape, so that it is now enclosed as it were in a rectangular tube with one side corrugated, two sides plane, and the fourth side (the bottom) plane, except at the place of the ridge. Next by means of a piston set the water gradually in motion in this tube. To begin with, the pressure on the lid will, in virtue of gravity, be non-uniform ; less at the high parts and greater at the low parts. If too great a velocity be given to the water by the piston the pres- sure will, in virtue of fluid motion, be greater at the high parts and less at the low parts. If the average velocity be made exactly U, the pressure will be uniform over the lid, which may then be dissolved ; thus the liquid is left moving steadily under the surface represented by equation (46) as free surface. But it is only in virtue of this motion being given to the fluid throughout an infinite length of the canal on each side of the ridge, that the motion can remain steady on each side of the ridge conformable to (46), except for the particular case of this general solution, corresponding to at (0 and C= fe a Ny. eee which reduces (46) to re) 0 x b= (o,N, sin ay +426Ne D) when 2 is positive 2 BS 0,0 and, v= fs OND when 2 is negative 2 | this being the practical solution for the case of water flowing from the side of x negative over the single ridge and towards the side of x positive. Itis the mathematical realization, for the case of a single ridge, of the circumstances described in Part I. No. 1 (ante, pp. 856-357), and is the mathematical e 3 Waves in Flowing Water. 529 solution promised in the last sentence of Part IJ. The demonstration that this is the practically unique solution for inviscid water flowing in a canal with a single ridge, and the explanation of how any other state of motion, such, for example, as that represented by (46) with any value of C and C', but given to the water throughout only a finite distance on each side of the ridge, settles into the permanent steady motion represented by (48), must be reserved for Part IV., which I hope will appear in the January number. _ Meantime the accompanying diagram represents by two curves two cases of the solution (46) for the particular value 2°456 of D/b ; thatis to say, for velocity =°6381 of the critical velocity “gD. The faint curve represents the solution (39) (41), or, which is the same, (46) with C=O and C’=0. The heavy curve represents the practical solution (48). These curves were drawn from calculations of a_ periodic solution, according to the first of the two methods indicated above, before I had found the analytical solution (39) by which the desired result could have been arrived at with much less labour. The faint curve was drawn first by direct calculation from the periodic solution : the letters 4 J, + J, —4/, —+1, &., show on the two sides of one ridge quarters of the distance from ridge to ridge in the periodic solution, one of the ridges being in the middle of the diagram. The heavy curve is found by adding to the ordinates of the faint curve the ordinates of a curve of sines, found by trial to as nearly as possible annul on the one side, and to double on the other side, the ordinates of the original curve. How nearly perfect was the annulment on the one side and the doubling on the other is illustrated by the small-scale diagram an- nexed (fig. 3), which has been drawn by | the engraver from a six times larger copy. } How nearly perfect the annulment and the | 530 Prof, R. Bunsen on the Decomposition doubling ought to be at any particular distance from a single | ridge is now easily calculated from the second line of equation (48), and will be actually calculated for the case of these curves, and probably also for some other cases for numerical illustrations, which I hope to give in Part IV. + LXI. Decomposition of Glass by Carbon Diowxide held in Solution in Capillary Films of Water. By Prof. R. BUNSEN *. [* an earlier publication | I have given my investigations of the phenomena which present themselves when carbon dioxide is allowed to act on capillary glass threads covered with an extremely thin film of moisture. According to these investigations, it appears that 49-453 grammes of such capil- lary threads are able in 109 days to take up so much carbon dioxide, that on heating not less than 236°9 cubic centim. of this gas is set free. The gas so retained in the water-film, showed towards pressure and temperature precisely the rela- tions which are presented in the ordinary phenomena of gas absorption by liquids. In these experiments, as in all that have been previously carried out, it has been assumed, both from the result of direct observation and on theoretical grounds, that the action of carbon dioxide on glass may be entirely disregarded. And, indeed, experiments were carried out in my laboratory