(Nii ws aati ‘ 4 aos Ses wine eae sear hee maw Sete . uy i 9 ABUT EA HE HD PREG A ; ' ‘ ine it Wheeden leibentacite he Hae Arie i Bite atta fr Re esi Re ee te al odie Ae Sew Th, Lee Be A TE - 4 eM TA: 4 ene ot Aa ee Me BOE time aed salrronoprates tbe sie Be 3 . PR eae On ae dale ; S, feed . Je roneine ne rarsct ni * ae y Y rs ein ‘ » y : ae Alas 7 q » oe ve, te " i ee k . Spe TOS e 3 c “a S PP al ‘ ; : 4 nl Seal cen ih pail! To ¢ ra 7 \ : Sap: asiort: ; » OT, er iaeitnd vigsay ces tntmerma-te divine ee oa FE Ie RR IED he cil Be ee + hy ich Ao Ft, NE Nhe Not rt BN f « RAG Rr hs pee Crash es Fh aod - i si ety toe eA Ss Rtn ; ib aap ' ~ meta bersoky A thekanoh pete aphid Speer anna Ve NNT Onan wee : 7 » re cain : : soo este : ; i . ‘S ee aoe Hrs 7 Ne ea a a al trea pe Mere ham Gah heen ett eM: Pe as coe Sede Rete Bh eM + 4 le Won) THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY LORD KELVIN, LL.D. P.R.S. &c. GHORGE FRANCIS FITZGHRALD, M.A. Sc.D. F.R.S. AND WILLIAM FRANCIS, Pu.D. F.L.S. F.R.A.S. F.C.S. ‘‘ Nec aranearum sane textus ideo melior quia ex sé fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib.i. cap. 1. Not. VOL. XXXIV.—FIFTH SERIES. JULY—DECEMBER 1892. LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD.; WIITTAKER AND CO. AND BY ADAM AND CHARLES BLACK ;—T. AND T. CLARK, EDINBURGH; SMITH AND SON, GLASGOW ;--HODGES, FIGGIS, 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 questionem, questio investigationem, investigatio inventionem.”—Hugo de S. Victore. “Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ccelo, Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J, B. Pinelli ad Mazuonium, CONTENTS OF VOL. XXXIV. (FIFTH SERIES). NUMBER CCVI.—JULY 1892. Mr. Carl Barus on the Fusion Constants of Igneous Rock.-— Part I. The Measurement of High Temperature. (Plate I.) Prof. W. H. Bragg on the “ Elastic Medium” Method of treating Electrostatic Theorems ...........:..-.00008- Mr. 8. U. Pickering on the Heat of Dissolution of Gases in obs. Ee eee oe eae a ee Mr. M. Carey Lea on Disruption of the Silver Haloid Molecule SORRENTO CO? aoe. asa seysmsveinje, mse age oye eis dine Dr. L. Natanson on the Probabilities of Molecular Configu- Ree i nyo. < fe is os iss Bos 4.8 mpd Myoleislen Baia ate oe Prof. H. A. Rowland on the Theory of the Transformer .. Mr. R. Inwards on an Instrument for Drawing Parabolic PO ocr raed, os ocala s see 0s ole wo a gee Lord Rayleigh on the Question of the Stability of the Flow [JOLLA .. $30 3S Cle ere rent Mr. C. Chree on Roiating Elastic Solid Cylinders of Elliptic ERNE ee CI Mey ais alithaph aid led alt I ashe ae eee ee Dr. W. Pole on some unpublished Data on Colour-Blindness. NN eh wae. sete scis os Co bw ass oh ee une’ Mr. F: J. Smith on an Air-mercury Pump, for raising Mer- cury in different kinds of Mercurial Pumps ............ Messrs. R: E. Hughes and F. R. L. Wilson on the Action of Dried Hydrochloric-Acid Gas on Iceland Spar .......... Mr. R: H. M. Bosanquet on the Calculation of the Ilumi- nating-Power of Hydrocarbons and their Mixtures ...... Proceedings of the Geological Society :— Dr. J. Prestwich on the Raised Beaches, and ‘“‘ Head” or Rubble Drift of the South of England.—Part I. Messrs. B. N. Peach and J. Horne on the Olenellus-Zone at the North-west Highlands :..2:52.3526.-6. 00.48) Dr. J. Prestwich on the Raised Beaches, and ‘‘ Head ” or Rubble-Drift of the South of England.—Part I]. .... Mr. C. Reid on the Pleistocene Deposits of the Sussex -- Coast, and their Equivalents in other Districts...... Mr. T. V. Holmes on the New Railway from Grays Page 1 18 35 46 Thurrock to —— Sections between Upminster _ Ne oaaRRO Nets eee oe 8 i a Mr eS a iv CONTENTS OF VOL. XXXIV.—FIFTH SERIES. ie Mr. T. Mellard Reade on the Drift Beds of the North Wales and Mid-Wales Coast ...0. 2... 20). 2 See Mr. E. Wethered on the Microscopic Structure, and Re- sidues insoluble in Hydrochloric Acid, in the Devonian Tamestone of South Devon’... 27... . 68... Mr. W. Gibson on the Geology of the Gold-bearing Rocks othe Southern Pransyaali TY I oe ee Mr. R. G. Mackley Browne on the Precipitation and Deposition of Sea-borne Sediment ................ Mr. HE. A. Floyer on the Geology of the Northern Etbai or Eastern Desert of Egypt; with an Account of the Emerald Mimes-)7.. 0.00: 03. oo a or ee Mr. A. Carson on the Rise and Fall of Lake Tanganyika Prof. T. G. Bonney on the so-called Gneiss of Carboni- ferous age at Guttannen (Canton Berne, Switzerland). Prof. G. A. J. Cole and Mr. G. W. Butler on the Litho- physes in the Obsidian of the Rocche Rosse, Lipari . . Experiments on Electrolytic Polarization, by L. Arons...... Note on the Absorption of Radiant Heat by Alum, by C. C. Hutchins On the Mechanical Equivalent of Heat, by M. C. Miculeseu . On the Theory of the Dissociation of Gases, by G. Jiger Auroras observed at Godthaab, by Adam W. Paulson On the Absorption of Light in Turbid Media, by A. Lampa . NUMBER CCVII.—AUGUST. Lord Rayleigh on the Instability of a Cylinder of Viscous Liquid under Capillary Force .. 2:5 a. se eee ss Mr. C. Chree on Rotating Elastic Solid Cylinders of Elliptic Section.— Part LU... 59s... ie os os ee Mr. E. Wythe Smith on the Measurement of the Internal Resistance of Cells .:....¢0.s..>.1+0+s0> ae Lord Rayleigh on the Instability of Cylindrical Fluid Surfaces Mr. W. B. Croft on Breath Figures |....:....296 5s geen Messrs. HE. Edser and H. Stansfield on a Portable Instrument for Measuring Magnetic Fields; with some Observations on the Strength of the Stray Fields of Dynamos. (Plates 108 Bee he ey ARR Prof, F. Y. Edgeworth on Correlated Averages........,.0% Profs. Liveing and Dewar on the Spectrum of Liquid Oxygen, and on the Refractive Indices of Liquid Oxygen, Nitrous Oxides and Ethylene, sey. sé ojo» siete ok se wie tee See Notices respecting New Books :— a oe Dr. T. 8. Hunt’s Systematic Mineralogy, based on a Natural Classification ....... +o se» Ponetarimhe punumieee Mr. A. B. Basset’s Treatise on Physical Optics...... 145 154 173 177 1380 186 190 205 210 5 5 ae CONTENTS OF VOL. XXXIV.-— FIFTH SERIES. ¥ Page Mr. E. J. Houston’s Dictionary of Electrical Words, Mernsyang Phrases’). oo. c.6 . =. eles ee gle aces we oe 215 Mr. L. Fletcher's Optical Indicatrix and the Transmission Grelenirimy Orystals 220.3... 1 ale oe ee ee es 217 On the Electrical Resistance of the Human Body, by M. von DET noc a) c0CGIRteIDI Raga ee cone nara nce ee eae ne 218 On the Electrical Force at the Electrodes, and the Electrifi- cation of a Gasin the Glow-discharge, by L. Warburg.... 219 On a New Method of Determining the Magnitude of Mole- TLE, 09) Cig PCS ot ae are iene 220 NUMBER OCVIII.—SEPTEMBER. Mr. A. B. Basset on the Difficulties of Constructing a Theory ormnucrCollapse Of Borler-fues i... ne ees le es ce 8 221 Mr. W. Williams on the Relation of the Dimensions of Physical Quantities to Directions in Space ............ 234 Drs. F. Bedell and A. C. Crehore on Equivalent Resist- ance, Self-Induction, and Capacity of Parallel Circuits with Harmonic Impressed Electromotive Force .............. 271 Dr. F, T, Trouton on the Theory of the Use of a perma- nently Magnetized Core in the Telephone .............. 276 Mr. A. A. Michelson on the Application of Interference Methods to Spectroscopic Measurements. — Part. II. ear atte. vie gis ajecachls 4+ aghohl ofers gp bueleia Sa sede 280 M. M. P. Rudski on the Level of No Strain in a Cooling Peermmmemeais Sphere wis). 0s. is vg'e som gucee ¢ eis thalels jar ee 299 Notices respecting New Books :— Prof. D. Mendeleeft’s Principles of Chemistry Proceedings of the Geological Society :— Mr. R. J. L. Guppy on the Tertiary Microzoic Forma- mons of Trinidad, West-Indies .........0.s++008 305 Rey. A. Irving on the Bagshot Beds of Bagshot Heath 306 EH. A. Johnson Pasha and Mr. H. D. Richmond on the Geolneyiot the Nile Valley... . ewe ene eee eee 306 On an Apparent Relation of Electromotive Force to Gravity, by Dr. G. Gore On the Mathematical Theory of Ferromagnetism, by H. E. J. CS. UTS IEXCHIS 8) 208 Sis pion tea Sent ne ee ir 2 er 307 rere 301 v1 CONTENTS OF VOL. XXXIV.—-FIFTH SERIES. NUMBER CCIX.—OCTOBER. | Page Lord Rayleigh on the Intensity of Light reflected from Water and Mercury at nearly Perpendicular Incidence. (Plate X.) 309 Prof. Ewing on Joints in Magnetic Circuits ............0. 320 Profs. J. Dewar and J. A. Fleming on the Electrical Resistance of Pure Metals, Alloys, and Non-metals at the Boiling- point of Oxysen: 2.5. sa hos eee op CR eae ee 326 Rev. O. Fisher on Theories to account for Glacial Submergence 337 Prof. E. B. Rosa on the Specific Inductive Capacity of Elec- trolytes: (Plate 1). . 2522 Ge) er 344 Mr. R. H. M. Bosanguet on the Calculation of the Dlumi- nating-Power of Hydrocarbons and their Mixtures.— Part IJ, Diluents, and general second approximation ........ 350 Prof. G. D. Liveing on Pliicker’s supposed Detection of the Line-spectrum of Hydrogen in the Oxyhydrogen Flame .. 371 Mr. C. Barus on the Thermoelectrics of Platinum-iridium and of Platinum-rhodium .. . sv sies os Gains ona 376 Notices respecting New Books :— J. Violle’s Cours de Physique’... . >. 1...) . eeu eee 381 Proceedings of the Geological Society :— Mr. J. Postlethwaite on the Dioritic Picrite of White Hause and Great Cockup s/.+.4.:.026 .) 215 > ee 383 Maria M. Ogilvie’s Contributions to the Geology of the Wengen and St. Cassian Strata in Southern Tyrol .. 383 Mr. J. Crawford’s Notes from a Geological Survey in Nicarapia tos. ose ap» age ene © sates aac en 384 Mr. B. Hobson on the Basalts and Andesites of Devon- shire, known as Felspathic Traps ..:.:-a, ss ogee 385 Mr. T. Tate on Recent Borings for Salt and Coal in the Tees Salt-Distrietane. 0). 2 sa ie ee 385 Treatise on Optical Physics, by A. B. Basset.............. 386 On the Determination of the Dielectric Constant of Conducting Liquids, by W. Stschegtiaeft ...'5...5. 15 lec eels ee 388 NUMBER CCX.—NOVEMBER. Dr. W. L. Robb on Oscillations that occur in the Charging of fi CONGENSET ga. ok es ee eve eas oe ie J 389 Lord Rayleigh on the Interference Bands of Approximately Homogeneous Light ; in a Letter to Prof. A. Michelson .. 407 Mr. F. G. Donnan on an Attempt to give a simple Theoretical Explanation of Raoult’s Law of the Lowering of Vapour- Pressure... sf wigs sos + OR eisls AS dos +. ee 411 Dr. G. J. Stoney on the Appreciation of Ultra-visible Quan- tities, and on a Gauge to help us to appreciate them...... 415 CONTENTS OF VOL. XXXIV.—FIFTH SERIES. vil Page Prof, F. Y. Edgeworth on the Law of Error and Correlated ‘ | STATE 22.8 26 Sieg tae Sein Cae PRG 08 eee eran 429 Dr. W. Pole: Further Data on Colour-Blindness.—No. II. 439 Lord Kelvin on Graphic Solution of Dynamical Problems .. 443 Notices respecting New Books :— Nietzki’s Chemistry of the Organic Dyestuffs ........ 448 Dr. Lodge’s Lightning Conductors and Lightning Guards 451 Dr. Hull’s Volcanoes, Past and Present .............. 452 Dr. A. Macfarlane’s Principles of the Algebra of Physics 453 Watts’ Dictionary of Chemistry, Vol. III............. 454 athe @olour of lons, by W. Ostwald .........6:.....-- 457 On the Physical Signification of 6 in Van der Waals’ Equation, NEI INN oy Sry kia s a iaes ee SEO wine ewes 459 On the Theory of the Use of a Permanently Magnetized Core in the Telephone, by J.\W. Giltay .......... ce cee eee ee 460 NUMBER CCXI.—DECEMBER. Mr. A. P. Chattock on an Electrolytic Theory of Dielectrics . 461 Lord Rayleigh on the Influence of Obstacles arranged in Rectangular Order upon the Properties of a Medium .... 481 Dr. Sydney Young on the Determination of the Critical EE ey ie ale ote els ce Oe als Petite te batho a ee» 503 Dr. 8. Young and Mr. G. L. Thomas on the Determination Meme Mine ICASILY.. 52h. tes Shea yes eee Seas 507 Dr. 8. Young on the Boiling-points of different Liquids at ORM IP Si alcatel Gv od aie Ss ha cies coe se at oe 510 Messrs. E. H. Griffiths and G. M. Clark on the Determination ot Low Temperatures by Platinum-Thermometers........ 515 Prof. F. Y. Edgeworth on the Law of Error and Correlated LOPES. ESS EAE rogers ono ee Air ie Alaa aa an 518 Notices respecting New Books :— Dr. E. J. Routh’s Treatise on Analytical Statics ...... 527 The Eifects of Self-Induction and Distributed Static Capacity in a Conductor, by Frederick Bedell, Ph.D., and Albert C. mn IPED se a a os vain a) v wi ajeia ce hulsiale's Hales oe Le 528 Experiments on the Conductivity of Insulating Bodies, by M. Bierceme neemmescrtilys ND a2 en bose ws wie as ne Vale dws Oe 530 On the Theory of Magnetization, by Prof. Dr. Wassmuth .. 5381 SME etter dis eis tas STE RO 532 PLATES. I. Illustrative of Mr. C. Barus’s Paper on the Fusion Constants of Igneous Rock. IT, Illustrative of Dr. W. Pole’s Paper on some bate Data on Colour-Blindness. III. & IV. Illustrative of Messrs. Edser and Stansfield’s Paper on a Portable Instrument for Measuring Magnetic Fields. V.-VIII. Illustrative of Mr. A. A. Michelson’s Paper on the Application of Interference Methods to Spectroscopic Measurements. IX. Illustrative of Prof. E. B. Rosa’s Paper on the Specific Inductive Capacity of Electrolytes. X. Illustrative of Lord Rayleigh’s Paper on the Intensity of Light reflected from Water and Mercury at nearly Perpendicular Incidence. ERRATUM. Page 319, line 16, read r= —2acos $0+ SSS fr prior to using are soldered together with felspar and the oxyhydrogen blowpipe. The bulb, edp, ends in a short neck, cdrs, just large enough to receive the stem, ef, snugly, and the canal, cd, through which the bulb is glazed, eventually becomes the prolongation of the capillary canal of the stem. Bulbs combining the features of figs. 2 and 3, being both reentrant and inglazed, were not in hand at the time. : Porcelain soldering is difficult, and calls for skill and patience. I found that after fastening the bulb to a rapidly rotating whirling-table, with the stem-axis vertical and coin- ciding with the axis of rotation, the operation was much facilitated. Symmetrical heating greatly diminishes the danger of breakage to which the bulb is liable because of the intense heat of the impinging flame. The machine is de- scribed in the Bulletin. Cf. § 18. At best, however, vacuum-proof joints are sparingly ob- tained, say one in every five solderings, and samples neat in external appearance are frequently worthless. If, therefore, Constants of Igneous Rock. 5 in the following pages I have proved that bulbs not glazed internally are quite available for the exact air-thermometry of high temperatures, one of the serious difficulties which the work entails will have been removed. 6. Metallic Capillary Tubes*.—The bulb and the mano- meter are joined by seamless metallic capillary tubes, about 200 centim. or more long, and ‘05 centim. in diameter within. These tubes are to be cut apart near the middle, and joined by a suitable three-way glass stopcock, through which the bulb may be exhausted and thoroughly dried at red heat, and both bulb and manometer finally charged with the necessary amount of dry air or other gas. Capillary tubes may be made of either copper, silver, or platinum. ‘The latter are preferable because they do not amalgamate in case of an accidental contact with mercury. I succeeded in making copper capillaries myself, by filling a thick copper tube with fusible metal, rolling and drawing the tube down to a wire, and finally melting out the fusible core under slight pressure applied at one end of the tube. Tubes of the kind necessary, however, are now in the market, and may be obtained in lengths not exceeding 5 metres. As employed in the work below, an external diameter of *1 centim. to a bore of ‘056 centim., or a volume of °0025 cub. centim. per centimetre of length, was found to insure a sufficiently rapid flow of gas from bulb to manometer, while the air con- tained in the capillary is nearly negligible in amount. 7. Manometer.—This is practically a U-tube, one branch of which is made of glass, 150 centim. long, accurately graduated in cubic centimetres, with a total capacity of about 300 cub. centim. Into the top of this tube, the available end of the platinum capillary, tapering to a fine conical point, is suitably soldered with cement. The point so adjusted is a convenient fiducial mark, the registration being made either by optical or by electric contact with the mercury meniscus of the manometer. The other branch of the U-tube is flexible, and communicates with a sufficiently large mercury cistern, which may be raised or lowered at pleasure. The whole is attached to a prismatic stand, 2 metres high, consisting of four uprights rigidly joined at the top and the bottom. Observations are made for equal heights of the mercury in both branches, by the cathetometer. A barometer is attached to the stand. Sensitive thermometers at the lower and the upper end of the glass measuring-tube show its temperature. Cf. § 18. * First used by Regnault (Relation des Expériences, Paris, 1847, p, 264). 6 Fusion Constants of Igneous Rock. 8. Revolving Muffie—In order that corresponding indica- tions of thermocouple and air-thermometer may be compared, both must be placed in a space at practically uniform tem- perature, which must be variable at pleasure from ordinary temperatures to extreme white heat. I constructed a special furnace for this purpose, in the way indicated in plan in the diagram (fig. 4). The body of the furnace is a thick cylin- drical box, B B, surmounted by a hemispherical lid, suitably Fig, 4.—Plan of the revolving muffle. Diagram. perforated. In this cylindrical enclosure a spherical muffle, C D, provided with hollow lateral arms or axles, E F, and placed symmetrically with respect to the centre of figure, is free to rotate around the horizontal axis of the arms. If the rate of rotation be sufficient, this insures constancy of tempera- ture within the muffle around the horizontal EF. Two blast- burners, G and H, purposely placed tangentially so as to be equivalent to a force-couple, blow a vortex of flame into this furnace, equalizing temperature around the vertical. Vir- tually, therefore, the muffle, regarded as a geometrical sphere, has two rotations, one round H F, and the other round the vertical through O. To make this apparatus theoretically perfect a third rotation, round a horizontal axis through O, and perpendicular to EF, would have to be supplied. This third rotation is a mechanical impossibility, bearing always in mind that cumbersome or complicated apparatus would rather detract from the end to be obtained than add to it. In the spherical space of constant temperature thus obtained is placed the bulb of the air-thermometer (not shown), with its stem projecting into or through the axle F; and the bulb is held in position, free from the muffle, by a clamp attached to the stem on the outside of the furnace. The thermo- couple is introduced through the opposite arm H, and its junction lies at the centre of figure of the bulb (fig. 2, above). It is an essential part of the construction of the furnace that during rotation the muffle touches neither the air-thermometer nor the insulator of the thermocouple, the latter being also supported by a clamp on the outside of the furnace. The practical form of this furnace, which after many trials =a ea A\ SAAT ATCO AWAY Fig. 5.—Sectional elevation of the revolving muffle-furnace, with the air-thermometer and thermocouple in place. Scale 1/5, ee Ws ee ceria I a ee te SS p NS SS mn | mi Y/ pal | “x Y in Yy (------- res AG @ 13 t= An — || — Ne Sy WLLL, fees ai SSSSS5S5= ceaae os Wh ra ee y cee ee Se pe ee ee reil a Saal ia pt Ll ET Sacra eases Y | Wh} SSSSqQS SS SSSSSSHNSS SESS SSS ! SSS SSS a Tas 1 : 2 > ‘ : VC on /; t (ce) foal 1 orl I OSE ae a rai aad Sa ee ea i YY SEY SSSA — ——— fhe eee h_ = I ' 1 ! | ma Sees See SeSeeaaeesa-|4|---—=-| LULL MLL MY] t : 1 Si a Sie 77 Te) eTaerHTzngzqae SLE, 8 Mr. C. Barus on the Fusion was found satisfactory, is shown in longitudinal - section (scale 1) in fig. 5, where BBB is the cylindrical body, and AAA the dome-shaped lid (perforated at aaa) already re- ferred to. The burners HH and GG, project into the furnace as far as the inner surface, and their construction is shown. Compressed air enters the tubes gg and AA, and the gas inlets are at cand ec’. Attached to the burners are the rectangular slides II and KK, which pass through guides Land M. The pins ee and dd further regulate the action, py which the burners can be easily inserted in or withdrawn from the furnace. Finally, the gas inlets, c and c’, are each in connexion with a graduated stopcock (half-inch clear bore), by which the gas-supply, and hence the furnace temperature, can be regulated at pleasure. In this way the torch of flame, fully two feet in length, is reduced to a blue cone about eight inches long by the blast, which may then be further reduced to a mere ribbon of flame by shutting off the gascocks. Burner explosions do not occur. The muffle proper is shown at ECF DE in fig. 5, and consists of two identical halves of refractory fire-clay, each of which is a hemisphere with two diametrically opposite gut- tered arms ; the two halves are placed together with their flat faces contiguous, but without cement, and secured by surrounding the axles with appropriate collars of iron, N N and N' N' (fig. 5), the outer edges of which are widely flanged. These flanges, P P and P’ P’, are turned circularly with their circumferences bevelled so as to fit the grooves of two pairs of friction-rollers, Q Q and Q’ Q', of which R and R’ are the respective axes, four in number. Set screws, T, T’, sink into the rib of the collar and fasten it to the axle, asbestos being - usually interposed. The four friction-rollers, Q Q', are adjustable (see figure), so that the muffle in its rotation never touches the walls of the furnace, but revolves on the rollers. Such an arrange- ment is essential, for the rolling parts must be placed at some distance from the hot parts of the axle to slide smoothly ; and the friction of an imperfect mechanism would, at high temperatures, tend to wrench off the axles of the white-hot muffle. Another essential adjustment may here be mentioned. After firing, neither do the axles of the muffle coincide in prolongation nor are they straight. Hence the friction rollers, Q Q’, rotate on cylindrical rods, R R’, along which the former are in some measure free to slide, their extreme positions being fixed by four adjustable collars, X X’. The belt pully, Z Z’, supplies the power. Finally, the figure shows the air-thermometer fkil in Constants of Igneous Rock. 9 position, supported by the universal clamp m™, attached to the vertical rod gq. A similar clamp n n, on the opposite side of the furnace, supports the insulator of the thermocouple p p. The ends of the wires of the latter appear at a and @, and pass thence to a petroleum bath at known temperature, where they are suitably connected with the terminals of the measur- ing apparatus (insulated mercury cups submerged in the petroleum). The figure shows that the thermocouples to be calibrated may be easily inserted or withdrawn while the air-thermo- meter remains in place. I will pass over the method of ad- justing the latter, remarking only that the collar P has a slit cut through it, from end to end, in the direction of a radial plane, and sufficiently wide to admit the platinum capillary s of the air-thermometer. This need not, therefore, be taken apart to remove it from the muffle. 9. Remarks on the Furnace-—The maximum temperature attainable is indefinitely high, much higher than can be measured by the porcelain air-thermometer. Measurements may be made either while temperature is slowly increasing or decreasing ; in the latter case the gas may be quite shut off below 500°, for the furnace itself then cools sufficiently slowly. My mean rate of rotation was about 50 per minute, but smaller rates are necessary at very high temperatures when the muffle begins to become viscous, and at low temperatures. During the heating to white heat, the parts of the axles within the furnace are liable to be hotter than the mufile. I will omit the method of remedying this defect, since it is quite absent on cooling, and since its effective value is almost negligible. Its harmful tendency is chiefly to bend the stem of the air-thermometer, an effect which can be counteracted by rotating the latter 180° from time to time. Rotation, moreover, keeps the mufile straight even when the clay is approaching the viscous stage. Protected from direct flame by the revolving mufile, the fragile porcelain bulbs are heated with great regularity, and the liability of the thermometer to break is therefore nil. The furnace has been so made that the zone of variable tem- perature which surrounds the stem of the thermometer is as narrow as possible. Indeed the correction is almost super- fluous. Were it desirable to use a compensator (a duplicate porcelain stem, closed at the bulb end, to gauge the stem error), this instrument may either temporarily replace the thermocouple, or be introduced through an independent lateral tube, corresponding to the axles F F. In order to obtain the zero or fixed point of the bulb, the 10 Mr. C. Barus on the Fusion latter is taken out of the muffle and submerged in water of known temperature; but check-readings may be obtained when the (cold) bulb is in place in the furnace, by inserting the reservoir of a sensitive mercury thermometer into the tube of the reentrant bulb, fig. 2. A special advantage of the furnace (which should always be heavily jacketed with asbestos and banded with iron) is the fact that the observer can get very near it without in- convenience. Rotating parts may be lubricated. An elliptic Fig. 6.—Plan of the elliptic revolying muffle, with the air- thermometers in place. Diagram. revolving muffle for the comparison of two gas-thermometers has also suggested itself, particularly in view of the fact that since the stem errors in this case vanish, comparisons may be made with great accuracy. This is shown in figure 6, where EC DF is the muffle and Om and Nn the two gas thermo- meters to be compared. THERMOELECTRIC APPARATUS. 10. Wires and Insulators.—The thermocouples used con- sisted of platinum combined with an alloy of platinum and 20 per cent. of iridium. ‘The wires were cut in lengths of a metre each, from large coils, and successive lengths showed the same electric properties. Hven when melted down on lime hearths before the oxyhydrogen blowpipe and drawn over again, the constants of the couples were not found to have appreciably qo per cent.) changed. The wires should be annealed at red heat before using. In order to keep the wires apart, I devised a machine by which doubly perforated insulators could be pressed, much in the manner of lead-pipe manufacture. The insulators were made of very refractory fire-clay, 25 or 30 centim. long, and *5 or *6 centim. thick, with two parallel canals, each *1 centim. in diameter, running from end to end. 11. Zero Method.—All my thermoelectric effects are mea- sured, as electromotive force, in terms of a given Latimer Clark’s standard cell. The connexions made are easily Constants of Igneous Rock. 11 understood from figure 7, where R and R’ are rheostats, having about 35,000 ohms available, and » is a bridge-rheostat con- taining a duplicate scale of 10, as high as 10,000. The cold Fig. 7.—Zero method of thermoelectric measurement. Diagram. 1 3) ] ends of the thermocouple communicate with the terminals at P, under petroleum. Two zinc-sulphate Daniells are inserted at H. The commutators A and B are useful in measuring small electromotive forces, since by simultaneously reversing both of them, all disturbing thermo-currents in the connexions are eliminated from the mean result. K is a duplex key con- structed as shown in figure 8, where two of __ the mercury cups (conveying the battery Fig. 8.-—Form of current from E) are filled to a_ higher ous Key level than the third. By aid of the keys oe at C the resistances + can be inserted either in parallel or in series. G is a sensitive Thomson galvanometer. The thermoelectric forces are directly expressed in terms of E, which is compared from time to time with the given standard Clark cell, the poles of which are temporarily put in connexion with P. The reduction of observations is facilitated by suitable tables, computed once for all. In this way my thermoelectric data have the same high degree of constancy in the lapse of time (years) as the standard cell in question. I have found this method, though it is somewhat more laborious, much more trustworthy than the torsion-galvanometer, with which I have also worked *., Even when the conditions are so chosen that the thermo- electric forces are expressed solely in terms of the torsion of the suspending fibre, it is not to be forgotten that increase of * See this Magazine, xxix. p. 146 (1890), 12 Mr. C. Barus on the Fusion the room temperature both decreases the effective magnetic field and increases the resistance of the coil, by a combined amount probably greater than 0°4 per cent. per degree. Hence in a room of variable temperature (the usual case), the torsion-galvanometer of unknown temperature-coefiicient is liable to lead to serious errors. RESULTS. 12. Method of Computation.—I will pass over this here, for the forms to be given to the cumbersome equations depend upon the special purpose of the observer. Suffice it to say that I divided the total gas volume into 4 parts, viz., the volume of the bulb and hot stem, the volume of the part of the stem where temperature falls off from hot to cold, the volume of the cold stem and capillary, and finally the volume measured in the manometer. The effect of each of these parts on the final result was carefully evaluated. 13. Bulb Volumetry.—The volume of the bulb at zero Centigrade, which enters fundamentally into the constant- pressure method, can be obtained by water calibration, ef. § 18. In the case of a manometer like the one described above, § 7, in which pressure can be varied over a sufficiently wide range and volumes read off with facility, the bulb- volume may also be obtained by airvolumetry. Jor instance, if v and p be corresponding values for the volume of air in the manometer and the pressure, the following values were obtained for the bulb volume, vp) :— p =715'24, 52°64; 75°22, 53°41 ; 75°54, 53°38 ; 63°38, 75°53 v= 40,1260; 4:0, 120°5; 3:0, 120°5; 120°5, 3:0 ——~ -—— } NS UW—~—~ W.-Y Ji= ooo 1c. e. 281°3 ¢. ¢. 279°9 c.¢. 281:3.c, c After much further experimentation I convinced myself that the differences here are simply thermal discrepancies, and are not due to the non-glazed interior surface of the bulb. If the high-temperature datum is to be measured with an absolute accuracy of 1° in 1000°, the v must be known with a degree of precision scarcely exceeding ‘02 per cent., i. e. to about 0°1 c. c.in the above case. Supposing sufficiently sharp means (accurate to 0°05 C.) for measuring or con- trolling temperature to be at hand (the above data were made with the manometer in air), the stated accuracy is attainable. 14. Coefficient of Expansion of Porcelain.—The porcelain of Bayeux, in addition to its refractory qualities, has an Constants of Igneous Rock. 13 advantage inasmuch as the (small) coefficient of thermal expansion is known*. The bulbs are supposed to have been so thoroughly fired at the outset that all permanent volume changes have vanished. It is clear, however, that by using the above manometer in case of the constant-pressure method, the coefficient of expansion of the bulb may also be measured by air volumetry. Let the manometer volume be changed while the (high) temperature of the bulb is nearly constant by varying the manometer pressure. Then, if Hy, Vy, Tj, and H,, Vz, T;, be two successive readings of the pressure, volume, and temperature, respectively, at the manometer, and if J(2) = (14+8T)/(1+aT), where @ is the coefficient of ex- pansion of porcelain, and a that of air, it follows that : ean iG F(1) +3(27@)) = =H, cg where T is the (hign) temperature and v the zero volume of the bulb, and where > contains the corrective members (stem volumes and temperatures). If H, be the barometric height for the day, T can be at once computed by the ordinary formula, and 8 may then be computed from 7(T) in (1). In the following table the measurements for T alternate with the measurement for 8 in time series, so that corresponding values are given. TABLE I.—Thermal Expansion of Porcelain. | Time. fi 6x 10°. Time. | Yi Bx10°. eT eae Bee | 568 eee eS Pes 8 (564) | 22 16 (998) 37 1s 560 a 724s bes, ba Oe 21 (564) 2% || 26 | (1004) 27 40 567 | 32 | 1006 In view of the fact that the quantities on which @ ultimately depends are of the same order of magnitude as the stem error > (of which more presently, §§ 17, 18), this method cannot * Deville and Troost, C. R. lix. p. 162 (1864). 14 Mr. C. Barus on the Fusion yet be looked to for close results. The data, crude as they are, however, show that 8 is determinable by this method with the same degree of accuracy with which it is to be applied, or that the stem error and 8 are determinable in terms of each other reciprocally. Jn this respect the constant- pressure method of high-temperature air-thermometry is unique, since 1t admits of easy modifications, by which the zero volume of the bulb, tts coefficrent of expansion, as well as all permanent changes of volume, may be evaluated without extra appliances. 15. Tables.—In making extensive comparisons between the thermocouple and the air-thermometer, I had three objects chiefly in view :—(1) To find out whether the temperature indications of the platinum-iridium thermocouple were regular and free from serious anomalies, in other words to calibrate the couple; (2) To compare the indications of different non-inglazed porcelain bulbs, for which purpose I compared two bulbs (Nos. 1 and 2) with the same thermo- couple under identical conditions ; (3) To find whether the flow of air between bulb and manometer was seriously retarded by the interposed lengths of metallic capillary tubes. This was discernible by comparing the data obtained when the furnace temperature gradually increased, with the data for decreasing furnace temperature (cooling). Accordingly, I have two sets of results in hand, the first obtained with bulb No. 1, and consisting of four series of data obtained on different days ; the second set obtained with bulb No. 2, and consisting of five series of results. Of these ‘I will only reproduce the latter (Bulb 2) here, inasmuch as these data are in many respects the more accurate. Correc- tions for the permanent volume-contraction of the bulb obtained from volumetric measurements made before and after each series of heatings are applied. In Tables II., III., and IV., “ No.” refers to the thermo- couple used, é is the thermoelectromotive force when the cold junction is at 20° C. and the hot junction at the air- thermometer temperature T. Constants of Igneous Rock. 15 TasieE 11.—Calibration of Platinum-Iridium Thermocouples. No. C50 | microvolt, 37 3440 37 3570 37 3700 37 3770 39 38850. 39 3900 39 7970 39 8340 39 8470 37 8670 37 8800 37 8800 Serizs I. €29 microvolt. | 10640 10850 11080 11090 11070 11050 12120 12290 12440 11720 11470 11400 972 1019 952 | €20 | microvolt. eeesesece Tasie II1I].—Calibration of Platinum-Iridium Thermocouples. Serizs II. No. Coy microvolt. 37 12650 37 12870 37 6| 13250 37 138360 37 11250 37 10150 37 9170 | 37 8240 37 7540 37 6780 37 6250 37 5660 37 5220 37 4820 = (ots a Series III. No. C5 | aT. microvolt. | SCk. Fi 39 | 11900 | 989 39 123870 | 1019 || 39 12590 1028 39 12850 _| 1051 39 | 12910 | 1054 39 11570 965 39 | 11080 929 | 39 9880 846 39 8880 778 39 8180 721 39 7350 664 39 6780 616 39 6170 573 39 5550 527 39 5150 494 39 4700 458 39 4350 426 CUR cocsrac. 403 | | Surizs LY. 12040 | 12060 het21t0 12140 11120 10040 8970 8110 7410 6740 6180 5640 5140 4740 4350 4140 microvolt. 413 16 Mr. C. Barus on the Fusion Taste LV.—Calibration of Platinum-Iridium Thermocouples. SERIEs V. i | No. | x9 T.. aio On iD, | No. Can ie. | | | microvolt. OR) microvolt. 0, II microvolt. “0, 38 | 5370 512 || 9030 774 | 38 | ‘7580 671 | 38 | 5560 526 || 38 11700 958° || 88 4 ” 6860 622 | 38 | 5900 565 388 11950 978 | 88 6300 579 38 | 5930 568 38 123800 | 1002 38 5740 539 388 6030 560 38 12430 1006 || 38 5290 501 38 6100 567 38 11470 935° || 38 4880 467 38 8340 734 3 10130 852 38 4450 435 38 8500 742 38 9190 790 || 88 4100 405 38 8850 760 38 8270 722 | 16. Chart.—The results of these three tables are repre- sented graphically by making e,) a function of Tin Pl. L., which may be said to be the final result of the calibration problem in hand. Caudal dashes, which point upward and to the right when the furnace temperature is increasing, and downward or to the left when it is decreasing, distinguish the different series of points. It is to be remembered that into this chart are crowded all the accidental errors and observa- tional errors of air-thermometer and thermoelectric measure- ment, when the whole work is done by a single observer. If the curve, Pl. I., be linearly prolonged above 1200°, then the junction of the couple eventually fuses at 1800°, as I find by experiment. Discussion. 17. Errors of Measurement—To interpret the chart, Pl. I., it is necessary to enter minutely into a consideration of the observational errors which affect the result 1° in 1000°. The divers quantities which enter saliently into the. equation for constant-pressure air-thermometry are— t, T,, Vi/2, v/v, H/h, a, B, where ¢ is the fiducial temperature and v the volume of the bulb, A the fiducial tension of the gas; where at the high tem- perature, V,, T,, H are the volume, temperature, and tension of the gas in the manometer ; and where finally vo’, a, 8 are the stem volume and the coefficients of expansion of gas and of porcelain respectively. I have done this both for the Constants of Igneous Rock... 17 above ratios and for the individual quantities, but the results cannot be reproduced here. After a minute and careful com- parison of all of this (see Bulletin, No. 54, pp. 227-238), I concluded that the values of high temperature obtained in ease of the two reentrant bulbs, Nos. 1 and 2, not glazed internally, were identical ; that by using the revolving mufile- furnace described above (§ 8) temperature (increasing or decreasing) changes slowly enough to permit the use of the above metallic capillary connexions (§ 6) between bulb and manometer with impunity; that the platinum - iridium (20 per cent.) thermocouple is free from serious anomalies ; that in using the furnace and the reentrant bulb, a virtual identity of environment for the thermocouple and the air- thermometer had actually been secured ; and that, finally, the methods preliminarily tested in the above pages are sutticient for the rigorous solution of the calibration problem up to an accuracy of 1° in 1000°. 18. Coneluston.—F or some years I have had the parts of a standard constant-pressure. high-temperature air-thermometer in the laboratory, but have been prevented, by duties of a more immediate geological bearing, from putting them together. I will briefly refer to these parts here. In the manometer, volumes are to be read to ‘05 cub. cm., temperatures to ‘07° C., and pressures to ‘01 cm. of mercury. Hence I surround both branches of the U-tube with a tubular cistern, through which water at constant temperature con- tinually circulates. Moreover, both branches are now made of glass and communicate below, by means of a suitable stop- cock, with an external flexible tube surmounted by a mercury reservoir, by which pressure may be changed at pleasure. A sufficient length of the metallic capillary between bulb and manometer is also submerged in the water, so that the fine current of air within may enter the manometer, already reduced to the temperature of the bath. Finally, instead of employing a cylindrical volume-tube in the manometer, as was done above, whereby the accuracy of measurement becomes rapidly insufficient in proportion as the air-thermo- meter temperature rises (see distribution of points on the chart, Pl. 1.), the first 100 cub. cm. of the tube are blown out into a spherical bulb, and the remaining length of 140 cm. of the tube utilized to measure the remaining 150 cub.cm. Thus even ‘01 cub. cm. is appreciable, and the tube is now available for accurate measurements between about 200° and over L500°. Again, since the normal volume of the gas actually expand- ing must be known to ‘01 cub. cm., it is necessary to reduce the stem volume of variable temperature to the smallest limits Phil. Mag. 8 9d. Vol. 34. No. 206. July 1892. C 18 Prof. W. H. Bragg on the “ Elastic Medium” possible. To accomplish this, a platinum capillary tube is to be cemented into the stem of the porcelain air-thermometer, by bringing atmospheric pressure to bear on the outside of the exhausted bulb, while the neck is being heated on the revolving table by the oxyhydrogen blowpipe, to the point of sufficiently reduced viscosity for adhesion. Experiment must ~show whether the bulb of the type figure 2 will ultimately answer all requirements, or whether a bulb which combines the features of figs. 2 and 3 will be necessary. Note that an independent method of standardization of the non-inglazed reentrant porcelain air-thermometer bulb, by thermal com- parison witha reentrant glass air-thermometer bulb, of known constants, is also feasible. Such a comparison is to be made © above 200°, to obviate all moisture and condensation errors, and either directly, in the elliptical revolving muffle (§ 9), or indirectly, through the intervention of the same thermocouple. I have constructed vapour baths (naphthalene, diphenyla- mine) for this purpose. I may add, in passing, that during my experiments with molten rock I was surprised by the tenacity with which the basic magma adheres to platinum and protects it. A platinum bulb covered with some refractory glazing may therefore be looked to, when temperatures beyond the reach of the porce- lain bulb (say about 1400°) are to be measured. I believe, however, that for the case of platinum apparatus, the method of absolute air-thermometry based on the high temperature viscosity of gases is more promising, if only some mathe- matical physicist would give us an expression for the depend- ence of gaseous viscosity on temperature. It seems strange that this important relation has thus far eluded search, and that little is known beyond the ingenious surmises of O. E. Meyer and the empiric law of my own. Inasmuch as the time of transpiration varies nearly as the 5/3 power of absolute temperature, the sensitiveness of the method is obvious. Phys. Lab., U.S. G.S., Washington, D.C. II. The “ Elastic Medium” Method of treating Electrostatic Theorems. By W. H. Braae, W.A., Professor of Mathe- matics in the University of Adelaide, South Australia*. T is usual to deduce the ordinary theorems of electro- statics from the law that two amounts of electricity repel one another with a force proportional directly to the product of these amounts and inversely to the square of the distance * Communicated by the Author. Method of treating Electrostatic Theorems. 1) between them. Faraday pointed out that the result so deduced, and confirmed by experiment, could be explained by assuming the existence of an elastic medium, the straining of which was electrification, and the consequent stress, elec- trical force. The elastic constant of such a medium Maxwell proposed to call the “ coefficient of electric elasticity.” Dr. Lodge has used the idea with the best of results in his book on ‘ Modern Views of Electricity.’ The reading of Dr. Lodge’s book suggested to me the working out of the results contained in this paper. The object of the paper is to draw attention to the advan- tages that are to be derived, especially by the student, from taking as hypothesis, not, as is usual, the law of the inverse square, but the existence of this elastic medium. The hypo- thesis may not be exactly true, but it is as nearly true, and in the same manner true, as that a wave of sound may be repre- sented by a wavy line. We have as much right to deduce theorems of electrostatics from consideration of the behaviour of this medium under stress, as to deduce the laws of the interference of sound from the summation of waves on a dia- gram. In both cases the whole truth is not presented, but sufficient to ensure the accuracy of the deduction. Suppose, then, the existence of an incompressible perfect fluid filling all space and all bodies—the ether. Suppose that in some bodies—copper, silver, &c.—this fluid finds no opposition to motion; but that in others—air, glass, silk, &c.—the particles of the body are so entangled or embedded in the fluid that the fluid cannot move without carrying along the particles with it ; and that whenever displacement of the medium takes place, a force of restitution is called into play, proportional to the displacement, to the amount of the moved particles, and to a quantity depending on the nature of the body in which the displacement occurs. This last quantity is Maxwell’s coefficient. In making these suppositions we are not laying upon that maid-of-all-work, the ether, any burden other than those she has for various purposes been already taught to bear. Suppose, further, that a positive charge of electricity cor- responds to the forcing an extra quantity of ether into the charged body, and a negative charge to the withdrawal of part of the ether the body contains. Then it is easy to show that phenomena similar to those ordinarily ascribed to the attractions and repulsicens of elec- trified particles must occur. 