seventeen years ago by Dr. Hmmerling, which showed that glass vessels in which an 11 per cent. solution of hydrochloric acid was boiled for hours together did not lose 0:0005 grm. in weight. If, in addition to this, we bear in mind that under ordinary atmospheric pressure, at 15° C., water dissolves only 0:2 per cent. by weight of carbon dioxide, an acid which is set free from allits compounds even | by the weakest acids, and, further, that repeated observations show that dry carbon dioxide has practically no action upon dry glass, then it must appear almost absurd to attempt to explain the gradual fixation of carbon dioxide on glass dried by calcium chloride by a chemical decomposition of the lass. | : But the matter presents itself under quite a different aspect when we have regard to the phenomena of absorption as occurring in capillary films. Water which at 15° C. and * Translated from Wied. Ann. x. pp. 161-165 (1886), by G. H. Bailey, D.8c., Ph.D. + Wied, Ann. xxiv. p. 321 (1885), of Glass by Carbon Dioxide. 531 0:76 metre pressure takes up about 0°2 per cent. of its weight of carbon dioxide behaves quite differently in capillary films, for it is not then under a pressure of one atmosphere, but under a very high capillary pressure, and so can take up’ so much more carbon dioxide, that, if we would study the decomposing action of the solution, we have no ground of comparison, and must solve the problem by direct experiment. Such an experiment could not be carried out either before or during the previous experiments without destroying the capillary glass thread; and thus it was not possible to proceed with the examination of this question till the experiments already proceeding were finished. The 49°453 grms. of glass used were, for this purpose, removed from the measuring tube and extracted with cold distilled water of such a purity that it left only 500,00 solid residue on evaporation. Tor the extraction, portions of 300 grammes of water were taken, and the whole 8000 grammes so used were filtered through a double filter and evaporated to dryness in a platinum vessel. The residue dissolved in hydrochloric acid with evolution of carbon dioxide, and con- tained 0°8645 grm. of sodium chloride and 0:0608 grm. of silica along with unweighable traces of calcium chloride or potassium chloride. From the composition of the capillary threads *, it appears therefore that there was not less than 2°882 grms. of the glass decomposed, or 5°83 per cent. of the whole quantity used. Wesee, then, that the chemical action of carbon dioxide under the influence of pressure in capillary films is far greater than we had any cause to expect. ‘The carbon dioxide had, in the course of the experiments, taken up from the glass a quantity of soda corresponding to 0°7841 grm. sodium car- bonate, and containing 0°325 grm. carbon dioxide. Since sodium carbonate is not decomposed, even at very high temperatures, the 236-9 cubic centim. or 0:4659 grm. of carbon dioxide set free on heating could not arise from this decomposition product of the glass thread. But sodium car- bonate takes up carbon dioxide and is transformed into the bicarbonate, and this carbon dioxide is set free again on heating, exactly in the same manner as in these observations. It is thus to be determined whether the phenomena ob- served in capillary absorption can be exclusively attributed to the formation of sodium carbonate. If we start from the most unfavourable supposition that all the sodium carbonate formed became bicarbonate, and that * Wied. Ann. xx. p. 545 (1883) [ Phil. Mag. March 1884]. 532 Decomposition of Glass by Carbon Dioxide. the temperature to which the glass was heated was sufficiently high to expel the whole of the carbon dioxide, then we can only account for 165:2 cubic centim. of carbon dioxide instead of 236°9 cubic centim. There must therefore, even under the most unfavourable conditions, have been at least 71:7 cubic centim. of carbon dioxide fixed on the glass otherwise than by chemical union. We cannot unfortunately determine how much, however, actually was due to the decomposition of the glass and how much to the capillary absorption. If, then, carbon dioxide, under the conditions described, can overcome the affinity of silica for soda, a similar action, although in a lesser degree, may be expected from pure water. That such an action really does take place may be expected, if one may draw a conclusion from the action at higher tem- peratures, as indicated by the following fact, which I had occasion to observe in the preliminary experiments on the esti- mation of the tension of water-vapour at very high temperatures. In these experiments I made use of narrow thick-walled tubes, sealed at the upper end, and attached at the lower end to a calibrated capillary tube 2 metres long, and which would withstand a pressure of 600 to 800 atmospheres. In the wider part of the tube containing air there was, standing over the mercury column by which the approximate pressure was measured, a column of water, and this was heated to 550° C. in the thermostat described *. At the part of the wall of the tube with which the water had been in contact there were alterations of a marked character. The glass was transformed to more than a third of its thickness into a hard white porcelain-like mass, and the inner cavity of the tube diminished to one-tenth of its original diameter. There can be no doubt therefore that glass and other silicates intended to be used in the examination of such questions are quite inapplicable. In order to obtain trustworthy results in absolute measure without the interference of chemical influence, there remains scarcely any other course than to repeat the whole of the experiments on capillary absorption with very fine gold or platinum wire, and allow for the chemical action on the relatively small surface of the glass measuring-tube. * Bunsen, /. ¢. [* 333° J LXII. Reply to the Observations made by Messrs. T. E. Thorpe and A. W. Riicker upon our Essay entitled “ Intorno ad alcune formule date dal Sig. Mendelejeff e dai Siggn. T. H. Thorpe e A. W. Riicker per calcolare la temperatura critica della dilatazione termica.’ To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, i the course of some researches we were engaged upon two years ago we found it necessary to determine the critical temperature of all, or that of the greatest possible number of liquids. Inasmuch as this had only been arrived at experimentally in a comparatively small number of bodies, it occurred to us to apply to the remainder the theories and formula of Van der Waals, and thus calculate the temperature from the expansion caused by heat. We had already com- menced our investigations when we became acquainted with the essay by Mendelejeff on the Expansion of Liquids published in the Annales de Chimie et de Physique, and with the summary of it in the essay by Mr. Thorpe and Mr. Riicker which appeared in the Bevbldtter, entitled ‘‘On the Critical Temperatures of Bodies, and their Thermal Expansions as Liquids.”” As the formula deduced by Messrs. Thorpe and Riicker from the formula of Mendelejeff involved very simple | calculations for determining the critical temperatures, we decided to adopt it. Before doing so, however, we were desirous of establishing to our satisfaction the relative correctness of the formula, and the limits within which it could be legitimately applied. This preliminary inquiry gave rise to an essay entitled “ Intorno ad alcune formule date dal Sig. D. Mendelejeff e dai Siggn. T. EH. Thorpe e A. W. Riicker per calcolare la tempe- ratura critica della dilatazione termica,” published in the Nuovo Cimento for July, August, and September, 1884, pp. 91-104. This essay was also published in the Gazzetta Chimica Italiana, vol. xiv. 1884; and a report of it appeared shortly after wards in the Beiblatter zu den Annalen der Physik und Chemie, and also in the Journal de Physique ded Almeida, and, lastly, at the commencement of the present year, in the Annales de Chimie et de Physique. Messrs. Thorpe and Riicker, in a note inserted in the ‘Philosophical Magazine’ for May, have done us the honour to reply to our essay. We feel convinced that these gentlemen have only read the abridged report of our Pra. Mags 8.5. Vol. 22: No. 139..Dee..1886. 20 5384 Critical Temperatures and Thermal Expansions. paper published in the Annales, or they would not have waited two years before replying; and we believe also that had they read the paper in its entirety they would not have found it necessary to send any reply. We had no intention of criticising the work of Messrs. Thorpe and Riicker, but merely of showing by numerous examples the limits within which the formulas of Mendelejeff, and of Thorpe and Riicker, were applicable. We believed that our inquiry would result in some utility, because it was not clear whether the formula of Mendelejeff, which was deduced from the comparison of the expansion of liquids, measured under a pressure of one atmosphere, could be applied to the expansion of liquids measured under a constant, or uniform, pressure, although not of one atmo- sphere. We proved that the formula of Mendelejeff is valueless for showing the results of the experiments of Hirn, who has carefully studied the expansion of certain liquids under a constant pressure of 11 metres of mercury. This fact appeared, and still does appear, to us to detract much from the general applicability of the formula of Men- delejeff for representing the expansion of liquids, which there- fore we cannot consider otherwise than as empirical, and applicable only within narrow