1. Since the ether is incompressible and fills all space, if zether be pumped-into one of the first class of bodies (con- C2 20 Prof. W. H. Bragg on the “ Elastic Medium” ductors) it must be withdrawn from some other part of space, and since the ether cannot be got out of bodies of the second class (dielectrics) it must be drawn from bodies of the first class. In other words, if the dielectric surrounding some conductor be pushed back to any extent, it will encroach to an exactly equal extent on some other conductor or conductors. This corresponds to the electrical law that, if any charge be imparted to a conductor, an equal and opposite charge will be induced on some other conductor or conductors. 2. Let us consider the strain phenomena resulting from the simplest case of charging, viz. pumping in or drawing out ether from a spherical conductor, the conductor being sur- rounded for a great distance by a medium uniform and free | from other conductors. Let Q be the quantity pumped in, i.e. the charge; let a be the radius of the sphere. Then all round the conductor the medium is pushed back a and at a distance » from the centre of the Q Tr Q Arra”’ sphere the displacement is n distance Suppose that, when a unit volume of the medium is dis- placed a distance z, the force of restitution is EH. z. 0 Draw a cone of very small angle from O the centre of the sphere. Let it intercept on the surface of radius x an area s, and on that of radius +67, dr being small, an areas’. The cone being of very small angle, and dr being small, s=s’. The force of restitution due to the displacement of the small a Aq? Hence if p be the pressure at the surface 7, and p+6p that at the surface v + 87, element of volume so intercepted is H.s. 67. HO), Sein : | Method of treating Electrostatic Theorems. 21 or he =C+ Anr- When r=, p is evidently zero ; aa «Agee 3. If we consider the case of a “charge” imparted to a conductor of any shape whatever, it will not be ordinarily possible to determine the pressure at any point in the sur- rounding medium. ‘The same difficulty arises of course in the ordinary method. But it is possible, as in the ordinary method, to map out the nature of the strain in the medium by drawing lines of displacement and surfaces of equal pressure. A surface of equal pressure corresponds to a surface of equal. potential. 3 Since there can be no displacement along a surface of equal pressure (otherwise the pressure must vary along the surface), displacementsalways take place at right angles to such surfaces. Hence a line drawn so that at every point on it the dis- placement is along the line is always perpendicular to the surfaces of equal pressure which it meets. Also if a number of such lines be taken forming a tube, there will be no displacement across the walls of the tube ; and hence if S be the area of any section, and « the displace- ment at right angles to it, S.« = constant along the tube. Such a tube can only, of course, proceed from a place where the dielectric is pushed back from the conductor to a place where it is drawn in; and if Q = total amount of charge at one end of the tube, —Q = charge at the other end. Suppose that the surfaces of equal pressure are so drawn that the pressure on any one differs by unity from the pres- sure on either of the adjacent surfaces ; and tubes of displace- ment are drawn to fill all space, and each of such size that the flux along it, S.a, is unity. Then it may be shown that each of the cells into'which space is divided contains half a unit of energy. For consider the cell in the figure (fig. 2). Let its bound- aries be a tube of flux 6/, and a pair of surfaces of equal pressure, the difference of the two pressures being dp. Both of and dp are to be so small that the cell Fig. 2. may be considered cylindrical. a Let d be the perpendicular. distance ae between the two surfaces, s the area of eee section of the tube, « the displacement. If a unit volume be displaced a certain distance x, the force 22 Prof. W. H. Bragg on the “ Elastic Medium” of restitution is Hw, and the energy of the displacement is ae aS - : In this case, then, the energy due to the strain of the matter . In the cell is +. Ha’. d.s. But considering the equilibrium of the element, (p+6p)s—ps=Hed.s ; Had=6p, and £80). Thus the energy =1.6p.6f. The energy, therefore, in a cell bounded by a unit tube and two surfaces drawn so that the pressure on one differs by unity from the pressure on the other, is 4. Hence in any particular case of strain, if we can count the number of cells into which space is divided by tubes and surfaces drawn as above, we can calculate the total energy of the strain. 4, Since the ether in a conductor is to be regarded as an incompressible fluid free from strain, it possesses the properties of a perfect weightless fluid, and the pressure is the same at every point of the conductor. If the charge on the conductor be Q, then Q unit tubes start from it. Ifthe pressure at the conductor be V, each of these tubes will cut through V sur- faces before it reaches the region of zero-pressure. Conse- quently the number of cells into which space is divided by the surfaces and tubes of a system of conductors containing charges Q,, Qo, Q3, &ec., the pressures at them being V,, Vo, Vs, &c., 1s QiV,+Q2Vo+..., QV. The energy of the system is therefore 32QV. 5. As a particular case, consider the energy of strain of a charge Q on a sphere of radius a, there being no other con- ductors near. The pressure at the surface of the sphere is or EQ ee) The energy is therefore ‘ee ela? 0, The energy in the space between the sphere and a con- Method of treating Electrostatic Theorems. 23 centric sphere of radius a’ is evidently se ED 6. We may define the ‘capacity’? of a conductor to be such a quantity C that if Q be the charge on it and V the resultant pressure, either the pressures or the charges on all neighbouring conductors being zero, then Q= VC ; or energy 2 of system =} ce Hence the capacity of the sphere in § 5a ao. Again, the capacity of a sphere surrounded by a concen- tric spherical conductor which is connected to the earth is, by § 5, Amr aa’ ) iG — a)’ a’ being the radius of the concentric conductor. For of course the effect of the extra conductor is simply to relieve all the strain external to it. The encroachment upon the interior wall of the conductor simply causes ether to flow away to the earth, for there is no opposition to this move- ment ; whilst force would be required to enlarge the outer boundary of the conductor (fig. 3). Fig. 3. 7. Next consider the case of two spheres containing charges Q, and Q,, the distance between the centres being d, and Fig, 4. Qa Q, ae Sa ae Pat 73 the radii 7; and 7, being small compared with d (fig. 4). 24 Prof. W. H. Bragg on the “ Elastic Medium” The pressure at the surface of the first sphere is made up of two Aqr charge of the other sphere. This is only an approximation to the truth, and the closeness of the approximation depends on aD) H Q: parts: —- “due to its own charge, and eae) due to the the smallness of 7 Thus we take the pressure at the surface of the first sphere to be BE (Q , Q’). maar that at the surface of the second, BE (Q | et oe The energy of the system is therefore Ef QP? | 2.Q:Q, = ce NR AB Na ih ile a 2" Al ry aN ak Oe 9 If we differentiate with respect to d, we see that there is a force of repulsion between the spheres equal to This is, of course, the “law of the inverse square.” ©The factor = corresponds to a K being the specific inductive capacity. In the ordinary statement of the law the K is often omitted, being arbitrarily taken as equal to unity. 8. So far the correspondence between the results deduced from the ordinary hypothesis and the hypothesis of the elastic medium has been obvious. It is not perhaps quite so plain what in this language corresponds to the “force at any point due to the attractions of an electrical system,” and in par- ticular to the law that “just outside a conductor this force (F') =47rp, or, more strictly, K fF =4zrp.” But it must be remembered that the “force at any point due to the attractions of an electrical system” is only a mathematical conception and not an actual physical quantity. It is the attraction on a unit of electricity, supposing the presence of that unit not to disturb the pre-existing distribu- tions of the system. ‘This condition is impossible. However, on the strain theory we have the law that just outside a conductor, where the medium is pushed back a dis- tance x, there is a force of restitution per unit volume equal Method of treating Electrostatic Theorems. 25 P) to Hz. Since E answers to a we evidently have here the correspondence we seek. And the force at any point 7’ corresponds to the force of restitution per unit volume of dielectric. 9. The case of two parallel plates may be very simply treated by this method. | Let S = the surface-area of either, d the distance between them, x the displacement. The plates are supposed to be so large compared with the distance between them that all the lines of displacement run- ning from one plate to the other may be considered straight, and w may be considered as the displacement of every particle of matter between the plates. Then the energy of the strain | Bee ialle, £°. The difference of pressures (V) is evidently the springback of the matter in a tube of unit area reaching from one plate to the other, and is therefore E.d.z. From the value of the energy, it is evident that there is a force urging the plates together equal to $.8.H.2”. Hence this force (I, say) eat . E.@ b S be written for H, is the usual formula for the —1 wiih or Agr which, if K electrometer. If Q be the charge on the positive plate, Q=8. x Hence VaHd ia _H.d.Q ere es: The capacity of the condenser is therefore aa H.d 10. Using our present hypothesis, the “ method of images” may be employed as in the ordinary hypothesis to solve the problems of the charged sphere near an infinite plate, two charged spheres whose radii are not small compared with the distance between their centres, and similar problems. As an example consider the first problem. 26 Prof. W. H. Bragg on the “ Elastic Medium” Obviously the charge, Q, on the sphere (O in fig. 5) Fig. 5. 5 i 1 ‘ i ! 1 ! t | I ' ‘ ' t ‘ t will force in the boundary of the plate on the side nearer O. As the plate is connected to earth this will cause no bulge on the other side of the plate, the excess of ether will run away. Suppose now a negative charge, numerically equal to Q, is placed at O’, the image of O, in the nearer surface of the plate. The plate being supposed for the moment exceedingly thin, this new charge will draw in the dielectric, and the amount of the drawing in at any point on one side of the plate is exactly equal to the encroachment on the other side of it. In fact, if the charges at O and O’ both existed, the dis- placement of dielectric would be everywhere the same as if the plate were ‘“ unearthed,” or, in fact, if it did not exist—it is exceedingly thin, it must be remembered. Thus the displacement at P, when there is a charge Q at O and an infinite non-insulated conducting plate, is the same as if there were charges Q at O and —Q at O/ and no plate at all. From the latter hypothesis we see the charge at P must be of the density Q.OB ~ Ir. OP* 11. Next consider the case of a charged sphere placed near the plane boundary between two dielectrics of different elas- ticities ; the radius of the sphere being small compared with its distance from the plane. Our method is well adapted for such a problem, as the physical significance of every step is clear. Method of treating Electrostatic Theorems. 27 Let the sphere bearing a charge Q be immersed in a dielectric of elasticity E,, the elasticity of the other medium Fig. 6. 1 1 1 F i i 1 1 r \ { i 1 1 1 i i r 1 i ‘ i 1 1 i i i 1 1 1 I 1 I 1 1 I t ! [ f L 1 I { being E, and H, being > F,. Consider the displacement at any point P. If E, were equal to Hy, the displacement across the surface would be as . If H, were zero, then, by the last section, 0: 05 the displacement would be 2 ‘TOP? OP" Since EH, lies between 0 and H, it is natural to guess that the actual displacement is (1+ y). ee , where yu lies between 0 and 1: and it is easy to show that this condition of things produces equilibrium everywhere. Under this condition of things the medium to the left of the plane of separation is strained as it would be if E, were equal to H,, and there were placed a charge —yQ at O! (the image of O) in addition to that at O. The medium to the right is strained as if H, were equal to H, and there were placed at O a charge (1+ ) Q. Considering the strain of the first medium, the pressure at P must be Ey (Gs—-s ..0P, OPS Considering the strain of the second medium, the pressure 28 Prof. W. H. Bragg on the “ Elastic Medium” at P must be EK, (+#).Q Aq” OP If there is to be equilibrium, these must be equal at every point of the surface of separation. +. Ey —p) =BA(1 +4) _ Ey —H, Py Bo Pu, By giving w this value, which is independent of the position of the point P, we make the pressures balance each other everywhere. We have therefore found that state of the medium into which it will settle under the given conditions. It is easy to write down the energy of the strain. For if d be the distance from the centre of the sphere to the plane of separation of the two media, the pressure at the surface of the charged sphere is approximately BE, (Q_ #Q Aq’ ( Pi Bae 7 being the radius of the sphere and small compared with d. Q * There is therefore a force urging the sphere towards the plane ; and the magnitude of it _ Kh, ».Q - Bart aris Slip | i,—E, Q’ To BE Tere 12. Let us next take the case of a sphere of one dielectric (elasticity = H,) immersed in another dielectric (elasticity = H,) in which there is a uniform displacement; or, as it is often termed, the sphere in the uniform field. Here again, I think, by using the present method, the problem is much easier to understand. To fix our ideas, suppose the displacement is to the left, and that E, is less than H,. Let the uniform displacement of the field =6, let the radius of the sphere =r. It is obvious that since the medium in the sphere is weaker than that outside, the sphere will yield as a whole and be dis- placed further in the direction of the existing distribution of the field. The lines of displacement behind the sphere will The energy is obtained by multiplying this by Method of treating Electrostatic Theorems. 29 converge on the gap left by the sphere, those in front will correspondingly diverge. We have to find the amount of the extra displacement at every point of the sphere. General considerations alone seem to show that the sphere will be undistorted, simply translated ; for from symmetry the positive charge on the one side of the sphere must be — numerically equal to the negative charge on the other side, peint by point, taking always two points the line joining which is parallel to the direction of the displacement of the field. The sphere is therefore uncontracted longitudinally, and therefore also laterally. But this is proved also in the following complete solution. Let us examine the effect on the external medium of shifting the sphere undistorted an extra distance a to the left. We shall find that if we give a certain value to a, the resulting strain of the external medium will cause a pressure across each unit of area of the surface of the sphere equal and opposite to the pressure across that area caused by the strain of the medium within the sphere. Hence there will be com- plete equilibrium, and we shall have found the nature of the state into which the sphere and medium will naturally fall. Fig. 7. a < pee SSS Now the state of strain into which the external medium will fall in consequence of the sphere being moved a distance a to the left is the same as would be produced if EH, were equal to H,, and charges 277? and —2zr* were placed at O! and O respectively. For the displacement at any point P Dany produced by such charges would be — along PO and Qarr® Or. sj hor. O'P? along O'P. Since a is very small, this amounts to a 30 Prof. W. H. Bragg on the “ Elastic Medium” 3 ; displacement = (ope = sy, along PO, and this is equal to acosd. But this is also the displacement along PO pro- duced by moving the sphere a distance a to the left. The surrounding medium therefore will take up that strain (in addition to its previous strain) due to charges —2zr* at O, and 27r* at O'. The difference of pressures at P and P! produced by such a strain is easily found. The original difference before the sphere was displaced was 2r cos @.H,b. In consequence of the new charges the pres- sure at P is lessened by the amount E, a ee am) to OP. OR” _H, 27r*.acos¢ ~ Agr’ 7 _ Kyracos ¢ | 2 5 and the pressure at P’ is increased by that amount The difference therefore amounts now to 27 cos 6H ,b—Eyr a cos ¢. The difference of pressures at P and P’ due to the strain of the medium within the sphere is 2(b+a)rcos$.H,. Equate these two, we have 2 ° Hi, (6 + a) — 2H,b—Hy,a es ee ae — 2b . EE Lg This value of a is independent of ¢; if then a have this value, the pressures balance everywhere. So the total displacement at B, which is sometimes called the density of the charge, =atb ey, 3H, ‘EE, +2H, If E,=E,, a=0, as of course it should. If E,=0, 2. e. the sphere is a conductor, the density of the charge at B=3b. ? 13. It is easy to find the loss of energy caused by the presence of the sphere in the field. We can first of all find the work that must be done to restore the external medium Method of treating Electrostatic Theorems. dl to its original state, 2. e. by pushing the sphere back to its old place, and then, considering the strain of the medium within the sphere, find the difference between the energy of a sphere of elasticity HE, displaced a distance 6, and a sphere of elasticity EK, displaced a distance a+b. This difference added to the first result will give us what we want. First, then, consider the external medium only. If the sphere have a displacement 6 + z, the external medium having a displacement 6, the difference of pressures at P and P’ is (by the last section) Hb. 2r cosp—H, re cos ¢. Let 6A be the small area of section of a tube running from P to P’. If we sum the expression H,r cos 6(26—~) . 6A over the central section of the sphere, we get the total resultant pressure which the external medium exerts on the sphere. (Of course when things are in equilibrium wz has such a value that this pressure is balanced by the spring-back of the medium within the sphere itself.) | Since 22rcos@.dA=V (the volume of the sphere) this sum becomes Luyey ey (26 —). The energy required is therefore { “Hy (25—2)80. 0 fed a? =B,V(ab— 4 ) Next consider the medium within the sphere. When the sphere is in the field we have a sphere of elasticity H, dis- placed a distance (a+); when it is absent a sphere of elasticity H, displaced a distance 6. The difference of the energies is 6 +6)? VE, — VE, 2+") b° | =VE, at i (a+b), (2 _ ‘) (by last section) a’ ab eae .. Total loss of energy caused by the presence of the sphere ab =o. V mies 82 Prof. W. H. Bragg on the “ Elastte Medium” Substituting for a, this becomes H,—E mo eowedl Sia mee ey he 2 EL POR, 207% To put this into the ordinary language we must write for E, and H,, = and io respectively, and for H,) we must write 1 9 F the “intensity of the field.” This gives us aie ' 2(2K,+ K,) This differs from the formula in Gray’s ‘Theory of Abso- lute Measurements in Electricity and Magnetism,’ in that it KF. has an extra factor = However, the K, has obviously dropped out by accident from Gray’s formula, as it is necessary to make the dimensions right. I cannot explain — the remaining difference. The formula used by Boltzmann involved the ratio of two expressions like the above, so that even if the formula obtained above is the correct one the = would divide out. We may examine by this method the effect of imparting a charge to an ellipsoidal conductor immersed in a uniform dielectric. The proof differs but little except in language from other proofs, so I will only state it briefly. It is easy to show that that state of strain, in which the particles on every ellipsoidal surface confocal to the con- ductor move normally outwards so as to lie on a new ellip- soidal surface similar to the old one, is a state which produces complete equilibrium everywhere. We must first see whether our suppositions are geometrically consistent with each other: it is not at once evident that the ellipsoidal confocals can expand similarly, so to speak, and at the same time the displacement be always normal to them. 2 2 wa Let = ah + 5 =1 be the equation of the ellipsoidal con- TS y a ductor. Let ie cay, a6 Pty is CEN, focal ellipsoids, and let similar equations with A,, A; in place of X, represent other confocals at right angles to each other and to Ay. Consider the tube formed by the intersection of the confocals Az, As, Ag+ OA and Ag+OA3. If prj, 22, 3, be the 2 =1 be one of the con- Method of treating Electrostatic Theorems. 3d perpendiculars from the centre on the tangent planes at Aj, Ag, Az, then the sides of the section of the tube by the surface ), are a— and—. : 2p. 2p Spe Ps The thickness, at the point A; Az Az, of the shell bounded by the ellipsoid A, and the similar ellipsoid into which ), expands under the given strain, is pp,, the linear dimensions of i, being supposed to expand in the ratio 1 to 1+p. Now the volume of this shell is equal to the charge on the conductor, Q. Hence we obtain easily The area of the section is therefore meee Oe Am V (a? +21) (0? +A) (? +24) Now Fee Ae UE Da) (e* V/ Naz - and p> p; have similar values. | Hence the flux along the tube, which is equal to OAg . OAs An» P3 is independent of d,, and therefore constant along the tube. Our suppositions are then possible. Perhaps it may make things clearer to state them in another way. If we suppose that tubes, like the one mentioned and occupying all space, act as guides to the particles as they are displaced outwards, and if the conductor itself expand “similarly,” then by what we have just proved the particles on any confocal ellipsoid are displaced in such a way that the ellipsoid expands “ similarly.” We have now to show that the supposed strain is one which ensures mechanical equilibrium. Let P be the point of intersection of Ay, A», and A3, P! of A, +5A,, Ay, and Az. Let 6dr, be small. Then the difference of pressures at P and P! P-fPi- =H. x PP! x displacement of medium at P. on =H x oni 8 oP or =H, — which is the same for all points on 4. Phil. Mag. 8. 5. Vol. 34. No, 206. July 1892. D 34 : On Electrostatic Theorems. So if one confocal is a surface of uniform pressure, so are those close to it. But the confocal at infinity is of course at zero pressure all over; so all the confocals are surfaces of equal pressure, and amongst them the conductor itself. The surfaces of equal pressure are therefore at right angles to the supposed displacement, and so the medium is in equilibrium. I should like to point out here the curious fact that in an ordinary proof of part of the above theorem, viz., that part which asserts the distribution of the charge on the ellipsoid to be represented by the expanding of the ellipsoid “ similarly,” the reasoning has to do with attractions znside the ellipsoid, and takes no account of actions outside, whereas the proof just given is exactly the reverse of this. It should be noticed that the ordinary proof is exceedingly defective, for it is upset entirely by supposing the presence of non-uniform dielectric in the interior of the conductor, which would make calculation of the attractions impossible ; also it is apparently unaffected by any want of uniformity of the external dielectric. The proof given above has neither of these faults. : In connexion with the subject it is worth noticing that the method of this paper leads to a most simple proof of the law that a closed conductor perfectly screens its interior from the action of external charges. For the only way in which the pressure of the ether in the conducting shell could affect the dielectric (uniform or not) in the interior would be by changing the shape of it, pushing it in in some places, and therefore allowing it to bulge in others. But since the pressure is uniform over the conductor and the interior of the shell is unalterable in volume, no work is done by any such deformation. Hence no energy can be imparted to the interior of the conductor, and consequently no changes can take place ; every forced change would of course require energy to produce it. I hope that I have made clear the value of this method of treating electrostatic theorems. It seems to me to possess several advantages. In the first place the student is able to form a mental picture of the physical meaning of every step in the mathematical reasoning, and he therefore finds it easier to understand the step, to remember it, and give it its proper relative importance. Moreover, the mathematical reasoning itself is much simplified in the case of some of the problems dealt with above. Again, the expression for the specific inductive capacity appears in every equation as it should do, The Heat of Dissolution of Gases in Liquids. 30 and its significance is at once comprehended: a student’s idea of this quantity, or of the electric elasticity E, is often very hazy, when formed from the definitions of the ordinary method. Lastly, the method offers the student a new stand- point, the view from which not only makes clear to him many things that were confused before, but greatly assists in giving him an insight into modern advances in electrical theory. Jil. Lhe Heat of Dissolution of Gases in Liquids. By Spencer UMFREVILLE Pickerine, /.2.S.* is solutions consist of chemical compounds of the solvent with the dissolved substance, the formation of these will nearly always be accompanied by the liberation of heat: this heat will, however, often be more or less masked by con- comitant actions of a reverse nature, especially in cases where the substance is a liquid or solid, the particles of which can- not become resolved into the quasi gaseous molecules which exist in solution without the absorption of a quantity of heat corresponding with that of their heat of vaporization, or of fusion plus vaporization. All such reverse actions, however, would be absent if the dissolved substance were taken in the gaseous state to start with, or, at most, there would only be a small absorption of heat due to a slight separation of the par- ticles of the solvent from one another; this in very dilute solutions would probably be negligible, and we should con- sequently expect, if the hydrate theory of solution is correct, that gases would dissolve with a considerable evolution of heat. The following determinations will show that this undoubt- edly is the case ; and whether this evolution of heat is due to the formation of definite compounds or not, the fact that it occurs must prove that solution is accompanied by a disap- pearance of potential energy, that the dissolved substance can no longer be represented as being in a condition of gaseous freedom, and that the solvent cannot be’ regarded as playing the part of ‘so much space.” The publication of these results (most of which were obtained some time ago) comes, I fear, rather late in the day ; for the “empty space” theory would seem to be already abandoned by some of the sup- porters of the physical theory of solution, and, indeed, has been categorically repudiated by van der Waals (Zetschr. f. phys. Chem. vi. p. 214, see ff. 188-222) in an important paper, in which be arrives at the conclusion that there must be a specific * Communicated by the Author. D 2 36 Mr. 8. U. Pickering on the Heat of attraction between the solvent and substance, even when the latter is supposed to be dissociated, which attraction, when satisfied, gives rise to a considerable evolution of heat. The following determinations, however, will, I trust, possess some interest beyond that of dismissing an already moribund theory. Indeed a knowledge of the thermal phenomena of solution must be of the highest importance for arriving at any true. theory on the subject, although, so far as I am aware, these phenomena have up to the present been entirely overlooked . by the advocates of the physical theory. As substances for investigation, I confined my attention to non-electrolytes, where the results obtained would not be com- plicated by any supposed dissociation ; and, by selecting those of which the heat of vaporization was known, it was only neces- sary to determine their heat of dissolution when in the liquid condition in order to arrive at their heat of dissolution in the gaseous condition. Water, benzene, and acetic acid were taken as solvents. The various results will be found in Table I. Three determinations were made consecutively in the same quantity of solvent, so as to obtain the values for the formation of solutions of three different strengths. The rise or fall of temperature noticed on adding the three consecutive quantities of 2 grams of the liquid is given under t/—¢; w is the water-equivalent of the calorimeter with its contents; D7; the heat of dissolution of a gram-molecular proportion of the liquid in the solvent the composition of which is entered in column II.; and ¢° C. the initial temperature of the deter- mination. In the case of volatile liquids the determinations are necessarily less accurate than in other cases, although the error was reduced as far as possible by using a closed calori- meter, and the experimental error may amount in extreme cases to five times that obtaining with non-volatile liquids (see Chem. Soc. Trans. 1890, p. 100), that is, to °005° on a single determination representing an error of 50 cal. in the value of D2’ where water was the solvent, 30 cal. with acetic acid, and 15 cal. with benzene. The temperatures of the added liquid and of the solvent were nearly identical when the two were mixed, and the heat-capacity of the solution _was taken as being equal to the sum of those of its consti- tnents. The details as to the substances used areas follows:— Acetic Acid ...... Mol. wt. =59°86. Sp. ht. =-46. F.p. =16°56° C. BCHZENO 4 iucs. ssc... a ==]fO2: So = 486. 4) sh eae Mileoliol Gasschs cies 3 =45°9. » = 60. Bp. =18:74at 77s GMOT vectensscnccs ee :. =73°84. » = O04. 5, .=0450 at (oe AGetOnGs...cs-ceeses 2 =57°87. i = "52... ; =—56:06 at: (62 Pyridine — .....-68 =Jeeo. ( ,, =457) 5, =1158 at vou Carbon Disulphide |, =7399. ,, = 24. ,, —4615at760-5mm. Dissolution of Gases in Liquids. 37 By plotting the results against percentage-composition and rawing a curve through them, I have obtained the value for the heat of dissolution of a gram-molecular proportion of the substance in 22,400 cub. centim. (gas strength) of the solvent (1244 H,O, 396 C,H,O, or 245 C,H,), and also of the heat of dissolution in an infinite amount of the solvent. In some eases these values can be but approximately correct, for the heat of dissolution often increases very quickly with the dilu- tion and renders extrapolation uncertain. In three cases, pyridine and carbon disulphide in acetic acid, and ether in benzene, there is no sensible alteration in the value with dilution; in every other it occurs in the above-named direc- tion, whether the heat of dissolution is positive or negative, and is particularly large in the case of alcohol in benzene ; a fact which is in harmony with the rapid increase of the molecular depression of the freezing-points of benzene by alcohol as the dilution increases (Phil. Mag. xxxiil. p. 449, AB, fig. 2, and p. 461), since both these phenomena point to the existence of alcohol aggregates in comparatively strong solutions, and the breaking up of these on further dilution. In Table II. I have collected the values for the heat of disso- lution of the liquids in infinity of solvent, and also that of the corresponding gases in 22°4 litres of solvent, the heat of vaporization at 18° being also given here. This latter is not known very accurately in many cases, and in that of the pyridine is not known at all. This Table shows at a glance that, whatever gaseous sub- stance or solvent we take, a considerable amount of heat is evolved during the dissolution, from 5000 to 10,000 cal., and, therefore, that there is a disappearance of a considerable amount of potential energy, so that we can but conclude that combination in some form or another, whether chemical or otherwise, must have occurred. Further, just as in the case of every known instance of chemical combination, the amount of heat evolved is not con- ditioned by the nature of one of the reagents only, but by that of both of them, so, here, the heat evolved by dissolving different gases in the same solvent is very different, showing a variation of as much as 6000 cal.; while the dissolution of the same substance in different solvents gives as great a di- versity of values, the variation in this case also amounting to some 6000 cal. Nor does there even appear to be any relation- ship in the differences ; for the differences between the values for the various substances are not the same in the case of the three solvents, as will be seen from the columns headed “ Diff.”’ in Table Il. The only general conclusion which can be 38 Heat of Dissolution of Gases in Liquids. drawn from the comparative magnitude of the various values is one which is decidedly in favour of the existence of chemical compounds in solution, namely, that more heat is evolved in every case here investigated by the dissolution of a substance in water than in the other two solvents examined, this being very significant in view of the fact that water undoubtedly has a much greater tendency to form definite molecular compounds than either acetic acid or benzene. It is scarcely necessary to mention that pyridine in acetic acid is not an instance of dissolution strictly comparable with the others, since the pyridine acetate which is formed is a compound of a different order of stability from the hydrates or analogous compounds which, I believe, are present in solutions. We have thus most of the main characteristics of chemical combination attending the dissolution of gases in liquids—a selective attraction, a considerable evolution of heat, and an action which is more energetic in the very case where che- mical combination is known to occur more readily ; and all that is wanted to convert this evidence of the truly chemical nature of the action into proof, is to find that the combination which does occur, occurs in definite molecular proportions. This last proof I have already obtained. Series of different strengths of all the solutions here dealt with have been examined as to their freezing-points, and in nearly every case changes of curvature, some of them very marked, have been found to occur at various definite and stimple molecular proportions. These results I hope shortly to publish. I believe that the chief objection now urged against the view that the nature of solution is truly chemical, is the ab- sence of any explanation of how chemical combination could lead to such a condition of the dissolved matter, that caleula- tions based on the idea that it is uncombined with the solvent, and even that it is split up into independently acting ions, should lead to so many correct results. This absence, | believe, no longer exists ; for in a recent communication to the German Chemical Society (Ber. xxiv. p. 3629) I have shown that this state of quasi-independence of the dissolved substance, and also of the ions of an electrolyte, would be a direct consequence of the existence of hydrates (or similar compounds) of a high degree of complexity in solutions. ~ TasLy I.—Heat of Dissolution of Liquids in various Solvents. Solvent for one mole- Dissolved substance. w. cule of dissolved (t'—Z)°C. Substance. 2 ee 5022 | 312 H,O. 4664 - ate ae ee 5021 | 312 H,O+ C,H,O 4500 eo 5044 | 322 H,0+2C,H,0 44h5 Shee PE EEO hy po, ¢ | -| Ce ai eee Geil a Liga erae 3819 | 660 H,O. 4563 4 ea deh epee 3741 | 674 H,0+ C,H,,0 4291 RE ens os wa 3881 | 650 H,0+2C,H,,0 “4251 . ) de PALHLOF | eee Se Ae Ra Sees Se ao. MCEVONG = (15............. 4-829 | 454 H,O. 3426 ae 4-880 | 405 H,O+ C,H,O. 3309 | a 4-900 | 403 H,O+2C,H,0. "3230 OS Se eee pi? oo * ae .. » ot oe eee ee Be Een 6" = ae LC VO 6196 | 435.H,0. 3226 ol eee 6189 | 435 H,O+ C,H.N. "3059 eed 6193 | 485 H,O+2C0,H.N. "2923 <4 go -dos Scone ee 122 TO. = ay ee | 5 EL EEO Es seca ACOHG ACIC”........-... 4-571 | 411 H,0. 0443 ee 4-534 | 414 H,O+ O,H,O,,. 0367 ae 4514 | 416 H,O+20,H,0,. 0354 i Saeee eee Lea Oe Sy, , See oy er On wt of eae U0) 5 3311 | 146 C,H,0,. —*1288 ot a le 3211 | 151 C,H,0,+ C,H,O. |—:1130 2 a 3183 | 152 C,H,0,+2C,H,O. |—-1057 ce Seo Cee Opaies po 4 #.'|. renee oo EC @ Shall Gis) 3 1) a Ra BOBS wee ccc wce cs. .3.. 0570) 140 C,H,O,. ‘0904 4 ee 5590} 189 C,H,0,+ C,H,,0.} 0872 3 5562) 140°C,H,0,+2C,H,,0.| *0828 oo Bob Cyl Oe i. | Sacer FER ccisacianc| "s+. Be Cre EL OR rey ck) GA bt EE oe PNEEUONG aceon. n es scse.. 4-220 | 145 C,H,0,. ‘0308 ee 4192 | 145 C,H,0,+ C,H,O 0252 ei cic ase. - 4183 | 146 O,H,O,+2C,H,O 0242 PM Pek cp icsigas ve. |, (4-20 BOGE IO see i Oy eee Oe ee Co HL Oe itis | yy pert IE ymedimes os ..00.... 4-036 | 206 C,H,0.. 1-0409 eee 01S 207'C5Hj0;-4- CoH.N. |’ T0134 Oy ee Jee eee 3°854 | 216 C,H,0,+2C,H.N. | 0:9591 a: See a Gla OF ls PO ee oe on Pacer — hb dae! Se, ees tiene ewe Oee s 4) |e meee Carbon Disulphide’®...|5°543 | 141 C,H,O.. — 2796 si 35 ..|5°948 | 140 O,H,0,+ CS — 2745 - a ..|5°039 | 141 C,H,0,+208, — 2653 As ot aay Wi eter SN OA 0 en | ieee ' 2) eaten E Sato Bogen Os.. 8. ER 1146 | 165 C,H,O,. ok Eoin. Mies nn. 1-274 | 149 C,H,C,+ H,O — 1808 at 1354 | 140 C,H,0,+2H,0 — 1805 | AS oS ee a GUOROREIGOL. °C uinanaeee | 28 Pee aeceere eee ot Ses 6 A ee a ee : Weight of water in the calorimeter, 599°5 grams. ”? 9 39 549° 9 : »» acetic acid “a 63152 9s Water-equivalent of the apparatus alone, 10°26 grams. Were | ons 616°35| 2626) 18-4 619°35) 2548) 18-9 622°35| 2593] 19-4 ete 2772| 18:9 Beast 2874) 18°9 615-4 | 5429) 18-4 617-4 | 5229)°°18°8 6195 | 5010|° 19-2 vieisee 5525) 138 epee 5633] 18°8 615°85) 2804) 18°5 619-40) 2431} 188 62195} 2372) 19-1 ae “| 3470) 188 ele 4900} 188 616715) 2530) 18:5 618-95) 2412) °18°8 b2rYor ala; 19k ee 2623; 188 ane 2678) 18:8 565°15) © 328) 18:5 567°23| § 275} 18:5 569-30) © 268) 18-5 naGace 399} -18°5 paced 455| 18:5 ‘| 803-75) —542) 18°5 30575} —494| 18-4 30765) —469| 18°3 seas —599| 18-4 mie nee | —680} 18-4 30475} 365) 18-7 307°75) 354] 18:8 310-75] 342) 188 semedt 372| 18:3 ieee 375| -18:8 30393} 126) 18-4 30611 107| 18:5 308°29} 103) 18°5 Deh a 148) 185 Jeetes 167} 18:5 303°55| 6174) 17-4 305°35| 6075} 18:5 307:07| 6027} 19°5 bepere 6262} 18:5 ae 6415} 18:5 303°07)/— 1129) 18:5 304°39)—1112) 18:2 305°71|— 1082) 17-9 veo —1150|} 18-2 nc. —1166) 18-2 302 90/— 851} 18°5 304:17|/— 775) 18:3 305°52|— 732) 18-1 Bogie — 938) 18°3 sens: —1040; 18:3 40 Mr. S. U. Pickering on the Heat of TaB1eE I. (continued). Solvent for one mole- Dissolved substance. w. cule of dissolved (¢'—t)°O.| W.? | Dm | #C, Substance. eS PANCOHOL? 7 d.ceeeencct ses 2°884 88:0... — 9539 | 199:42'—3026) 18°6 AS ieee ae 2°818 90 C,H,+ O,H,O —°7414 | 201:10|\—2428) 183 Be Bitte. Lenco ere sne.ss 2°684 95 C,H,+2C0,H,O. |—°5345 | 202°72/—1811| 18:4 “Vijog it, 2k) aa io A ae octet 1a tice |i. saemeke 85) Tae — 3380) 18:4 jaw yg. ae CONN Ee cca wp. PP aa gecce nl ee — 3565) 18°4 LE ese ae oh 4°435 92 C,H,. —0170 |200:06— 57) 183 Be Sele Bre eiettadas ds - 4:443 92 C,H,+ C,H,,O. |— ‘0151 | 202°44;— 51) 182 Bane ee Ne sain’ 4°450 92 C,H,+2C,H,,O. |—°0178 | 20482/— 60) 181 toe ee ee ee DAO he , ee te |) eects Sv eee — 57) 182 in, al ah. a CO eA i Rae ecines See — 57| 182 Acetone’ .....2:........ 3°526 Di-C-H... —°0756 | 199°50)-- 248) 18:5 Se fu Mat ERT tote is 3510 91 C,H,+ O,H,O —0688 | 201°32;— 229) 18-4 SH lye Se ee 3514 91 C,H,+2C0,H,O — ‘0637 | 203°14;— 2138) 18:3 yl see PAG Hoe, oe eee — 282) 18-4 i yy See eh p/p. These are the general equations. In order to apply them to the present problem of small disturbances from a steady motion represented by u=0, 0=0, w= We where W is a function of 7 only, we will regard the complete motion as expressed by uw, v, W+w, and neglect the squares of the small quantities uw, v, w, which express the disturbance. * “On the Circulation of Air in Kundt’s Tubes,” Phil. Trans. November 1883. + Basset’s ‘Hydrodynamics,’ § 470. the Stability of the Flow of Fluids. 63 Thus du du _ dQ 1 »] Ti + ae — Tae? e e ° . e ( ) dv yw dW _ dQ a regia: ~ aia dW dQ u—— at +w = ae mer 8") * (3) which, with the “ ae of continuity,” d(ru) , dv dw | a kay ae ~ Oe 27 (4) determine the motion. The next step is to introduce the supposition that as func- tions of t,z,@, the variables wu, v,w, and Q are proportional to pilnt+ke-+ 68), We get pee EW) 22 =) (n+kW)v=-Q, . Meee re ake W whee. « (6) = (ru) +isu+tikrw=0. . ... (7%) From these equations three of the variables may be elimi- nated, so as to obtain an equation in which the fourth is isolated. The simplest result is that in which Q is retained. It is = 1 dQ 2k dW dQ _ ae ia ea kt) — n+kW dr dr ae r dr But the equation in w lends itself more readily to the impo- sition of boundary conditions. If s=0, that is in the case of symmetrical disturbances, the equation in wis obtained at once by differentiation of (8), and substitution of u from (5). After reduction it becomes (9) If the undisturbed motion is that of a highly viscous fluid in a circular tube, W is of the form A+ B>”, and the second 2 tas, an <1 ie i y dr 64 Lord Rayleigh on the Question of part of (9) disappears. There can then be admitted no values of n, except such as make n+W=0 for some value of r included within the tube. For the equation au 1du det ee being that of the Bessel’s function of the first order witha purely i imaginary argument, admits of no solution consistent with the conditions that w=0 when r vanishes, and also when ry has the finite value appropriate to the wall of the tube. But any value assumed by —kW is an adinissible solution for n. At the place where n+kW=0, (10) need not be satis- fied, and under this exemption the required solution may be obtained consistently with the boundary conditions. It is included in the above statement that no admissible value of n can include an imaginary part. If s be not zero, we have in transforming to u to inglade also terms arising from the differentiation in (8) of —Qs?/r°, that is + 5 —Fu=0,. + . . (10) for the second of which we substitute from (5), and for the first from (8) itself. The result is ae U 1 du 3s? + kr? yr dr s? + kr? 2s? Re —s+ ea 2b] a few... rewire = hu OE ep GY From (11) we may fall back on the case of two dimensions by supposing 7 to be infinite. But, in order not to lose generality, we must at the same time allow s to be infinite, so that, for example, s=r. Thus, writing x for r, and y for r?, we find for the differential equation applicable to the solution in which all the quantities are Preppnnonas to eee ey) (n+kW) (+k) d Wu — Wu b= we . (12) agreeing with that formerly discussed i for a figs difference of notation. the Stability of the Flow of Fluids. 65 We will now consider ae in the abbreviated form, | (n + kW) +2 - do “he Wiku, where a is a positive number not less than unity ; or, again, 7 ( _ U bi ee, _ kureWy, dr\ dr atkw The question proposed for consideration is whether (138) admits of a solution with a complex value of n, subject to the conditions that for two values of 7, say 7, and 1, wu shall vanish. This represents the flow of fluid through a channel bounded by two coaxal cylinders. Suppose, then, that n is of the form p+igq, and u of the form «+72, where p,g, a, @ are real. Separating the real and imaginary parts in (13), we get kr GEE Ep (e+eW a+ 8h (14) (13) id e(n= br a= = Uae kr* ans et B= (et AWB qu}; . (15) and thence a ( , de AB) _ be Wa a+ 8)-9 AG =) 4 a dr —(ptkWy+q? 2 « (16) We now integrate this equation with respect to r over the space between the walls, viz., from 7, to 72. The integral of the left-hand member is da d Bi ie —ar® = : and this vanishes at both limits, 8 and « being there zero. The integral of the right-hand member of (16) is s accordingly zero, from which it follows that if Wy be of one sign through- out, g must vanish—that is to say, no complex value of n is admissible, The general value of W,, viz., PW dW kh? p? — $? dr” r dr Pr +s” ats . (18) reduces in the case of two dimensions to @W /dr*, or, as we may then write it, d’?W/dx’. Instability, at any rate of the Phil. Mag. 8. 5. Vol. 34. No. 206. July 1892. ir 66 Lord Rayleigh on the Question of full-blown exponential sort, isthus excluded, provided PW [dec is of one sign throughout the entire region of flow limited by the two parallel plane walls. | Commenting upon this argument, Lord Kelvin * remarks that the disturbing infinity, which arises in (13) when n has a value such that n-+kW vanishes at some point in the field of motion, “ vitiates the seeming proof of stability.” Perhaps I went too far in asserting that the motion was thoroughly stable ; but it is to be observed that if be complex, there is no “disturbing infinity.” The argument, therefore, does not fail, regarded as one for excluding complex values of n. What happens when n has a real value such that n-+kW vanishes at an interior point, is a subject for further examination. The condition for two dimensions that d?W/da? is of one sign throughout is satisfied for a law of flow such as that of a viscous fluid, and we shall see that the corresponding condi- tion for (17) in the more general problem is also satisfied in the case of the steady flow of a viscous fluid between cylin- drical walls at 7, and 72. The most general form of W for steady motion symmetrical about the axis is T W=Ar?+Blogr+C, . . . . (9) in which the constants A, B, C are related by the conditions 0=Ar,?+B log 7,4+0, O=Ar,?+B log ro +C. From the last two equations we derive A(r.?—1,?) + B log 7,/7;=0, . . « (20) so that A and B have opposite signs. Introducing the value of W from (18), we obtain as the special form here applicable 4s?A —2k?B which is thus of one sign throughout the range. A small disturbance from the steady motion expressed by (19) is therefore not exponentially unstable. The result now obtained is applicable however small may be the inner radius 7, of the annular channel. But the exten- sion to the case of the ordinary pipe of unobstructed circular section may be thought precarious, when it is remembered that provision must be made for a possible finite value of wu when r=0. But although a and @ may be finite at the lower * Phil. Mag. Aug. 1887, p. 275. + Basset’s ‘Hydrodynamics,’ § 614. the Stability of the Flow of Fluids. 67 limit, the annulment of (17) is secured by the factor 7%; so that complex values of n are still excluded, provided W, be of unchangeable sign. In the present case the B of (19) vanishes, and we have CW 1dW eet de so that (18) gives As?A W, ky? + s® satisfying the prescribed condition. The difficulty in reconciling calculation and experiment is accordingly not to be explained by any peculiarity of the two- dimensional motion to which calculation was first applied. It may indeed be argued that the instabilities excluded are only those of the exponential type, and that there may remain others on the borderland of the form ¢ cos ¢, &. But if the above calculations are really applicable to the limiting case of a viscous fluid when the viscosity is infinitely small, we should naturally expect to find that the smallest sen- sible viscosity would convert the feebly unstable disturbance into one distinctly stable, and if so the difficulty remains. Speculations on such a subject in advance of definite argu- ments are not worth much; but the impression upon my mind is that the motions calculated above for an absolutely inviscid liquid may be found inapplicable to a viscid liquid of vanishing viscosity, and that a more complete treatment might even yet indicate instability, perhaps of a local cha- racter, in the immediate neighbourhood of the walls, when the viscosity is very small. It is on the basis of such a complete treatment, in which the terms representing viscosity in the general equations are retained, that Lord Kelvin + arrives at the conclusion that the flow of viscous fluid between two parallel walls is fully stable for infinitesimal disturbances, however small the amount of the viscosity may be. Naturally, it is with diffidence that I hesitate to follow so great an authority, but I must confess that the argument does not appear to me demonstrative. No attempt is made to determine whether in free disturbances of the type e” (in his notation e*) the imaginary part of n is finite, and if so whether it is positive or negative. If I rightly understand it, the process consists in an investigation of forced vibrations of arbitrary (real) frequency, and the con- clusion depends upon a tacit assumption that if these forced * Phil. Mag. Aug. and Sept. 1887. F 2 68 Lord Rayleigh on the Question of vibrations can be expressed in a periodic form the steady motion from which they are deviations cannot be unstable. A very simple case suffices to prove that such a principle could not be admitted. The equation to the motion of the bob of a pendulum situated near the popes point of its orbit is 2 oe _ mt i) re where X is an impressed force. If X=cos pt, the corre- sponding part of is weg CORE 99 c= ie but this gives no indication of the inherent instability of the _ situation expressed by the free “ vibrations,” ahev sn Be, ses i. ces As a preliminary to a more complete investigation, it may be worth while to indicate the solution of the problem for the two-dimensional motion of viscous liquid between two parallel planes, in the relatively very simple case where there is no foundation of steady motion. The equation, given in Lord Kelvin’s paper, for the motion of type gilnttke) Fs ip (Si 2 TS + lw) 0 vou) = 0... (24) The boundary conditions, say at a=-+a, are that u, (v), and w shall then vanish, or by (7) that u=0, du/dx=0. The following would then be the proof from the differential equation that for all the admissible values of n, p is zero and q is positive. Writing as before, w=a+if, and separating the real and imaginary parts, we find —u(a-#) 8 +p (Fa—Ha)—9 (FE—B)=0, « (28) (Se —i)a+p (Ge—H8)+4( (5S Ka )=0.. (26) - Multiply (25), (26) by a, 8 respectively, add and integrate with respect to 2 over the range of the motion. The coeffi- cient of g is \{eae- aa. the Stability of the Flow of Fluids. 69 and this is equal to zero in virtue of the conditions at the limits. In like manner the coefficient of w is zero, as appears on successive integrations by parts. The coefficient of p is da’? (dB? -\4(@) +($) + Hat +1262 bde 5 so that p=0. Again, multiply (25) by 8, (26) by a, and subtract. On integration as before the coefficient of g is {{ S “+f (Fy + he2q? + x? } dx, and that of pw is ue war (a BY? (Z) (e : aaa {{ “) +(Z5) + 2k ap + 2k =| + k*a* + k*8 hae. Hence g has the same sign as p, that is to say, ¢ is positive. That n in é”* is a pure positive imaginary is no more than might have been inferred from general principles, seeing that the problem is one of the small motions about equilibrium of a system devoid of potential energy. Since (24) is an equation with constant coefficients, the normal functions in this case are readily expressed. Writing it in the form d an d? {ia- PFS aa pum0, ke we see that the four types of solution are Bee. Cae. ene. ete. where —k? =k? + in/y; tae (28) or, if we take advantage of what has just been proved, =a, 6 Peete ei «| (29) where g and p are positive. It will be seen that the odd and even parts of the solution may be treated separately. Thus, for the first, u=A sinh kx+Bsinkiz, . . . . (80) and the conditions to be satisfied at z= tu give 0=A sinh ka+B sin Ka ; (31) O=kA cosh kat+H#B cosklaf’?** ° 70 Mr. C. Chree on Rotating Elastic so that the equation for x’ is | tan Ka _tanh ka Ka ka = Again, for the solution involving the even functions, u=C cosh kv +D coskiz, . . . . (38) (32) where cot a _ __coth ka Ba aaa Paste ° . ° . . (34) Equations (32), (384) give an infinite number of real values for k’, and when these are known g, and n, follow from (29). The most persistent motion (for which g is smallest) corre- sponds to a small value of /, and to the even functions of (33). In this case from (34) a=, 2, ar, &e., the first of which gives as the smallest value of ¢ ge prle 6 one ea The corresponding form for u is u=e*—t(1+cos(ma/a)). . . . . . (86) This type of motion is represented by the arrows in the following diagram :— \ es t On the other hand the smallest value of g under the head of the odd functions is = par (14808) and the motion is of the type f ay Y = Terling Place, Witham. IX. Rotating Elastic Solid Cylinders of Elliptic Section. By C. Cures, WA., Fellow of King’s College, Cambridge*. - Part 1.— The Short Elliptic Cylinder or Disk. N the ‘ Quarterly Journal of ... Mathematics,’ vol. xxiii. pp. 16-33, I considered various cases of isotropic elastic solids rotating with uniform angular velocity about an axis * Communicated by the Author. Solid Cylinders of Elliptic Section. 71 through the centre of gravity. Amongst the cases treated was. that of a thin elliptical disk. The solution obtained for this case * was, as explicitly stated in my paper, only approxi- mate, certain surface conditions not being exactly satisfied. In a criticism of solutions hitherto proposed for thin rotating circular disks, Professor Pearsont referred to mine in language which, though comparatively flattering, implied doubts as to its trustworthiness, on the ground that it did not satisfy certain surface conditions he regarded as essential. I have since shown that my method can supply a solution{t for a thin eircular disk which satisfies all the conditions held essential by Professor Pearson, and that this only adds to my original expressions for the displacements certain terms of the therd power of the thickness. The first object of the present paper is to show that a similar unimportant addition meets Professor Pearson’s objection in the more general case of an elliptic disk. My principal reason, however, for returning to the subject is that in my previous paper no attempt was made to evolve the physical conclusions latent in the somewhat complicated mathematical formule. This deficiency will, it is hoped, be met by the present paper, more especially by the tables of numerical results. Very likely the mathematical problem never has its conditions exactly realized in practice, but its solution may nevertheless prove a useful guide and auxiliary to experiment. In treating elastic solids whose surfaces consist of different portions intersecting at finite angles, it has in general been found impossible to satisfy all the conditions which the ordinary theory regards as holding at every point of a surface between the stresses in the material and the applied forces. Saint-Venant §, however, and other eminent authorities, have held that when a dimension of a body is very small, as the thickness in a thin disk, all that is required, at least for practical purposes, is an equality between the statical resuitant of the stresses and that of the surface forces applied over the small dimension. The applied forces may in fact be replaced by any statically equivalent system, and, according to the authorities above referred to, the displacements given by the mathematical theory which so replaces the actually existing * Quarterly Journal, J. c. pp. 27-28, equations (125)-(128). t ‘Nature,’ vol. xliii. (1891), p. 488. t Cambridge Philosophical Society’s ‘ Proceedings,’ vol. vii. pp. 201- 215, 1891. § See Pearson’s ‘ Elastical Researches of Barré de Saint-Venant,’ arts. 8 and 9. 12 Mr. C. Chree on Rotating Elastec surface forces can differ appreciably from the theoretically perfect solution, even when the surface forces are large, only at distances from the surface which are comparable with the small dimension. My solution in the Quarterly Journal contained a number of arbitrary constants determined by the surface conditions. The number of constants being insufficient to satisfy all the surface conditions of the exact mathematical theory, the conditions whose failure was selected in my paper as of least importance were those which signified the vanishing of the stress components parallel to the faces at every point of the rim. While the other stresses vanished as required at every point of the flat faces and the rim, these stress components on the rim were taken to vanish only in the central plane -=0, and elsewhere they were of the order <’ of small quantities. Professor Pearson’s objection is that these stresses remained unequilibrated, i.e. when treated by ordinary statics led to a resultant force on every generator. The present solution 1s free from this objection, and so presumably will be recognized as a final solution for a thin disk by all who share Professor Pearson’s views. § 2. As in my previous paper the origin is at the centre of the disk, the axis of z being normal to the flat faces, while the axes of # and y coincide with the major and minor axes of the central section parallel to the faces. The peri- meter of this section is the ellipse oe byte 1 So te whose major axis is 2a. The displacements are, as before, «, 8, y, and the dilatation A, where _da , dB , dy nar i Te , Lt ee For the stresses I shall employ the symmetrical notation of Todhunter and Pearson’s ‘ History,’ so that the P,Q, R, 8, T, U of my previous paper are replaced respectively by «xz, yy; 22, yz, ze, and ay. Also, instead of Thomson and Tait’s elastic constants m and n, I shall in general employ Young’s Modulus HE, and Poisson’s ratio* y, as more serviceable for practical applications. The equations which ought to be satisfied when the right cylinder (1), supposed of uniform density p and length 21, rotates with uniform angular velocity about its axis are nine * In what follows 7 is assumed to lie between 0 and ‘5; see Phil. Mag. vol. xxxii. p. 236 (1891). Solid Cylinders of Elliptic Section. 73 in number. There are three internal equations to be satisfied at every point within the solid, viz. :— ~~ dA @ &@ . @ ON.) ye 9B) 4 where V=o'p(2’?+y’*)/2. Of the six surface conditions, three apply at every point of the plane faces z= +1, viz. :— Res Na oy: 5 ee 3 4) oe eB) SEOUL, ORNS The remaining three apply at every point of the cylindrical surface (1), usually denoted here the rim. Denoting the direction-cosines of the normal to (1) by » and yw, these conditions are : Met pig ahan eet) en (7) eer ey es Pee as SS (8) ae is 024. i Megs + ~ ks) My previous solution satisfied exactly the internal equations (3), and also all the surface conditions except (7) and (8). Instead of these it gave F=02, G=0'2, where C and C’ are independent of z. Thus over the length 21 of a generator, there -remained a resultant force, whose components parallel to the axes of # and y were +1 2 +1 2 { Fdz =, | Gigs ces. 4 140) Professor Pearson’s objection to this solution as applied to a thin disk is not that it fails exactly to satisfy (7) and (8), but that it leads to an unequilibrated resultant, given by (10), for the stresses over a generator. Tio remove this objection all that is necessary is to determine two of the arbi- trary constants of my general solution from the equations 74 Mr. C. Chree on Rotating Elastic +1 +1 . ‘6 Fae=0, |’ Gdz=0;.....°. = instead of as previously from F=0, G=0, when z=0. §3. We may spare ourselves the trouble of redetermining the constants, as the requisite changes are easily hit on and still more easily verified. In the formule (125), (126), and (128) of my previa paper for a, 8, and A, we have only to replace 2? by z?—/?/3, and in the formula (127) for y we now write z (z?—/*) for 23. The verification is easy if it be noticed that this adds to the previous values of «, 8, 7; and > terms [?e/x, ?B’y, l’y/z, and A’ respectively, where a’, B’, y A! are constants satisfyi ing a’ + Bl+q'=Al, The internal equations (3), above, contain only second differential coefficients of the displacements, and so are unaffected. Also, it is easily proved that in our new solution we have everywhere Y aa a cee A i coon yemena ae Q..').) demi: 3) De Thus the conditions (4), (5), (6), and (9) are identically satisfied, and, lastly, it is easily verified that the new solution satisfies (11). The solution is the following :— EA(8a* + 2a7b? + 3b*) + {w?p(1—2n)} = (a? +8’) fat +07b?(1 +) +54} — wv fat +.a°b? + 24 + nb?(a? + 3b?) —y"} 2a* + al? + bt + na? (8a? + b*)} a wee +7) (4P —2")(8a* + 2a7b* +- 3b"), . so. Ha (3a* + 207b? + 3b’) /w’p = x(a? — nb”) {a* + a7b?(1 +) +6°} —lr {a +07? + 0 + nb? (a? —b?) —397b'} — ay? fa' + (a° —b?) (2a? + 6?) — 97a? } + (4? —2*) en{a' + ab? 4-264 + nb?(a?+3b")}, . 2 . . we s (14) BA (3a! + 2028? + 3h*) op = y(B2—na®) {a* + a°*(1 +) +54 —1y fa‘ + a?b? + b*—na?(a? —b”) — 3na*} — a?y {b*—9 (a? —b”) (a? + 26?) — 17070} + (42? —2”)yn{2a4 + 070? + b4 + a°(8a7 + B?)t, 2 2 ww (8) Ey(8at + 2070? + 364) /w?p = — 2 (a? +b”) {a*+ a°b?(1 +) +0%} + 2a?n{at+ cb? + 204+ b?(a? + 8b?) } + zy?n{2a* + ab? + b* + na?(3a? + 0’)! —12(P—2*) mee (Sat + 20°? + 3d*)...) 5. a . . Solid Cylinders of Elliptic Section. 75 The three stresses which do not everywhere vanish are given by the formule :— ta(3a + 2a°b? + 3b') /w?p =(a? — 2”) fa +.07b?(1 + 9) +0*} —y’a* (1 +387) — +@P-2); ~ { (at + 020? + 2b!) (1 +9)? + 2?(a'—b')}, yy(Ba* + 2070? + 3h*)/o? p= (B— y?) {a + o20?(1 +) +b} —2°S* (1 + 37) + (4P—2") 5 ia §(Qa* + ab? +b‘) (1+)? —2n?(a'—B')}, . my (BU + 20°? + 3b) wp = —ary(a' +L‘ — 20°"). . § 4. When 7=0 the surface conditions (7) and (8) are exactly satisfied, and the solution is thus in this case complete. For other values of 7 the solution is only approximate, but accord- ing to the theory of statically equivalent load-systems, it can differ appreciably from the complete solution only at points whose distance from the rim does not exceed a few multiples of 1. Elsewhere our solution would, according to this theory, seem to be correct so far even as terms of order /? in the strains and stresses. In a thin disk, however, such terms are very small and in practical calculations may be neglected. I shall thus frequently omit them, speaking of the solution when they are neglected as the jirst approximation. This first approximation is of course identical with that supplied by my previous solution, as the expressions for the strains and stresses in the two solutions differ only by terms independent of x, y, or z and of the order /?. It should also be noticed that the mean value of every strain or stress in the new solution taken throughout the thickness of the disk is the same for any given value of x, y as if terms of order ? and Bin the displacements were non-existent. In other words, the strains and stresses supplied by the first approximation are for every value of x and y the mean values of those sup- plied by the complete solution. § 5. It is most convenient to consider in the first place the stress system. From (12) we see that one of the principal stresses is everywhere zero and has for its direction the parallel to the axis of rotation. The principal stresses parallel to the faces vary in direction from point to point. They are parallel to the axes of the ellipse only when zy=0, 2. ¢., at points in these axes themselves. We shall confine our atten- tion to the stresses given by the first approximation. Putting for shortness | Sat + a202(1 +7) +04} + (304+ 20%? +3b4)=K, . (20) (17) (18) (19) 76 _ Mr. C. Chree on Rotating Elastic we may resolve the stress system at any point into the fol- lowing simple systems :— (i) A uniform normal tension w?pa?K parallel to the major axis. 7 Gi) A uniform normal tension o%pb?K parallel to the minor axis. (iii) A normal pressure — w%o7?K directed along the radius r of the circle concentric with the disk which passes through the point in question. (iv) A normal pressure directed along the tangent at the point considered to the ellipse which passes through the point and is similar and similarly situated to the rim of the disk. If a', b' denote the semi-axes of this ellipse, p’ the perpendicular from the centre on the tangent, this pressure is — cilia (=) o*p(1+3n) 34 2eR +a GT) + (21) At the rim of the disk the principal stresses in the plane through the point considered parallel to the faces are directed along the normal and tangent to the rim respectively. The = former stress is everywhere zero, and the latter, it, is given by a* + b4+—2na?b? ( ab y y) 304 + 20°? + 804 \ p where p is the perpendicular from the centre on the tangent to the rim at the point considered. Since 7 cannot exceed *} this is everywhere a tension. Its maxima are found at the ends of the minor axis, its minima at the ends of the major axis. Without entering on a discussion of the terms of order /? in (17) and (18), it is worth noticing that they indicate that for a given value of wz, y the traction, algebraically considered, is greatest in the central plane and diminishes from thence to the faces of the disk. This seems to indicate that the tendency in the material to retire from the axis of rotation is greatest in the central plane. § 6. One of the most important points in such problems as the present is the determination of the greatest speed .con- sistent with safety. Unfortunately, our knowledge of the conditions requisite for safety is very incomplete, and it is at least doubtful whether the question comes in general within the scope of the mathematical theory. But admitting the doubtful character of existing theories, it is desirable to examine the conclusions they lead to, if only for the reason that fresh light may thus be thrown on the question of their a it=wp . (22) Solid Cylinders of Elliptic Section. 77 validity. Here attention will be directed only to the stress- difference and greatest-strain theories”. In applying these theories we may confine our attention in the first place to the first approximation. Doing so, it may be proved by somewhat laborious analysis, omitted here as in itself of no interest, that the greatest values both of the stress-difference and greatest strain occur at the centre of the disk. They are respectively the values of #x# and = for w=y=0. Denoting the maximum stress-difference and great- est strain by S and § respectively, we find eerie at Ae IS tc) ee 2B) 8 =7pa? (1—nb?/a?) K/E = (1—nb?/a2) S/H,. . (24) where K is given by (20). § 7. We may attach to the limiting speed at least two different values. ‘Taking, for instance, the stress-difference theory, and measuring force in tons weight and length in inches, we may regard 8 as the greatest longitudinal traction in tons weight per square inch under which the material in question, in the form of a bar, satisfies Hooke’s law “ stress proportional to strain’’ with sufficient exactness for the legitimate application of the mathematical theory. We may, however, regard § as the number of tons weight per sq. inch which engineers consider the safe working limit in the material, provided this be taken sufficiently low to satisfy the linearity of the stress-strain relations. The former view would unquestionably be theoretically the more satisfactory if experiment showed a distinct point in stress-strain diagrams where Hooke’s law ceases to hold. Even, however, if such a point did exist when the load on the bar was gradually raised in a particular way, it might be rash to assume that the same definite point would present itself in the material of a rotating cylinder whose speed was being gradually increased. An objection to the second view is the wholly arbitrary nature of the limit it assigns to 8, and the fact that the safe load varies with the engineer, and is likewise an unknown function of the contingencies to which he surmises the parti- cular structure may be exposed. It thus appears best to present the results in such a way that a reader may attach his own values to S or s, and may have a minimum of trouble in deducing numerical results. — ah * See Phil. Mag. September, 1891, pp. 240-241. 78 Mr. C. Chree on Rotating Elastie § 8. By reference to (23), (24), and (20) it will be seen that in a disk of g¢ven material * and eccentricity 8 and s both vary as (wa)?.. Thus wa is a more convenient quantity to tabulate than w. Also as a basis for the comparison of limiting speeds in disks of different eccentricities, it is con- venient to suppose that the major axis has in all the same length 2a. Ina given material E, 7 and p are constants, and so are the greatest values permissible theoretically in 8 and s. Materials may, however, have the same value for 7 and yet differ widely in their other properties, and this should be borne in mind in applying the following results. Table I. is based on (23) and Table II. on (24). For the purpose of distinguishing between the two theories of ‘‘rupture,” the angular velocity is termed @, in Table I., and w,in Table II. If we ascribe to S and sin the tables the limiting values permissible to the stress-difference and greatest strain in the material under consideration, then the values we deduce for w, and @, are the limiting speeds according to the respective theories. If, on the other hand, we ascribe to @, and w, a given value, we deduce the corresponding maxi- mum stress-difference and greatest strain occurring in the disk. TABLE I, Value of aa + J Sip. b/a qe 0 2. 25 3 5 0 1732 1-782 1-732 1-732 1-732 2 1-721 1-714 1-718 711 1-705 4 1-693 1-670 1-665 1-659 1-638 6 1-661 1-622 1-613 1-604 1569 8 1-639 1-591 1-579 1:568 1525 1-0 1-633 1-581 1-569 1:557 1-512 * For convenience this is here employed to signify absolute identity in physical properties. Materials nominally the same, e.g. steels of dif- ferent brands, or even the same brand under varied treatment, may differ in the values of E and p, and in the values permissible to S or s. Solid Cylinders of Flliptic Section. 79 TABLE IJ. Value of w,a + V/Hs/p. b/a. = 0. ‘2. "25. 3. 5. 0 1732 1-732 1-732 1-732 1-732 2 1-721 1-721 172k 1°7215 1-722 4 1°693 1-698 1-699 1701 1-708 6 1661 1:684 1691 1698 1°732 8 1639 1°708 1-723 1-744 1:849 1:0 1°633 1-768 1812 1851 2°138 The results are only approximate*. ‘The value 0 of b/a represents a disk whose minor axis, though extremely small relative to the major axis, is not absolutely zero. Assuming that the stress-difference and greatest-strain theories really apply to all distributions of stress, the values to be attached to S and Hs in determining the limiting speeds are the same in any one material. § 9. A few hints on the practical application of the tables may assist the reader. If S and E be given in absolute C.G.S8. units, z.¢. dynes per sq. cm., then substituting for p the number which denotes the ratio of the density of the material to that of water, we deduce from the tables the values of the velocities w,a and @,a in centimetres per second. If S and E be given in grammes weight per sq. cm., then multiplying their numerical measures by 981 we obtain their values in dynes and so find the velocities as before. To reduce the velocities in these two cases from centimetres per second to feet per second, we divide by 30°480. If S and E be given in tons weight per sq. inch, the reader may avail pameelt of the following result. Let S = Hs = ¢ tons weight per sq. inch, and let p =d times the density of water, then the velocities @,a and w,a may be found in feet per second by replacing _ J Sie S/p and / Bs/p eu in the tables by 4078 x ¥ t/d. * Numerical results will here be termed exact or approximate according as they are the complete values supplied by the formule, or only the first figures of a decimal. ‘The decimal has in every case been calculated to a greater number of figures than appear in the text. 80 Mr. C. Chree on Rotating Elastic § 10. For a comparison of the results of the stress-differ- ence and greatest-strain theories we have only to remember that S = Es. We thus see that the limiting speeds allowed by the two theories are the same when either 7=0 or b/a=0, and that in any other case the limiting speed allowed by the greatest-strain theory is the greater. The difference between the two theories is the more conspicuous the larger the values of 7 and b/a. For values of b/a other than 0 the limiting speed according to the stress-difference theory falls as 9 increases, whereas according to the greatest-strain theory it increases with 7». It is also noteworthy that while the limiting speed on the stress-difference theory falls as b/a increases, there is on the greatest-strain theory, unless 7=0, a value of b/a less than 1 for which the limiting speed is a minimum. This critical value of b/a is less the greater the — value of ». § 11. A considerable amount of caution must be observed ~ in applying the results of the tables to disks in which b/a is very small. Jor, in the first place, no point in such a disk is very far from the rim, where the accuracy of our results is somewhat doubtful : and, in the second place, if the velocity in such a disk were to alter, there would arise at every point a reversed effective force. proportional to the rate of change of the velocity, directed approximately at right angles to the major axis, and there would be a tendency for the elongated disk to snap in two, just as if it were exposed to flexure in a plane perpendicular to the axis of rotation. §12. When the rim is so nearly circular that we may neglect the fourth power of the eccentricity e, we find from (23) and (24) S'=0"pa* (38 +-9)/8;.2 (2 bs Lo S== wpa? (3+) (L—m) {1+en/(l—n)}+8H. . (26) Thus when ¢é* is negligible the maximum stress-difference is the same as in a circular disk whose diameter equals the major axis. Also, since 7/(1—m) cannot exceed 1, the greatest strain, though greater than that in the circular disk, cannot stand to it in a greater ratio than 1+e?:1. Thus, according to either theory, the limiting speed is but little affected by the substitution for a truly circular form of a slightly ellip- tical. This constitutes a strong a prior: probability that any slight want of uniformity in the length of a disk’s radius, which does not remove the centre of gravity from the axis of rotation, nor involve any sudden discontinuity in the — = ee ee ee a ore Solid Cylinders of Elliptic Section. 81 curvature at the rim, will have but a small effect on the limiting safe speed. §13. Taking into account terms of order /?, we find for the corrections 6S and 65 to be added to the values (23) and (24) : | (L—7)(1 +387) G0: \ éS= ~— 61—7) © 1— TWA +9)? ~—-Ba® Wah? 30* J” (eo. ___ w”pl*n (1 +7) 1+38n a*—b* Ba aay Ui tag See C8) According to the theory of statically equivalent load-systems, these corrections should make the values of S and 5 com- plete so far as terms of order /?, for the measurements of these quantities, being taken at the centre of the disk, are free from the uncertainties which attend the application of our solution to points near the rim. The corrections vanish if 7=0, and even for ordinary values of 7, such as *25, they are small fractions of w?p/?. They are always positive, increasing as the square of the disk’s thickness, and are independent of the absolute lengths of the axes of the elliptic section. They are greatest in a circular section. The corrections, if trustworthy, lead to the following law: Unless n=0, the limiting safe speed diminishes as the thickness increases, but this diminution is but trifling while the thickness ts a small fraction such as so, or even zy, of the major axis. With ordinary values of 7 the corrections would become appreciable if values such as 1 or 4 were assigned to //a, but the application of our solution to disks as thick as this is not warranted. For thick disks with ordinary values of 7 the only conclusion we can fairly draw is that our solution raises a probability that the limiting speed is less, perhaps con- siderably less, than in a thin disk of the same section and material. §14. A general idea of the nature of the deformation pro- duced by rotation is most easily derived from a consideration of how the displacement y varies throughout the disk. According to the first approximation we find from (16) : cei NG ee Oye E10 ive ens: y eet teiiaeaomis> ei 2e) where, for shortness, ear ola? 0 \yieter te. ee Ty (0) ay? = (a* + 0?) }a* + .a7b?(1+) +0*}=+ {a* + 07h? + 26+ + nb? (a? +3h?)}, (31) b,?= (a? + 6?) {a* + a2b? (1 +7) + O*} = {2at + 07h? + U4 + na?(3a? +0?) $. (82) The value of K is given by (20). Phil. Mag. 8.5. Vol. 84. No. 206. July 1892, G 82 Mr. ©. Chree on Rotating: Elastic The simple relation A i_ lta 1 _alltaje’p G70? _ a? eee He will subsequently be found of service. It is easy to prove for all permissible values of b/a and 9, ay > a, b, = b. Thus by (29) every point in the disk approaches the central plane z=0. Those points originally in a plane section parallel to the faces, which are also during rotation equi- distant from the central plane, lie on one of aseries of elliptic cylinders whose common axis is the axis of rotation, and whose cross sections are similar and similarly situated to x?/a,? +y?/b? ie 2: ie « Re (34) (33) The common eccentricity ¢, of these sections is given by } e2=1—(by/a,)2=2(2—e2) (L +3) +{4(14+9)—2(3 +0) +e}. (85) The major axis of (84) lies along the major axis of the disk, but e, is easily proved less than e for all permissible values of eand 7. Ellipses similar and similarly situated to (34) drawn in the central plane may, for the sake of reference, be termed ellipses of equal longitudinal displacement. §15. For points originally in a plane section parallel to the faces z is constant, and thus by (29) such a section is, according to the first approximation, deformed into a para- boloid whose axis is the axis of rotation. The paraboloid is elliptic for all permissible values of b/a and y, and its con- cavity is directed away from the central plane. Over the faces themselves the terms in (16) neglected by the first approximation vanish; thus the paraboloidal form of the faces is exact so far as our solution goes. The curvature in any plane through the axis of rotation at the vertices of the para- boloids into which the plane sections transform is, by (29), directly proportional to z, and so is greatest in the faces of the disk. Also, as 6;a’b?(1+5n) +304. (70) Unless this inequality hold, 1/R, is a minimum at the end either of the major or minor axis. The maximum value of 1/R, always occurs at the end of a principal axis, and this is the major or the minor axis according as (a—b)*(a? +ab +b") > or < nab(3a? + 2ab +307). § 33. The curvature 1/R; may be regarded as measuring the tendency of the material in the central plane at the rim (69) 100 Dr. W. Pole on some unpublished to be driven by rotation nearer to the longest diameter than is the material in the faces. Its maximum in the positive quadrant occurs at the point where = Vab sate , 4) OQ i.e. tand= Vb*/a> where @ is the vectorial angle. The corresponding maximum is given by a _@pa, a—b at*—2na’b?+l* R WY geen oe oe This vanishes of course in a circular disk for all values of 7 and for 7=0 in all forms of disks. § 34. In a circular disk every line originally parallel to the axis of rotation has a plane containing that axis for osculating plane. Calling the radius of curvature in a line at distance r from the axis of rotation R,, we easily find from (67) eer a/R, = (w*pa®/E) x (1+) Comparing (73) with (88) and (37), we find between the curvature in a generator of the rim and that at the centre of a face of a circular disk the simple relation io). 24... 3, +. ne a/R,= (wpa?/E) x (r/a) x n(1 +7) ; — Thus in a thin disk the curvature produced by rotation in a rim generator is much greater than that produced at the centre of the faces. ‘The curvature in the rim generator might perhaps be measured by an optical method, at least in a circular disk. X. Some unpublished Data on Colour-Blindness. By Dr. Witttam Pots, F.R.S.* [Plate II.] ae SE data have reference to a paper “On Colour- Blindness ”’ which the Royal Society did me the honour to publish in the Philosophical Transactions of 1859, and a Report on which, by Sir John Herschel, was printed in the ‘ Proceedings,’ vol. x. p. 72. There were some points in this * Communicated by the Author. Data on Colour-Blindness. 101 paper left incomplete, and I was advised to add supplementary explanations, but, for various reasons, this was not done. As, however, there is still much uncertainty and misunderstanding on the subject generally, it appears to me that, even after this lapse of time, it may be useful to put on record any facts or observations which may tend to throw light upon it. The paper contained, so far as I know, the earliest com- plete quantitative demonstration of the general facts of dichromic vision, made, at the suggestion of Prof. Stokes, by Clerk Maxwell’s elegant device of the colour-top, then newly introduced. I carried out the necessary series of experiments, first on my own vision, and afterwards on that of some other persons reputed to be colour-blind ; and I was surprised to find in the latter an anomaly which led me to add the following postscript, dated October 1859 :— “Since the foregoing paper was written I have had the oppor- tunity of examining three colour-blind persons, and have found that the vision of each is perfectly dichromic, corresponding pre- cisely in general character with my own. But the remarkable feature has presented itself, that the coefficients of the colour-top equations vary considerably in the different cases. Thus, for example, although my equation XY. will always hold in its general form, : m Vermilion + Ult. =p Black +q White, yet the values of m, n, p, and qg will vary for different individuals.” The observations themselves were not given, but I propose to add them now, with others, as being in many respects of permanent interest. Two of the persons examined were brothers, named Nicoll- Carne, gentlemen of high position and education, residing near Cowbridge in South Wales ; and they had already, at a much earlier time, acquired some public celebrity in regard to this matter. On the 25th of June, 1816, a paper was read before the Royal Medical and Chirurgical Society of London by Whitlock Nicholl, Hsq., M.D., of Cowbridge, entitled “ Account of a curious Imperfection of Vision;” and it referred to “a boy 11 years old,"clever, lively, and healthy,” who made the kind of mistakes which are now so familiar to us, but which were then thought so strange and unaccountable. He was said also to have “an infant brother,” of whose vision nothing could then be known. Now these two brothers were the gentlemen whom, forty-three years afterwards, I accident- ally heard of, and who kindly gave me permission to examine their vision. I may give the following extracts from letters received from one of them :— 102 Dr. W. Pole on some unpublished “Nash Manor, June 1859.” “The great peculiarity of my case is that it has existed in my family, as far as we have been able to trace, for generations, and has always skipped over the female branches, although, as in my own case, it has been transmitted through them. ‘My brother has the same peculiarity of vision. “T have long been convinced that I really see but three colours, viz. blue, yellow, and white, and the absence of all colour, black. By taking the water-colours gamboge, Prussian blue, and Indian ink, I can produce all the colours I see in Nature. I form my reds, greens, and browns by combinations of yellow and black ; pinks and crimsons by light and dark blue. The rainbow appears to me to be composed of but two well-defined and distinct bands, stripes, or colours, the one shaded from light to dark blue, and the other shaded from light to dark yellow.” “R. C. Niconi-CaRne.” I visited these gentlemen on the 14th of July, 1859. They had never before been subjected to any precise or quantitative experiments, and I am glad to have the opportunity of record- ing the actual facts of their vision. Both brothers took part in the examination, and their colour-impressions, so far as { could tell, appeared precisely similar. I also examined a Mr. Lloyd, a middle-aged gentleman of good intelligence, but not pretending to scientific knowledge ; and a fourth example was Mr. Parry, a professional chemist, whose case was peculiarly useful as forming the greatest con- trast with mine. In all these cases I followed the same plan. After a pre- liminary examination with the Chevreul circle and “gammes,” I obtained their matches with the revolving-disk tests, in the manner described for myself in par. 19 of my paper. I have collected the whole of the equations at the end of this article, and I have added the case of Mr. Simpson, who was tested by Prof. Clerk Maxwell about the same time, with, I believe, the same colours. I am also able to add to these, by indirect evidence, another case of great celebrity, namely that of John Dalton. Sir John Herschel was good enough to sugmit for my examination the data he had obtained from Dalton, as mentioned in the well-known letter of 1833; and I had no difficulty in estab- lishing the general similarity between Dalton’s vision and my own, subject to certain variations of the kind I am now considering *. And, further, I found some data by which to estimate roughly the nature and extent of these variations. He had pointed out samples of red and green silk, which * See ‘Contemporary Review,’ May 1880, for further particulars. Data on Colour-Blindness. 103 seemed a very near match to him; while to me there was the difference that the green was the darker by two numbers of Chevreul’s yellow gamme. This fixed Dalton’s variation as about intermediate between my two extremes, probably corresponding pretty nearly with Mr. Lloyd’s. The quantitative data here collected, and all comparable, furnish very important materials for inference and reasoning as to the still open question of the varieties of colour-blindness. It is not my object to enter further on that argument now, but it is necessary to show that, in order to make use of the facts by proper classification, more detailed attention must be devoted to the minute peculiarities of dichromic vision; and this opens another point in which my original paper was deficient. I had given a general comparison between the normal and the dichromic hues seen, referring briefly to various tints and shades of them; but it is now necessary to examine these tints and shades more accurately than has, I think, hitherto been done. For the proper understanding of what we really do see depends almost entirely on this point, and it is often much misunderstood. Although the dichromic patient, as his name implies, sees only two varieties of hue, yet the number of colowr-impressions he derives from these admit of great diversity of character ; and this is the reason why his defect so often escapes notice. He hears of the variety in colour presented by nature ; and he knows that nature also offers great variety to him: what he does not know till he is taught, is the different nature of the variety in the two cases. It is this we have now to explain. The colour-impressions received in dichromic vision may be classed in eleven species, seven of which further admit each of infinite variety in degree. And in describing them I shall adhere, without discussion, to my original names of yellow and blue for the two hues seen. Then we have :— 1. The impression of the full yellow colour in its most powerful form. 2. The corresponding impression of the full blue colour. 3. Then there is the impression of what Sir John Herschel called “the equilibrium of the two,” or full white. And 4, There is what he called the “ negation” of the two, i. e. black. According to the method of testing by pigments, these four form what we may call the fundamental colour-impressions of dichromic vision, by combinations of which all other colour- impressions are produced. These combinations form by far the most important part of the visual phenomena, and require 104. Dr. W. Pole on some unpublished therefore special study. They belong, as already stated, to seven species. , 5. An infinite variety of impressions may be given by diluting the fundamental yellow with different proportions of white. The painters have a practical name for these, namely tints ; but we want, for quantitative purposes, another form of expression. The freedom of a colour from white is usually called, in this country, “ purity;”’ but it will suit our present object better to adopt the equivalent of the German term Sdttegung. Helmholtz, for example, repeatedly uses such expressions as “ héchste Sittigung,” or “bei sehr gesiittigten Farben ;” and I propose to define these mixtures as degrees of saturation. If to y parts of yellow we add w parts of white, then the resulting “ degree of saturation” is = rar the full saturation of the yellow colour being =1. | 6. Then we have an infinite variety of impressions of a different kind, by darkening the fundamental yellow with black. These modifications painters call shades, and we may term them degrees of brightness or degrees of lumi- nosity. Thus, if to y parts of yellow we add 6 parts of black, the degree of luminosity will be expressed by on the original luminosity of the yellow colour being =1. 7. But there is another infinite variety of impressions obtainable by mixing yellow with different proportions of both white and black, by which its saturation and luminosity are both reduced, and the character of the impression is materially changed. If y parts of yellow are mixed with w parts of white, and 6 parts of black, the degree of saturation will be ey as i oS iain oe, Ea ree and the degree of luminosity Tiyan 8, 9, 10 are corresponding varieties of impression for the blue colour. a 11. Then, finally, there is an infinite variety of impressions of “grey,” formed by mixing white and black in different proportions. The luminosity of any such mixture will be w nate : The mixtures of yellow with blue produce no new impres- sions—none but what are included in thg above classes. It is worth while also to define another quantitative function of the dichromic mixtures, 7. ¢. the strength or power of the impression of the colowr proper, in relation to the full power of the hue from which it is derived. I propose to call this the Data on Colour-Blindness. 