limits, but having with regard to the formulas more commonly adopted the merit of greater simplicity. We were well aware that the formula of Mendelejeff and that of Messrs. Thorpe and Riicker (which is applicable within the same limits as that of Mendelejeff) were not applicable to water, and we especially called attention to the fact in the note at page 98 of our article, and also at page 102. The formula of Messrs. Thorpe and Riicker for calculating the critical temperature leads to results which agree exactly with those obtained from experiments at temperatures lower than the normal boiling-point. We have already applied that formula for the calculation of critical temperatures of all liquids of which the thermal expansion has been studied. We shall feel obliged by your kindly publishing the above statement. Yours &ec., A. BaRTotui. Hi. STRACCIATI. [ 535 J LXIII. Electromagnets—V1. The Tension of Lines of Force. By R. H. M. BosanqueEt. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, 4 ae experimental determination of the magnetic attrac- tion corresponding to measured values of magnetic induction has long been in my mind. It has some bearing on my theory of permeability (see Phil. Mag. ser. 5, xix. 1885, p. 87 (note) and p. 94, in both of which places B, is inadvertently printed for 33”); but its independent interest is considerable. After the following experiments were arranged, a paper on a somewhat similar subject was read to the Royal Society by Mr. Shelford Bidwell. From the accounts which have been so far published, the method of this paper appears to be entirely FIXED ELECTRO-MACGNET SMALL COIL FOR MEASURING INDUCTIONS MOVEABLE ELECTRO-MAGNET Jf GOUNTERPOISE SCALE PAN %& different from mine, as the magnetizing force and the weight were alone directly measured: and in other respects the method differs essentially from that which I have adopted. 536 Mr. R. H. M. Bosanquet on Electromagnets. The sketch represents the instrument constructed for these experiments. This consists of two cylindrical iron magnets, each 20 centim. long and ‘526 centim. in diameter. Hach is wound with 1096 coils of wire. One magnet is fixed in a wooden frame, while the other moves exactly under it smoothly and without friction, between two brass collars (not shown), through a vertical distance of about a quarter of an inch. The magnets were at first ground together so as to ensure a perfect contact. The lower or sliding magnet has attached to its lower end an iron hook, which passes through the table and sustains the wooden pan for the reception of the weights. To this magnet also is firmly attached a brass collar, in which are two holes for the pivots of the beam. The beam which carries the counterpoise is about 18 inches long ; it moves on knife-edges, and has at one end a half-round fork, which em- braces freely the collar on the lower magnet, and is pivoted thereto. On the other end of the beam is hung a lead coun- terpoise exactly balancing the wooden pan and magnet. The magnetizing current was sent through both electro- magnets, while the induction was measured* by small coils placed round the point of contact of the two magnets. The inductions were measured by reversal. Weights were care- fully added until the contact was broken, and the greatest weight sustained was taken as the observed number. This could be observed with certainty to the nearest ounce. The first result is that the formula, S33? (W,) weight in grammes = 3a. 981 (S= section of contact), derived from (vol. ii.) Clerk Maxwell’s ‘ Electricity and Mag- netism,’ p. 256, represents the values well on the whole ; but for very small inductions the weight sustained is several times as great as it should be according to this theory. And this did not arise from residual induction ; for when the magnetizing current was interrupted, no power was left to sustain any weight at all. I therefore framed the empirical formula W , involving both % and 9%’; this represents the lower and middle values fairly, but not the highest. It cannot, however, escape notice that the observed values are best satisfied by adding about 4 oz. to all weights calcu- lated from W,. Since, however, this addition must vanish for %=0, we cannot thus obtain a general formula. Note * For the method see Phil. Mag. ser. 5, xix. p. 75. Mr. R. H. M. Bosanquet on Electromagnets. 537 that the induction has been forced up to the high value of 18,500. I have shown in recent papers that this quantity has no fixed limit in bar-electromagnets. This accords to some extent with the conclusions of Mr. Shelford Bidwell in the paper before referred to. The table of experiments is followed by one containing the means of the errors of the formule W, and W, for the groups into which the experiments are divided. Finally there are set out a few experiments made with the electromagnets not in contact, but separated to the distances mentioned by slips of card and wood. ‘These experiments are not approximately satisfied by the foregoing formule ; but they would agree to some extent with W, if, instead of the coefficient = 00004056, 87.981 we took about half that quantity, say ‘000026. The experi- ments at distance *2 would require a formula of the type of We. At present I am unable to give any explanation of these numbers. ; The general truth of the law Tension «SB? is sufficiently established. In my theory of permeability I have assumed the tension within the magnetic body « %. Itis clear that this assump- tion leads in that case to consequences which correspond with reality, and that the assumption tension a 8%? does not do so in that case. I am at present unable to explain the diver- gence between the two points of view. The weights have all been reduced to Ibs. and ozs. to facili- tate comparison with the actual experiments. Batteries were used for weights up to about 5 lb.; the dynamo from about Z lb. upwards. Weights W, and W,. aesk ey Wit ga, log S= 14671. W.=S(«8? + 138) ; log «=5°55391; log 1=9-76660. W obs. | _ W;. Diffs WwW, Diffs Ib: ozs ibsiez Ib. oz. | lb. oz. lb. oz 0 3 0 12/1 -—0O 18! 0 33 +0 03 0 4 0 16/—0O 24! 0 40 0 5 0 13) —0 37| 0 3:4) 20 Mae 0 5 0 138/;-0 37} 0 34 ]-0 16 0 6 0 14) —-0 46! 0 386 /—0 92-4 0 6 0 47} —0 13] 0 86/40 26 0 8 0 68] —O 1:2] 011-4 +0 3-4 0 10 0 56} —0 44] 0 97 |—0 03 FAG 0 13:0 | —O0 30} 1 30/+0 3-0 1 6 1 0O —0O 6 1 6 0 1 6 1 4 —0O 2 aie +0 5 ihe) 3) —0O 6 110 + /S20 iealo 1 10 —O 2 al SS). Zales 2 9 —0 4 3.2 |FOnS 36 3 —0O 4 3 11 +0 5 on 9 3 5D —0O 4 31d |40 6 3 13 on —0 6 4 4 /40 7 Bye Ths 3 6 —0O 7 3 1D. ee Ome rae o 15 —0 5 4 7. £653 5 11 6 4 +0 9 6°22 & eer aay 5 il irate: —O0 3 6 1 | 4056 DalZ, 5 12 0) 6.4. |2oNs 5 138 DD —0 8 5 13 0 6 1 5 14 —0 3 6% (0 Se 6 2 5 F —O ll 6 0 /|—0O 2 6 8 6 5 —0 3 4 9 |e 6 10 6 15 +0 5 i 8 Bone 6 12 63 +0 1 aaa +05 6 12 6 2 —0O 10 6.10 | =p Vem eS) —1 8 6 1 —-1 0 74 Wei 0 (Me eis i ee 6 14 —0O 4 Lie hey es 8 8 S12 +0 4 9 4 |4+0 12 10 O |10 10 4-0 10 | 11) 0) 1S ag 10 2 9 6 —0O 12 9158 |—{0 5 Li eo i hOe 75 —l 2 /|1011 —0O 12 wales: eevee 0) 16) 0. 2203 ZA. 218 1 ae CNIS AS ome 21 3 )19 14 =H 05 19 13° 3) as 21S S air (aan Le Yas —t 0 24 15° |p 28 2 |25 18 —2 5 25 6 | |—oRp 30 12 | 29 13 —015 |}29 2 |-1 10 384 5 |382 2 =? 8) PSbh 4 aay 39, 1, .|.38-11 —-i, 6 (\82 11, }—~9. 6 oD 4 oo b4 —] 6 |38214 |-~92 6 35 15 | 33 11 —2 4 /32 11 —3 4 36. 5. dks +1 3 /36 4 /=6 1 386 9 |40 1 +3 8 /38 9 2 0 36 14 |39 6 +2 8 ss 0 |e 3 BO Ae lon nes +041 136 8° |=—0 16 ote eo LO SL Wet 4 Ae ae 41 3 | 387 14 =—3 )0 186 9 /=410 41 9 |48 5 +112 /|41 9 0 12) oe) ta FD 3. ta flee fea: Bates 44 3 |47 12 +3 9 | 42 11 —1 8 44.5 |40 38 —4 2 |38 11 —5 10 0 | 45 138 —1 8 /41 8 —3 13 Mr. R. H. M. Bosanquet on Electromagnets. Arithmetic mean of errors of W, and W.. Number of W obs. ; : experiments. We: W;- lb. oz lb. 02 lb. oz. From 0 8 , < ee \—o 321 | 0 0:34 10 ‘Brom, 21 G95} , ; Ah 51 |f—-9 81 | +0 56 10 Brant {5,11 ; ; s ee \_o 56 | —0 12 10 From 7 1 ; ; lege ee } -0 95 | —0 57 10 From 26 7 , aoe ead } -0 41 | -1 48 10 From 36 138 : pa ee } -0 10 | —2 132 7 Weights sustained by Magnets when separated. W K= y ae Sw Distance Date W obs 3G. K. of magnets| Mean c. apart. lb. oz. (| 2 0 | 48035 | -00002805 |) 5 0 8022°1 | 00002514 1886 | 10 12 12667 00002168 in. A t 3 S 4| 16 4 13893 00002724 |$ -028 | -00002638 aon ee | 17 14 | 14588 ‘| 00002718 1a 2 14858 00002803 Lin22d as 17464 00002738 ( By 3736°4 | -00003042 | ee 6330°2 | 00002675 MRE 9709°6 | 00002510 ord & 4th. {| 10 9 11743 00002478 |} -060 | 00002584 12 5 12449 00002571 | 14 5 13878 00002404 Mion ek 15415 00002408 | ) (| 0 7 | 1611-8 |-00005449 | 0 10 25451 | 00003122 bd 4371°3 | 00002222 4th & 5th. ¢< 2, 60956 |-00002014 |} -2 2 14 6909:°7 | 00001948 | 3 12 8034-0 | 00001880 \| 5 9 | 10437 | -00001652 | ) 539 [ 540 J LXIV. Intelligence and Miscellaneous Articles. SILK v. WIRE, OR THE “ GHOST” IN THE GALVANOMETER. BY R. H. M. BOSANQUET. [HE ballistic galvanometer which I used for some years for the measurement of induction currents consisted of a small astatic pair of needles with mirror, surrounded by a small coil of very low resistance. ‘The suspension was from a silk fibre about six inches long, the fibre being left just stout enough to carry the weight. This whole combination was extremely sensitive and for the most part convenient to work with. The silk suspension, however, has certain troublesome properties. I shall not here enter into the manner in which it gradually untwists itself when stretched, or into its property of taking a set from any change of position; but shall confine myself to the appearance which we called the ‘ ghost.” At certain times the needles of the galvanometer