105 chromic strength. In class 5 this will be the same thing as the degree of saturation ; in class 6 it will be the same as the es. ber “fie hai’ degree of luminosity ; but in class 7 it will be = ree: Now, having defined quantitatively the different varieties of dichromic colour-impressions, we ought to be able to repre- sent them in graphic form ; and this is another point in which my original paper required supplementing. Professor Clerk Maxwell, who was kind enough to take much interest in my case, pointed this out to me, referring me to the explanation he had given on the subject in the Phil. Trans. for 1860. The system he there laid down, founded on Newton’s law for colour mixtures, has since become well known to earnest students. For normal vision an equilateral triangle is drawn, and on its three angles are supposed to be located the three fundamental colour-sensations—red, green, and violet, by combinations of which, according to Young’s theory, all hues are formed. Then every possible hue will be repre- sented by a point within the triangle, which can be determined by Newton’s rule. The application of this, however, to dichromic vision has never yet received proper attention. Maxwell, in his fig. 11, gave an example, the object of which was to compare di- chromic with trichromic vision, and for which, therefore, he used the normal triangle. But, to show simply the pheno- mena of dichromic cases, or to compare them with each other, this becomes unnecessarily complicated, and makes a clumsy figure. We know nothing of red, green, or violet (none of them existing in our system), and have no inducement to base our triangle upon them ; and, moreover, we have none of the varieties of hue, to depict which the normal triangle is formed. For these reasons, availing myself of a suggestion made to me by Lord Rayleigh, | have adopted a different arrangement of the diagram, which, while founded on the same principles, shall be specially adapted to the facts of our vision. I retain the equilateral triangle, but I place on its three angles the three chief sensations received by dichromic eyes; ?.e. the warm colour, the cold colour, and the sensation of their “equilibrium,” white. I then find an additional point for their “ negation,” black, and thus I get the four starting-points of the colour- impressions which it is the object of the diagram to portray. The construction will be readily understood from fig. 1, where Y is the warm colour, yellow ; U is the cold colour, ultra- marine blue; and W is white ; the point for black is deter- mined for each special patient by equation I. For my vision 106 Dr. W. Pole on some unpublished the line YU must be divided at O in the proportion of 613 to 384, and a line must be drawn from W through O and pro- longed downwards. The distance WO being taken as 64 parts, the prolongation must be made 36 parts, which will give the point Bk. Then the lines from this point to Y and U will complete the figure. It will now be easy to see how all the varieties of our colour-impressions, above described, may be represented on the diagram. The simple impressions 1, 2, 8, 4 are on the angular points. The line Y W contains the whole series of class 5; the line U W the whole series of class 8; the line Y Bk contains the whole series of class 6; the line U Bk, that of class 9; the line W Bk contains the whole series of greys in class 11. The position of a point representing any given mixture along these lines is fixed by the usual Newton’s rules; thus a mixture of half yellow and half white would lie halfway between Y and W. A mixture of one part yellow and 3 parts black would lie at a distance from Bk of one fourth the length of the line. The impressions in classes 7 and 10 are represented by points lying in the interior of the figure ; No. 7 (yellow) lying all on the left-hand side of the grey line; No. 10 (blue) all on the right-hand side. The mode of finding the position of any point will be best shown by an example: take the emerald green as defined by my equation IX. First divide the grey line into (58+19=) 77 parts, of which set off 19 at the lower end, giving a point a. From this draw a line aY, divide it into 100 parts and set off 23 of these froma. This will give the point G, representing my emerald green. The various quantities relating to this impression may be determined geometrically as follows :—Draw a line from W through the colour-point G, cutting the line Y Bk in s ; then the degree of saturation=1— - . Similarly draw a line from ay Bk through G, cutting WY in J, then the degree of luminosity — = , and the chromic strength = aG If a line be drawn from vermilion, or from any red colour, to U, the point where it crosses the line W Bk will represent a colour which, though a powerful crimson or “ purple ” to the normal eye, will be grey to dichromic eyes, or, in other words, a red invisible to them. And, similarly, in a line GU, the point where it cuts the grey will be an “ invisible green.” In this way, the figure, when completed for the various Data on Colour-Blindness. 107 colours used in the tests, will give, at a glance, a perfect idea of the phenomena of vision of the dichromic patient ; and by the simple application of a pair of compasses, the quantitative determinations may at once be found. It will now be easily seen that the difference between this dia- gram and Maxwell’s normal one exactly expresses the essential difference between normal and dichromic vision. In the former the most salient feature is the immense variety of hues ; the normal-eyed person has not only six chief colours, each one differing essentially from all the rest, but their com- binations with each other afford an infinity of additional varieties of hue. This variety is wanting in dichromasie, but its place is supplied by an infinity of varieties of tint and shade. Now in both diagrams the area of the figure may be filled with innumerable points, each of which will express a different variety of colour-impression ; but while in the normal case these are varieties of hue, in the dichromic case they are varieties of tone. The want of appreciation of this difference becomes con- tinually evident in communications between normal and colour-blind people, the two parties being frequently quite at a loss to understand each other ; and it accounts for some of the mistaken views held by normal-eyed persons, even those who haye studied colour, as to what our impressions really are. The errors often made by them in describing our sensations seem quite -as ludicrous to me, as any attempts of mine to identity the normal colours would be to them. The same consideration influences the discussions on the the alleged danger with railway signals. The alarmists think it quite sufficient to say that ‘‘ the colour-blind are unable to distinguish between red and green.” But this is speaking in an unknown tongue. The point to test is, ‘‘ whether the peculiar variety of colour-impression made on the dichromic eye by the red-lamp rays is so different from that made by the green-lamp rays, that the distinction between them is easily perceived.” This is a form of the problem not often thought of. Fig. 1 is the diagram of my own vision. Fig. 2 is that of Mr. Parry, which forms the greatest contrast to mine. It is not necessary to give the intermediate ones ; but the im- portant results are recorded in the following table, which will be a useful addition to the simple elementary data. 108 Dr. W. Pole on some unpublished TABLE of the Comparative Colour-Impressions produced by the rays of various coloured pigments upon several indi- viduals, each having Yellow and Blue Dichromic Vision. The full Saturation, Luminosity, and Chromic Strength of the fundamental colour-disk used are each denoted by unity. : and ue , : rothers| Parry. Pole. | Simpson probably bre ae ‘ arry Dalton. By Rep Cotours (Hue yellow). Carmine. Sa tunahOnenn.:..6000°, an unexpectedly high number, which is in disaccord with the ordinary views.—Beblatter der Physik, No. 5, 1892; from Wiener Berichte, November 12, 1891. AURORAS OBSERVED AT GODTHAAB. BY ADAM W. PAULSON. The author publishes the journal of the observations of all the auroras which appeared at Godthaab (South Greenland) during the winter of 1882 to 1883, and develops the general considerations to which they lead. Aurore boreales are divided into two distinct categories. When the phenomenon is feeble they simply appear as luminosities or Juminous clouds. The more marked aurore boreales affect the arc shape. These arcs may present various shapes known as curtains, bands, zones, and corone. The two latter, no doubt, do not differ 144 Intelligence and Miscellaneous Articles. from each other ; the difference of appearance arises from a variation in the direction of the corona in respect of the observer. ‘The aurore in the form of curtains or of draperies are very long and high, but have no appreciable thickness. They may sink to within 600 metres from the ground. Zones and corone, on the contrary, are always at very great heights and may exceed 320 kilometres. Thus, in the south of Greenland, the region in which the aurore are produced extends from the highest regions of the atmosphere to the surface of the soil, while in temperate countries the phenomenon takes place simply in the higher regions of the air. If, then, we admit the electrical origin of the phenomenon, this electric current circulates in low latitudes in the higher regions of the atmosphere and produces the well-known appearance of rarefied gases, while in the zone proper of auroree boreales it descends to the ground. The vertical direction of this current in arctic countries, together with the variations of density of the air traversed, must be the cause of the great difference of this phenomenon in arctic and in temperate regions. The usual colour of the aurora borealis is white with a faint tinge of green or of yellow. The cloud forms have a greyer colour. The edges of the draperies are often momentarily coloured red or green. At Godthaab the greatest daily activity of the aurore is at 9 p.m., and the annual maximum is about the winter solstice. The series of the observations in Greenland show a maximum frequency at the periods when the number of sun-spots is smallest, and conversely. The law enunciated by M. Trombolt, according to which the auroral zone moves in the course of twenty-four hours so that during the night it is directed to the north, seems contradicted by experience. Lastly, a well-defined distinction has been observed between the greatest frequency of aurore boreales in arctic and in temperate countries. If this is a general fact, it will prove that a more active evolution of the aurora borealis in low latitudes enfeebles the auroral activity in the characteristic region of the aurora.—Journal de Physique, February 1892; from Observations internationale polaire ; expedition danoise: Copenhagen 1891. ON THE ABSORPTION OF LIGHT IN TURBID MEDIA. BY A. LAMPA. In front of the collimator-slit of a Glan’s photometer a glass trough with parallel sides was placed, and the lower half filled with an emulsion of a definite composition of an alcoholic solution of mastic (4:062 to 5°592 gr. in 100 gr. absolute alcohol) in water. ‘The absorption was examined which the paraliel rays experienced on passing through the emulsion, and in twenty places in the spectrum. The observations agree very well with Lord Rayleigh’s formula. The author estimates the diameter of the emulsion at less. than 0-24. On comparing emulsions in which the product of thickness and density was constant, the weaker media seemed to absorb more strongly.—Beiblitier der Physik, No.4, 1892; from Weener Berichte. a ee ee a ee THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIES.] WG Wis 1 892: XVI. On the Instability of a Cylinder of Viscous Liquid under Capillary Force. By Lord Rayueiew, Sec. R.S.* HE main outline of the theory of the instability of a long cylinder of liquid is due to Plateau, who showed that if the equilibrium surface r=a be slightly deformed so as to become Mite OOS UE iene sala te sia hse! (A) in which z is measured parallel to the axis, the deformation is stable or unstable according as ka is greater or less than unity; that is, according as the wave-length 2 of the varicosity is less or greater than 27ra, the circumference of the cylinder. The solution of the merely statical question is, however, insufficient for the application to the important problem of the disin- tegration of a jet of liquid. A deformation of any wave- length exceeding 27a increases exponentially with the time (e%) ; and what we require to know is the relation between q andr. 1, as was to be expected. In the former case the values of in are numerically greatest when ka=17/4°5. In the other extreme case where inertia may be neglected in comparison with viscosity, we have ie tra) dy = pat wlp-Jo so that the instability is greatest when ka has the same value as in the first case. The general form of the quadratic is (tn)? +in.p'/p+H(k’a?—1)=0, . . . (12) where H is positive. If ka <1, both values of in are real, one being positive and the other negative. The displacement is accordingly unstable, and the greatest instability occurs with the above-defined value of ka. If, on the other hand, ka >1, the values of in may be either real or imaginary. In the former case both values are negative, and in the latter the real parts are nega- tive, so that the deformations are stable. ee The investigation applicable to a real viscous liquid of vis- cosity 4, or py, is much more complicated than the foregoing, mainly in consequence of the non-existence of a velocity- potential. But inasmuch as the motion is still supposed to be symmetrical about the axis, the equation of continuity gives ee Pe ree oye e e e e (13) where yy is Stokes’s current function. For small motions y satisties the equation * Aa daa a dt ee zat capa aes at In the present question yf as a function of z and ¢ is propor- tional to e!+*), and it may be separated into two parts, Ww . Camb. Trans. 1850. See also Basset’s ‘Hydrodynamics,’ vol, ii. p. 209. of Viscous Liquid under Capillary Force. 149 and 2, of which yy satisfies ete ay 40 4) gto, cs dr? yr ar- r ar and fy satisfies ate I aNe (pe ae 1 “¥2) /2 mae ge har e(5 ip) ea OO where [i =) tO ale ae a a ae I LL) At the ae we have to consider the normal stress P, and the tangential stresses. Of the latter one vanishes in ile of the symmetry, and the other is to be made to vanish in con- formity with the condition that there is to be no impressed tangential force*. Thus du dw ae ae ee ° ° ° ° (18) or in terms of y by (13) | ae ae pears reclame (19) Introducing Wi, W2, and having regard to (15), (16), we may express this condition in the form Bie Ney (kk? )abo—= 0, 2.26 - « (20) which is to be satisfied when r=a. Again, for the normal stress, P= —pt+ 2Qudu/dr Be ap 2s ue =0(G"— 3 Boge w) +20 o = {57(7 ot) + 2% ing,(¥) b "re —_ ) ei Herein wo )= ld re (ie) =o r dr) rdr dr\r ar r dr’ For Ww, 3 ld d oe) KP aN _ r dr “Fiee dr y dr * It is here assumed that there is no “ superficial viscosity.” 150 Lord Rayleigh on the Instability of a Cylinder for ro, 1d/,d1d) _Bdfs_ MPa, r dr (- drr dr Pot ek ge Cas so that hak Kars = : oe Ste: ikr ar + r kp dr > The variable part of the capillary pressure is, as we have already seen, TE(k?a?—1) a? : in which 2 na Thus, the condition te be satisfied when r=a is _ 42,2 J2__ 72 - TL — Ka?) ky _ f k? dps op i np dw (23) a na tka dr dy ka dr * The forms of yr, W, are to be determined by the equations (15), (16), and by the conditions to be satisfied when r=0. It will be observed that y, satisfies the condition appropriate to the stream function when there is a velocity-potential. This would be of the form p= Jo(tkr), os ee so that rd i=\(ru)dz= oS = re Jy (zkr). Thus di =Ards (thr) 2.5 ee is the most general form admissible, as may be verified by differentiation. In this Jo(kr) satisfies the equation Jo’ (thr) + Is (ihr) +Jo(tkr)=O0. 8°. Gaeta Since (16) differ from (15) only by the substitution of k’ for k, the general form for wy, is = Berds GET Oe 5 ern By use of these values the first boundary condition (20) becomes 2A Jo (ha) + (k? +k) BJ, (tk'a)=0. . . (28) We have next to introduce the same values into the second boundary condition (23). a this OW = ihr [ Jo" (ihr) + Ju (itr) ]=—Athad (ha) i ee Oe ee ee ee a oe of Viscous Liquid under Capillary Force. 151 by (26). In like manner, Te — —Bik'aJd(ck'a). Thus Tal eee rae ay CAS (ka) + BI Cha) 2 = —»[B pes hia) + 212A IT," (ika) + 2EKBIg" (ik a) | a = [Atha J (tka) +Bik'ad,(ik'a)]. . . . (29) Between (28) and (29) we now eliminate the ratio A/B, and thus obtain as the equation by which [in conjunction with (17) | the value of nis to be determined Td — ka?) kak’? — pa on Fe ae (:ka) Qhk’ Jn! (ika) kK? +k? Jo (tka) hi (hl? —2) S((ika) ~ 2, ~ ETE) Ty ikay 20 a) en Bee Nha), = Fag thao (ia) — ae IJ Gkay ah iad (ik a) b (30) We shall now apply this result to the particular case where the viscosity is very great in comparison with the inertia. The third part of (30) may then be omitted, and we have to seek the limiting form of the remainder when xX’ is nearly equal to k, as we see must happen by (17). In the first part, jl? — fe? es ok pee = = 214 Tne Tika) In the second, Qkk' JIo'(tka) ikaf Io"? — Joo" 38k Fe eee) P= kay and _ B(k?—7) Jo (tka) vn ae J) dk HEP) Iya) KO = — Thus the limiting form is T(1—k?a?) hiss = ike oe ie ole Pe a0 oe pa. n 152 Lord Rayleigh on the Instability of a Cylinder in which, however, we may effect further simplifications by means of the properties of Jy. We find by use of (26) Jo 2@—J Sl" — ee 5/5 24 w(1 = a 3) so that, finally, fe Hite 4") Os an 2pa . ka? §J 92/392 +14 1/k2a*} In (31) the argument of Jo, Jo’ is cka, or z as we will call it for brevity. And by a known property J) =—J,; Now ~2 Z Z Jo(@=1-s =F oy Ca )=541-g4 arr = ae x OP Ae sce ; tv a? at Jt) {1+ God uae. a These functions have been tabulated by Prof. A. Lodge” under the notation I,(2), 1,(z), where so that if z=ka ax? Jo(tx) =1+ 92 + a? at t I,(a) =J (ev) =1+ gat 2 get pile ke: “le aie. aren erates (32) I, (x) = — iJ, (iz) =3{1+55 eee He raat} . (33) In this notation x75 Jo? (tx) /J 1? (2x) +14 1/22} =a? + L—wv7l,?(~)/1,2%(@), . (84) and we have to consider the march of (84) as a function of «. When z is very small, Ij(a)=1—}02, T(z) = hat qhe’, so that (34) = —3+terms in 2*; and then from (31) n= ae See * Brit, Ass. Report, 1889, p. 28. : of Viscous Liquid under Capillary Force. 153 We shall see that this corresponds to the maximum insta- bility, and it occurs when the wave-length of the varicosity is very large in comparison with the diameter of the cylinder. The following table gives the values of (34) for specified values of w:— x, (34). a, (34). 0-0 —3°0000 1°0 — 3°0188 0-2 —3°0000 2°0 —3°2160 0°4 — 3°0004 4:0 —4:458 0-6 | —3-0023 | 60 | —6-247 It will be seen that the numerical value of (34) is least when «=0, which is also the value of w for which the numerator of (31) is greatest. On both accounts, therefore, in is greatest when # or ka=0. But over the whole range of the insta- bility from ka=0 to ka=1, (84) differs but little from —3, so that we may take as approximately applicable aba ear) =. Umea ea ile cath (36) The result of the investigation is to show that when vis- cosity is paramount long threads do not tend to divide themselves into drops at mutual distances comparable with the diameter of the cylinder, but rather to give way by attenuation at few and distant places. This is, I think, in agreement with the observed behaviour of highly viscous threads of glass or treacle when supported only at the ter- minals. A separation into numerous drops, or a varicosity pointing to such a resolution, may thus be taken as evidence that the fluidity has been sufficient to bring inertia into play. The application of (31) to the case of stability (ka>1) is of less interest, but it may be worth while to refer to the extreme case where the wave-length of the varicosity is very small in comparison with the diameter. We then fall upon the particular case of a plane surface disturbed by waves of length X. The result, applicable when the viscosity is so great that inertia may be left out of account, is the limit of (31) when a, or 2, is infinite, while & remains constant, or s eb imnet (ae) eae. By means of the expressions appropriate when the argument is large, the limit in question may be proved to be —1; so ~ that in= Tk in=— — a . (87) 154 Mr. C. Chree on Rotating Elastic If gravity be supposed operative in aid of the restoration of equilibrium, we should have to include in the boundary con- dition relative to pressure a term gp& in addition to ThE; so that the more general result is obtainable by adding gp/k? to co Thus k inz—3°(T EY - 82. giving the rate of subsidence of waves upon the surface of a highly viscous material. It could of course be more readily obtained directly. When gravity operates alone, ye te ae ela LO: n= uk Din rae (39) which agrees with a conclusion of Prof. Darwin*. A like result may be obtained from equations given by Mr. Basset fT. XVII. Rotating Elastic Solid Cylinders of Elliptic Section. By C. Carus, V.A., Fellow of King’s College, Cambridge. Part Il.— The Long Elliptic Cylinder. § 35. B* a long cylinder is here meant one whose length 21 bears to its greatest diameter 2a a ratio such as is required for the legitimate application of Saint-Venant’s solution for beams. What this ratio may be depends on the degree of accuracy aimed at, but the best authorities seem satisfied with values of //a which are not markedly less than 10. The cylinder is supposed to be rotating uniformly, and to be free from all but “ centrifugal” forces. In the paper in the Quarterly Journal, already referred to, I obtained a solution for a rotating elliptic cylinder, but its length was supposed to be maintained constant by the application of suit- able forces over the ends. This is a totally different case from the present, in which the cylinder is supposed free from all sur- face forces and capable of altering alike in length and diameter. The present solution is thus completely new, except for the case of a circular section which I have already treated else- where §, and for the limiting value 0 of 7 when the alteration * Phil. Trans. 1879, p. 10. In equation (12) write z/a=k, and make 20. + Hydrodynamics, vol. ii. § 520, equations (21), (27). See also Tait, Edinb. Proc. 1890, p. 110. { Communicated by the Author. § Cambridge Philosophical Society’s Proceedings, vol. vil. part vi. Solid Cylinders of Elliptic Section. 155 in the length of the cylinder is zero*. It seems, however, unnecessary to reproduce the algebraical work of determining the arbitrary constants occurring in my general solution of the elastic solid equations. The work is of no interest in itself, and the accuracy of the solution may be easily tested by reference to the internal equations and surface conditions it has to satisfy. It satisfies exactly the internal equations (3), the three conditions (7), (8), and (9) over the curved surface, and the conditions (4) and (5) over the flat ends. There remains only the last surface condition over the ends, f= +l, viz. eo. If this were satisfied the solution would be absolutely exact, and applicable however great or small //a might be. This condition is satisfied when 7=0, but otherwise we have to avail ourselves of the principle of statically equivalent load systems, replacing the above condition by b a aNaz—aF ii { ZA Oye Sra. ise eg MO) -—av_6 a Nat—2 This equation the following solution will be found to satisfy for every cross section, and not merely for the ends. The notation is the same as in Part I. EA (1—4) (8a* + 2070? + 30*)/o2p = (1—2n)| (a? +0) {at + ab? + bt — 4 (Bat — 20°? + 364) —49?(a2 —2)?} (1+1)a?{a*t + ab? + 2b4—n(at—b*)}— (149) y2{2at + a%b? +04 4+y(at—b*)], (76) Ha(1—1) (at + 2a%b? + 804) /o2p = xa*(a* + a*b? + b4) —n(a° + atl? + 2a7ht + b°) —1n?(a? —0?) (at + 30%) + 27°(a? —0")%( a? + 0") —3(14+7)a* {at + a2b? + b'—n(2at + ab? + 3b4) +7°2(a*#— 04) } Bh ger oie ee) eat ks eS * In this case the solution is the same as for the thin disk, and also as equations (131)—(183) on pp. 81-82 Quarterly Journal, vol. xxili., when m is put =n, and an obvious printer’s error interchanging x and 2° in (181) is corrected. 156 Mr. C. Chree on Rotating Elastic EB(1—7) (3a* + 2020? + 304) /w2p = y (0? (at + a2b? + b*) —n (a + 2a4b? + abt +B) + 1y?(a2— 02) (Bat +b?) + 44? (a? —b?)?(a? +0) } —3(L+)y*{at+ 0%? + b4— (3a! + ab? + 204) —42(at—)4)} — (1+) ya2{A—qaX(a +2) 47a}, 2. 2. eee (TB) y= —0'pn (a? +6*)e— (4B) ie oS eee (79) wa (1.—m) (8a! + 2a°l? + 3b") /o%p = (a2 — 22) {at + 0212+ b4—n(at +34} —ya(l +24), 7 yy(1—n) (Bat + 20°02 + 3b4) 0% = (B®?) {at + al? +4 —n(a!t +84} —«*b(142n), . . 2 (ee 2z(1—7) (3a* + 2a°L? + 304) /w®p =] n(a2 + B2) {(a2 +02)? — (a2 —02)2} — nx*{at + 070? + 2b*— (a*—b*) } —ny?{2a* + ab? + bt +n(a*—5*)} (82) ay(1—n) (3a! + 20°? + 304) /o*p= —ayfat + b'—n(a2+07)%}, . . (88) prt MN cS yemeneQ SO (84) § 36. As already stated, zz vanishes when 7=0. For other values of 7 the solution applies only under the same restrictions as Saint-Venant’s solution for beams, and portions of the cylinder immediately adjacent to the ends should be excluded from its domain. At every point zz is one of the three principal stresses, but the other two vary in direction from point to point of the cross section, being parallel to the axes of the ellipse only at points which lie on these axes. The stress system other than 22 may be conveniently analysed into a series of simple systems. Tor shortness let {at + a0? + b4—n (at + 4)} + {(1—n) (Ba! + 20°? + 3d4)} = K’, (85) then the simple systems are as follows :— i.) A uniform normal tension w?oa?K’ parallel to the p p major axis. Gi.) A uniform normal tension w?pb?K’ parallel to the minor axis. Solid Cylinders of Elliptic Section. 157 (iii.) A normal pressure —w*%pr?K’ directed along the radius 7 from the axis of rotation. (iv.) A normal pressure directed along the tangent at the point considered to the ellipse which passes through the point and is similar and similarly situated to the cross section. If a’, b’ denote the semi-axes of this ellipse, p’ the perpendicular from its centre on the tangent, this pressure is 9. L+2n ab? a’b! y Caio ea a p ee) On the curved surface of the cylinder, of the two principal stresses other than zz one, directed along the normal, is zero, and the other, whose line of action is the tangent to the cross section, is given by the equation _ateb—n(t +2) fad gy Taya + 204 BH (> ) ieee where p is the perpendicular from the centre on the tangent to the elliptic section. ‘This stress vanishes when the section is circular for the limiting value *5 of 7. Under all other conditions it is a tension. This analysis of the stress is very similar to that given for the thin disk in § 5. When 7 is so small that its square is negligible the two analyses are abso- lutely identical. tt=w*p § 37. The stress zz vanishes over the cylindrical surface Dean Ae ye Oa | artes Ons ta we ee A GO) where Ay; =} (a? + 0?) { (a? +07)? — (a? —b?)?} = fat + ab? + 2b* —n (at —B*) },(89) by? =1(a? +b?) { (a? + b?)? — 9 (a? — b?)?? + {2a* + a2b? + b* + n(at—5*)}.(90) It is a tension at points within, a pressure at points outside this surface. If e,, denote the eccentricity of (88) and e that of the rotating cylinder, €11° /e? = (1 + 2) a? (a? + 5?) + {2a* + ab? + b+ n(at—d*)}. (91) When 6=a and n='5 then e,=e. But under all other conditions C11 < é, or the surface of no longitudinal stress and the surfaces of equal longitudinal stress, which are similar to it, are of 158 Mr. C. Chree on Rotating Elastic smaller eccentricity than the surface of the rotating cylinder. It is also easy to prove 3 a41 0. When this inequality holds the portion of the cross section wherein the longitudinal stress is a pressure forms a complete annulus limited externally by the surface of the cylinder and internally by (88). When, however, the above inequality does not hold—and by taking 0/a small enough it can always be reversed even when 9=*5—the portion of the cross section wherein zz is a pressure consists of two detached areas surrounding the ends of the major axis. § 38. When 7=0 the maximum stress-difference is always CoN correctly given by the axial value of wz. ee ut for other values of 7 it is given by the axial value of wx—yy or by the axial one value of wx-zz according as b/a is less or greater than a certain value. ‘This value of 0/a increases with 7, being approximately *277 when »=*25, and 511 when n»="5. The greatest strain is always correctly given by the axial value of da da and greatest strain s may easily be found from equations (77)-(82). It seems unnecessary to write them down. The limiting safe speed cannot be determined solely by reference to a limiting elastic stress or strain on account of a species of instability which may arise. This question of instability will be discussed presently, but in the meantime it is convenient to record results from which the limiting speed, according to the stress-ditterence and greatest-strain theories, might be simply derived when the circumstances are such that these theories apply. In Table XIII. the angular velocity is termed »,, and in Table XIV. it is for the sake of distinction termed @;. Assigning to 8 and s in these tables their limiting values for the material under consideration, we obtain the limiting speeds according to the stress-difference and greatest-strain theories, while by assigning a given value to w,a and @,a we obtain the corresponding maximum stress- difference and greatest strain. The tables should be compared with Tables I. and II. The expressions for the maximum stress-difference 8 rs f ; Solid Cylinders of Elliptic Section. 159 TaBLeE XIII.—Value of w,a— V Sip . ne ha 70: 2, 4, 6, 8. 10. 0 1-732 1721 1693 1661 1639 1°633 25 ieee aoe ior. F70Gae alan ps2 5 reo 724 1-784 1-784 1848, 20 TaBLE XIV.—Value of w,a—./ Hs/p . n. wae. -0. 2, 4, 6. 8. 1-0. 0 1732 1721 1698 1661 41-639 1-633 25 1746 1784 41710 41704 1745 1-852 5 1789 1775 1762 1:822 2066 2-828 The last result in Table XIII. is ewact, the restare approximate. When 7=0 the results are the same on the two theories, and apply, as already stated, to cylinders ofall lengths. For large values of 7 the greatest-strain theory would allow a con- siderably more rapid rotation than the other theory for all values of b/a. But for values of 7 such as -25 the difference between the two theories is remarkably small. Fora given value of 7, other than 0, the limiting speed has on both theories at least one minimum as b/a increases from 0 to 1 ; but for ordinary values of » the limiting speed depends wonderfully little on the value of b/a. § 39. We shall next consider the principal displacements in the cylinder. The longitudinal displacement vanishes when 7=0, and for any other value of 7 the cylinder shortens under rotation. Hach cross section remains plane, which is perhaps the most striking difference between the phenomena and those in thin disks. The reduction in length per unit length (—6///) varies directly as 7, so it will suffice to give its value when n="25. TaBLE XV.—Value of (—8l/l), 4=°25. b/a= | Onn 2. dye Ge 28a TO): (—00/l) + (w2pa2/E) = 0625 065 0725. «085 = 1025-125 160 Mr. C. Chree on Rotating Elastic The results are all exact. This table should be compared with Table III. § 40. In the cross section the most important displacements are the alterations in the lengths of the principal axes. The major axis always lengthens under rotation. The increase da in the semi-axis is got by putting «=a, y=0 in (77). The alteration 5b in the minor semi-axis is got by putting =O, y=b.in (78). When 7=0 it is easily seen that the minor axis lengthens under rotation for all finite values of 6/a. For other values of 7, however, the minor axis shortens when 0/a is less than a critical value b,/a. The value of b,/a increases with 7 ; thus, answering to 7='25, °3, °5 we find approwimately b,/a='584, *663, °783 respectively. These do not differ very much from the corresponding results in the case of a thin disk (see Table VJ.). In the following table of values of da/a and 6h/b no sign is attached to the former quantity as being always positive. Taste XVI. Values of (6a/a)—(wpa?/E) and (6b/b) +(w®pa?/B). C—O bla= | 0 2 4, 6 8 1-0 ‘n=O (| 2 225 988 242 248 “95 (Sa/a) + (w2pa2/E)= } n="25| -224 226 +229 +2299 216 +1875 n='5 | 229 280°: -9381 220 187 ia n=0 | O +:0090 +-0372 +-0870 +1587 +-25 25 |—-0885 —-0794 —-0496 +0052 +-0858 +°1875 (8b/b) + (wpa?/E)= 4 n= =-5 |—1875 —-1780 —-1458 —-0840 +-0087 +-125 The results for }/a=1 are exact, the others are nearly all only approximate. This table should be compared with Table IV. § 41. The expressions for the displacements and strains in the long cylinder are more complicated than in the thin disk, and their full consideration would require more analysis than the interest of the results seems likely to warrant. I shall thus merely call attention to the more striking features of the radial strain along the principal axes of the cross section. Along the major axis the radial strain is the value of = with y=0. In acircular section when =:'5 the radial strain Solid Cylinders of Elliptic Section. 161 has a constant value along the radius. But for all other possible combinations in the values of b/a and 7 the radial strain along the major axis continually diminishes algebraically as the distance from the centre increases. At the centre the radial strain along the major axis is always positive, but under certain conditions it maybe negative, z.e. a com- pression throughout a small length at the ends of the axis. These conditions are most easily investigated by determining the points where ae vanishes in the major axis. From the dx symmetry we need only consider the point on the positive side of the axis of y,and we shall denote its abscissa by x. When &,>a the radial strain is an extension along the whole semi- axis; but when 2, ay RLF ait Ss Pipa (93) The radial strain is a compression throughout this very small length a—2 of the major semi-axis and elsewhere is an extension. When 7 and a are constant, a—, hasa maximum value na/3 when b/a=1. When 0/a is very small as well as 7, we find that the radial strain is an extension over the whole major axis when b/a<4 /3n approximately. When 7, though no longer very small, remains less than -3, the radial strain is an extension throughout the whole major semi-axis when 0/a is small, but is a compression over a small portion a—a, at the end when 6/a exceeds a certain value Phil. Mag. 8. 5. Vol. 34. No. 207. Aug. 1892. M 162 Mr. C. Chree on Rotating Elastic increasing with 7. Thus for y=*25 this critical value of b/a is approximately °4034, and the approximate values of a—a, answering to the values °6, °8, and 1 of b/a are respectively 0045 a, :0150 a, and :0339 a. When 7="3 it is obvious from (92) that 2=a when b/a=1. Also the coefficient of a*— 6? in the right-hand side of (92) is easily proved to be positive for this value of 7 ; thus 2) >a for all other values of b/a, or the radial strain is never a compression at any point of the major axis in any form of elliptic disk, though in a circular disk it just vanishes at the rim. As approaches close to ‘5 the coefficient of a*—8? in (92) may become negative, but it is easily shown that the term containing 87—1 is then always sufficiently great to keep the right-hand side of (92) positive. Thus when 7>°3 we have >a for all values of b/a, or the radial strain is an extension at every point of the major axis. § 42. Along the minor axis the radial strain is the value of - with e=0. When 7 is less than 4(713—3), or °3028 approximately, the radial strain along the minor axis diminishes algebraically as y increases for all values of b/a. For larger values of » the radial strain diminishes or increases algebraically as y increases according as b/a is greater or less than a certain value depending on 7. This critical value of 5/a increases from 0 when 7=*3028, and approaches 1 as 7 approaches °5. For »='3 the critical value is tn/ V/13—3, or ‘3891 approxi- mately. For ba=1, with n=°5, the radial strain has every- where a constant value. When 7=0 the radial strain is for all values of b/a an extension at every point of the minor axis except ‘the ends, where it vanishes. When 0/a is very small the radial strain is a compression throughout the whole minor axis unless 7 be very small.- When both 0/a and 7 are very small, we find for points in the minor axis the approximate formula d @? 7 aE Ov). oY ae So in this case the radial strain is a compression along the whole minor axis when b/a< “7, but for greater values of b/a it is an extension between y=0 and y= /b?—n2a?- Solid Cylinders of Elliptic Section. 163 When 7 is not very small it may be proved that = is nega- tive along the whole minor axis when 0/a is less than a certain critical value 6;/a depending on 7. The approximate values of 6;/a answering to the values °25, °3 and ‘5 of 7 are re- spectively °523, °628, and °577. As b/a increases from 03/a the radial strain becomes an ex- tension throughout a portion of the minor axis, and this portion expands until eventually when 7>°3 it includes the whole axis. The approximate values of yo, the distance from the centre of the points where the radial strain vanishes in the minor axis, are given in the following table :— TaBLeE XVII.—Value of y0/d. b/a = 25. 3 5 6 633 ae 963 8 909 926 507 1:0 966 1-0 =e In the case of the first blank the radial strain is a com- pression, in the case of the second an extension at every point of the minor axis. When 7 has the values °25 and °3 the radial strain is a compression in the length b—y), an extension in the length y; but when 7="5 it is an extension in the length 6—yo, a compression in the length yp. § 43. As already indicated in § 38 an elastic theory of rupture, such as the stress-difference or greatest-strain theory, even if satisfactorily established, would not necessarily suffice in the present case to fix the limiting speed. This failure arises from the possible occurrence of instability, consisting in a tendency in the axis of the cylinder to bend, it being driven as it were by “ centrifugal” force from coincidence with the straight line joining its ends. This question has been investigated by Prof. Greenhill*, who has deduced formule for the limiting speed in isotropic cylinders. Before applying his formule we shall briefly consider what is the actual problem he has solved. He takes the cylinder, with its axis displaced by rotation into a curved line, and supposes * Institution of Mechanical Engineers, Proceedings 1883, pp. 182 et seq. M 2 164 Mr. C. Chree on Rotating Elastic the action of the “ centrifugal” forces the same as if the mass were collected in the axis. ‘This axial distribution of force is then supposed in equilibrium with the elastic forces, the dis- tribution of stress over each cross section being assumed to give a couple as in the ordinary Bernoulli-Eulerian treatment of beams under flexure. This leads to a differential equation of the form | Pe hy 0 (95) ge Pg 0) 2 ee where x is measured along the line joining the ends of the cylinder’s axis, y is the distance from this line of a point in the axis in its displaced position, while wu is a constant depend- ing on the velocity, material, and dimensions of the cylinder. Supposing the origin at an end of the axis, we may represent the solution of (95) by y=A sinh wx +B coshua+C sinpa+Dcospx, . (96) where A, B, C, D are constants depending on the terminal conditions. Prof. Greenhill takes two alternative sets of conditions :— (1) y=0= at both ends, .. ; .~...% d d2 (2) es Z , in These lead to different results for the limiting speed. § 44. The condition y=O merely fixes the origin of co- ordinates, assuming the line joining the ends of the axis to be os fixed in space. The condition ante 0 at an end signifies that 2 the direction of the axis is there fixed, while S%=0 signifies the vanishing of the elastic couple given by the Bernoulli- Hulerian method of treatment*. This latter condition is thus required by Professor Greenhill’s theory when no applied couple acts over a terminal section. Now, supposing Professor Greenhill’s method of reducing the physical problem to a mathematical form sufficiently exact, so that (95) is satis- factory, it is clear that the reliance to be placed on his results * Prof. Greenhill introduces the condition 54 =0 on his p. 200, with- out explicit reference to his elastic theory, but the above is, I believe, the explanation he had in view. Solid Cylinders of Elliptic Section. 165 must largely depend on how closely actual conditions are represented by the assumed terminal conditions. This point is not one which a mathematician is competent to decide, but I do not see that either set given by Professor Greenhill seems more likely a priori than the combined set nt Y =7= age Oe gpa ts (99) Thus, suppose the bearings on which the axle rests each of breadth c, where ¢ is, as must happen in practice, finite, though small compared to the length of the cylinder. Further, suppose the bearing to consist of a hollow circular cylinder, inside which the circular axle can move freely without appreciable friction. If the axle fitted the bearing exactly, dy dx and when #=c, with similar conditions of course at the other end. ‘Thus, without introducing any elastic principle, we | dy dy : should deduce that y, ae ie and, perhaps, some higher then y and— would each require to vanish both when «=0 differential coefficients must vanish at each end. We cannot, however, satisfy all these conditions with a solution such as (96) without having each of the 4 constants identically zero, in which case we reach no conclusion whatever as to instability. If we accept Professor Greenhill’s elastic theory, the con- as dition ==) must hold over the terminal sections, unless their faces are in contact with some stops or held in some way. Since the cylinder naturally shortens under rotation, it is difficult to conceive how any system of support which did not introduce very large frictional forces when the rota- tion was slow could leave the terminal sections anything but free when the rotation was rapid. Thus, if it be possible for the supports to keep ot 0 at both ends, as Professor © Greenhill ‘supposes in his first set of terminal conditions, it seems doubtful whether his theory leads to any conclusions whatsoever as to instability. § 45. It is only fair to recognize that the hypothesis that the axle exactly fits the bearings—though apparently implied in Professor Greenhill’s first set of terminal conditions—is hardly possible in the strict mathematical sense, and some of 166 Mr. C. Chree on Rotating Elastic the above difficulties might be avoided by a judicious use of this fact. Thus, if we suppose the radius of the bearing to exceed that of the axle by 67*, we could satisfy the set (98) of terminal conditions by putting in (96) A= be). Quls 97,2"... oa whence yee sn (wa/ 2s... 2 =e The constant OC is not, however, altogether arbitrary. The axle must press against the bearing at the section e=c, and it seems most reasonable to suppose it would also press against it at the section z=0, though this is not apparently absolutely necessary. On the first view we must have C=Alér/(me) ; on the second view C is only limited to being less than this value. This solution, it will be noticed, answers to one definite value of pw, z.e. to one given angular velocity. With either a smaller or a greater angular velocity—with definite exceptions in the latter case to be noticed presently—the solution (96) can satisfy the terminal conditions only when C vanishes as well as the other three arbitrary constants. It would thus appear that the meaning of our solution is as follows :— Supposing the angular velocity gradually to increase from zero, the axis of the cylinder must remain straight until the velocity is reached for which 2u/=7. The axis may remain straight when the velocity passes through this value, or it may not. If it remains straight for this critical value it must con- tinue straight while the velocity continues to increase until the value is reached for which 24¢/= 27, when again it is possible for it to bend. It may, however, happen that as the velocity is attained for which 2uJ= the axis bends. The bending will take place suddenly, the consequence being an impulse between the cylinder and the sections c=c, e=2/—e of the bearings. This may suffice of course to smash the bearings or the cylinder, in which case the instability theory may per- haps be considered satisfactory. If, however, this impact does not smash either the bearings or the cylinder, the danger seems to be passed unless the rotation be kept exactly at the critical velocity. Now it is clear that if l47/c be very small, the resultant “ centrifugal’ force answering to the displace- ment (101) is small, and the impulse on the bearings may be but trifling. Thus the danger which the instability theory * This implies, however, that when the axis of the cylinder bends, the line joining its ends ceases to be fixed in space. Solid Cylinders of Elliptic Section. 167 recognizes may conceivably be very small with many forms of support. There are obviously a series of other critical velocities answering to EM — a Oya. ty sos where 7 is any positive integer. The larger the value of 1 the greater would be the danger attending the bending of the axis. Since oc o?, the corresponding angular velocities are as the squares of successive integers. The previous considerations, it must be clearly understood, are intended not to throw doubt on the existence of a species of instability, such as Professor Greenhill imagines, but merely to give a general idea of the uncertainties attaching to any numerical details to which his theory leads on account of possible divergences between the terminal conditions he assumes and those existing in practice. The only positive conclusion we have come to is that a tendency to instability may be expected to show itself by a want of smoothness in the motion and an undue wearing away of the inner edges of the bearings. This tendency to instability might be seriously increased by a slight departure of the centre of gravity of the cylinder when at rest from its axis. § 46. We have next to consider the nature of the hypotheses by which the equation (95) is obtained. The assumptions that the action of the “ centrifugal” forces may be calculated by collecting the mass of the cylinder into its axis, and that the elastic stresses over a cross section give origin to a couple proportional to the curvature of the axis, are certainly not more exact, even near the centre of a long cylinder, than the Bernoulli-Kulerian treatment of the flexure of a rod under its own weight. Near the ends of the cylinder the strain and stress must differ widely from that assumed by Professor Greenhill, as he takes no account of the displacements which exist in the absence of instability in a rotating cylinder. These considerations show that while it is quite possible Professor Greenhill’s formule may lead to correct results for short cylinders, there is no apparent reason from an elastic- solid point of view why they should. My own view is that the application of these formulze to cylinders whose length is less than 8 or 10 times their diameter is certainly not justifiable—an opinion in which I hope Professor Green- hill will concur—and that in longer cylinders the application of either formula can hardly be considered satisfactory unless some definite evidence exists that the terminal con- ditions it supposes are approximately satisfied. § 47. Considering the uncertainty which prevails, it will 168 Mr. C. Chree on Rotating Elastic suffice to indicate how these instability formule may be applied to check the application of the elastic theories of rupture without entering on any elaborate calculations. In applying his first set of terminal conditions* Professor Greenhill makes a numerical slip. His amended formula for the limiting speed in a cylinder of length 21 is w*p/Ex? =(2°36502/l)4, . . . . (102) where « is the radius of gyration of the cross section about an axis through its centroid perpendicular to the plane of bending. The corresponding formula for the second set of terminal conditions * is opin —=(n/2i*. |. As we require. the least velocity for which instability may arise, « is the least radius of gyration obtainable, 2. e. is b/2 in an elliptic section. The corresponding plane of bending con- 2 tains the minor axis. The terminal condition a =0 of the second set of surface-conditions depends on Professor Green- hill’s elastic theory, which does not seem a close representa- tion of matters near the ends. Thus, in selecting one of the formule for illustration I have preferred the first, as based only on geometrical considerations. The results it leads to in a cylinder of length L are, however, the same as the second formula would give for a cylinder of length wL/(4°73). ‘Taking then (102), suppose we determine by means of our previous formule the value of the maximum stress-difference or greatest strain answering to the velocity which this formula allows in a given cylinder. Then if this value of the stress- difference or greatest strain be within the limits allowed by the elastic theory of rupture, the greatest safe velocity is that assigned by the instability formula, assuming of course that the theory it is based on is satisfactory. Take, for instance, the greatest-strain theory, and suppose our formula gives for the greatest strain s Hs/w*pa?=N, where N is a certain function of n and b/a. Then ascribing to w the value given by (102), and putting «=0/2, we find $= (1°18251)!(20/a)2(a/l)$N. . . « (104) This shows how very rapidly the greatest strain answering to the limiting velocity of the instability theory diminishes as J/a * Ic. pp. 198-200. Solid Cylinders of Elliptic Section. 169 increases. Also the value of 5 in (104), while varying with n, is quite independent of the density or of Young’s modulus. Taking 7='25, we deduce from (104) the following values os —— TasLeE XVIII. Greatest Strain answering to Instability Velocity. bla = 2, 4. 6. 8, 1-0. (//a)*s= "104 428 ‘970 1644 2281 In a material such as steel or good wrought-iron the strain given by this table for a circular cylinder in which /=10a would answer to a longitudinal load of some 3 tons per square inch, and a slight ellipticity in the section reduces this but slightly. ‘Thus in circular, or nearly circular, cylinders whose length is not decidedly greater than 10 times their diameter, it would certainly be only prudent to consider the magnitude of the stress-difference and greatest strain before applying so rapid a rotation as the instability theory allows. In cylinders in which 0/a is as small as *2, instability may be expected to arise under quite a slow rotation, and to attempt to rotate such cylinders with the velocity allowed by the elastic theories would be extremely rash. § 48. As regards cylinders whose length is less than 8 or 10 times their greatest diameter, the results of the instability theory are hardly likely to prove satisfactory ; but there can be little doubt of the generai truth of the conclusion the theory leads to, viz. that the tendency to instability diminishes rapidly as the ratio of length to diameter is reduced. On the other hand, while the application of the results deduced from our elastic equations is not legitimate in short cylinders unless n be zero, or at least very small, there is no reason to suppose that the maximum stress-difference, or the greatest strain, will be either very much greater or very much less than in long cylinders under similar conditions. One of the strongest reasons for this statement is derived from the comparison of Tables I. and II. with Tables XIII. and XIV. According to these tables the limiting speeds allowed by either elastic theory are fairly similar for long cylinders and for thin disks, and it seems most unlikely that any disproportionately large difference will exist in cylinders of intermediate length. The greatest difference between disks and long cylinders occurs 170 Mr. C. Chree on Rotating Elastic in a circular section with 7=°5, when the velocity allowed by the stress-difference theory in a long cylinder is approximately 1°323 times that which it allows in a thin disk of the same radius. For ordinary values of 7 the differences are always much less than this. Suppose for instance »=*25, then, employing the suffixes 1 and 2 as before, and using © for the limiting angular velocity in the thin disk, » for that in the long cylinder of the same cross section and material, we deduce the following results :— TABLE XIX. Ratio of Limiting Velocities in Cylinder and Disk. b/a= 0. 2, “4. 6. 8. 1-0 w,/2,= 1 1019 1-043 1-058 1-078 1-104 w,/Q,= 1008 1007 1:007 1-008 1-013 1-022 § 49. The solutions we have obtained both for thin disks and long cylinders are, unless 7=0, only approximate, and the principle ef statically equivalent surface-forces on which they ure based is one as to whose degree of accuracy opinions may differ. It is thus very desirable to subject our results to some independent test. If we take the value of (—r) in (16), integrate it over a cross section of the disk and divide the integral by zrab, we obtain the mean approach to the central plane of points originally in a plane section at distance z. Representing this mean value by (—8z) we easily find wo { (tay Se { a8 +124 Ho (2 —22) i: _ (105) (B= WE PH). os es | Now for the change in length of a long elliptic cylinder of the same section as the disk we find by (79) nw : (SS)= +h). But the right-hand sides of (106) and (107) are identical, or we have (—dl/l) =(—8l/l)=nw?pe?H, . . (108) Solid Cylinders of Elliptic Section. 74 where « is the radius of gyration of the elliptic cross section about a perpendicular to its plane through its centre. Again, let A=7ab denote the area of the section of an elliptic disk, and let 5A, be the increase produced by rotation in a section at distance z from the central plane. ‘Then denoting the direction cosines of the outwardly directed normal to the rim by ) and p, we have SA, =| (ru +n8)as= ||($ “+ 7, eds . (109) where « and £ are given by (14) and (15). The line-integral is to be taken round the entire rim, the double integral over the whole cross section. Hmploying the double integral we easily deduce Aw’p BAs= SOP §(1—n) (a2 +82) + 49(1 +n) (232%) }; « (110) whence, if SA be the mean value of 5A, between +1, SOA/A=(—n)w*px?/H, . . . . (111) where « has the same meaning as in (108). Employing the values of « and 8 given by (77) and (78), we obtain precisely the same result as (111) for the change of cross section in the long cylinder. Finally, let v=2z7rabl denote the volume of the disk, and dv its increase under rotation, then Sue \\Adrdy de MS. 4 nie (tay the integral extending throughout the entire disk. From this we easily deduce by means of (13) OU U—(t—2ajecpe ler a. |. (PLS) or | Duo wee Py met Oe 128s VELT A) where k=E~{3(1—2n)} is the bulk modulus, and I the moment of inertia of the disk about the axis of rotation. Taking the value of A given by (76), we arrive in the case of the long cylinder at precisely the same formula (114). § 50. Now these coincidences it must be admitted are of a striking character, and if it can be shown that they are not the outcome of mere accident, but the exact results to which the complete theory of rotating elastic solids is bound to’ lead, this would seem strong evidence of the trustworthiness of the present solution as regards both its soundness of basis and its 172 On Rotating Elastic Solid Cylinders. accuracy of detail. That the results are exactly true is easily shown as follows :— Consider a right cylinder of any length or form of cross section, rotating round the axis formed by the centroids of the sections, taking as before the centre of the cylinder for origin and the axis for axis of z. We may write the internal equa- tions in the form :— dex , dey , dew de * dy t de +0 92=0, » i day , dyy , dyz Ee tag ag ter =0 os atate =0, /... ~ a the notation being the same as before. Now multiply (115) by nz and (116) by ny; then from the sum of these two equations subtract (117) multiplied by z, and integrate throughout the volume of the cylinder. Integrating the terms involving the stresses by parts, we find the surface-_ integrals vanish in virtue of the surface-conditions at the free surface of an elastic solid, and thus obtain (\s2—n (ee + Gy) }dadyde + no°p[\\(0" +y%)dedyde=0. Now by the ordinary stress-strain relations, a eer ee dy. ee—n(vat+yy) =H wae {lj - dx dy dz= —nw*px* . 2A 1/E, where A is the area of the cross section, and « its radius of gyration as in (108). But thus d | Gac=p—y=2n, where yz is the value of y at an end of the cylinder, and \\yde dy=6l. A. Thus the formula (108) is exactly true for all forms of cross section in all right cylinders long or short. Measurement of the Internal Resistance of Cells. 178 Again, multiply (115) by x, (116) by y, (117) by z, and add. Then integrating throughout the entire volume, we find —\\\(acoe +yy 422) dx dy dz+ «?\\\p(2? +y?)dx dy=0, the surface-integrals vanishing as before. But by the ordinary stress-strain relations, ava + yy +22 = 3k, where & is the bulk modulus, and thus we get dv =(\fAde dy dz= w*1/3k, where I has the same meaning as in (114). Since Y= 2A, we have §A/A=68e/v—S81/l, = (1—7)w?px?/E. Thus (111) and (114) are also proved to be absolutely true in all right cylinders rotating about their axis of figure. The preceding formule by which the solution has been tested are particular cases of certain much more general results*, to whose discovery the author was led by the recognition of the coincidences pointed out in § 49. XVIII. Note on the Measurement of the Internal Resistance of Cells. By EK. Wyte SmitHf. la order to determine the actions which take place in an accumulator during charge and discharge, it is necessary to know the working electromotive force at the different stages. This might be observed by breaking the circuit ; but immediately on doing this the electromotive force varies at a very rapid rate, so that if only four or five seconds be occupied in taking the measurement an error of 25 per cent. may be made in the difference between the electromotive force and the terminal potential difference. If time-readings be taken after breaking the circuit and a curve drawn con- necting H.M.F. and time, this curve may be produced back in the way described by Prof. Ayrton and others in a paper * Cambridge Philosophical Society’s Transactions, vol. xv. part iii. + Communicated by the Physical Society: read June 24th, 1892. 174 Mr. E. W. Smith on the Measurement at the Institute of Electrical Hngineers. But, as this method has its objections in addition to that of interrupting the circuit, it is very desirable to determine the actual working E.M.F. in some other way. The E.M.F. could be readily obtained from the terminal P.D. and the current if the internal resistance were known. It is for the determination of this latter quantity that I have devised the following modification of Mance’s method. Of course this method is applicable to the measurement of the resistance of other forms of battery besides accumulators. Fig. 1. E ----—-40 i Siete al | In fig. 1 let the cell of E.M.F. e, and internal resistance, d, be the one experimented upon, 7 being the resistance of the external circuit which may contain an H.M.F’. H, for example that of the dynamo used to charge the cell. This circuit is connected at O with an auxiliary circuit, in which the resist- ances m, and 7; are so adjusted that the points A and B are at the same potential, the resistance of the cell of E.M.F., e,, being included in n,. Suppose a current, C, from some ex- ternal source to- pass through both circuits in series. The P.D. between A and B will now be V. Let P.D. between A and O be », and that between O and B be vg, (v,; +v.)=V. We have ease myt+H wu—-ey rs b r ny mM,” therefore _ brC—re—bH aaa sia and al mynC + ghey, ae My, + ny oy b+C—re—bE 9 mnC+ mye, b+r My +n4 of the Internal Resistance of Cells. 175 But when C=0, V=0, | —re—bk mye, * 0= —__ F b+r m+n 5 e br Mm Ny 5 v=C(s7, i aa If we measure the apparent distance R, between A and B by any convenient method, and if C be the current sent through this circuit between A and B by the testing arrange- ment, we get R, equal to Ni ; or M42, . b+r " m+n, Now if we have three circuits (fig. 2) connected together fs Fig. 2. A DOO0000 e 1) (a1) 2 0 “2 7, : n, DOV OQ QY 10990000 My M B Cc at the point O, the cell of H.M.F. equal to e and resistance b being the particular one whose resistance is required, and if the resistances m,, 7m, 7, N2 be so adjusted that the points A, B, and C are at the same potential, the apparent resist- ances R,, R,, R3 between the points A and B, A and ©, and B and C, will have the following values :— br MN, R,=-— ) b+r mtn, br MgNq ase => >; ) b+7r — Mz+MNg MyNy MoNg My +N, Mtn,’ 176 Measurement of the Internal Resistance of Cells. < br sap R, + R,—R; on 2 then the required resistance of the cell, , Say, equals 2, 2 23 bane = ee + &e. r If, as in the case of an accumulator, 2 is small compared * b=2+—: r When an accumulator is discharging, taking b=wz gives usa a value for 6 about 2 per cent. too low. If the P.D. at terminals of the cell under test is greater than the E.M.F. of a single balancing cell (as is the case Fig. 2a. bi during charge) then the circuit must be modified as shown (fig. 2a). Ifa Wheatstone’s bridge be employed to measure R,, Ro, and R;, there will be no necessity to employ any special instrument in the testing for equality of potential of the points A, B, and C. For all that need be done is to remove an infinity plug on the bridge, close the galvanometer circuit, but not that of the testing battery, and adjust m,, m,, n,, ng until the galvanometer remains at zero. Elie: | XIX. On the Instability of Cylindrical Fluid Surfaces. By ULorp Rayueies, Sec. R.S.* i es former papers{ I have investigated the character of the equilibrium of a cylindrical fluid column under the action of capillary force. If the column become varicose with wave-length A, the equilibrium is unstable, provided »% exceed the circumference (27a) of the cylinder; and the degree of instability, as indicated by the value of g in the exponential e% to which the motion is proportional, depends upon the value of A, reaching a maximum when A=4'51 x 2a. In these investigations the external pressure is supposed to be constant ; and this is tantamount to neglecting the inertia of the surrounding fluid. When a column of liquid is surrounded by air, the neglect of the inertia of the latter will be of small importance ; but there are cases where the situation is reversed, and where it is the inertia of the fluid outside rather than of the fluid inside the cylinder which is important. The phenomenon of the disruption of a jet of air delivered under water, easily illustrated by instantaneous photography, suggests the con- sideration of the case where the inside inertia may be neglected; and to this the present paper is specially directed. For the sake of comparison the results of the former problem are also exhibited. Since the fluid is supposed to be inviscid, there is a velocity- potential, proportional to ¢* as well as to e#, and satisfying the usual equation vd ldo 2 or Reh Tea ae io amare oe ar ual) If the fluid under consideration is inside the cylinder, the appropriate solution of (1) is p=Jo (ikr) =I, (kr) 3 . . ees (2) and the final result for q? is oT A=k’a*) tka. Jo’ (tka) 1 5a Jo (tka) _ T (Ka?) ka Ip! (ka) 3 ve ads I, (ka) oe * Communicated by the Author. + (1) “On the Instability of Jets,” Math. Soc. Proc. November 1878. (2) “On the Capillary Phenomena of Jets,’ Proc. Roy. Soc. May 1879 (8) “On the Instability of a Cylinder of Viscous Liquid under Capillary Force,” supra, p. 145. Phil. Mag..S. 5. Vol. 34. No. 207. Aug.1892. N 178 Lord Rayleigh on the Instability in which T represents the oa tension, p the density, and, as usual, Iq (2) =Io(éx) =1+4 5 ly alan » (4) 2 a pt Pe ra mE it) ae av L{e)=a, (iw) = 5 toqgtoge gt . (5) But if the fluid be outside the cylinder, we have to use that solution of (1) for which the motion remains finite whenr=00. This may be _expressed in two ways*. When r is great we have the semi-convergent form =— (ae 1° 1’. 3" es = ¢ 1a ae + 56 12. 82,5? — yeep tf oi and for all values of 7 the fully convergent series 1 h2y? ky p=(yt log kr) Iy(kr) — “Si ge pe a in which y is Buler’s constant, equal to °5772..., and ic 1 S,=1+ 5 T 3 Ser + a, ° e ° ° (8) In this case the solution of the problem becomes T (k’a@?—1) ka $' (ka) = pa ble) d being defined by (7). In (9) p represents the inertia of the external fluid, that of the internal fluid being neglected, while in the cor responding formula (3) p is the inertia of the internal fluid, that of the external fluid being neglected. There would be no difficulty in writing down the analytical solution ap- plicable to the more general case where both densities are regarded as finite. The accompanying Table gives the values of { ne ae to which g in (3) is proportional, and of {te <00t | * See the writings of Sir G. Stokes; or ‘Theory of Sound,’ § 341. of Cylindrical Fluid Surfaces. £79. corresponding in a similar manner to (9). In the second case we have (0) = (y+ log $2) To(a) 6 Z a’ a —pSi— gph — eggs + (12) o$'(v)=1,(2) + (7+ log $2) 2 ,(2) The ae ze — 9 Sim gag 82— ge ga gba ° (13) | or I,(x). | xI,(x).| (10). || —(z). | eo). | (11). 0:0 1:0000. | -0000 0000 0 1:6000 | -0000 Ol 10025 | -0050 "0703 2°4270 "9854 | °6339 0:2 10100 | :0201 "1382 1°7527 ‘9551 | °7233 03 10226 | -0455 ‘2012 13724 O1GIE |) 7795 O-4 1:0404 | -0816 2567 11146 8738 | “8113 0-5 1:0635 | :1289 3015 "9244 8283 | ‘8198 0-6 0-7 0-8 0-9 1-0 10920 | *1882 | -3321 Sy ISA || B22 11264 | :2603 3433 6607 7353 | “7585 11665 | -3463 3269 0654 ‘6894 | °6625 1:2130 | -4474 "2647 ‘4869 6449 | °5017 ousted bs EOE ‘0000 ates viNeh 0000 On account of the factor (1—.*) both (10) and (11) vanish when #=Oand whenz=1. Beyond «=1, (10), (11) become imaginary, indicating stability. It will be seen that when the fluid is internal the instability is a maximum between «="6 and w="7; and when the fluid is external, between «=°4 and e='d. That the maximum instability would correspond to a longer wave-length in the case of the external fluid might have been expected, in view of the greater room available tor the flow. The same consideration also explains the higher maximum attained by (11) than by (10). In order the better to study the region of the maximum, the following additional values have been calculated by the usual bisection formula Gol aati Gon B18). ze 16 x, (10). v. (11). 65 3406 45 ‘8186 70 3488 50 8198 15 ‘3397 55 ‘8147 N 2 180 Mr. W. B. Croft on Breath Figures. The value of x for which (10) is a maximum may now be found from Lagrange’s interpolation formula. It is Tees OSI corresponding to Nae etl 20, ae oe and agreeing with the value formerly obtained by a different procedure. In like manner we get for the value of « giving maximum instability in the case of the external fluid, (ieee Roli and N=OASXBa, 60. Se Some numerical examples applicable to the case of water were given in a former paper. It appeared that for a dia- meter of one millimetre the disturbance of maximum insta- bility is multiplied 1000-fold in about one-fortieth of a second of time. This is for the case of internal fluid. If the fluid were external, the amplification in the same time would be more than one-million-fold. Terling Place, Witham, July 2. XX. Breath Figures. By W. B. Crort, V.A., Winchester College*. Ei bospians years back Prof. Karsten, of Berlin, placed a coin upon glass, and by electrifying it made a latent impres- sion, which revealed itself when breathed upon. About the same time Mr. W. R. (now Sir W. R.) Grove made similar impressions with simple paper devices, and fixed them so as to be always visible. A discussion of Karsten’s results occurs in several places, but I have not been able to find details of his method of performing the experiment. During my attempts to repeat it some effects have appeared which seem to be new and worthy of record. After many trials I found the following method the most successful:—A glass plate, 6 inches square, is put on the table for insulation : in the middle lies a coin with a strip of tinfoil going from it to the edge of the glass: on this coin lies the glass to be impressed, 4 or 5 inches square, and above it a second coin. It is essential to polish the glass scrupulously * Communicated by the Physical Society : read June 24, 1892. Mr. W. B. Croft on Breath Figures. 181 clean and dry with a leather: the coins may be used just as they usually are, or chemically cleansed, it makes no differ- ence. ‘The tin-foil and the upper coin are connected to the poles of a Wimshurst machine which gives 3 or 4 inch sparks. The handle is turned for two minutes, during which one-inch sparks must be kept passing at the poles of the machine. On taking up the glass one can detect no change with the eye or the microscope ; but when either side is breathed upon, a clear frosted picture appears of that side of the coin which had faced it: even a sculptor’s mark beneath the head may be read. For convenience those parts where the breath seems to adhere will be called white, the other parts black. In this experi- ment the more projecting parts of the coin have a black counterpart, but there is a fine gradation of shade to corre- spond with the depth of cutting in the device: the soft undulations of the head and neck are delicately reproduced. he microscope shows that moisture is really deposited over the whole surface, the size of the minute water granulation Increasing as the point of the picture is darker in shade. There seems to be no change produced by the use of coins of different metals. If sparking is allowed across the glass instead of at the poles of the machine, traces of metal are sometimes deposited beyond the disk of the coin, but not within it. Around the disk is a black ring + inch broad: sometimes the milling of the coin causes radial lines across this halo. If carefully protected there appears to be no limit to the permanence of the figures, but commonly they are gradually obscured by the dust gathered up after being often breathed upon : some of the early ones, done more than two years back, are still clear and well defined in the detail. It is possible to efface them with some difficulty by rubbing with a leather whilst the glass is moist. They are best pre- served by laying several together when dry and wrapping them in paper: they are not blurred by this contact. It is a curious fact that certain developments take place after a lapse of some weeks or months. The dark ring around the disk gradually changes into a series of three or four, black and white alternately ; other instances of such a change will be noted below. Let it be noticed that in coin pictures the object is near to, but not in contact with, the glass: for in the best specimens the rim of the coin keeps the inner part clear of the surface. Obviously a small condenser is made by the coins: it is not essential ; at the same time images made by a single coin, put to a single pole, are inferior. 182 Mr. W. B. Croft on Breath Figures. The plan which gives the surest and most beautiful results is to place five or six coins, lying in contact side by side in a cross or star, on either side of the glass: it is not necessary that each coin shouid exactly face one on the other side. There has not appeared any distinction between the figures made by positive and negative electricity. When several coins are side by side, touching one another, there appear in the spaces between them, which are mostly black, well-defined white lines, common tangents to the cir- cular edges of the coins. If these are of equal size the lines are straight; otherwise they are curved, concave towards a smaller coin. They seem to be traces in that plane of the loci of intersection of equipotential surfaces. Similar effects are obtained when coins and glasses are piled up alternately, and the outer coins are put to the poles of the machine. With six glasses and seven coins perfect images have been formed on both sides of each glass. With eight glasses the figures were imperfect ; but there is little doubt this could be improved by continued trials as to the amount of electricity applied. If several glasses are superposed and coins are applied to the outer surfaces, there are only the two images at the outside. After the electrification there is a strong cohesion between the plates. It requires some practice to manage the electrification so as to produce the best results, There are two forms of failure which present interesting features. Sometimes a picture comes out with the outlines dotted instead of being continuous. At other times, if the electrification is carried too far, the impression comes out wholly black ; but on rubbing the olass when dry with a leather the excess is somehow removed, Naturally it is dificult to rab down exactly to the right point, but I have succeeded on several occasions in developing from a blank all the fine detail of elaborate coins. Here, again, we have another instance of the development by lapse of time, for an over-excited piece of glass usually gives a clear picture after an interval of a day or two. Impressions from stereotype plates have been taken of which the greater part is legible: the distinctness usually improves atter a few days. In default of a second plate, a piece of tin-foil about the same size should be put on the opposite side of the glass. Sheet and plate glass of various thicknesses have been used without any noticeable change either in the treatment or the results. I have put an impressed @ olass on la photographic plate in Mr. W. B. Croft on Breath Figures. 183 the dark, but did not get any result on developing: my im- perfect skill in photographic matters leaves this experiment inconclusive. Probably all polished surfaces may be similarly affected: a plate of quartz gives the most perfect images, which retain their freshness longer than those on glass. Mica and gelatine g give poorer results: it is not possible to polish the surface to the necessary point without scratching it. On metal surfaces fairly good impressions can be produced if,as Karsten advises, oiled paper is put between the coin and the surface. In the order of original discovery the figures noticed by Peter Riess should come first. He discusses a breath-track made on glass by a feeble electrical discharge ; as well as two permanent marks, noticed by Httrick, which betray a disin- tegration of the surtace. ‘T have found that when a stronger discharge is employed more complex phenomena of a similar kind are produced. A 6-inch Wimshurst machine is arranged with extra condensers, as if' to pierce a piece of glass. If this is about 4 inches square the spark will generally go round it. For a day, more or less, there is only a bleared watery track, 33; inch wide, when the glass is breathed upon; but after this time others develop themselves within the first, a fine central black line with two white and Lao black on either side, the total breadth being the original 43; inch. These breath-lines do not precisely coincide in ‘position with the permanent scars, but the central one is almost the same as a permanent mark, which the microscope shows to be the surface of glass frac- tured into small squares of considerable regularity : vn either side is a grey-blue line always visible, which Riess ascribes to the separation of the potash. After several months I found two blue lines on either side, which I believe were not visible at first. Of course these blue lines may be seen on most Leyden jars, where they have discharged themselves across the glass. In 1842 Moser, of K6nigsberg, produced figures on polished surfaces by placing bodies with unequal surfaces near to them: the action was ascribed to the power of light, and his results were compared with those of Daguerre. Moser says, ‘‘ We cannot therefore doubt that lght acts uniformly on all bodies, and that, moreover, all bodies will depict themselves on others, and it only depends on extrane- ous circumstances whether or not the images become visible.” In general, the multitude of images would make confusion ; : it can only be freshly polished surfaces that are free to reveal 184. Mr. W. B. Croft on Breath Figures. single definite impressions. However great Moser’s assump- tion may be, there are many achievements of modern photo- graphy that would be as surprising if they were not so familiar. I have not the means of knowing the precise form of Moser’s methods: in the experiments which follow there is usually contact and light pressure, and if they are not wholly analogous, they may for that cause help to generalize the idea: in none of these is electricity applied. A piece of mica is freshly split, and a coin lightly pr essed for 30 seconds on the new surface: a breath-image of the coin is left behind. At the same time it may be noticed that the breath causes abundant irideszence over the surface, whilst it is in a fresh state. It is not clear how the electricity of cleavage can have an active agency in the result. It is familiar to most people that a coin resting for a while on glass will give an outline of the disk, and sometimes faint traces of the inner detail when breathed upon. An examination-paper, printed on one side, is put between two plates of glass and left for ten hours, either in the dark or the daylight: a small weight will keep the paper in continu- ous contact, but this is not necessary if thick glass is used. A breath-impression of the print is made, not only on the glass which lay against the print but also on that which faced the blank side of the paper. Of course the latter reads directiy, and the former inversely; the print was about one year old, and presumably dry. More often both impressions are white, sometimes one or other or both are black. At other times the same one may be part white and part black, and they even change while being examined. During a sharp frost with east winds early in March, 1890, these impressions of all kinds were easy to produce, so as to be quite perfect to the last comma; but in general they are difficult, more especially those from the blank side. At the best period those from the blank side of the paper were white and very strong ; also there were white spots and blotches revealed by the breath. They seemed to correspond with slight variations in the structure of the paper, and suggest an idea that the thickness of the ink or paper makes a minute mechanical indentation on the molecules: the state of these is probably tender and sensitive under certain atmospheric con- ditions, as happens with steel in times of frost. The following experiments easily succeed at any time :— Stars and crosses of paper are placed for a few hours beneath a plate of glass: clear white breath-figures of the device will appear. A piece of paper is folded several times each w ay Mr. W. B. Croft on Breath Figures. 185 into small squares, then spread out and placed under glass : the raised lines of the folds produce white breath-traces, and a letter weight that was above leaves a latent mark of its cireular rim. Some writing is made on paper with ordinary ink and well dried : it will leave a very lasting white breath-image after a few hours’ contact. If, with an ivory point, the writing is traced with slight pressure on glass, a black breath-image is made at once. Of course this reads directly, and the white one inversely. It is convenient to look through the glass from the other side for inverse impressions, so as to make them read direct. Plates of glass lie for a few hours on a table-cover worked with sunflowers in silk: they acquire strong white figures from the silk. In most cases I have warmed the glass, primarily for the sake of cleansing it from moisture ; but I have often gone to a heat beyond what this needs, and think that the sensitive- ness has been increased thereby. It is not not easy to imagine what leads to the distinction between black and white, different substances act variously in this respect. I have placed various threads for a few hours under a piece of glass, which lay on them with light pressure: wool gives black, silk white, cotton black, copper white. A twist of tinsel and wool gives a line dotted white and black ; after a time these traces show signs of developing into mul- tiple lines as in the spark figures. Two cases have been reported to me where blinds with embossed letters have left a latent image on the window near which they lay ; it was revealed in misty weather, and had not been removed by washing. JI have not had a chance to see these for myself, but both my informants were accustomed to scientific observation. A glass which has lain above a picture for some years, but is kept from contact by the mount, will often show on its inner side an outline of the picture, always visible without breath. It seems to bea dust figure easily removed : possibly, heat and light have loosened fine paint particles, and these have been drawn up to the glass by the electricity made in rubbing the outer side to clean it, ‘The picture must have been well framed and sealed up; most commonly dust and damp get in and obscure such a delicate effect. Iam at a loss to imagine simple causes for these varied effects. J am not inclined to think, except in the case of water-colours, which is hardly part of the inquiry, that there is a definite material deposit or chemical change; one 186 Messrs. Edser and Stansfield on a Portable cannot suppose that imperceptible traces of grease, inera- dicable as they may be, would produce complete and delicate outlines. The cleaning off of impressions may at first seem to indicate a deposit ; but this renewal of the surface might rather be like smoothing out an indented tin-foil surface : such a view might explain the case where a blank over- electrified disk is developed into fine detail. The electrified figures seem to point to a bombardment, which produces a molecular change, the intensity of electricity bringing about quickly what may also be done by slow persistent action of mechanical pressure. At present it seems as if most of the phenomena cannot be drawn out from the unknown region of molecular agency. While experimenting I was not within reach of references to former researches, but I have since done my best to find them out, and to indicate all I have learnt in the body of my paper. Poggendorff, vol. lvii. p. 492; translated in Archives de Llectricité, 1842, p. 647. Riess’ Llectrische Hauchfiguren in Repertorium der Physik; translated in Archives de Hlectriaté, 1842, p. 591. Riess’ Die Lehre von der Reibungs LMlectricitdt, vol. i. pp. 221-224. Mascart, Electricité Statique, vol. i. p. 177, Taylor’s Scientific Memoirs, vol. ili. XXI. A Portable Instrument for Measuring Magnetic Fields. With some Observations on the Strength of the Stray Fields of Dynamos. By Kpwiy Evser and HERBERT STANSFIELD*, [Plates III. & IV.] BS instrument was constructed for the purpose of giving direct readings tor the strength of magnetic fields, such as are found in the neighbourhood of Dynamos ; thus avoid- ing the inconveniences attending the Ballistic method. Porta- bility, a considerable range, and a fair degree of accuracy were the qualities sought. The instrument, as now con- structed, whilst satisfying the first of these conditions, may be used to measure any field from 1 line per centimetre up- wards, with an error of only about 2 per cent.; the only accessories required being a dry cell and a resistance-box. * Communicated by the Physical Society: read May 13, 1892. Instrument for Measuring Magnetic Fields. 187 In principle it is the inversion of the D’Arsonval galvano- meter ; the torsion necessary to restore a coil, through which a constant current circulates, to its normal position, parallel to the direction of the lines of force, furnishing readings pro- portional to the field at the position of the coil*. A diagram of the instrument is shown in fig. 1 (Plate ITT.). A B is a small coil, oblong in shape, wound of No. 44 B.W.G. copper wire, and suppported half on each side of a sheet of mica. It is suspended from each end by strips, 10 centim. long, of rolled German-silver wire, each strip having a loop which is passed over a small brass hook riveted on to the mica, and in electrical communication with a terminal of the coil. The strip CA is in electrical connexion with the case of the instrument at C, whilst the strip D B is insulated from it at D by an ebonite plug, attached to the torsion-head H. Inside this latter is a commutator for automatically reversing the current, so as to take readings on each side of the zero. It consists of four semicircular strips of copper, cross connected as shown in fig. 2, a and b being connected to the two battery terminals. Two springs, one soldered to the case, the other insulated from it, but connected to the end of the suspending strip DB, press on these semicircular strips. When the torsion-head is at zero no current passes, the springs then being at ¢ and d respectively (fig. 2). To take a reading the torsion-head is turned, thus sending the current through the coil. Should the latter be deflected in the wrong direction, the current can be reversed by means of the plug contact, P, attached to the battery leads. Readings are taken on each side of zero in order to eliminate any error due to imperfect balancing of the coil; an aluminium pointer G, attached to the coil, being always brought by the torsion to the zero position on a small scale. In order to obtain at once a spring-suspension and an adjustment for the torsion of the strip, a particular form of geometrical slide is used. A A’ (fig. 3) is a thick brass tube, turned at BB’ toa slightly conical plug to fit the tube of * After the completion of this instrument our attention was called to some experiments of Messrs. Siemens and Halske, in which the same principle was used. “In order to measure the intensities of the rotary field, a coil was hung in the centre of the ring, in such a way that its magnetic axis was perpendicular to the measured direction of the resulting magnetic axis of the ring. The coil was then excited by a continuous current, and was kept in position by a spring. The torque uf the spring served as a measure of the intensity.” “Deduction and Experiments on Rotary Currents.” A. du Bois Reymond, ‘ Electrical Review,’ June 5, 1891, vol. xxviil. No. 706. 188 Messrs. Edser and Stansfield on a Portable the instrument (fig. 1). Two grooves (seen in plan at CC’) are made along this tube, a cross head F (fig. 1) on the screw HF fitting into them. This screw is drilled along its whole length to admit a thick wire HI, the latter having a cross head I, also fitting into the longitudinal grooves. These two cross heads are then connected to the two ends of a spiral spring, in such a manner that they are pressed by it against opposite sides of the grooves. ‘The suspending strip being connected to the central wire HI at C, its tension can be increased or diminished by means of the nut K, without altering the position of the coil. Any sudden jerk will also be taken by the spring, thus obviating the risk of stretching the suspending strip. Scratches on the wire HIC near H indicate the tension used. : As a source of current a Hellesen dry cell is used. When joined up through 50 ohms the H.M.F. of the cell is practi- cally constant, whilst its internal resistance is negligible*. The resistance of the instrument having been made up to 50 ohms, it follows that its sensitiveness can be varied by introducing an independent resistance in the circuit. Let C = constant of instrument (7. e. field for 1° of torsion, with no external resistance in circuit) ; n = multiple of 50 ohms in circuit, exclusive of resist- ance of instrument ; ? = mean angular torsion ; then Field in C.G.S. measure = C (n+1) 8. C was determined, and the instrument calibrated, between the coils of a galvanometer of the Gaugain type through which a known current was passed. For an H.M.i. of 1:45 in the dry cell it was found to be +293. The error shown in the calibration was always below 2 per cent. By permission of the Committee of Hxperts, and of a num- ber of the firms exhibiting, a series of measurements were made at the Electrical Exhibition, Crystal Palace. The results obtained are shown in the remaining figures. Fig. 1 (Plate IV.) shows the fields measured at various distances from different dynamos, the distances plotted as abscissee and the fields as ordinates. It is noticeable that machines of the multipolar type show a much steeper curve than other dynamos. ‘This is especially noticeable in the case of the Gulcher Dynamo curve (G). * See Electrotechnische Zeitschrift, August 1, 1890, vol. 11. No. 51. Republished in pamphlet form by Siemens Bros. and Co., Ltd. We have independently verified these results. ee ee et Instrument for Measuring Magnetic Fields. 189 Fig. 4 (Plate III.) shows the fields round Mr. Kapp’s 8-pole machine. They are noticeably small. Fig. 2 (Plate IV.) shows the effect of edges, corners, &c. on the strength of field. On the flat surface of the pole-piece the field was about 600 (varying between 517 and 690), on the edges increased to about 1000, whilst on the corners it reached a strength of over 1100 C.G.NS. lines per centimetre. Fig. 3 (Plate 1V.) shows the deformation of the stray field produced by the armature reactions. The measurements were made on an Elwell-Parker Motor. The strength of field on the trailing edge was about 460, whilst that on the leading edge was about 500. Fig. 6 (Plate III.) shows various measurements made on the Gulcher Dynamo; Fig. 7 (same Plate) gives the field near one of Mr. Kapp’s Dynamos ; Fig. 4 (Plate IV.) shows the fields at two positions of a Laing, Wharton, and Down shielded dynamo. Some curious effects of armature reactions are noticeable on the Thomson-Houston Dynamo (fig. 5, Plate III.). As the bars in this machine act as a yoke, the result is due to combined magnetic leakage and armature reaction. By the kindness of Mr. Harrison we were enabled to make some experiments on a watch, previously unmagnetized, which he lent us. We found that a field of about 10 had no appre- ciable effect on its rate of going, but that after being subjected to a field of about 40 it lost about 8 minutes per day; and even after being demagnetized in an alternating field it still continued very erratic in its actions. Of the dynamos whose fields we have measured, with the exception of the Thomson- Houston, Ship’s Dynamo (Laing, Wharton, and Down), and Mr. Kapp’s large Multipolar, it would not be safe to go nearer than about 2 feet*. Moreover, with a watch with a steel balance-wheel (the one experimented upon had a brass one) even greater precautions might have to be observed. Finally, we wish to record our thanks to Mr. Harrison for allowing us to experiment on his watch; to Mr. Barton for his assistance in constructing the instrument ; and to Messrs. Crompton, Kapp, Laing, Wharton, and Down, the Gulcher Company, the Electric Construction Corporation, and Major- General Festing, for permission to experiment on their various dynamos, and also to publish the results. 