would move about with sudden and capricious movements, the mirror often traversing several degrees of the scale. The decision and sharpness of the movements were very remarkable, and we habitually spoke of their cause as the ‘ ghost.” The ghost used to visit us mostly in summer between the hours of nine and eleven in the forenoon, and about six in the evening *. When these movements began it was no use attempting to work with the galvanometer. ‘There can be no doubt that the movements were due to the solar heat falling more or less directly on the instrument and causing hygroscopic changes in the silk fibre. In the early summer of this year I found it necessary to free myself from this source of interruption, and constructed a galvano- meter with a wire suspension. The difficulty consists in combining a needle system large enough to vibrate very slowly on the wire suspension, with a coil having sufficient power over the needles, and at the same time a low enough resistance. The needles are stout knitting-needles seven inches long. They are hung from a support fastened to the wall by a very fine wire about 5 feet long. The needles are very nearly astatic, and the complete double vibration takes a little over half a minute. The coil consists of about 500 turns of No. 20 B. W.G. ‘The resistance is much greater than that of the old instrument and the loss of sensitiveness is an inconvenience; but the instrument works well in connection with our large earth induction-coil of 250 turns of | the same wire, and it is entirely free from the visits of the ghost. It is my conviction that silk and thread suspensions are sources of error and inconvenience to an extent that has been imperfectly realized; and that they ought to be entirely banished from all instruments of precision. * The aspect of the galvanometer-room is north and east. E54 | INDEX to VOL. XXII. ——f>——_ Actps, on the electrical conduc- tivities of the, 105. Ackroyd (W.) on an electric-light fire-damp indicator, 145. AAthereal physics, tests of Herschel’s, 255. Aitken (J.) on dew, 206, 363. Amagat (F’.) on the measurement of very high pressures, and the com- pressibility of liquids, 384. Amalgamation, on the expansion produced by, 327. Ayrton (Prof. W. E.) on the expan- sion of mercury between 0° C. and — 389° C., 325; on the expansion produced by amalgamation, 327. Baily (W.) on a theorem relating to curved diffraction-gratings, 47. Bartoli (A.) on the critical tempera- tures of bodies and their thermal expansions, 553. Basset (H. B.) on the induction of electric currents in an infinite plane current sheet rotating in a field of magnetic force, 140. Battelli (A.) on Peltier’s phenomenon in liquids, 231. Bidwell (S.) on a modification of Wheatstone’s rheostat, 29; on the maenetic torsion of iron and nickel wires, 251. Bodies, on the critical temperatures of, and their thermal expansions as liquids, 533. Boltzmann (Prof.) on some experi- ments relating to Hall’s phenome- non, 226. Bonney (Prof. T. G.) on rock-speci- mens from Pembrokeshire, 74. Books, new :—Milne’s Volcanoes of Japan, 463; Croll’s Climate and Cosmology, 464. Bosanquet (R. H. M.) on electro- Phil. Mag. 8. 5. Vol. 22. No. 139. Dec. 1886. magnets, 298, 035; on permanent magnets, 500; on the “ghost” in the galvanometer, 540. Buchheim (A.) on the extension of a theorem relating to matrices, 173. Bunsen (Prof. R.) on the decompo- sition of glass by carbon dioxide, 530. Busch (M.) on an electrical experi- ment, 388. Capillary phenomena, on the appli- cation of thermodynamics to, 230. Carbon, on the electrical resistance of soft, under pressure, 358, 442. dioxide, on the decomposition of glass by, 5380. Chase (Dr. P. E.) on Herschel’s ethereal physics, 255, Chree (C.) on bars and wires of vary- ing elasticity, 259. Collie (Dr. N.) on the preparation of tin tetrethyl, 41; on the salts of tetrethylphosphonium and _ their decomposition by heat, 183. Conductors, on the self-induction and resistance of compound, 469. Copper, on the electrolysis of, 389. Cornu (Prof. M. A.) on the distine- tion between spectral lines of solar and terrestrial origin, 458. Coulombmeter, on a, 96. Cunynghame (H.) on a new hyper- bolagraph, 138. Deeley (R. M.) on the Pleistocene succession in the Trent basin, 72. Dew, on the theory of, 206, 270, 363, Diffraction - gratings, on a theorem relating to curved, 47. Doumer (M.) on the measurement of pitch by manometric flames, 309. Ze 542 Duhem (P.) on the application of thermodynamics to capillary phe- nomena, 230. Durham (J.) on the volcanic rocks of North-eastern Fife, 77. eae on the equations of the, 288. Earth, on the constitution of the erust of the, 1; on the physical structure of the, 233, 3381; on the annual precession calculated on the hypothesis of the solidity of the, 328. Edgeworth (F. Y.) on some problems in