3 * One could not safely go within three feet of the Elwell-Parker Continuous Current Transformer. - [190 J XXII. Correlated Averages. By Professor F. Y. Epgeworrs, M.A., D.C.L.* PFAHE “correlation”? t between the members of a system such as the limbs or other measurable attributes of an organism may in general be expressed by the formula T=Je-"* dz, dz Gz, 7c. 5 ~ where 3 R=p,(X,— 2)? + po(&,— 42)? + &e., + 2q12(%1 — 21) (K— #2) +2913 (XK — 21) (Kz — #3) + Ke; Xj, X2, X; Mc. are the average values of the respective organs ; &, Ya, &e. are particular values of the same ; p,, Po. ++ G12) 713 are constants to be obtained from observation ; J is a constant deduced from the condition that the integral of II between extreme limits should be unity. The expression II represents the probability that any particular values of 2, x,, &c. should concur. It enables us to answer the questions: What is the most probable value of one deviation 2, corresponding to assigned values «y', ay/ &e. of the other variables? and What is the d¢spersion of the values of xv, about its mean (the other variables being assigned) ? This general formula for the concurrence of particular values of several organs is deducible from the proposition, proved by theory and observation, that each organ considered by itself assumes different values according to the exponential law of error. Ina subsequent paper I hope to justify this principle ; at present, assuming the propriety of the above- written formula, I propose to show how the constants Pry P2+++ 912) 9i3+-- are calculated. This problem has been Solved by Mr. Galton for the case of two variables. The happy device of measuring each deviation by the corre- sponding quartile taken as unit enables him to express the sought quadratic in terms of a single parameter ; as thus :— 2 : 2 ree 7 Ys Xo ei aE a SE where our p is Mr. Galton’s r, and the x,, x, of our general formula are zero. The parameter is found by observing the * Communicated by the Author. + See Galton, Proc. Roy. Soc. 1888, “ Co-relations and their Measure- ment;” and Weldon, Proc. Roy. Soc. 1892, “Certain Correlated Varia- tions in Crangon vulgaris,” Prof. F. Y. Edgeworth on Correlated Averages. 191 value of x,, say &,, which most frequently corresponds to an assigned value of #, say #3 (or vice versd). From the equa- tion = =() we have dx, ee / RIES AY Ep=ply, p=oirty. It should be observed that for the purpose of this calcula- tion it is not necessary, as Mr. Galton has done, to pick out ihe values of x, corresponding to each value of a. It is sufficient to take the sum, or the mean, of all or some of the positive, exclusive of negative, values [or negative, exclusive of positive | and the sum or mean of the corresponding values, not exclusive of negative [or positive]; and to equate Sfi=pSa,’, p=S&+S2,’; or Baie poet A Ber, Sze. CoM Lt aie od doe: omitting perhaps the extreme observations, with respect to which the law of error is liable to break down. For example, let it be required to find the coefficient of correlation between the stature and left cubit of adult males, from the data utilized by Mr. Galton in Table II. of his paper on “Co-Relations”” (Proc. Roy. Soc. 1888, p. 138) ,without the trouble of the detailed selection, the elaborate “dépowillement,” which the construction of his table requires. Take the cubit as the independent variable, 2, of the last paragraph, and write down all the deviations of the cubit (from its mean value) which are above zero and short of the extremity. There are ninety-three such instances—between 18°5 and 19°5 inches—among the materials which Mr. Galton has employed in constructing his Table II. (op. cit.). The sum of these deviations is 1762-75 inches; the sum of the ninety- three concurrent deviations of stature is 6422 inches. Hach of these has to be divided by the corresponding quartile ; °56 inch in the case of the cubit, 1°75 inch in the case of the statures. Thus Mere lioAdo . 6422" ee OO aR a Which is the value for the coefficient found by Mr. Galton. In working this example I have taken the figures from the compartments of Mr. Galton’s Table II. ; for instance, reckon- ing that there are 55 cubit-deviations “ 18-5 and under 19-0,” 8. 192 _ Prof. F. Y. Edgeworth on 2. €. 18°75 nearly, and similarly 38 cubit-deviations measuring on an average 19°25. But it is not necessary in general (the prevalence of the law of error being presumed) for the pur- pose of calculating the coefficient to work up the materials as in Mr. Galton’s Table II., or Table III. It suffices to take the figures from the original observations pell-mell as they come ; in fact it is better to proceed thus, as we avoid the inaccuracy involved in assuming that all the observations in each small compartment, e. g. between 18°5 inches and 19:0 inches, have the same measure—18°75 inches. I take an example from some of Mr. Galton’s unmanipulated observa- tions which he has kindly submitted to me. The raw material consists of a table like the following, each column giving the measurement of one organ, each row the measurements on one man,—in the random order in which the visitors to the Anthropometric laboratory presented themselves. Here isa specimen :— TABLE i. Statures. Left Cubits. fig Mali san Sore, gate Sa Eb = fe nS Mea Rak oh att OT cE fe a Nay Aik se rege ec 70 At NER eee ie aia 66 Se eek or aes (05D GAS 4 ene To eee ne ee ee 69 ee rer tA Woes Re 68 Re RAO Wei Bas uc = 64°4 18°2 65°95 16°38 6671 Wie | 71°6 19-3 68:4 18:2 65 17°3 63°7 16°8 62°3 16°5 (There are other columns with which we are not at present concerned.) From the right column I pick out all the figures which are above 18:05 (the mean yalue of the cubit) and write them down in a new column, as thus :— ee Correlated Averages. 193 Tass II. Left Cubit Corresponding (above 18:05), Stature. 19°3 (24 Lice BL Vigles) 18°8 715 18:4 70 18°6 70°3 19°6 12:2 184 69 18°3 | 68 19°3 CEG 18:2 68°4 ier 68°3 ibe) Ch | 68-4 aillte) 70:2 19-4 (gal 18°38 68°3 15)280°6 | 15)1051°6 18707 | 70°107 The mean deviation of the cubits, in absolute measure, is 18-707 —18:05 "36 : Be 7 (5 statures 1s 0 sae 5 et2 ; assuming from Mr. Galton’s tables in Proc. Roy. Soc. 1888 that 18°05 and 67:2 inches are the respective mean values of the cubit and stature, 56 and i°75 their respective quartiles. For the coefficient of correlation we have The corresponding mean deviation of the mean deviation of cubit in absolute measure + corresponding mean deviation of stature = 1:17 + 1°66 =77 nearly. The truer value is ‘8, as determined by Mr. Galton from some hundreds of observations ; but we must not expect an accurate result from a few observations here instanced by way of paradigm. In the work of this example I have purposely allowed one mistake not largely affecting the result to stand by way of warning ; the 18 marked with an asterisk in the second table which, being Jelow the mean value of the cubit, has no Phil. Mag. 8. 5. Vol. 34. No. 207. Aug. 1892. O 194 Prof. F. Y. Edgeworth on business in a column furnishing positive values of the devia- tion wv. It is clear that, if we admitted such values indis- criminately, we should obtain an indeterminate result for p. Of course, the more elaborate tabulation employed by Mr. Galton in the paper referred to and elsewhere is useful for the purpose of verifying that the law of error is indeed fulfilled, of testing up to what distance from the mean it holds good, and for other purposes. It may be observed that the calculation of the coefficients of correlation is not affected if we take as the unit, not the quartile, but the same +476..., the modulus of each set of observations. In what follows it is to be understood, when the contrary is not expressed, that the variables 2, 2, &.— representing deviations of different organs or attributes from their mean value—are thus measured. To extend Mr. Galton’s method to the case of three vari- ables x,, #2, v3; first determine the coefficients of correlation for each pair (2%), (#3), (€_ 23) 3 SAY, Pir, Piz, Pu» Lhus the probability of any particular 2, and z, concurring is Ke-*dz,dx2, where K is a properly taken coefficient, and Q a a yy 2 = 2 2 Q° 1 —Pi2 1 — Pie 1 — P12 (See above, p. 190, and Mr. Galton’s work there referred to.) ow the expression above written must be deriv r Now tl on ab tt t be d ed from the sought expression Je~® dw, da, da3, where R= pyey? + potty? + pgt3? + 29498 Lo + 29130123 + 2qo3o's, by integration with respect to «3; between extreme limits + and —«». If we watch the process of integration, we shall find tat, the coefficient of «,? (in the exponent of the integral) is P2pa— G3" Ps : the coefficient of 2, x2 is 9 712 Pa 713 728 :; Ps and the coefficient of x,? is Pi Ps 923 qos” Ps These coefficients are to be equated respectively to 1 — 2 Pip : | ; ' | Correlated Averagese ABE the coefficients of 2,?, w,v,, and «x,*, in the quadratic above written considered as derived directly from the Galtonian coefficient pj, Hmploying the quadratic to determine the most probable w, corresponding to any assigned 2, (cf. above, p- 191) we have | — 913 923 Via Ps Paps— Hs. ° Pig This may be written Sal) | Oe P y) where (),. and P, are certain minors of the determinant A, which forms the discriminant of R ; A=| Py Uy 4% Gia, P2723 Pig. 23, Pa If we carry the integration one step further, integrating Ke-® with respect to x, between extreme limits, we shall find for the probability of any particular 2, the expression ae Le-*"p, day. But by convention the modulus of the probability-curve under which the values of w, range is unity (above, p. 194). There- tore A= P,. By parity of reasoning, A = es = P, 4 pees 13 ie E, eB Ps Therefore Que = App 2 i= Apis ee Apo;. Thus we may write the reciprocal A! of the determinant A :— A'= | A, Apis, Api: Ap», A, Apo: Api, Aps, A But, by a well-known theorem, each first minor of A’=A multiplied by the corresponding constituent of A. Thus ee A’3”) = Ap, 3 ote Piss A? p12) = At. 196 Prof. F. Y. Edgeworth on Whence Pi =A(1 —po3”) 912 = A (P23 Pis— Piz) 3 with corresponding values for the other coefficients. It remains only to determine A. ‘This is effected by the equation A=P,. Employing the values of py, ps Ke., gis, &., which have just been found, Pe | A(1 — 3?) A (p12 P23— pis) | A(P12 Pos— pis) AA — pi”) whence A=A*{(1—p13°) (1 — pis*) — (P12 P23 — Ps) *5 3 and A is determined in terms of the p’s, of which the numerical values are supposed to have been ascertained. The whole system of coefficients is therefore determined numerically, Heample. —Let 21, &2, v3 respectively represent deviations of stature, cubit, and height of knee. The coefficients of cor- relation for each pair are pyp='8, pi3="9, P23=°8 3 as ascer- tained by Mr. Galton (Proc. Roy. Soc. 1888, Co-relations, Table V.). To find the coefficients of the expression Pky? + Poo? + pas? + 2428 Lo + 29130103 + 2403002, which is the exponent of the expression for the probability that any particular values of 2, 2, #3 should concur; or, in other words, the equation of the ellipsoid of equal probability (the final constant being omitted). Here for the reciprocal of the discriminant A we have Alea 4A, ome at 8G) | Be aN A'8 PRR ND, all | W hence 1, 8 pe =: A(1~-°64) =A‘36, Bye Nut 18; 0D | eS —A(-72—-8) =—A-08, tas By parity pp=A°19, ps=A°36, qi3= — A'26, qo3= —A°08. | A:36, —A°:08, —A°:26 | ThusA=| —A08, 19, —A-08 |. | —A:26, —A°08, A°36 | ; ‘ % Correlated Averages. | 197 And to determine A we have Peg "08 | Aveaya> == A*("36 xa 087) —°08, "36 = A?-(620 (a result which may be verified by observing that the other two principal minors afford, as they ought, the same equation for A). Whence A=16'129. Substituting this value in the values of the coefficients above found, we have for the sought expression 5°806x,2 + 30640? + 5°806x3?—2 x 1°2902,.7, —2x4:1942,7,—2 x 1'290a,273. Thus we see that the dispersion of «#, corresponding to ae ee Te | V7 5806 most probable deviation of one organ, e.g. the cubit, corre- sponding to assigned deviations of the two other organs is found by differentiating with respect to , the expression above written and equating to zero. Thus, if x,', v,! be the assigned deviations of stature and height of knee, and & the most probable corresponding deviation of cubit, 3°064€,=1:2902/' _ 1:290(x,' + a!) +1 290.2). e— 3-064. I have verified this deduction by actually observing the value of cubit-deviation which on an average corresponds to assigned values of the other deviations. For this purpose I employ a table such as Table I. at p. 192, with an additional eolumn for heght of knee. This material, consisting of uu- manipulated observations on the stature, cubit, and knee-height of three hundred men, was kindly furnished to me by Mr. Galton. Out of these three hundred triplets I pick out ninety-six which have the stature and knee-height above the respective means of those organs; and proceed in a manner analogous to the simpler calculation discussed at p.193. The columns for stature and cubit will be nearly the same as those of Table II. in that passage; but not quite the same. For negative values are admissible there for the deviation of stature which is there the dependent variable, but not here, where both stature and knee-height are treated as independent variables. On the other hand, negative deviations of cubit are admissible here, but not there. The rationale of this dis- tinction is sufficiently explained by the remark at p. 194 (top). Here is a specimen of the calculation, the columns tor assigned values. of 2, #3; has for modulus = 42(«,! a Bal). 198 Prof. F. Y. Edgeworth on cubit and stature being here, as for the most part they must be, identical with those used for the simpler calculation (at p- 193). Tasue ITI. Height of knee Statures. Corresponding | (above 20°5). (above 67:2). cubit. | ee eS a ee 29-9 72-4 193 | 21-4 719 18-7 | 216 715 188 21 70 18-4 | 21°F 70°3 18-6 22 722 | 196 | 211 69 18-4 20°6 68 183 216 716 193 21-4 68-4 182 20°6 63:3 | 18:1 208 68-4 18°7 21 70-2 18 22:3 711 194 21-4 683 18:8 ———————_ ——— — O —————— 321-4 1051-6 | 280°6 The mean deviations of cubit and of stature are as before (in units of the quartiles) 1:17 and 1°66. The mean deviation on ~ 205) +8 (20°5 in. being the mean and *8 in. the quartile for the height of knee ; Galton, Proce. Roy. Soc. 1888, Co-relations, Table I.)=1:16. According to theory the sum of the mean positive deviations of the stature and knee-height divided into the corresponding mean devia- tion of cubit should yield a quotient ‘42. In fact 1:17+(1°16 + 1°66) =-415. of knee-height is ( This consilience between fact and theory will appear all the more striking when it is mentioned that the anthro- pometrical observations from which Mr. Galton educed his coefficients *8, *9, and *8 were not coincident with those by which I have verified the theory built upon those coefficients. This theory is readily extended to the case of many variables. One way of looking at the whole matter is as follows :— Beginning with the case of three variables—the familiar wv, y, <—let us suppose that w, the probability of particular values of 2, y, < concurring, =Je-®dwx dy dz, where R=ax? + by? + ez* + 2fyz + 2gaz + 2hxey ; and let us inquire how, Correlated Averages. 199 given the coefficients in this expression, we could deduce the coefficient of correlation between any two of the variables, @ and y. The sought coefficient, say 7, is such that to any assigned value of w, e.g. x’, there corresponds ra’ as the most probable value of y: that is to say, the y which in the long run—the long run whose stages are different values of z—most frequently occurs. The direct method is to substitute 2! for 2 in w, integrate between extreme limits with regard to z, differentiate with regard to y,and equate to zero. Geometri- cally we may imagine a surface in the fourth dimension of space at a distance w, from the plane «=w’ ; where w,is what w becomes when a! is substituted for «. The annexed diagram is intended to assist the imagination by representing the curves of probability projected on the plane yz. The point O, is the centre of that system of ellipses which is formed by ~ ~ - Se ae the section of the plane e=w! with the ellipsoid an? + by? + cz? + 2fyz+ 2guz + 2hay =const. The surface w, is evidently symmetrical on either side of planes parallel to the axes y and ¢ through the point Oj. Thus the sought y to which corresponds the greatest number of instances in the long run, the y for which the strip +00 j w, dz is greatest, is the ordinate of O,; which is also —o 200 Prof. F. Y. Edgeworth on the y for which the corresponding w-ordinate is greatest. It is given therefore by the equations dw, oe dw, 0 Pa eae .w, being what w becomes when w! is substituted for #) ; or by the equations dR, dR, fw y a dz ae that is hz! + by +fz=9, ge +fy+cz=0. Whence BT ae ae ia y= Hes v =i"; where H and A are minors of the discriminant we Ekg (Cah ay 2 Cs be Thus 7= . as before. By parity of reasoning the most probable « corresponding to an assigned 7/ is ss . Thus A=B. And by parity the coefficients of correlation between the other pairs xz and yz are respectively proportional to G and F; and C=B=A. From the same point of view it appears that the coefficient we i opr e of x? in the exponent of the expression \\wdy dz is For suppose for » moment that there were added to w a term —2xwa; the centre of the surface being thus tranferred from zero to x. The most probable « is found by solving the system ax + hy +9z=x, ha+by+fz =0, ge+fy +ez =0. A : Solving for 2, we have Avc=Ax, «= —x; and accordingly i (Vw dy dz is of the form Je~ + (ex). Correlated Averages. 201 But, since by convention the modulus of the variable is unity, A=A (=B=C); a result which continues to hold as x is supposed to vanish. This reasoning is quite general; and accordingly, re- placing the symbols x, #,, #3, we may extend to four and higher numbers of variables the solution which has been given above for the case of three variables. In the case of four variables py, p13, &C. p24... being as before the coefficients of correlation for each pair, the reciprocal of the discriminant A’= A, Apiz, Apis, Api. — App, A, Apes, Apos Apis, Apes, A, Apis. Apis, Apos, Apss, A. We have merely to border with a new row and column the determinant used for the case of three variables. But in forming the first minors of the reciprocal in the case of four (and similarly for any even number) it must be observed that, according to the rule of signs*, the minor of the reciprocal which is equated to A’q,, (in general A"~?q,.) is not now A? (p23 P34, P14), but —A* (p23 ps4 P14) 3 OF, as it might be more elegant to write, —A*(px3 p34 pi). By the same rule, Api3= =F A’ (Pos P31 P42); Apis= — A? (p21 32 Pas), Example.—To exhibit the correlation between four quan- tities, of which the first, 2,, is formed by taking at random the sum of ten digits (say from a page of mathematical tables); the second, v,, is formed by adding to the first 2, another random decade; 73=., + another random decade; w,=.#3+ another random decade. The coefficients of correlation between the pairs, which have usually to be ascertained by observation, are here deducible aprion. EE. 9. py= of . For, putting & and &, for the actual deviations of the first and second quantities (the sum of ten and the sum of twenty digits), we may regard &, and (€,—€,) as fluctuating independently, according to a modulus which is the same for both, say ¢; being that which apper- * Cf Salmon’s ‘Higher Algebra,’ chap. 1, arts, 6 and 8. 202 Prof. F. Y. Edgeworth on tains to a sum of ten random digits (itis 165). Accordingly the equiprobable curves are _ £2 S + (G1 &2) = = const. ; or hee 2 261 — 2b iba by Siena = const. Reducing to units of modulus by putting Beate 3 v= Fs ; t= W5e ’ we have $ 20? —2V 22, ty + 2ao3 1 : | whence pj.= a, In like manner it can be shown that 1 il 2 3 Pi3s>= Pitt q? P23= 3» P= - P34= a Thus, for the determinant which is the reciprocal of the discriminant, we have EES NOE ay/}, Aine Ne A A, me A,/2 3 3 A A ee “vi A, a/3 oO , J arp d/o ar/t tes i =) 0) 7s “ dD 9 di aye ape ha | did eae Se HI ne) G2! pol HSI O98 = Correlated Average . 203 Jo VE VS 3 4’ y = 2 A aE. 3 if — aes 1, hy fe pe oe 3 I Jeb fh 2 E | 7 ee Pe i 2 3) 3 : i li y We fe. Ke. . &e. P2=—A The determinant A, thus determined in terms of itself, may be written :— 1 tN ee a i 0 ie i 1 ie het 1 iva Bo) tog See ge 3. at 1 To determine A we have 1 Reet Pe aer °° icin 1 fet Sree | 3 GAs aise a Ta on or Re ae =k moyh A= A’ x. | GY 4 4/3 O16 eo 1 aes fo G 204 On Correlated Averages. When A?=576, A=24. Substituting the value of A in the expressions for the coefficients, we find for the sought quantic R, D2, + dare? + 6x9? + 40 2—2V 2x, ty—2V 6 02 3—2V 12 03.2%. To verify and illustrate this result, let it be employed to answer the question, What is the most probable value of «, corresponding to assigned values of #, v3, #4. Differentiating R with respect to «and equating to zero, we have 2v,— V9 2.=0; importing that to any assigned deviation of the second variable the most probable corresponding deviation of the first variable is “2 times less; each deviation being reckoned in units of its own modulus. The truth of this follows at once from the datum that p= — To give another proof of that proposition :—the mean positive error of the actual deviations of the first decade of digits from its mean value 45 is the modulus for such a decade, say c, +V7. Now let there be added a second random decade. Its deviations being in the long run as often positive as negative, the sum of the two decades is the same as that of the first (the deviations of the first being exclusively . . . . . . c positive). Thus to an actual mean positive deviation ye of , uy the first quantity corresponds the same actual mean deviation —_ of the second quantity. But the modulus of the second Vo quantity (the sum of two decades) is “2 times that of the first. Therefore, in units of modulus, to 2, assigned corre- sponds, as the most probable value of «2, a deviation less in ie . : the ratio a Whence, by the Galtonian theorem, to 2, assigned corresponds as the most probable value of 2, a devia- . . . 1 . . . . tion less in the ratio va Which is the proposition here deduced from our formula. The case of jive variables follows in respect of the signs of the first minors the analogy of the case of three above illus- trated ; the case of six is analogous to that of four. And soon. ; 4 [ 205 ] XXII. On the Spectrum of Liquid Oxygen, and on the Refrac- teve Indices of Liquid Oxygen, Nitrous Oxide, and Ethylene. By Professors Liveine and DEwaAR*. i September 1888 were described in this Magazine (p. 286) the absorption-spectrum of oxygen gas in various states of compression. At lower pressures the absorptions known in the solar spectrum as A and B were most conspicuous, and as the pressure increased the other bands described by Jannsen came out with increasing intensity. The former appear to be due to the molecules of oxygen, and increase in intensity directly with the mass of the oxygen producing them ; while the latter appear to arise from the mutual action of the molecules on one another, since their intensity is dependent on the density as well as the mass of the oxygen producing them. With the small dispersion employed in these observations the absorptions A and B were not resolved into lines as in the solar spectrum, but they had otherwise the same general characters: A consisted of two bands, and both A and B were sharply defined on the more refrangible edge and gradually faded out on the less refrangible side. Considering how much more diffuse the lines forming these groups in the solar spectrum become as the sun gets nearer the horizon (see M*Clean’s photographs), it is probable that, under the cir- cumstances of our experiments, they would not have been resolvable into lines even with higher dispersion. Subsequently, in a paper read at the Royal Society (Proc. Roy. Soe. vol. xlvi. p. 222), we described our observations on the absorption of a thickness of 12 millim. of liquid oxygen. We noticed, as Olszewski had done, the strongest three of the diffuse bands seen in the spectrum of the compressed gas, but could not detect A. The mass of oxygen in 12 millim. of the liquid was not enough to make A visible. We have since made observations with larger quantities of liquid oxygen. For this purpose we have used a glass tube of the form shown at a in the annexed figure, about 3? inch in diameter and 3 inches in length. This tube had the ends blown as flat and clear as possible, and it was enclosed in a box with glass sides bcd, and the air in the box well dried, in order to prevent the deposition of hoar-frost on the tube. The liquid oxygen was poured into the tube at the pressure of the atmo- sphere, and at first, of course, boiled violently, until the tube was reduced to the temperature of boiling oxygen, —181°, * Communicated by the Authors. 206 Professors Liveing and Dewar on the after which the liquid boiled slowly and quietly. Through the length of the tube (that is, a thickness of about 3 inches of liquid oxygen) we viewed the hot pole of an electric are with a spectroscope having two calcite prisms of 30° and one of 60°. As reference-rays we used the red potassium-lines, of which the positions with reference to A and B were well determined by Kirchhoff, and confirmed by our own obser- vations. These lines were easily obtained by dropping a little of a potassium salt into the are. The diffuse bands previously seen both in the gas and liquid were all of exceptional strength, but we did not notice any addition to their number except a faint band just above G. In place of A we observed a band, but different from A in the following remarkable particulars. Instead of having a sharp edge on the more refrangible side and fading gradually towards the less refrangible side, its position appeared to be reversed; the sharp edge was on the less refrangible side, and it faded away gradually on the more refrangible side. More- over its sharp, less refrangible edge did not coincide with the sharp edge of A, but reached very nearly to the more refran- gible of the twe potassium-lines, that is, had a wave-length of nearly 7660. At the same time the band extended beyond the sharp edge of A on the more refrangible side. There was no indication that it was resolvable into lines, or even into two bands. Turning to the place of B in the spectrum we were not able, with that thickness of oxygen, to detect any band in that place. Olszewski (Wied. Ann. xlii. p. 663), with a thickness of 30 millim. of liquid oxygen, observed a somewhat faint band corresponding to A, which with a Rutherford prism was not resolvable into lines, but he has not noticed the reversed position of the band. Using a similar tube for the liquid oxygen, but six inches long, the band at A came out very much stronger and ex- tended much further on the more refrangible diffuse side, but was not conspicuously expanded on the other side, and did not hide the potassium-line. At the same time a fainter band Spectrum of Liquid Oxygen. 207 appeared at the place of B. This had precisely the same ™ character as that at A ; that is, it had its sharp edge on the less refrangible side and faded gradually on the more refran- gible side. Its sharp edge also did not coincide with the sharp edge of B, but reached nearly to the red potassium-line 6913. By estimation, using the potassium-lines for com- parison, we put the wave-length of the less refrangible edge at about 16905, while its diffuse side was visible to about 6870, that is, barely to the place of the strong edge of B. It is plain that these two bands are related to each other in the same way as the solar groups A and B are related, and we cannot avoid the conclusion that they represent A and B, but modified by the change of the absorbent from the gaseous to the liquid state. If, as there is good reason to think, A and B are the ab- sorptions of free molecules of oxygen, the persistence of these absorptions in the liquid seems to show that the molecules in the liquid are the same as in the gas. At the same time the changes they undergo ought to throw some light on the nature of the change in passing from the gaseous to the liquid state, as well as on the causes which produce the sequences of rays which are called channelled spectra. We have noticed, as Olszewski also has noticed, that liquid oxygen is distinctly blue. This is, of course, directly con- nected with its strong absorptions in the orange and yellow. On looking at a mass of liquid oxygen through a direct-vision spectroscope in any direction the scattered light shows the strong bands plainly. Indeed they remain visible when the oxygen has evaporated to the last drop, and they increase in intensity as the liquid is cooled, so that when the pressure on the liquid is reduced and the oxygen cooled by its own evaporation to —200° they become exceedingly black. Olszewski states that this blue colour is not, so far as he could make out, due to ozone, and we are of the same opinion. Ozone dissolves easily in liquid oxygen and imparts to it an indigo-blue colour. Such a solution when poured into a saucer of rock-salt assumes the spheroidal state, and as the oxygen evaporates becomes more concentrated, and finally explodes with considerable violence. In the dilute solution we could not detect any absorptions due to the ozone. We attempted to obtain a larger quantity of liquid ozone, or of a concentrated solution, for the observation of its spectrum. Oxygen oxonized in a tube cooled by solid carbonic acid gave small beautiful cobalt-blue drops of liquid, but when a few of these drops collected together in a tube immersed in liquid oxygen to cool it to —1381°, they exploded and blew the whole 208 Professors Liveing and Dewar on the apparatus to pieces, comminuting the tube to fine powder. This instability of ozone, equally at very low and at high temperatures, is a significant fact in regard to the form of chemical energy. It seems probable that it is connected with the great absorbent power of ozone. The radiant energy absorbed must give rise to molecular movements which may, we conceive, set up disintegration. The determination of the refractive index of liquid oxygen, at its boiling-point of —182° C., presented more difficulty than would have been anticipated. The necessity for enclosing the vessel containing the liquid in an outer case to prevent the deposit of a layer of hoar-frost which would scatter all the rays falling on it, rendered manipulation difficult ; and hollow prisms with cemented sides cracked with the extreme cold. It was only after repeated attempts, involving the expenditure of a whole litre of liquid oxygen on each ex- periment, that we succeeded in getting an approximate measure of ihe refractive index for the D line of sodium. The mean of several observations gave the minimum deviation with a prism of 59° 15! to be 15° 11’ 80”, and thence w=1'2236. The density of liquid oxygen at its boiling-point of —182° C. is 1:124, and this gives for the refraction-constant, ae = 1982; and for the refraction-equivalent 3°182. This corresponds closely with the refraction-equivalent deduced by Landolt from the refractive indices of a number of organic compounds. Also it differs little from the refraction-equivalent for gaseous oxygen, which is 3°0316. This is quite consistent with the supposition that the molecules of oxygen in the liquid state are the same as in the gaseous. eae (2+ 2)d we find the value of it for liquid oxygen to be *1265, and the corresponding refraction-equivalent 2024. These are exactly the means of the values found by Mascart and Lorenz for gaseous oxygen. The inherent difficulties of manipulation, and the fact that the sides of the hollow prism invariably became coated with a solid deposit, perhaps solid nitrogen, which obscured the image of the source of light, have hitherto prevented our determining the refractive indices for rays other than D*. If we take the formula tor the retraction-constant * This will be prosecuted further, however. The refractive index of oxygen has an important bearing on the electro-magnetic theory of light, considering that we are dealing with a magnetic liquid. The polarizing Spectrum of Liquid Oxygen. 209 The determination of the refractive indices for liquid nitrous oxide did not present so great difficulties. The minimum deviations for the rays C, D, F, G, and for the lithium ray % 6705°5, and the indium ray % 4509°6, were found to be, respectively, 22° 53’, 23°, 23° 18’, 23° 33’, 22°52’, and 23° 28’, The corresponding values for » are 1°329, 1°3305, 1°3345, 1°3378, 1°3257, and 1°3368. The specific gravity of liquid nitrous oxide at its boiling- point of —90° C. was found, by weighing 100 cubic centim. of the liquid, to be 1:255. This gives, for the D ray, = =(°2634 and for the mole- cular refraction 11:587. Or, if we take the other formula, jl (uw? + 2)d 7163. Subtracting the refraction-equivalent for oxygen we get for the molecular refraction of nitrogen 8°405 or 5°139 according to the formula used. Mascart’s determination of the index of refraction of gaseous nitrous oxide for the D ray was 1:000516 and the corresponding molecular refraction 11°531, or 7°69, according to the formula used, and in this case the older formula for the refraction-equivalent satisfies the condition of continuity between the gaseous and liquid states better than the newer. It was more difficult to obtain the refractive indices for liquid ethylene on account of its irregular boiling. Liquid oxygen and nitrous oxide boil steadily, but ethylene in sudden bursts of large volumes of vapour. The minimum deviation for the D ray was found to be 25° 29’, approximately. This gives w= 1°3632, and, since the density of the liquid at its as ; SAV he Sas ous. Ew?—13 boiling-point of — 100°C. is 0°38, 7 = 0°627 and (42d = 0°384. The corresponding numbers for gaseous ethylene, according to Mascart, are 0°578 and 0°385. The agreement for the second formula is close, but we doubt if much stress can be laid on this, inasmuch as we know that the liquid ethylene contained a small quantity of ether. =°'163 and the corresponding molecular refraction angle corresponding te the index of refraction found above for liquid oxygen is 50° 45’, and one of us has found that when liquid oxygen is cooled to —200° by its own evaporation at reduced pressure so as to present a steady surface, and the image of a candle is viewed by reflexion at that surface, the light is very completely polarized when the incidence is at that angle. Phil. Mag. 8. 5. Vol. 34. No. 207. Aug. 1892. P f 210 4 XXIV. Notices respecting New Books. Systematic Mineralogy, based on a Natural Classification. By T.S. Hunt, .A., ZL.D. 8vo. Pages i-xvil, and 1-391. Scientific Publishing Co.: New York, 1891. NDEAVOURING to determine a Natural System in Minera- logy, the author of this work carefully studied the minera- logical systems proposed by Berzelius (1815-24), Werner (1817), Mohs (1822-24), Shepard (1835-44), Jameson (1820), Breithaup (1836-46), Weisbach (1875-84), Rammelsberg (1841), Dana (1837-82), Naumann (1849 ?-85), Groth (1882). The chief features of these systems are carefully defined, and the author observes that at present “the results of the chemical analysis of species are generally considered as of paramount significance ; while hardness, specific gravity, crystalline form, and optical characters assume a secondary value in classification, and are regarded as important chiefly in connexion with determinative mineralogy. The conception of a true natural method, which, although but partially understood, was at the basis of the system of Mohs, has been generally lost sight of; the order which the naturalist finds in the organic is no longer apparent in the inorganic world, as presented in modern mineralogical text-books; and this state of things has contributed not a little to the comparative neglect into which systematic mineralogy has of late years fallen.” et es aren “‘ There exist, in fact, inherent and necessary relations between the physical characters and the chemical constitution of inorganic bodies, which serve to unite and reconcile the natural- historical and the chemical methods in mineralogy. A physico- chemical study of the mineral kingdom, having in view these relations, will enable us, while remaining faithful to the traditions of Werner and of Mohs, to frame a classification which it is believed will merit the title of a Natural System in Mineralogy.” The chemical and natural-history methods being both defective, Dr. T. S. Hunt has endeavoured to reconcile the two, observing @ strict conformity to chemical principles, and at the same time retaining all that is valuable in the natural-history method. ‘¢ Chemical and physical characters (he observes) are really depen- dent on each other, and present two aspects of the same problem, which can never be solved but by the consideration of both. Differences in specific gravity between two or more solid species do not become intelligible until we know the equivalent weights of these species as deduced from chemical investigation. It is not the specific gravity itself, but the relation of this to the equivalent weight which must be taken into account.” Hence in the intro- ductory chapters the author has explained ‘‘in a somewhat elementary manner such principles of physics and chemistry as seem necessary for a clear understanding of this relation. The nature of chemical change, the connexions between gases, liquids, and solids, the periodic law, the principles of progressive series ee _ ee ee ee ee oe eee a ee Notices respecting New Books. 211 and of chemical homology, of solution, and those of polymerization or rather of intrinsic condensation, and the relation of condensa- tion to hardness and to chemical indifference, have therefore been dwelt upon at some length. The problem of determining the co-efficient of condensation in liquid and solid species has moreover been discussed in a separate chapter; while a new and simplified chemical notation, which is believed to be advantageous for the purposes of the mineralogist, has been set forth. The subject of mineral constitution, and the theoretical questions therein involved, are treated in Chapter X.” A new mineralogical classification is propounded in Chapter XL., the mineral kingdom being divided on chemical grounds into four classes:—I. Metallacee; IL. Halidacee; IIl. Oxydacew; IV. Pyricaustacee. The Metallaceee comprise the non-oxydized metalline minerals, including the metals, their alloys and all their compounds with sulphur, selenium, tellurium, phosphorus, arsenic, antimony, and bismuth. The resemblance, it is noted, between the typical and malleable metals, such as gold, silver, lead, copper, nickel, and iron, and the elementary metalline species, tellurium, arsenic, antimony, and bismuth, are such, Dr. Hunt states, that the com- pounds of these with the metals above named cannot well be separated from alloys. In this Class the author enumerates 22 genera with 167 species with latin names either indicative of their composition, or having reference to their common accepted names. Subclass A consists of the Metallometallata. Order I. Metal- linea; Genus 1. Metallum, native metals and alloys; 2. Metallinum, semimetals. Order II. Galeninea; Genus 1. Thiogalenites, metallic sulphids; 2. Salenogalenites, metallic selenids; 3. Yellurogalenites, metallic tellurids. Order III. Diaphorinea; Genus 1. Arsenodea- phorites, metallic sulpharsenids; 2. Stibiodiaphorites, metallic sulphantimonids ; 3. Bismutodiaphorites, metallic sulphobismu- thids. Order IV. Pyritinea; Genus 1. Pyrites, metallic sulphids (of iron, cobalt, nickel, copper, chromium, and ruthenium, having a hardness above 5 of Mohs’ scale); 2. Pyritinus, metallic sul- phids (Giron, nickel, copper, and tin with hardness from 3°5 to 5). Order V. Chloanthinea; Genus 1. Phosphocloanthites, a metallic phosphid; 2. 0. If this condition is not satisfied, the equilibrium will be unstable, and a disturbance will cause the flue to collapse. If the flue be regarded as of finite Jength J, a correction is required whose importance depends upon the magnitude ratio of a to l. Taking the simplest case of an indefinitely long tube, the problem of determining F may be attacked in two different ways. In the first place, let the flue be slightly deformed, and let the period of the small oscillations be found; then * Lord Rayleigh, Proc. Lond. Math. Soe. vol. xx. p. 225. Basset, tbid. vol. xxi. pp. 33, 53; Phil. Trans, 1890, p.-433. + Proc, Lond. Math. Soc. vol. xx. p. 373. { Phil. Trans. 1890, p. 487. ~ R 2 224 Mr. A. B. Basset on the Difficulties of Constructing the condition of stability requires, that the periods should be real quantities. In the second place, let the potential energy in the deformed state be found ; then the condition of stability requires, that the potential energy of the flue when in equilibrium should be a minimum. Hither of these methods will determine the mathematical form of the function F. In order to apply the first method, it might be supposed that we might start with the general equations of an elastic solid, and calculate the values w,, v, of the radial and tan- gential displacements when the flue is in equilibrium. Having done this, let w)+w 1, v)+v, be the complete values of the displacements when the flue is performing small oscillations. The values of all these quantities can be obtained from the equations of motion; and by means of the boundary con- ditions, some of the arbitrary constants which occur in the solution can be determined, and the rest eliminated ; and the - resulting equation will give the period. But in attempting to apply this method, it will be found that the pressures [1,, II, disappear; consequently the periods of vibration are the same as the free periods, when the flue is not subjected to any surface-pressure. This result is ob- viously wrong. The reason is, that the general equations of an elastic solid in their ordinary form are linear. In order to solve the problem by the first method, it would be necessary to take account of certain quadratic terms, in which quantities upon which the motion depends enter into com- bination with the surface-pressures. The problem may be illustrated by considering the pro- pagation of waves in a liquid. When the liquid is initially at rest, all quadratic terms are to be neglected ; but when the liquid possesses an independent motion, all quadratic terms which depend upon this independent motion must be retained. The application of the second method involves a knowledge of the expression for the potential energy due to deformation. The form of this function is known when the surfaces of the flue are free from external pressures*; but this expression, as we have already pointed out, is inapplicable when there are external pressures. 6. In order to understand more clearly the necessity of retaining the above-mentioned quadratic terms when applying the first method, let us consider the stability of a rod of length / which is subjected to a tension T,. Then employing the notation and method explained in my ‘ Hlementary * Phil. Trans. 1890, p. 443, equation (24). a Theory of the Collapse of Bovler-flves. 225 Treatise on Hydrodynamics and Sound,’ ch. viii. p. 147, we have ACB J Bag. qu’ whence ‘ = =(+ t+ a) ae = (i+ soa) Accordingly ve @tTile) (Lt hie) _ SS aia | = as ale Cae PP whence I REY Gi i ae | G=eo(9+ tee « Sea ae (1) When no forces other than stresses applied to the ends of the rod act, the equations of motion—§ 135, equations (7)—are ar N oe oe | dN on re ae = —odol, Pi ° e . : < (2) a =oKrwp. | When the rod is naturally straight and is free from stress, T, N,and G are zero in equilibrium ; consequently, when the rod is performing small oscillations, these quantities will be linear functions of the displacements and their differential coefficients ; accordingly N/p', T/p’ are small quantities of the second order, which are to be neglected. But when the rod is subjected to a tension 'T’,, the value of T is equal to T,+ T’, where I’ depends upon the motion; accordingly the quadratic term T/p! cannot be neglected, but must be replaced by T,/p’. Writing —dx«=ds, and recollecting that I dw. dv Gan eee ae (1) becomes eo 226 Mr. A. B. Basset on the Difficulties of Constructing whence the last two of (2) become aN dw f - a5 + Ty 73 = 70, T,\ av dv 2 ob hele Bea pe Ae aw ree K o(g+ a) -+-N=cK oF? whence, eliminating N, we have T)\dae ids dw d‘v 2 A) ee ell See ep é - (a+ \om oda to dio" dpde 7) which is a well-known result. Let J be the length of the rod; then we may take t+emz/lt pec teeny. where m is an integer. Substituting in (3), we obtain >a ea} m? 2 fh Km an At , o(1+ Era 1 (9+ Z) toh. (4) @ When T, is positive, so that the force is a tension, p is always real; but when T; is negative and equal to —P, so that the force is a pressure or thrust, p will be imaginary unless K’wg © SEE Enea The least value of the expression on the right-hand side occurs when m=1; the condition of stability is therefore gen’ /l? P le re2/* e . ° . e . (5) If the length of the rod is large in comparison with the radius of its cross section, the term 7«?//? in the denomi- nator may be neglected, and we obtain Huler’s law of thrust. 7. We must now return to the subject of boiler-flues. This problem has recently been discussed by Mr. Bryan by means of the energy method*; but his work, as distinguished from his result, is vitiated by the assumption that the potential energy per unit of length of the cross section due to bending is equal to Amnh? /d’w : LO 5 meieege ee Eo aEee 3R(Sp—")?, or Bai (m+n) (ap +w) ) which, as we have already pointed out, is only true when the surfaces of the flue are free from external pressure. * Proc. Camb. Phil. Soe. vol. vi. p. 287. a Theory of the Collapse of Boiler-flues. 227 Mr. Bryan has also attempted to introduce the external pressures into his expression for the total potential energy due to strain, by taking into account certain terms depending upon the extension of the middle surtace; but this does not help the matter, for if the pressures II,, II, were suitably chosen, it would be possible to prevent the middle surface from experiencing any extension, but the expression for the potential energy would still contain terms depending on IT, and II,. 8. ‘he necessity for introducing the external pressures may also be seen as follows. Let the flue be regarded as infinitely long, so that the problem is one of two dimensions. Let o be its density, 2h its thickness. ‘hen the equations of motion are* EN 71 ds p —= f4V, | an, -'T ji c: —— +1L(1— =) Ha (1+ =) =pib, > . (6) iG | where p is the radius of the deformed middle surface, and = 2heo. Now if we knew the correct expression for the total energy of the system and were to apply the Principle of Virtual Work, we should obtain (i) two equations connecting the dis- placements v andw; (ii) the values of T, N, and G in terms of vand w. And if we were to substitute these values of T, N, and G in equations (6), the last one would reduce to an identity, whilst the first two would reproduce the equations of motion in terms of v and w, which we had already obtained. Since the second of (6) contains the pressures [T,, II,, it follows that the expression for the energy must also contain these quantities. From (6) it appears that the problem of the collapse of a boiler-flue can be solved, provided we can obtain the value of the flexural couple G, and provided also we assume that the extensibility of the middle surface may be neglected. 9. ‘The expression for this couple in the case of a strained cylindrical shell is given in my paper in the ‘ Philosophical * If the extensibility of the middle surface istaken mto account, the right-hand sides will contain certain additional terms—see Phil. Trans. 1890, p. 489, equations (11); but the existence of such terms does not affect my argument. 228 Mr. A. B. Basset on the Difficulties of Constructing Transactions’ ; but the surfaces of the shell are expressly assumed to be free from external pressures and tangential stresses, although the extension of the middle surface is taken into account. As the analysis in the general case of three- dimensional motion is somewhat long and complicated, it will be desirable to reproduce in a concise form so much of the original work as is necessary to obtain the value of the flexural couple when the motion is supposed to be in two dimensions. We shall neglect the extension of the middle surface, but shall suppose that the shell is subjected to external pressure. Let v', w' denote the tangential and normal displacements of any point of the substance of the shell; let o,/, o3' be the extensions in these directions, and a’ the shearing stress. Also let the unaccented letters denote the values of the quantities at the middle surface, where r=a; and let the values at the middle surface of the differential coefficients of the various quantities with respect to r be distinguished by brackets. Let r=a+h’, and let 2A be the thickness of the shell. Then, employing Thomson and Tait’s notation for stresses and elastic constants, we have 1 /de oo rs + w'), ¥ ue ee | aa ae oT » Ot Rar he ada, Tide ca ghee aera Now R= (m+n)o'3+(m—n)o', =(m+n)o3+ £ (m+m)(F) a (m—n)(F2) bars ahiae (8) for since the middle surface is supposed to be inextensible, go=90, Since R’ is some function of h and A!, it follows from Taylor’s theorem that R=A+ A,h'+ A,h!, + cnatic ‘ ° * (9) where A, A, ... are unknown functions of the displacements and the thickness. Accordingly, equating coefficients and putting Be (m—n)/(m+a),7. 52-1, eo a Theory of the Collapse of Boiler-flues. 229 we get o,=A/(m+n (=) = A, = 5 (=) ° ° ° (11) dr m+n dr] From (7) we get (= Gea) *} WD, a | ae Saat + oo -i et Seah oho? If G be the flexural couple about a generator, measured in the direction in which the curvature diminishes, G= =p Qhidh'. —h! But Q! = (m+n)o',+(m—n)o! =(m—n)o3 + { (m-+n)( (=) + ci (Ft h! Amn Werk GV, =HA+|/EA,+ (m+nja aaa + — rT i=) HEP accordingly, mee 2 3 _8mnh*® (do | A i Ld?" =~ aoa 3(m+n)a\dd gr ae sag) Cg) Now wh may be neglected when the surfaces of the shell are free from tangential stresses ; and A,h® and Ah? may be neglected when the surfaces are free from normal stresses or pressures. Under these circumstances we obtain 8mnhi (d?w US eek SE Fer 8) ue (14) which is the well known result in this case, and shows that the flexural couple is proportional to the change of curvature. But when the surfaces of the shell are subjected to external and internal pressures II,, II,, the terms involving A, A, cannot be assumed to be negligible when multiplied by jee The quantity A is the value of R at the middle surface of the shell ; and since R varies from —II, to —II,, as we pass from the interior to the exterior of the shell, it is evident that A must have -some value intermediate ‘between these two quantities. It therefore follows that under these cir- cumstances the expression (14) is erroneous; the correct 230. Mr. A. B. Basset on the Difficulties of Constructing expression is given by (13), which contains the three unknown quantities A, A,, and a. 10. Although the existence of external pressures renders it inadmissible to treat Ras zero, yet the stresses S and T still vanish at the surfaces of the shell. I have shown in my papers that the terms of lowest order which these stresses contain are quadratic functions of h, fh’; and the arguments by which this result was obtained are unaffected by the existence of external pressures. Since the problem we are discussing is one of two dimensions, T will be accurately zero ; also since h N=| Ndi’, Sah and §! is a quadratic function of h and h!, it follows that N must be proportional to h*, whence by the third uh (6), dG/ds ts proportional to h’. 11. We have now to consider how A and A, depend upon h. Since the quantity A is the value of R at the middle surface of the shell, it cannot contain any negative powers of h, otherwise R would increase indefinitely as the thickness of the shell diminishes indefinitely, which is impossible, since R must lie between —II, and —II,. It is, however, possible for A to contain a term independent of the displacements ; but such a term need not be considered, inasmuch as it disappears on differentiation. The quantity A, might, however, contain a term involving h-; for the value of R at a point near the middle surface is A+ Ahi + = eae When the thickness of the shell diminishes indefinitely, i! approaches the limit , but in such a manner that A! is always less than h ; if themerone A, contained a term involving ha the quantity A,h'’ would not become infinite in the limit. On the other hondl if A, did contain such a term it would necessarily be independent of the displacements, inasmuch as the term in question must disappear on differentiation, otherwise dG/ds would not be proportional to h’. It therefore follows that the only portions of A, A, which it is necessary to consider are those portions which are independent of h. [regret to say that I have not succeeded in discovering any method by which their values may be rigorously deduced ; but although as a rule I distrust argu- ments founded on general reasoning, yet the difficulty of obtaining the values of these quantities by a rigorous mathe- matical investigation having hitherto proved insuperable, a Theory of the Collapse of Boiler-flues. 231 renders it necessary in the present case to have recourse to such arguments. 12. When a flexural couple is applied to a thin shell, it produces extension of the middle surface and change of curva- ture. Under these circumstances the most natural hypothesis to make is, that this couple is a linear function of the exten- sions and the changes of curvature, together with a constant term depending on the external pressures. When there are no external pressures this hypothesis may be proved to be true by the method of my former paper ; but when the shell is subjected to such pressures, every circle whose plane is perpendicular to the axis of the cylinder is elongated or ‘contracted, independently of the strain produced by the couple. Accordingly we should anticipate that the coefticients of the extensions and the changes of curvature would contain terms depending on the pressures; and we shall therefore assume that qa 8h mn \(Se+ | a el vk ay de? w) Ah? ( mn(m—n) a & eae Gr ea TOK ae +w) +y,. (15) where a, 8, y are quantities which depend on the pressures. When there are no external pressures, a, 8, y are each zero, and the above expression for G reduces to that given in my former paper—see Phil. Trans. 1890, p. 441, equations (14) and (16). It will be noticed that this hypothesis is consistent with (13), as it involves nothing more than the assumption of certain definite values for the undetermined quantities A, Aj. The preceding argument though plausible is {not fentirely free from danger. When a thin shell is free from surface forces, it might be argued that the effect of any stress is to produce extension, change of curvature, and torsion ; and that consequently the expression for the potential energy must bea quadratic function of the quantities by which these three states are specified. If, however, the expressions for the potential energy of a thin cylindrical or spherical shell be examined— Phil. Trans. 1890, p. 443, equation (24), and p. 467, equation (16)—it will be found that they contain certain terms depending on the differential coefficients of the extensions. 13. One further difficulty still remains. The equations (6) for determining the small oscillations of the flue involve the two displacements v and w, and the stresses T, N, G. By means of the above value of G these stresses can be eliminated, aT a ie —— Ga wa20. i 232 Mr, A. B. Basset on the Difficulties of Constructing and a single equation will remain connecting v and w. This is insufficient to determine the period, and the only course open to us is to assume that the middle surface is inextensible, which gives a further relation between v and w, and furnishes a sufficient number of equations. I am inclined to think that the neglect of the extension of the middle surface is not likely to lead to any serious error in the final result ; but at the same time the solution cannot be regarded as perfectly rigorous, for when the surfaces are subjected to unzform pressure "(which is the case we are considering), an extension _ of the middle surface must necessarily take place unless the pressures are specially adjusted. 14. We are now in a position to obtain a solution of the problem of the collapse of a boiler flue, which though imperfect, is probably substantially correct. When the natural form of the cylinder is circular, N is zero in equilibrium ; accordingly we may write N/p=N/a. Also if the extension of the middle surface be neglected, ds=add¢, #) r= a(S ) ao ede 8) Mae dv — +w=0. apt" J The equations of motion (6) now become +N=pav, 1 214 (il, =1h)o ME. 4 Te (SF +10)= par; SAD ag? J From the first two we get aN dg? +N+(I1,— Th) 5 a 4 tw) = =n0( g ®). (18) Let D=d/d?; then, since the extension of the middle surface is neglected, we shall have G=1(D?41)w,. . >... ae 8h? ( mn =aalara ts) hts bideb ier a jus (20) where —s a Theory of the Collapse of Boiler-flues. 233 Accordingly from (18), (19), (20), and the last of (16) and (17) we get {Ta-1(D? +1) + (1; —T,)} (D? +1) D°w + wa(D?—1) w=. To solve this assume Mi Oe, CTE LD. where s=2.3.4..., then ; {Ta—!(s?—1) — (II, —II,) }(s?—1)s’ =pa(s?+1)p?; (21) and therefore p will be real provided II,—II, < I(s?—1)/a. The least value of the right-hand side occurs when s=2, in which case Ii,—II, <3] /a 8h? ( mn < 5 ( +2), . ° : ° . (22) This is the condition for the stability of the flue. In the particular case considered by Mr. Bryan, I,=0, and since it is practically certain that « is a linear function of II,, we may write «=£II,, where & is some constant which is independent of h. Whence (22) becomes 3 3 Il, (i- “) 2 Shimn ae a(m+n) Since powers of 4 higher than the cube are to be neglected, the condition of stability becomes 8h3mn Hs a?(m+n)’ which is Mr. Bryan’s result. Although I am disposed to think that this result is rigorously true as a first, and probably for practical purposes as a sufficient approximation, yet a rigorous mathematical theory requires a knowledge of the correct expression for the energy. An inductive process, by means of which a function is shown to satisfy all the requisite conditions, would prob- ably be the simplest method of discovering its value. August 9, 1892. ‘i XXVII. On the Relation of the Dimensions of Physical Quantities to Directions in Space*. By W. WittiaMs, Assistant in the Physical Laboratory, Royal College of Sciencet. {> a paper read before the Physical Society, Nov. 24, 1888, Prof. Riicker showed that in the dimensional formule of electromagnetic quanties w and k the two specific capacities of the medium are generally omitted, or rather their dimen- sions are suppressed, and that the consequence of this sup- pression is to give rise to two artificial systems of dimensions, the electrostatic and the electromagnetic. That these systems are artificial appears when we consider that each, apparently, expresses the absolute dimensions of the different quantities, that is their dimensions only in terms of L, M, and T; whereas we should expect that the absolute dimensions of a physical quantity could be expressed only in one way. Thus, from the mechanical force between two poles, we get 1 /m?\ Le eS f= ("), ata USSU A/ Hy and this, being a qualitative as well as a quantitative relation, involves the dimensional identity of the two sides. If now pis put =1, we either zgnore its dimensions or assume that it has none, being but a mere number. In the former case the dimensions deduced for m under such circumstances must be admittedly artificial, since if the suppressed dimensions of p were inserted, those of m would be different, and would involve a different physical interpretation. In the latter case the dimensions deduced for m must be its absolute dimensions and must therefore involve its physical interpretation. But if we start with the relation jet, oe where 7 is now the force between two charges gq, and if we treat k as a pure number, we obtain for m by means of the dimensions of g deduced by this relation absolute dimensions again, but different now to what we had before. In this way we gettwo different absolute dimensions for the same physical quantity,—each of which involves a different physical inter- pretation. If, however, w and k& be not mere numbers, but concretes expressing physical properties of the medium, as is * See Note at the end of the Paper. + Read before the Physical Society, June 24, 1892, and communicated by A. W. Riicker, F.R.S. On the Dimensions of Physical Quantities. 235 indeed proved by the relation [wk] = (L°T")*, then the fact that the dimensions of electrical and magnetic quantities are different in the two systems arises only from our ignorance of the dimensions of w and k, which must be such as to bring the two systems into accord when ultimately expressed in terms of L, M, and T. Prof. Riicker therefore suggested that w and & should be admitted into the formule along with L, M, and T, and that, since nothing is definitely known as to their physical nature, they should be regarded as secondary fundamental units, additional to L, M, and T, but which must be ultimately expressed in terms of them. There is thus, in each formula, an indication that the irrational and unsuggestive dimensions are not absolute as they stand; each formula is made to express a physical reality, and each con- tains in the proper form the factor necessary to render the formula absolute. This system has been adopted, among others, by Prof. Gray in his smaller edition of ‘ Absolute Measurements in Hlectricity and Magnetism,’ and by Prof. Hverett in his last edition of ‘ Units and Physical Constants.’ Prof. Fitzgerald has pointed out (Phil. Mag. April 1889) that if a system be taken ‘‘in which the dimensions of wu and k are the same, and of the dimensions of a slowness, that is the inverse of a velocity (L~'T), then the two systems become identical as regards dimensions, and differ only by a numerical coefficient just as centimetres and kilometres do.” But although pw and & are quantities of the same order, both being capacities, their dimensions need not necessarily be the same, unless electrification and magnetization be phenomena of the same order as well. If, however, these be different orders— as they almost certainly are—the one possibly a strain, the other a vortex motion, then itis unlikely that £ and yp, bearing, as each does, similar relations to two dissimilar phenomena, should have identical dimensions. In one of the Appendices to ‘ Modern Views of Electricity ’ Dr. Lodge develops a system in which mw and &! have re- spectively the dimensions of density and rigidity. It is then found that the dimensions of ail the other quantities become unique, and capable of purely dynamical interpretations. Thus, magnetic moment becomes linear momentum; magnetic induction, linear momentum per unit volume; magnetic force, velocity; electrical force, pressure ; current, displacement x velocity ; self-induction, “inertia per unit area,” &. Ina * See also “ On the Dimensions of a Magnetic Pole in the Electrostatic System of Units,”’ Dr. Lodge, Phil. Mag. Nov. 1882. 236 =©Mr. W. Williams on the Relation of Dimensions similar manner other such systems may be developed. Thus, take the equation where F is an impressed force, M the mass of a moving body, v its velocity, and & a coefficient of resistance. Compare this with C B=L on +R, where E is the voltage of a closed circuit, L its self-induc- tion, C the current, and R the resistance. Let us identify EK and F dimensionally, and work out the analogy in detail. To do this, we have only to equate the dimensions of E and F, and find what value of w satisfies the relation. Then sub- stituting this value of w in the formule of the other quantities, we are able to complete a connected dynamical analogy of electromagnetism by starting with voltage as a force. In this case, we find that electrification is a displacement ; current, a velocity ; electrical potential, force ; quantity of magnetism, linear momentum; self-induction, inertia, &c. In a precisely similar manner, by starting with the equation 1 @ Gal oe VR where G is a couple, I a Moment of Inertia, and » an angular velocity, we get: —electrification, a strain; electrical potential, work; electrical force, force; current, angular velocity; quantity of magnetism, angular momentum ; self-induction, moment of inertia, &c. In all cases we are able, by means of the dimen- sional formulz expressed in terms of w and &, to work in detail any dynamical analogue we may choose to take as a starting-point. Of course all dimensional values of uy, together with the corresponding ones for k, must render electromagnetic dimensions unique. It is only some of these, however, that give rise to formulze capable of dynamical interpretation, and it will be found on examination that the number of such values is small. Now, if electromagnetism is ultimately dynamical, the dimensions of electromagnetic quantities must be of the same kind (ultimately) as those of ordinary dynamical units. Hence, by examining all the possible cases in which the dimensions of mw and & lead in the case of the other electro-magnetic quantities to dimensions of the dynamical order, we may be able to obtain some light as to the nature of w and k themselves. To show how this may be done, and the kind of results obtained will be the object of the following paper. of Physical Quantities to Directions in Spavé. 237 Before doing this, however, it becomes necessary to examine in more detail the real nature of dimensional equations, and to determine how far they are capable of giving reliable results when used in the way here suggested. Primarily, a dimen- sional formula expresses only numerical relations between units, and for the purpose of the present paper is defective from the fact that different physical quantities have the same dimensional formule. For example, couple and work, as pointed out by Prof. 8. P. Thompson in the course of the discussion on Prof. Riicker’s paper above referred to; or, again, an area and the square of the same vector-length ; pres- sure and tangential force per unit area, &c. If, therefore, di- mensional equations are to be used at all in the sense above indicated, it becomes necessary, in the first place, to be assured that the process is valid, and, in the second place, that no contradictory or unintelligible results arise from causes such as the above. The dimensional formula of a physical quantity expresses the numerical dependence of the unit of that quantity upon the fundamental and secondary units from which it is derived, and the indices of the various units in the formula are termed the dimensions of the quantity with respect to those units. When used in this very restricted sense, the formule only indicate numerical relations between the various units. It is possible, however, to regard the matter from a wider point of view, as has been emphasized by Prof. Riicker in the paper above referred to. The dimensional formule may be taken as representing the physical cdentities of the various quantities, as indicating, in fact, how our conceptions of their physical nature (in terms, of course, of other and more fun- damental conceptions) are formed—just as the formula of a chemical compound indicates its composition and chemical identity. This is evidently a more comprehensive and funda- mental view of the matter, and from this point of view the primitive numerical signification of a dimensional formula as merely a change ratio between units becomes quite a dependent and secondary consideration. The question then arises, what is the test of the identity of a physical quantity? Plainly, it is the manner in which the unit of that quantity is built up (ultimately) from the fun- damental units L, M, and T, and not merely the manner in which its magnitude changes with those units. Thus, the unit couple and the unit of work both change in the same manner with the unit length, but they are physical quantities of dif- ferent kinds. Their nwmerical dependence upon L, M, and T is the same, as expressed by the formula (MLT~’)L, but Phil. Mag. 8. 5. Vol. 34. No. 208. Sept. 1892. S 238 Mr. W. Williams on the Relation of Dimensions the manner the unit length enters their definition is different : in the case of work the two units of length involved are in the same direction ; in the case of couple they are mutually at right angles. The absolute measure of a force is the work done through unit linear displacement ; similarly, the absolute measure of a couple is the work done through unit angular displacement. Hence the relation between couple and work is similar to that between force and work, the difference being that angular displacements are considered in the former case, linear displacements in the latter. But the measure of an angular displacement is independent of the unit length. Hence, in expressing the numerical dependence of the unit couple and the unit of work upon the fundamental units we get the same formula ML?T~*, although the difference in the physical nature of the quantities is of the same kind as that between force and work. And generally, when used in the purely numerical sense above indicated, the dimensional formule fulfil all requirements; it is only when endowed with the higher function of defining the physical identities of the various quantities that they are found to fail. That the dimensional formule are regarded from this higher standpoint—that is, regarded as being something more than mere ‘change ratios” between units—is shown by the fact that difficulties are felt when the dimensions of two different quantities, e. g., couple and work, happen to become the same. If, however, the numerical dependence of the units of the two quantities upon the fundamental units be the same, and if the formule are to express nothing more, then the two quantities must have the same dimensions, and from this point of view we are not entitled to feel any difficulty in the matter. That such difficulties are felt arises therefore from the more com- prehensive signification which is attached to the formulee—a signification which obviously includes all the numerical consi- derations which alone constitute the more restricted one. Let us, therefore, for the purpose of the present paper, regard the dimensional formula of a quantity as the symbolical expression of the physical nature of that quantity, so far, of- course, as it depends upon the fundamental conceptions of mass, space, and time (and, in the case of thermal and elec- trical quantities, of secondary conceptions also ultimately dependent upon mass, space, and time). To obtain the formula for any quantity, it is only necessary to express how the unit of the quantity is built up from the fundamental and secondary units. Now, the units of mass and time, and all secondary units, are involved in all physical units in a simple manner. They are raised to different powers. - But, owing of Physical Quantities to Directions in Space. 239 to the dimensions of space, the unit of length is involved in different ways, according to the different relative directiuns in which it may be taken. In all cases, however, the unit of any quantity can be completely expressed—so far as it in- volves the unit length—by taking the unit length along one or more of three mutually rectangular directions Oz, Oy, Oz, whose absolute direction in space is of course determined by the nature of the physical relation into which the quantity enters. The dimensional formule can therefore be expressed in terms of M, T, and X, Y, Z, where M is the unit of mass, ‘I’ the unit of time, and X, Y, Z the unit of length taken respectively along the directions Ow, Oy, Oz. Thus, if MXT~ is the unit of force, MX?T is the unit of work ; MXY “ZT”, energy per unit volume ; MXZT~’, a couple in the plane XZ, &. The above quantities are, of course, the same in kind but different in direction from MYT’, MZT~, &c. This method of expressing the dimensional formule is nothing more than a modified extension of Prof. S. P. Thompson’s suggestion as to the use of “ —1 in dimensional formule, as will appear more fully later (see discussion on Prof. Riicker’s paper above referred to). When it is necessary to determine the numerical dependence of the unit of a quantity upon the fundamental and secondary units, the distinction between units of length taken in different directions must be suppressed, and the unit of length in what- ever direction taken must be represented by a single symbol L. The index of L will therefore be the sum of the indices _of X, Y, and Z, and the fermule thus simplified will indicate the changes in the magnitude of the unit when the funda- mental and secondary units are themselves changed. Thus, we can immediately deduce from any formula the particular and simplified form it assumes when only the numerical rela- tions between units are to be considered. It will be convenient to designate these simplified forms of the formule the “ change ratios ’’ of the units, and to reserve the term “ dimensional formule ” for the more general forms in which the identities of the quantities are primarily concerned. The dimensions of the ordinary dynamical quantities may now be expressed :-— * Inertia = 2M. length. = =x V¥ gor 2. . Area = Mg A, or ZoX. ; for a dimensional formula expresses the nature of a quan- tity, not its magnitude, and the same formula must therefore apply to both and 9@. The dimensions of 6 and Q@ are therefore YX~’. : 14. Angles = -XY~, (Direction of radius Y, of the are X) &e. 15. Angular Velocity = XY~'T™, &e. of Physical Quantities to Directions in Space. 241 16. Angular Acceleration = XY~'T~*. 17. Moment of Inertia = MX”, MY’, MZ?. Moment of couple ni MXYT™ = MY?®, &e. Angular acceler. XY 'T- 18. Angular Momentum = Iw = MY?(xXY)T~ =MXYT™, &. 19. Energy of Rotation = 41? = MY?(KY")T~ =MX°T, &e. Bie Couple: = 122 = MY(XY-)T? = MXYT™, 21. Work done by a couple=G0=(MXYT~”)(XY~’) =—MXeT. 22. Solid angle = QQ = - where Oa is an element of area on the surface of a sphere radius 7. Taking in- stantaneous axes, X being always along the radius, and Y, Z in the tangent plane at a point, this becomes dimensionally, [9Q]=(YZX"”), The quantity 7 may enter into physical relations in two different ways. It may enter in its purely geometrical or trigonometrical capacity as a definite number of radians (or, what is equivalent, as the ratio of the circumference of a circle to its diameter), and be thus definitely related in a physical sense to the other quantities ; orit may enter in its numerical capacity as part of a numerical coefficient determined by higher abstract analysis. Cases of the former kind only have to be considered in the following paper. In these cases a enters the relations (ultimately) as a definite number of plane or solid angles, and it therefore consists of two factors, namely, a numerical factor, 8:14°°', and a concrete factor, the unit plane or solid angle. We may therefore speak of the dimen- sions of 7, meaning thereby the dimensions of the plane or solid angles which it 2n such cases implies, and the dimensional identity (in the directional sense) of the relations into which it so enters cannot be complete if the concrete factor is neglected. Thus, let A=4zr?, where A is the surface of a sphere of radius yr. ‘Taking axes as in 22, this becomes dimen- sionally YZ=(YZX~*) X’, and the relation is not true if 47 be treated as a pure number unless an area is represented by the square of a vector length, instead of by the product of two different vector lengths as in vector algebra. Similarly, in the relation V=477°, V being a volume, 7 is of the dimen- 242 Mr. W. Williams on the Relation of Dimensions sions of a solid angle, while in /=2ar, where / is the circum- ference of a circle, it is of the dimensions of a plane angle. The significance of w as it occurs in electromagnetism in connexion with radial and circuital fluxes will be considered later. The following examples serve to illustrate this method of expressing dimensional formule. 23. Radius of Curvature——Bending per unit length of a curve. Let AB be an element of a curve, and let the tan- gents at A and B intersect at O, making an angle @¢. The angle O¢ is ultimately =i. If AP be taken along the normal X at A, and OA along the tangent Y, then AB is also ultimately along Y, and may be written Oy. Hence the curvature becomes Se or dimensionally (KY~*)Y"=XY ~~, where XY‘ are the dimensions of 9¢, and Y those of AB. The radius of curvature at O is therefore of the dimensions eGo Gane 24, Centrifugal Force-—Let a particle describe the path 2 AB with velocity V, the centrifugal force is B=(m —). Taking instantaneous axes at A, as in 23, this becomes dimensionally [Fl SMe *) RX) = Mie, Thus the force is directed along the radius X. 25. Compressibility.— Hydrostatic pressure is of the dimen- sions MXT-*(YZ)~*, according to the direction of the plane of reference YZ. Strain is here of no dimensions, being the ratio of two concretes of the same kind. Hence the dimen- sions of compressibility are M~*X~*YZT”. | 26. Rigidity—‘A simple shear is a homogeneous strain in which all planes parallel to a fixed plane are displaced in the same direction parallel to that plane and therefore through spaces proportional to their distances trom that plane” (Kelland and Tait’s ‘Quaternions,’ p. 204). Let the planes of displace- of Physical Quantities to Directions in Space. 243 ment be the planes XY, and the direction of displacement X. The applied stress (tangential force per unit area) is The shear is XZ~*. Hence the rigidity is of the dimensions MMe 5 ga = (MXM -ZT): The velocity of propagation of a wave of distortion is — 7a n= rigidity. Hence p —Inp—2 iv] = MZ(YZ) a M(X YZ) Thus, the applied stress acts over the face XY, and the dis- turbance is propagated along Z. : 27. Torsion..—Let OABC be the section of a cylinder, axis Z and radius Y, subjected to torsion; aA Bb the face of an elemen- tary cube edge 97, bounded as in the figure; and @ the twist per —Z7T. Fig. 2. CB A \ ee eee eee Ze D c unit length. The shear experienced by the elementary cube is ultimately as =7r@, or dimensionally XZ~* (for 6 is of the dimensions KY Ley, and r=Z). Let P=nré be the applied stress (tangential force per unitarea). Then[P]=MY~‘T-*, The tangential force over ring of which aABéd forms part = (2rror)nrO =27r'nOOr, or dimensionally MY~*T~?(XY) = MXT™, and the moment of this round O = 2rr°néOr, or dimensionally MXYT~*. The moment due to the whole face 4 = QarnOD,. ror= — 6, where R=OA, and this is of the same dimensions as 2anr°69r, for the r* in the former expression is of the same dimensions 244 Mr. W. Williams on the Relation of Dimensions 4 as 7°Q@r in the latter. Thus a5 : is of the dimenstens Moxey *. : To determine from this the dimensions of 7 we have [arnr46|=MXYT*, t= KY 2", Gal =Y", [n] =MZ(XY)T; [a |MZ (Re (XY) 7 ee oa Faw. Ge Thus 7 in this case is of the dimensions of a plane angle in the plane XY. 28. Viscosity.—The viscous resistance between planes moving with relative velocity u= ge is given by F= (Noo Ly) (Clausius). Taking the components of w and L along Z (normal to direction of motion) this becomes dimensionally 8In( th) Gyo( Aer AE a tangential force in the plane of motion (XY). The co- efficient of viscosity is EF MYT : ght n= Bu = 7-1 = MZ YZ)" °T dz = Tangential force per unit area + shear per unit time. 29. Surface Tension.—Tangential force per unit length normal to itself. Let Y be the direction of force in the plane YZ, and K the surface tension. Then [| i= MYA T= va = Energy per unit area. Fig. 3. a Let AOB be a normal section of a cylindrieal liquid film, of Physical Quantities to Directions in Space. 245 axis Z, radius X. The normal pressure due to the curvature is K ( =): or dimensionally MYZ 1 ? (XY )=MX(YZ)"'T"; as should be the case. The above examples are sufficient to illustrate the method of expressing dimensional formule in terms of X, Y, Z. In all the above cases isolated quantities only are dealt with, and therefore the relation of the directions of directed quantities to the dimensional axes is a matter of indifference. This is not so in the case of equations between quantities. Here, however, we have only to transform the equations to Carte- sian co-ordinates and then express the dimensions of each term by means of the above. Thus, let A, B,C, ... be quantities connected by the equation Poe tol) — 0 Expressed in Cartesian co-ordinates this becomes Pee ee) (B+ Bt B)+(Ct+C4+C)4+ ... =0; and we can now immediately express the dimensions of every term. We thus get the dimensional expression of the equation itself. A physical equation implies that the quantities con- nected are the same in kind. There must, therefore, be a dimensional identity between the various terms. ‘This holds, however, in its wider or directive sense only of component quantities in the same direction, for if we equate any term in the above to all the rest we always get a term such as A, equated to another such as A,, the same in kind, but different in direction. If, however, we write the equation Cee Ui+.. A+ Bete) (A, +B +...) =0, there must now be a dimensional identity in the directional (X, Y, Z) sense between the terms in the same bracket. Up to the present the dimensional formule have been re- garded as purely conventional. It is now necessary to ex- amine whether the conventions made use of can be justified in any other way ; and it becomes important to have a clear physical distinction between pure numbers and concretes, for, according to the above conventions, dimensional formule are now extended to include quantities hitherto regarded as pure numbers. Hvery concrete quantity should have definite physical di- mensions in the extended sense of the term; only pure numbers should be quantities of no dimensions. For physical 246 Mr. W. Williams on the Relation of Dimensions purposes, a pure number may be defined as the ratio of two concretes of the same kind. If the concretes are directed quantities, their ratio is not a pure number unless they are similarly directed. Thus, the ratio of a force along X to that along Y is of the dimensions (MXT ~*)/(MYT~*)= XY *; that is, of the dimensions of an angle, or rotation, indicating that to compare the scalar magnitude of the two forces, they have to be rotated into coincidence. And, generally, the ratio of any two directed quantities of the same kind isa pure number together with a quantity of the nature of rotation, the latter being the versor part of the quotient of two vectors. Since the measure of a rotation is independent of the unit length, the versor part of the quotient is a pure number so far as the scalar unit lengthis concerned. It cannot, however, be truly a pure number, for different degrees of rotation are compared in terms of definite units of rotation. Hence, a versor, or its equivalent, an angle, or an angular displacement is a concrete quantity equally well with mass, length, and time, and should have its own proper dimensions. It is only because of the restriction of the term “concrete” to quantities the magni- tudes of whose units change with the units of length, mass, and time that concretes of the nature of angles, and angular dis- placements, come to be regarded as purely numeric. From this point of view, therefore, 2 pure number is always of the nature of a tensor, or rather, unity must be regarded as the dimensional formula of a tensor, and of physical quantities of like nature, e. g. volume-strains *. Let a, 8, y be three vectors (non-coplanar). Then 1. a, B, or y=Linear displacements, or lengths in magni- tude and direction. 2. V(a8) &.=Area of parallelogram bounded by a, ~. 3. S(aBy)=Volume (neglecting sign) of parallelopiped defined by a, 8, ¥. A. US Veen =Angular displacement required to bring @ parallel to a. If a, 8, y be vectors mutually at right angles, it is un- necessary to distinguish between scalar and vector products, -and if they be also equal, between tensors and versors, for the product of any two vectors is now a vector, and of three a scalar: the quotient of any two is a versor, since the tensor =1; while the product of parallel vectors is always scalar. We may therefore take «, «8, «By, 3? and : (or unity) as * See “The Multiplication and Division of Concrete Quantities,” Prof. A. Lodge, ‘ Nature,’ July 19, 1888. of Physical Quantities to Directions in Space. 247 the typical representatives of the respective ideas of vector lengths, areas, volumes, angles, and pure numbers (tensors). If, now, we call these the dimensional formule (that is, the physical representations) of these ideas, we see that the tensor, although a quasi-physical quantity like the versor, becomes physically represented according to the above conventions by unity, while the versor is represented by the ratio of two vectors. Thus, the meaning of the fact that tensors and phy- sical quantities of like nature are of no dimensions, is that their dimensional representation according to the above con- ventions simplifies down to unity. No other physical quan- tities are, however, dimensionally represented by unity. We may, therefore, say that tensors &c. are also physical quan- tities, and therefore concrete, and that since every physical quantity consists of two factors, namely a pure number and a concrete, we may regard a pure number when occurring alone in a physical relation as a physical quantity whose concrete factor is a tensor. Since, however, the term ‘ concrete” is reserved for those quantities which cannot be completely specified by pure numbers, we come ultimately to regard a tensor as a pure number, and the versor as a concrete quantity. If we substitute for a, 8, yin the above X, Y, Z respec- tively, we get the same expressions for lengths, areas, volumes, and angles as already obtained in the dimensional formule. Hence, we see that the conventions made as to the dimen- sional representation of the various quantities, namely, that areas, volumes, and angles should be represented by products and quotients of dzfferent vector lengths instead of by powers and quotients of a given one, are justified by the fact that they are consistent with the meanings of products and quotients of rectangular vectors. For the products and powers of parallel vectors can never represent areas or volumes, and their ratios can be nothing but pure numbers. Diniensional formule may be conveniently expressed in this way, that is, by taking three rectangular vectors in space as axes of reference. We may, of course, suppose X, Y, Z to be the vectors. Then the formule already deduced will be unaltered, except that the proper signs have to be inserted according to the order the products and quotients are per- formed. The great advantage of thus supposing X, Y, Z to be vectors instead of mere Cartesian lengths is that the dis- tinction between scalar and vector quantities is made apparent by the formula. Thus, X’, Y’, Z?, or R? (R being any vector) me scalars; X,Y, A, XV,NA, AX, XY", ¥Ze* ZX and their reciprocals are vectors; and (XYZ) a scalar. To make 248 Mr. W. Williams on the Relation of Dimensions the formule exhibit at the same time the numerical depen- dence of the derived upon the fundamental units, we may put X=2L, Y=jl, Z=kh, 2,7, & being quadrantal versors and L the scalar unit length. Thus, neglecting signs, we have :— Force=MXT?=M(iL)T ”. Work=MX?T ?=M(iL)*T ?=ML?T~’. Couple=MXYT?=MyLT ?=MALPT™. Energy per unit volume, = (ML2T)L?=ML Tt, for volume= (7jkL>)=L’, yk=1. MXP. MELT Pressure = ~ age Sete =ML'T, (i=jk). Tangential force per unit area= MYT? =MGgL)T”. Angle= KY =(iL)(jL) = ; a 2 he Work done by a couple=MXYT“*(XY~") =(MyL?T ”)k =MI2T™, (yjk=1). Thus, the indices of M, L, and T indicate the numerical dependence of the derived upon the fundamental units, the formule being identical in this respect with the ordinary ones, while the directional properties of the quantities are clearly differentiated by 2, 7, k. Multiplying the numerator and denominator of a formula by X, Y, or Z makes no change in its physical interpretation. For physical quantities differ from each other only in their numerical dependence upon the fundamental units, and their ‘space relations,” the former being expressed by the scalar part of the formula L, M, T, the latter by the vector part, J, k. But if X, Y, Z be equal rectangular vectors, the tensors of ROY FZ be ays change can be made in the scalar or vector part of any formula into which they may be multiplied. Thus :— are unity, and their versors are zero. Hence, no onm—2 Force = MXT-?= — = Space rate of variation of energy along X. | ny—2 Force= MXT-?=——p— = Couple per unit area. ‘ : j | of Physical Quantities to Directions in Space. 