probabilities, 371. Elasticity, on bars and wires of vary- ing, 259, Electric current- and potential-me- ters, on the application of electro- lysis to the standardizing of, 389. currents, on a new instru- ment for recording the strength and direction of varying, 96; on the induction of, in an infinite plane current sheet rotating in a field of magnetic force, 140. Electric-light fire-damp indicator, on an, 145. Electrical conductivity of gases and vapours, on the, 387. experiment, on an, 388. residue, simple demonstration of the, 80. ——- resistance of carbon, under pres- sure, 308, 442. Electricity, ona mode of maintaining tuning-forks by, 216. and magnetism, notes on, 175, 469. Electro-chemical researches, 104. Electromagnet, on the formule of the, 288. Electromagnets, 555. , on the law of similar, 298. Electrometer, on an absolute spheri- eal, 79. Electromotive forces, on the seat of the, in the voltaic cell, 70. Electroscope, on the gold-leaf, 228. Emmott (W.) on an electric-light fire-damp indicator, 145. Fire-damp indicator, on a, 145. Fisher (Rey. O.) on the variations of gravity at certain stations of the Indian are of meridian, and on the constitution of the Harth’s crust, 1. Force, on the tension of lines of, 535. INDEX. Frolich (Dr.) on the theory of the dynamo, 290. Galvanometer, on the “ghost” in the, 540. Gas, on the law of attraction amongst the molecules of a, 81. Gases, on the electrical conductivity of, 387. Geological Society, proceedings of the, 72, 220. Glass, on the decomposition of, by carbon dioxide, 530. Gravity, on the variations of, at cer- tain stations of the Indian are of meridian, 1. Gray (T.) on a new standard sine- galvanometer, 368 ; on the electro- lysis of silver and copper, and the application of electrolysis to the standardizing of electric current- and potential-meters, 389. Hall’s phenomenon, on some experi- ments relating to, 226. Heaviside (O.) on the self-induction of wires, 118, 273, 332. Hennessy (Prof. H.) on the physical structure of the Karth, 233 ; on the annual precession calculated on the hypothesis of the Harth’s solid- ity, 328. Herries (R. 8.) on the Bagshot beds of the London basin, 78. _ Hicks (Dr.) on the Pre-Cambrian age of certain rocks in Pembrokeshire, 73. Hughes (Prof.) on some perched blocks, 220. Hyperbolagraph, on a new, 138. Integrator, on a form of spherical, 147, Iron, on the energy of magnetized, 175; on the magnetic torsion of wires of, 251; on the change in ae moment of, upon torsion, Keller (M.) on the increase of tem- ee produced by a waterfall, 312, Kolacek (F.) on the gold-leaf electro- scope, 228. Kriiger (R.) on a new method for determining the vertical intensity of a magnetic field, 311. Langley (Prof. S. P.) on hitherto un- recognized wave-lengths, 149. Laurie (A. P.) on the electromotive force of voltaic cells having an INDEX. aluminium plate as one electrode, 213. Letts (Prof. E. A.) on the prepara- tion of tin tetrethyl, 41; on the salts of tetrethyl phosphonium, and their decomposition by heat, 81. Lippmann (M.) on an absolute sphe- rical electrometer, 79. Liquids, on Peltier’s phenomenon in, 231; on the compressibility of, 384. Lodge (A.) on a new representation of moments and products of inertia in a plane section, and on the rela- tions between stresses and strains in two dimensions, 453. Love (E. F. J.) on magnetization, 46. Luvini (J.) on the electrical conduc- tivity of gases and vapours, 387. Magnetic field, on the determination of the vertical intensity of a, 311. researches, 50. — rotation of mixtures of water with some of the acids of the fatty series, 467. Magnetism, notes on, 175, 469. Magnetization, on, 46. Magnets, on the specific-induction constants of, in magnetic fields of different strength, 386; on per- manent, 500. Manometric flames, on the measure- ment of pitch by, 309. Matrices, on the extension of a theo- rem relating to, 173. Maurer (M.) on the constant of the Sun’s heat, 312. Mendenhall (J. C.) on the electrical resistance of soft carbon under pressure, 358. Mercury, on the expansion of,between 0° C. and —-39° C., 325. Monckton (H. W.) on the Bagshot beds of the London basin, 78. Naccari (E.) on Peltier’s phenome- non in liquids, 231. Nickel wires, on the torsion of, 251. Nicol (W. W. J.) on the vapour- pressures of water from salt-solu- tions, 502. Ostwald (Prof. W.) on the seat of the electromotive forces in the voltaic cell, 70; electrochemical re- searches by, 104. ours phenomenon in liquids, on, 231, 543 Perkin (Dr. W. H.), on the magnetic rotation of mixtures of water with some of the acids of the fatty series, 467. Perry (Prof. J.) on the expansion of mercury between 0° C. and —389° C., 325; on the expansion pro- duced by amalgamation, 327. Pitch, on the measurement of, by manometric flames, 309. Pressures, on the measurement of very high, 384. Probabilities, problems in, 371. Ramsay (Dr. W.) on some thermo- dynamical relations, 32. Rayleigh (Lord) on magnetism and electricity, 175, 469. Rheostat, on a modification of Wheatstone’s, 29. Rutley (F.) on eruptive rocks from St. Minver, Cornwall, 78. Sack (H.) on the specific-induction constants of magnets, 586. Salt-solutions, on the vapour-pres- sures of water from, 502. Shida (Prof. R.) on a new instru- ment for recording the strength and direction of a varying electric current, 96. Silver, on the electrolysis of, 389. Sine-galvanometer, on a new, 368. Smith (F. J.) on a form of spherical integrator, 147. Solution, on the theory of, 514. Spectral lines of solar and terrestrial origin, on the distinction between, 458. Stenger (F.) on a simple demonstra- tion of the electrical residue, 80. Stracciati (E.) on the critical tem- peratures of bodies and their ther- mal expansions, 533, Strahan (A.) on the glaciation of South Lancashire, &c., 75. Stresses and strains in two dimen- sions, on the relations between, 453. Sun, on the constant of the heat of the, 312. Sutherland (W.) on the law of at- traction amongst the molecules of a gas, 81. Tetrethylphosphonium, on the salts of, and their decomposition by heat, 183. a a tg relations, on some, 544 Thermodynamics, on the application of, to capillary phenomena, 230. Thompson (Prof. 5S. P.) on a mode of maintaining tuning-forks by electricity, 216; on the formule of the electromagnet and the equa- tions of the dynamo, 288. Thomson (Sir W.) on stationary waves in flowing water, 353, 445, 517. Tin tetrethyl, on a new method for the preparation of, 41. Tomlinson (C.) on Aitken’s theory of dew, 270. (H.) on sources of error in connection with experiments on torsional vibrations, 414; on the effect of stress and strain on the - electrical 442. Torsional vibrations, on, 414. Tuning-forks, on a mode of main- taining by electricity, 216. Turbines, on, 313. Vapour-pressures of water from salt- solutions, on the, 502. resistance of carbon, INDEX. Vapours, on the electrical conducti- vity of, 387. Voltaic cell, on the seat of the elec- tromotive forces in the, 70. cells, on the electromotive force of, having an aluminium plate as one electrode, 213. Water, on stationary waves in flow- ing, 353, 445, 517. of crystallization, observations on, 467. Waterfall, on the increase of tempe- rature produced by a, 312. Wave-lengths, on hitherto unrecog- nized, 149. Waves, on stationary, in flowing water, 353, 445, 517. Wheatstone’s rheostat, on a modifica- tion of, 29. Wiedemann’s (Prof. G.) magnetic researches, 50, Wires, on the self-induction of, 118, 273, 332. Woodbridge (J. L.) on turbines, 313. Young (Dr. 8.) on some thermody- namical relations, 32. END OF THE TWENTY-SECOND VOLUME. Printed by Taytor and Francis, Red Lion Court, Fleet Street. ; i 5 5 22. F The ; Phil. Mag.S.5. Vol. 22. Pl. 1 Fig), 7. = 300 A al | E : 380 LED } ve | 370 : 80 - = — 360 ale : 5 ale Saale el ma Ay: is Soft tron} CHING) eae ng linafly | PSY | Zz 0 30 co 90 120 150 180 210 w + 7 Peet Deed a ae 20 + 2. ; ° + les Slee be th | 20 rl — 4 40 “ ate = Pi > Minune temporartly njagnelixed tn [novepssely aly | BO “I 3; { 100 = el i = ee pak | 120 hot eh =| 350 “IF = “all li vir} bare Lojgitirclinielly Wecieang aaa j a i 0 7] 7 a0 aT 320 10 + = =i vi °) ¥ -— —_t—}, = —_ == IN f) 30 GO 30 720 150 130 210 zr T | " uf id. 8. ates a Fig). 4: Fees oe oe Se —— + — SS eee ee Re Near 80 eae ee t Ce Sees oT = I= GO 4} — + - + 70— =I ) [Neckel Vee, slepilea lon gitindinelby aes 40 -+ | - 0 30 60 90 120 150 150 210 20 -——+} = é ae 20 [eee Hal =I * oi L eS Nick ts only aged, Freel x 60 1 — pisalls | 80 + +—— 5 et 30 60 90 Lo 150 20} 30 gence ae , Phil. Mag. S.5. Vol. 22. Pl Il. A AA i) B i i | ii (BO' mil le | | j \ c a le HW ma (| = = 2 I (ll r i Wa | Mi Mintern Bros. lith Phil. Mag. S.5. Vol. 22. Pl ia Line of no aurren& = = i —— : | aa i i ! Hl muy js Phil. Mag, 8. 5, Vol. 22. Pl. IV. Vig. 1. —_— ¢ “i ASSIS FSS SS SSS SSS SESS SAAS yy» SSN LLL LLL LLL Lal ‘ Bolometer and Protecting Case. M Phil. Mag, 8. 5. Vol. 22. Pl. V. Apparatus for determining Relation between Index of Refraction and Wave-length. Phil, Mag, S. 5. Vol. 22. Pl. VI. n > 158 [ j | : | 1.57 ———— 2 | 1.56 | oe 1.55 — i 1.54 pe aa L 1.53 | 1.52 : eaerezo} Msirvation 9X0, 151 [ee i VISIBLE | | H 4 r O 5.0 6.0 n=4 (X) for Rock-salt Prism. Phil.Mag.S.5 Vol. 22. Pl. Vil. ( y ( Annales de Chimie et de Thysique. 6° Serie, Tome VII. ( Janae 1886.) | Cf a 5 ide Leo ep TLE: ted %, Bel A de Sfectie ys ips YE = YW. Goma : O30 G32 Phil. May. 5. vol O36 ail siete ve, PC PTT. 1 | | P Vol. 22. JULY 1886. No. 134. Published the First Day of every Month.—Price 2s. 6d. THE LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE, AND JOURNAL OF SCIENCE. 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