249 = Tangential force per unit area= MY “T° = 2s —2 = Force along X in plane XY = a = Force along Z in plane YZ. These are the two components of a shearing- = stress referred to a unit cube. Multiplying by rhe xX MZ? xz we get xyz = Couple per unit volume. Now, a shearing-stress must be of the nature “4 of a couple, for a shear is of the nature of an ¥ 7 angle, and the product of the stress into the shear is work done per unit volume. DEN oe MXC TS Surface tension = Sagas aXaV 1: = Hnergy per unit area. P Peni? Mer aga eine ee ressure = —y7— = —yy7 = Energy per unit volume. Since, as above shown, we obtain the same dimensional formule by Cartesian and vectorial methods, we may attach to X, Y, Zand their products and quotients in these formula either purely vectorial or purely Cartesian meanings just as we please: in both cases, the formulz represent ihe same physical identities. When X, Y, Z are vectors the formule express the directional properties of the corresponding quan- tities. [An inspection of a formula shows this, for the pro- ducts and quotients of two rectangular vectors are vectors directed normally to the planes containing the original vec- tors, and the products and quotients of parallel vectors are scalar}. Thus: Pressure MX(YZ)~'T™, scalar, for YZ is directed parallel to X, hence 2 is scalar. Couple MXZT~°, a vector directed along Y, &c. In deducing the dimensions of electromagnetic quantities it will be necessary to start with the dimensions of energy, or energy per unit volume. Using Cartesian units of length this is MX°T*, MY?T-’*, MZ?T’, or MR?T”, as the case may be; and since we shall have to deal with connected equations between the various quantities, the particular form to be used becomes of importance. Of course, energy being a scalar quantity, the above do not differ essentially: the difference arises only from the different ways (the different dynamical reactions) by which the expressions are derived. 250 Mr. W. Williams on the Relation of Dimensions It might be more convenient to put X=iL, Y=jL, Z=kL, then the dimensions of the energy of the medium become MI2T~’. There is thus nothing to indicate how or with reference to what dynamical reactions the expression is derived. In what follows, however, Cartesian expressions will still be used though more cumbrous and involving more explanation. They may be immediately converted into the above. The energy of the medium will be expressed in terms of an instantaneous linear displacement R upon which our conceptions of the electromagnetic displacements at a point, whatever their nature may be, must ultimately depend. B keeping R, at first, separate from X, Y, and Z, the formule gain in generality, and by retaining R, X, Y, Z instead of their equivalent vector forms rL, 7, yl, kL, the dynamical connexion between the various quantities is. more clearly expressed. ‘The quantity 7 enters prominently into electromagnetic relations, and it becomes necessary at the outset to determine in what manner it is to be dealt with. This subject has been discussed by Mr. Oliver Heaviside (Elec. October 16th and 30th, 1891), and his conclusions may be briefly summarized as follows :— If m and g be point sources of induction and displacement respectively, the measure of the induction and displacement at a distance r from the source (if the fluxes emanate isotro- pically) is i eS Boh) GS i= me) aa Ce 2) where B and D are the densities of the fluxes over spherical surfaces enclosing the sources. And, similarly, the density of any radial flux at a point should be estimated by the total flux through a spherical surface having its centre at the source and passing through the given point divided by the surface. Writing B=wH, and D=kE, where mw and & ex- press physical properties of the medium, we have nea m Urea g ae Aiea) aE aCe Now, H and E express the strengths of the fields produced by the fluxes m and g at distances r from the source. Hence ~ and : express the strengths of the sources. Again, multi- plying the above by m and q respectively we get fen? 7107 7 1 yg? mH = a cc) gk in eee! of Physical Quantities to Directions in Space. 251 But mH is the force experienced by a pole m when placed in a field of strength H, and similarly for kE. Hence Li mm ae g? pentta}(t,) Frege }(,t) where F', and F, are respectively the forces between two poles m, or two charges q, at distances r apart. In other words, since in expressing the force between two poles or two charges we have to regard each pole or charge as an ?solated point source of displacement, we should regard the one pole or charge as producing radially a field of given strength, and then express the force experienced by the other when placed in this field. If, now, the unit pole and the unit charge be defined re- spectively by the relations m=? V4arpl, a q—F VArkk., meena of, as usual, m=? / wk q=r VkF,, the effect is to redistribute m in electromagnetic equations as a whole. It is found, however, that all those relations into which it is now made to enter depend upon and involve the consider- ation of circuital or radial fluxes, and m obviously enters as a plane or solid angle in connexion with circles and spheres of reference. It has thus a definite physical meaning, and is always definitely related to the other quantities in the relation. On the other hand, in the case of the relations from which it is removed, it previously entered only because of its suppression elsewhere, and had no fundamental connexion with the other quantities. The relation between the new and the old units thus defined are given in the papers above referred to. For the purpose of the present paper it is sufficient to notice that the relation ATudy 0z= =e oF Nay 02 nv) becomes = fO'y 5) Udy OZ = ey ae Oyoz: p=4mm becomes p=m: B=pH+4z7I becomes B=,H +I, so that B and I are identical ; the electrostatic energy of the ED medium Ge 7 Since, in these relations, 7 essentially preserves its primi- ) becomes ED, thus harmonizing with BH. 252 Mr. W. Williams on the Relation of Dimensions tive geometrical or trigonometrical meaning, its insertion or suppression implies the insertion or suppression of a concrete quantity, and not merely a number, so that the natural relations between the various quantities are affected. For this reason, therefore, the “rational units,” given by Mr. Oliver Heaviside, in which z is primarily associated with the radial and circuital fluxes of the field, will alone be used in what follows. The relations made use of in deducing the dimensions of electromagnetic quantities are of three kinds. I. Relations between quantities of the same kind, either purely electrical or purely magnetic.— These are :— (a) Electrical. 1. D=A&E. 3. e=Surface-integral of D. 30 4. C= ys D. 5. H=Line-integral of E. Where D=eleciric displacement ; k=specific inductive capa- city ; E=electrical force ; C=electric current ; e=quantity of electricity ; C=current density ; H=electromotive force or voltage of a closed circuit. (b) Magnetic. 1 3 Se. . CC tg 3. m = Surface-integral of B. es. 4, Cc = ay BB: 5. EH, =Line-integral of H. B=magnetic induction; sm=specific magnetic capacity ; H= magnetic force; m=quantity of magnetism. Mr. Oliver Heaviside designates E,, the Gaussage of a closed magnetic circuit to correspond with H, the Voltage of the corresponding electrical one, C,, and C,, being the magnetic currents. II. Relations between quantities of different kinds.—These relations are embodied in the two laws of circuitation: (a) Cirewtation H=C=H,, = 2 .€} of Physical Quantities to Directions in Space. 253 or, the Gaussage of a closed magnetic circuit measures the total electrical current through it. (b) Circuitation E =C,,=H= 2. Mm 3 or, the Voltage of a closed electrical circuit measures the total magnetic current through it. The corresponding relations as given by Maxwell are fy — 92) 7 (a) wy Bz=(ST —S= Jovds, u being the component current-density along «, and £, y the components of magnetic force along y and z respectively, the whole being referred to an elementary magnetic circuit (Oyd0z). [The 4a in the expression 4%u is dropped, as previously explained. | — (OY — 1,92 \ ”) P= (os! ° 3) neglecting A and y. P is the component electrical force along xz, and 6 and ¢ the components of magnetic induction along y and z respectively. I1f. Dynamical Relations.—These are relations between quantities whose product express the energy, or energy per unit volume of the medium. mai: mM) : pC... Energy. DE : BH: AC... Hnergy per unit volume. But D=ZE, and B=ywH. Hence ke? : wH? . . Energy per unit volume. By means of these relations we can express in terms of M, X, Y, Z, T, and one selected quantity the dimensions of all the rest. The only useful cases, however, are those in which the selected quantities are either mw or k, for these express physical properties of the medium at a point, and are in- dependent of the electromagnetic reactions going on there. The dimensions in terms of mw are obtained by starting. with the relation wH’=Energy per unit volume; and similarly for the dimensions in terms of f. Since the above dimensions have to be deduced by means of a connected system of equations, it becomes necessary to make a suitable choice of axes of reference. Let X be the axis of the electric displacement, and Y that of the magnetic displacement at a point in the medium. For -.an isotropic medium (the only case we have at present to consider) these Phil. Mag. 8. 5. Vol. 34. No. 208. Sept. 1892. T 254 Mr. W. Williams on the Relation of Dimensions are mutually at right angles, and Z is at right angles to both, being the intersection of the electric and magnetic equi- potential surfaces. Let this relation between the directions of the axes of reference and the displacements hold for every point of the medium, so that the axes constitute an instanta- neous system at every point. In passing therefore from. point to point in the medium, and for different epochs at the same point, the axes and the displacements preserve their relative directions, while their absolute direction in space in general alters. Fig. 5, Let AO’ be a closed electric circuit, and BO a correspond- ing closed magnetic circuit, both being circles in planes at right angles to each other. Taking instantaneous axes as above, every element of the circuit AO’ is Ow, and of the circuit BO is Oy, while an element of the intersection of the planes of the circuits is Oz. The length of the circuit AO’ is >O2, and of the circuit BO, 2dy, while the surfaces of the circuits are ultimately %(O7dz), and >(dydz). We have therefore :— 1. Circuitation H=2(Hoy)=C. 2. Circuitation E=>(EQx) =H. 3. Surface-integral of D==(Ddydz) =e. 4. Surface-integral of B= (BOxOz) =m. To express these dimensionally, we have to neglect the summation >, and substitute for Oz, Qy, Oz respectively X, Y,Z. The relations then become :— 1. Dee 2. Cia=el-t 3. =D YZ) 4, CP oT" 5. B =E(X)=C,, =mT=B(XZ)T-. of Physical Quantities to Directions in Space. 258 . CO, =mT=E. . m= BAL). mole UE): aes ia, — a Y) — C—el 4 — (YZ) Ts : The energy of the medium at any point may be expressed y eH aI -Sm(a2?+y?4+2)=>(mr’), where r is the instantaneous linear displacement upor which both the electric and magnetic displacements at that point depend, for the two laws of circuitation express that the electric and magnetic displacements at a point in the medium are not independent, but originate from the same dynamical cause. Hxpressed dimensionally, this becomes M(X?+ Y?+ Z?)T- or MR*T-?, and the dimensions of energy per unit volume are MOXA A Since the axes ef reference (X, Y, Z) and the displacement R are both instantaneous with respect to any point, the direction of RK must be definitely related to the axes, the relation being dependent upon the dynamical mechanism of the field. It will be convenient, however, to express the dimen- sions of electromagnetic quantities, first, in terms of M, X, Y, Z, T and R; and afterwards to determine the relation between the direction of R and the axes. The fact that the - formulze of some of the quantities involve R, while others de not, then simply means that in the former cases the corre- sponding quantities depend upon and involve the instanta- neous linear displacement specified by R ; while in the other cases they are independent of the displacement and express only physical properties ef the medium. I. Electromagnetic System :— eH? = MRT (XYZ). Hence 1. H=p *[MPRT (XYZ) =). 2. B=pH=p3| M@RT (XYZ) *]. 3. m =p=B(XZ)=p2[M*RT '(X?Y-=Z')]. 4, B=mT7=4y*[ MRT? (X?Y 222) J. . E=E(X") poe ee 2). Oc 256 Mr. W. Williams on the Relation of Dimensions 6. A =ET=p3[MPRT | (X-?Y 22?) ]. 7. C=HY=p *(MPRT | (X?Y'Z *)]. =O(YZ) =p 3 [MRT (XYZ 4)]. 9, ¢ =CT=p *[MR(X *Y'Z >) |. 10. D = CT=e(YZ) =n *[M?R(X *Y ?Z*)]. From 5 and 10 we have pe ts —1>7—2'712 and substituting k~*[Z°T?] for w all through in the above we get for each quantity its electrostatic dimensions, the results being identical with those obtained directly as below. II. Hlectrostatic System :— kE? = MRT? (XYZ) 1. E= «3(MERT (XYZ) ]. 2. D =kE = [MRT (XYZ) “]. 3. e = D(YZ)= P[M RT (XK “Y'Z)]. 4, CO =eT*= #[MPRT "(X *Y2Z*) ]. 5. € =O(YZ) t= eee 6. H=C(Y") = [MRT (XY 2). i, B= EOO = k(M?RT lxty 47, Fa 8. nm =p=hl— k[ MPR ( xtyZ-4)]. 7. =m(XZ)~* = k~*{ M?R( XK 4Y?Z74)]. 10. A =ET =k ?[M?R(X dey From 6 and 9 we have 2 —B _p-arp7—2e) . h=_ak | Arey s and by substituting ~—'[Z-°T?] all through for & in the above, we recover the electromagnetic dimensions previously deduced. The following are examples of the manner these formule work out :— of Physical Quantities to Directions in Space. 207 1. The velocity of propagation of an electromagnetic dis- turbance in the medium is given by Vv 1 OTT which is of the dimensions of a velocity along Z, the normal to the plane of displacements (XY). 2. The flux of energy at a point is given by W =(EE) ; and expressing E and H dimensionally in terms of p» or k we get =ZT- (dimensionally), Les aes Z _owy, 1S lag SG CA e's the bracketed factor being of the dimensions of energy per unit volume, and the other a velocity in the direction Z, the axis of flux of energy. 3. The force between two poles m is given by , 1 ae ice F,, = (mH) Hires : where H is the strength of the field produced at the one by the other. Expressed dimensionally, this becomes MRT? ‘Bs — ( Y 3 which is of the dimensions of the space rate of variation of energy along Y, that is, the force along Y, as is evident from the equivalent algebraic expression oie: e. (mr2). Putting R=rL, and Y=jL, where r and 7 are unit vectors in the directions R and Y, and L the scalar unit length, we have OD ise a ee eae j SoM (ds) P= MES. Thus, the force between two poles is in the direction of magnetization. The reason why the force is expressed in terms of the energy of the system is that it is a mechanical force arising in some way from the mutual reaction between matter and the medium. ‘The quantities m and H in terms of which the force between the two poles is expressed refer to the medium alone, and since nothing is known as to the rela- tion between the medium and matter, the relation above MRT? M(rL)t- MLeT — MLT 258 Mr. W. Williams on the Relation of Dimensions expresses only the resultant reaction taking place, namely that there is a space rate ef variation of the energy of the system along Y, the direction of the force. The same re- marks apply to the cases below. 4, The force between two charges q is given by t 2 F,= (8) =; (75), a Oe Nee or dimensionally [F,] ear 3 the space rate of variation of the energy of the system they constitute along X, the direction of the force. 5. The force per umt volume between two elements of conductors carrying currents is given by F,=BC, where C is the current-density in the one, and B the induction at this one due to the other. Hxpressed dimensionally, this becomes pe Be MATA eee ree. a: ys cum aw. Og BA: which is. of the dimensions of space rate of variation of energy per unit volume along 4, the direction of the force, for the conductors move at right angles to their lines of force which are circles about them. Whatever be the dimensions of # those of k must be given by 2 [Z-*T’]. Now, the ratio of the dimensions of the same quantity expressed in the two systems is a power of uk[Z*T-*]. Hence, all possible dimensional values of p together with the corresponding ones for k will, when sub- stituted in the formule, bring the two systems into accord. Of these, however, there must be one pair, and only one pair, which give rise to dimensions whose interpretations are physically real. It is, of course, impossible to determine from purely dimen- sional considerations what this particular pair must be, for this would imply a knowledge of the dynamical nature of electromagnetism. It zs possible, however, to assign to ~ and & dimensions fulfillmg certain assumed conditions. The dimensions of the remaining quantities then become unique, and we may, by deducing the physical interpretations of the formule, pass on to the quantities they then represent. In this way, the formule in terms of # and & may be utilized as means for tracing out in detail the various analogies between electromagnetism and dynamics. or every dimensional value of » and & we thus obtain a perfectly connected dy- namical analogue of electromagnetism, which may or may of Physical Quantities to Directions in Space. 259 not be rational and intelligible as a whole, and whose relation to physical reality depends upon the imposed conditions. For the purpose of the present paper, let us impose upon the dimensions of ~ and & the condition that the indices of the fundamental units in their formule are not to be higher or lower than those found in the formulee of ordinary dynamical units. This is, of course, a purely arbitrary condition, and simply expresses that the dynamical analogues to be traced out are restricted to those which are of a simple, natural, and intelligible character. All other cases are purposely excluded, not, of course, as a matter of necessity, but as a matter of convenience, for nothing would be gained by the introduction of formule whose interpretation would be obscure and un- intelligible in our present state of knowledge. Let us there- fore proceed to deduce from dimensional considerations the various analogies between electromagnetism and dynamics, subject to the arbitrary condition imposed above. In the case of ordinary dynamical units we notice that 1. No fractional powers of the fundamental units occur. 2. The indices of M are never higher than +1. 3, Ph 9) us Y, Z 1% >] >] +2. 4. >) 99 fi 1) 9 ? +3. This, therefore, is the range within which » must be found subject to the above conditions. [Of course, if the fundamental units be not L, M, T, but some of the now derived units, then fractional powers may occur. Thus, if V (volume) be a fundamental unit, the dimensions of length are Vs, and of area V%, &. So long, however, as L,M,T, are fundamental units, we cannot ex- pect fractional powers to occur. For length, or space of one dimension, is the simplest conception of space which we can form, while #me and mass (not necessarily that of matter, but tangibleness in general) are fundamental conceptions beyond which we cannot go. Now, all dynamical concep- tions are built up ultimately in terms of these three ideas, mass, length, and time, and since the process is synthetical, building up the complex from the simple, it becomes ex- pressed in conformity with the conventions of Algebra by entegral powers of L,M,T. The analytical process, that is, splitting up a complex conception into its ultimate consti- tuents (evolution in Algebra), becomes expressed according to the same conventions by fractional powers, e.g. L=(V)8, [Area]=(V)’, &. But, obviously, if mass, length, and time are to be ulttmate physical conceptions, we cannot give interpretations to fractional powers of L, M, and T, because 260 Mr. W. Williams on the Relation of Dimensions we cannot analyse the corresponding ideas to anything simpler. We should thus be unable, according to any physical theory, to give interpretations to formule involving frac- tional powers of the fundamental units. Our only hope in such cases would be that the units themselves might cease to be fundamental. On the other hand, the buildmg up of a derived from fundamental units is always a simple process, nothing but whole units of the latter kind being mvolved. We should thus expect, both from the algebraical meanings of integral and fractional powers, and from the manner physical. units are derived, to find the dimensional formulee of physical quantities free from fractional powers. That the formule of all dynamical quantities, that is all the absolute formule we are acquainted with, are of this kind, isan argumeni in favour of this view. A vector length has two properties, direction and mag- nitude. When squared, a vector length becomes scalar. A vector enters into physical relations either as a vector (having direction and magnitude) or as a scalar (having magnitude only). Hence we may say that the first and second powers of a vector, X and X? (say), represent the two different ways in which it enteys all physical relations. For all other integral powers differ from these only in scalar magnitude. If the dimensional formule are to express dif- ferent natural relatwns between quantities, then, so far as the unit length is concerned (the representative of space), all these relations are involved in X and X?, X beinga vector. If this be true, then the fact that the indices of vector lengths, in the formule of dynamical quantities, are never higher than +2 becomes explained. Ina similar manner, we may explain the fact that the indices of M are always +1, for here no new idea is introduced by M? asin the case of vectors: it differs from M only in scalar magnitude, and still conveys the same physical idea, namely mertza. It is widely different in the case of time, for every new negative power of T introduces a new physical idea with respect to time—that is, it introduces the conception of a new time-flux. It is not necessary, however, to postulate anything as to these matters. It is sufficient for our present purpose to know what the limits of the indices of the fundamental units are in the case of known dynamical quantities, without having to account for the same. The above are only sug- gestions. | Let the dimensions of w be [v]=M” R(X", YZ"), ee ee ree elle | of Physical Quantities to Directions in Space. 261 then those of & are pee i PCR Vo eee Since the indices of M, X, Y, Zin the dimensional formule are odd multiples of 4, their indices in the dimensions of » must be odd, otherwise, on substituting for w and & in these formule their dimensional values, the formule would not be rational- ized. For a similar reason, the indices of R and T must be even. Hence, the indices of M, X, Y, Z must be +1, and of R, 0, or +2. In the case of Z, +1 is not admis- sible, otherwise we should have Z~° in the case of k. Thus, % must enter both w and kas Z'. In the case of Il’ the possible values are 0, +2, +4,....+2n. These values may be tabulated thus :— [ev] Lk] (a) (0) (a’) (0') 0 en +2 Os +2 —2 Oe: +4 +4 —4 —2 +6 +6 —6 —4 +8 +2n —2n —2(n—1) +4+2(n+1) Under a are given the positive possible values for ¢ in w, under a' the corresponding values in the case of k. Under b, the negative values in the case of w, and under 0’ those correspond ing in the case of k. The only cases necessary to be considered are T° and T?, for, as seen from the table, all other possible values of T lead to dimensional formule involving powers of T higher than +4. The dimensions of w and & involve only X, Y, Z, Mand T. This is of course obvious if KR has to coincide ultimately with X, Y,or Z. If Ris not ultimately to coincide with X, Y, or Z, it cannot enter into the formule of wand k. For KR can enter into the formula of « only as R? or R~*. In the former case, pw contains R, and w= contains R™*. But pw? is a factor of the formule of m, B,E, E, and pu? of e, D, C,C, H; and since R is a factor of the formule of all the above quan- tities, we should have :— i, m, B, E, H, containing R in their dimensional formule, li. ¢, D, H, C, C, containing R°: thus indicating that some of the forces and fluxes of the field would be dependent upon KR, the others independent of it. But this is obviously impossible, since NR is the instantaneous 262 Mr. W. Williams on the Relation of Dimensions linear displacement at a point in the field upon which depend all the electromagnetic forces and fluxes at that point. Ina similar way, if « involves R~*, m, B, E, E would be indepen- dent of R, and e, D, H, C,C dependent upon it, leading to the same conclusion as before. Thus, « and & must be quan- tities independent of the electromagnetic reactions (specified by R) which may be going on in the field, as is otherwise evident since # and & express physical properties of the medium. We have therefore to consider the different ways the dimensions of mw (say) can be built up from M, X, Y, Z and T, subject to the conditions already laid down. The possible dimensional values for » are to be obtained from M*! X** Y*' Z7' T°*? by forming all the possible com- binations of the quantities taken all together. The possible cases are :— °) Lol. MXaaeaae Ll... 1 MRS eo 2: MX Va 2. MXV Te a } MY Ye 3. MXY 72 er A MX ONG. a MX Ye Mit MOXY eZ AVS Mei pel ers eG Aaa 2) i eNo e 3, MG Nee: 3. 0M AX Yo a 4avlee 2 ee ee There are thus sixteen cases, and since for each value of w we have up to the present supposed that R may ultimately coincide with X, Y, or Z, or with neither, the different sys- tems which may be deduced from the above are sixty-four innumber. There are other conditions, however, which enable us to reduce this number. The quantities m, H, C, ande are scalar, und B, E, C, D vectors. Hence, the dimensional values of m must be such that when substituted in the dimensional formule, the scalar quantities remain scalars, and the vector quantities vectors properly directed with respect to the dimensional axes. Since all quantities are scalar so far as M and T are concerned, it is only necessary to examine how the scalar and vector character of the quantities depend upon X,Y, Z. To do this, we may take the relation [m]=y?/MRT(X?Y °Z?) J, and substitute for w successively the four factors (XYZ), (XYZ7'), (XY7Z"),\ (Kk YZ—). ...The conclusions des of Physical Quantities to Directions in Space. 263 duced for m must obviously hold for the other quantities. Thus :— 1. Let p? contain X-?Y=Z"?, Then [m] contains Cael) b( OY 37) — RY This becomes scalar only when R coincides with Y. 2. Let uw? contain X?Y?Z~*. Then [m] contains (X?Y2Z *) R (X?Y Z?) = RX. This becomes scalar only when R coincides with X. 3. Let uw? contain X?Y@Z"*. Then [m] contains Cer Aen v7 | Rx Your This becomes scalar when R coincides with Z. 4, Let w? contain X~=YZ"*. Then [m] contains i CS wi k Perey =k, This cannot become scalar. | Thus, in order to render the scalar quantities scalar, R must ultimately coincide with X, Y,or Z. There are thus three cases to be considered :— 1. When R coincides with Y, » contains (XYZ)~"- 2. 9 yr) ” X, 9 (XYZ) e 3. 3° be) 9) Z, 99 (AV UAT) f But, according to the electromagnetic theory of light, R cannot coincide with Z, for XY is the wave-front of an instan- taneous plane-polarized disturbance at a point in the medium— the disturbance ultimately originating in the displacement R, and since the medium is isotropic, the linear displacement specified by R can have no components along Z. We are thus restricted to the two cases where R coincides with Y, and the dimensions of « involve (XYZ)~, or where R coin- cides with X, and the dimensions of w involve XYZ. In the former case, the instantaneous linear displacement R of the medium at any point is in the plane of polarization (the plane of magnetic displacement), magnetic energy is kinetic, and electrical energy potential. In the latter case, the dis- placement R is at right angles to the plane of polarization, magnetic energy is potential, and electrical energy kinetic. Physicists are not yet agreed as to the interpretation of Weiner’s results, so that no arguments as to the direction of the instantaneous displacement of the medium at a point can be based upon his experiments. We have, therefore, to suppose 264 Mr. W. Williams on the Relation of Dimensions that Ii} may lie along either Y or X, and the possible systems become reduced to eight, as below :-— 1. MCQwe 2. M XVZ ee 3. M(XYZ"). A, MO Vee 5. MARAT 6. eye 7. Meera SMe eee If the dimensions of pw be (1) those of & are (2), and if those of k are (1), the dimensions of mw are (2), and similarly for the other three pairs. The dimensional formule 5 and 7 are unintelligible, for we have no dynamical units involving positive powers of both M and T, and it is difficult to give to such formule an un- telligible interpretation. The same difficulty appears in 6 and 8, for if one of these be the dimensional formula for pu, the corresponding one for kis 5 or 7. Apart, however, from this difficulty, these formule 5, 6, 7, and 8 lead to fluid theories of electromagnetism. Thus, from 5, we get for the dimensions of m:— [m] =*(M?RT—(X?Y 23) ]=M?(X ?Y 24) R(xPY 2?) MT *T?=MRY"*=M (since R here must coincide with Y). Similarly, from 7 we have [m]=MX’. Again, from 6 we get [e]=M, and from 8, [e]=MY’. Thus, we have to suppose m to be a quantity of the nature of mass or moment of inertia, and similarly for e. In both cases, mand e would be pro- perties of the medium depending upon its inertia, instead of being parts of the electromagnetic reactions going on in the medium,—an electrical current would thus be a quantity of the nature of momentum, which is inconsistent with Maxwell’s theory of electromagnetism. Other and not less serious difficulties will be met with if an attempt is made to develop their interpretations more fully. Similar remarks apply to the systems arising from 38 and 4. Thus, from 3 we have fond Ee Mee Woe YAEL CL le]=u *[MR(X*Y°Z *) | Se SSS eee poet: Wee Oe) Feta bk =M?X *Y *Z2MeRX *Y¥*Z *#=RX =I, since R here coincides with X. Thus, e is of the di- mensions of a volume-strain, the ratio of two identical concretes. Similarly, we get from 4, [m]|=1, also a volume- strain. Again, in the former case we have D=e(YZ)~* =volume-strain per unit area. The two currents C and C of Physical (Quantities to Directions in Space. 265 become volume-strain per unit time (I~, or Y'Z TI’). Electrical foree E becomes MXT*, and voltage HE =MX?T~ (energy). Since m=ET, m becomes MX?T™’, - and magnetic moment ml = MX°YT™. Again, B magnetic eos becomes m(XZ)~' = MXZT™; and H magnetic force (Y'T~*). Again, in the second case, we have B=m(XZ)*=volume-strain per unit area; H=MT™ = 0 *=yolume-strain per unit time ; e=MY'T: a Oe MYT *(energy) and C (current density) =MY?2(YZ) "TI. These formule, although it may be possible to give them in- terpretations, are at present unintelligible, and do not suggest any connected relation between the various quantities. The cause of the unintelligible character of these latter formulz lies in the fact that the formula MXYZ™ is not sym- metrical with respect to the dimensional axes X, Y, Z. Since the above formula is independent of T, the corresponding quantity must be of the dimensions of some property of the medium depending ultimately upon the znertia of the medium alone. It is difficult, however, to conceive what property depending upon the inertia of the medium can involve the three dimensional axes unsymmetrically except the rotational inertia, or moment of inertia of a unit volume. The above formula, however, does not admit of such an interpretation. Thus, whether MX YZ be the dimensional formula of yu or &, the results are either incapable of interpretation (as, for ex- ample, » and k), or the interpretations are unintelligible. Thus there are left only cases 1 and 2. If the dimensions of pu be those of 1, that is M(XYZ)*, uw becomes the inertia of unit volume, or the density of the medium, and magnetic energy is kinetic. If the dimensions of pw be 2, those of k are 1,and & becomes the density of the medium, and electrical energy is kinetic. There are thus two cases to be discussed. I. Magnetic Energy Kinetic.—p the density of the medium. 1. Magnetic foree=H=p “*[M?RT (XYZ) *]=RT = YT’, since R must now coincide with Y. This is the linear velocity directed along Y the magnetic axis. 2. wien induction=B=Intensity of magnetization, Teer 15 =(xyz)5 = Linear momentum per unit volume. 3. Magnetic moment=ml=I(XYZ) =MYT~™ = Linear momentum. 4, Current strength=C=HY=YT. 266 Mr. W. Williams on the Relation of Dimensions 5. Current density =C=C (Y4) =YZ> 1 >= Angus velocity. The velocity is instantaneously directed along Y, the magnetic axis, and the axis of rotation is X, the axis of the electrical displacement. If the angular velocity be that of a vortex filament coinciding with the current, the strength of the current, C, becomes the strength of the vortex—product of angular velocity into cross-section of filament. 6. Electric displacement =D=CT= YZ *=an angular dis- placement. [According to the elastic solid theory, D would be of the dimensions of a shear, the planes XY being displaced parallel to each other in the direction Y. | @. Electrical force=E. Since [ED] =MY(XZ)"T?, an [D]=YZ", E=MX“T™*. Taking D as an angular dis- placement, E must be of the dimensions of a torque, for ED is of the dimensions of work done per unit volume. [Leta cylinder rotate in a resisting medium. The resistance to the rotation is a tangential force per unit surface of the cylinder, directed everywhere parallel to the motion. If the axis of the cylinder be X (the electrical axis), and the radius Z, an element of its surface is of the dimensions YZ. Hence, the resisting stress is of the dimensions MX~*I*=[E]. Thus E is a tangential force per unit area. Now E has a moment round X. Hence we have DY Ee? Speragow e 82 indicating that the resisting stress over an element of surface is inversely as the distance of the element from the axis of rotation, and its moment directly as that distance. | —2 8. Voltage—Writing [E] = Bie we get ee [EB] =(—7— Now H, the line integral of E, is fe source of the disturbance in a closed electrical circuit. Hence, taking the above mechanical illustration, E becomes the terminal torque re- quired to maintain the motion of the cylinder against the resistance. “The terminal torque corresponds to the im- pressed voltage. It should be so distributed over the end B” (in this case a rotating tube) “that the applied force there is a circular tangential traction varying inversely as the dis- tance from the axis” (Mr. Oliver Heaviside, Hlec. Jan. 23, 1891). In the above case, an element of the tangent at a point is Y, and of the radius Z. So. of Physical Quantities to Directions in Space. 267 Bs) 9. Vector potential = A= ET = MX UT t= ss = angular momentum per unit volume. 10. Specific resistance. 1) | aad a = coefficient of viscosity = tangential force per unit area + shear per unit time. 11. Self-induction.—Let a linear conductor carry a current C. Its self-induction is given by L =e or dimensionally MY’. If L be defined as the self-induction of the conductor for a current of unit density, ng gat —1 pee ty SEZ? b= 2=MT'(Y2Z- Ty '=MZY'= yy . s . M =moment of inertia per unit area; or = = Z? = moment of inertia of a disk of unit mass per unit area. 12. Specific inductive capacity = k =|MZ(XY)'T-?]-". On the elastic solid theory —! would be defined as the rigidity of the medium. But k—!1=RT—!, where Ris specific resistance. Thus the interpretations of k and R go together. If R (as above) be a coefficient of viscosity, k—1 becomes “a quas: rigidity ” of the medium “arising from elastic resistance to absolute rotation.” (Mr. Oliver Heaviside, Elec. Jan. 23, 1891. 15) Blectrical charge = g = D(YZ) = Y?= (Product of angular displacement into area). 14. Magnetic pole = m = ee = Magnetic moment per unit length = Linear momentum per unit length. The connexion between Vortex Motion and Hlectromag- netism is shown in Basset’s ‘ Hydrodynamics,’ from which the following are extracted (vol. 1. chap. iv.) :— 1. Vortex filament = Electrical current. 2. Velocity of liquid (linear velocity, components u, v, w) = Magnetic force (components a, 8, 7). 3. Molecular rotation (angular velocity, components &, 7, %) = Current density (components u, v, w). 4, Velocity potential due to vortex (f) = Magnetic po- tential of current (Q). 5. Doublet sheet = Magnetic shell. 6. Circulation (£) = Work done in moving a magnetic pole once round current, 208 Mr. W. Williams on the Relation of Dimensions 7 Flux through vortex = Potential energy of magnetic shell, 8. The action of a vortex filament upon the surrounding liquid is determined by the quantities L, M, N. These cor- respond to the components F, G, H of electromagnetic momentum (A). Thus, when u is of the dimensions of density and magnetic energy kinetic, the interpretations of the dimensional formule of electromagnetic quantities are identical with those of the _ corresponding quantities in the case of vortex motion. In the Phil. Mag. vol. xxxi. p. 149, Prof. J. J. Thomson develops a method of representing electromagnetic effects by means of tubes of electrostatic induction distributed throughout the field. The axes of these tubes coincide with the axes of electric displacement, and the tubes terminate normally upon the surfaces of charged bodies. By means of these tubes it is possible to picture the changes going on when electricity passes through electrolytes and conductors, and when changes are produced in the electromagnetic field. If, now, we suppose these tubes of electrostatic induction to be vortex filaments, this method of representing electromagnetic phenomena becomes identical with that indicated by the interpretations above given to the dimensional formule. The two rotational theories of electromagnetism which have been hitherto put forward are discussed by Mr. Oliver Heavi- side (Hlec. Jan. 23, 1891). The first of these is that already deduced above from dimensional considerations, by supposing pv. to he of the dimensions of density. Here the axis of ro- tation at any point is the axis of the electric displacement ; the angular velocity is the current-density ; the velocity is magnetic force ; the torque called into play when the angular velocity of a vortex changes is the electric force; the terminal torque, the source of the disturbance, is the voltage; the relative angular displacements of the vortices when their velocities change is the electric displacement; the linear momentum of the irrotationally moving liquid is magnetic in- duction or intensity of magnetization; the angular momentum is the vector potential ; the strength of a vortex is the strength of the current ; specific resistance is a quantity of the nature of viscosity, and k—! of the nature of rigidity. In the other system things are inverted. Here & is the density of the medium, electrical energy is kinetic, and the axes of the vortices coincide with the directions of the magnetic dis- placements. The interpretations of the formule of the various quantities are as below. of Physical Quantities to Directions in Space. 269 II. Electrical Energy Kinetic—k the density of the medium. 1. E=k [MRT (XY #Z-4)] Mi ( XYZ) WRT (xX *Y *74)— ei xT (since R here must coincide with X). Thus, E is the velocity of the medium. 2. #4 =EX=X°T™. Magnetic current. (See below, its interpretation depends upon that of m and B.) oom — BT = X?, 4. B=m(XZ)“=XZ" = an angular displacement. Thus, m = product of angular displacement into cross section. 5. Cm=BI“XZ'T = Angular velocity. Thus, H the strength of the magnetic current corresponds to the stren eth of a vortex, C,, being the density of the magnetic current. 6. e=MT™. This corresponds to magnetic pole in the previous case. =i 7. D=e(YZ)'=M(YZ)"T*= eee = Linear momen- tum per unit volume. \—2 8. C =0I'=(y7-)= Force per unit volume. 9. C=MT’. This corresponds te voltage in the previous case. 10. H=C(Y")=MY"T™. Tangential force per unit area, a quantity of the nature of a torque, corresponding to E in the former case. It is thus seen that this system is simply the inverse of the other, and it is unnecessary, therefore, to go into more detail. To summarize, therefore, we may say that, of the eight possible systems previously mentioned, only two give rise to dimensions whose interpretations are intelligible, natural, and connected as a whole; and these interpretations accord with the two rotational theories of electromagnetism which have been hitherto put forward. Of the other six, it was shown that four necessitated, in some form or other, fluid theories of electricity or magnetism, and that the interpretation of the formule of w and £ and other quantities are difficult. These remarks also apply to the other two cases, in which electri- fication and magnetization became respectively volume-strains. All other cases were disposed of by excluding :— (1) All values of y leading to dimensions involving higher Phil. Mag. 8. 5. Vol. 84. No. 208. Sept. 1892. U 270 On the Dimenstons of Physical Quantities. er lower (fractional) powers of the fundamental units than those encountered in the case of ordinary dynamical quantities- (2) All values which while satisfying (1) rendered scalar magnitudes vectors, or vectors scalar. . (3) All values which while satisfying (2) rendered the direction of the instantaneous linear displacement at a point in an isotropic medium parallet to Z, and therefore normal te the plane of displacement (XY). In the foregoing discussion I have attempted to generalize the ordinary dimensional expressions for physical quantities, and to carry somewhat further than he did the suggestion of Prof. S. P. Thompson that the idea of direction may be associated with the symbols as well as that of numerical magnitude. This is clearly possible—as the examples givers show—in the case ef ordinary dynamical quantities, and the method certainly enables us to distinguish between things which are physically different, though (if the ordinary system: in which the idea of direction is suppressed be used) the dimensions are the same. ‘The dimensional formula thus becomes for a physical quantity the analogue of a structural formula for a chemical compound.. Applying these ideas to the more difficult case of electrical and magnetic quantities, with the actual nature of which we are imperfectly acquainted, I have tried to use the method as an instrument for discriminating between probable and im- probable hypotheses. It is at all events, interesting to note that we are thus led to two possible dimensional systems which agree with the two principal groups of analogies by which electrical and magnetic facts have been illustrated by those who have most deeply studied these from the dynamical point of view. Between these two I do not attempt to decide ; but I cannot but hope that the methods I have suggested may make the study of dimensional formule not merely a con- venient method of expressing numerical relations between fundamental and derived units, bat a means of seeing more deeply into the physical facts they represent. Notre.—I regret that when I communicated the above paper to the Physical Society I was unaware of an important article by Prof. A. Lodge on “The Multiplication and Divi- sion of Concrete Quantities” (‘ Nature,” July 19, 1888). In this article Prof. Lodge discusses the general question of the products and quotients of concrete quantities, and of the meaning of physical relations between concretes of different — kinds. He clearly points out that the dimensions of quan- tities do not always afford a test of their identity, and that in Equivalent Resistance §¢. of Parallel Cirewtts. 271 particular concretes of the order of angles and angular dis- placements are treated as pure numbers. The present paper may be regarded as an attempt to remove such difficulties by taking as fundamental in our theory of dimensions the vector instead of the cartesian unit length. By expressing the for- mul according to the conventions of vector algebra, we thus assign dimensions (in the extended meaning of the term) to all quantities which are physically recognized as concrete, the only quantities having unity as their dimensional formula being pure numbers and quantities of the nature of tensors, that is, ratios between similar and similarly directed concretes. Since, as pointed out by Prof. Lodge, the cause of the above difficulties lies in the neglect of the directional relations of quantities, these modified formulze approximate more closely to what they are sometimes, and conveniently taken to be, namely conventional expressions of the kinds of different physical quantities. XXVIII. Eyguivalent Resistance, Self-Induction, and Capacity of Parallel Circuits with Harmonie Impressed [lectromotive Force. By FReperick Brepeut, PhA.D., and ALBERT C. CreHorek, Ph.D.* N a paper on “Forced Harmonic Oscillations of various Periods,’ in the Philosophical Magazine for May 1886, Lord Rayleigh derived analytically expressions for the equi- valent resistance and self-induction of parallel circuits in terms of the resistance and self-induction of each branch. The object of this paper is to obtain by other methods similar expressions for the equivalent resistance, self-induction, and capacity of any number of parallel circuits in terms of the resistance, self-induction, and capacity of each branch. From these general results, particular expressions for the case of circuits with resistance and capacity alone, and for the case of circuits with resistance and self-induction alone, are readily derived. The results in this latter case are identical with those given by Lord Rayleigh in the paper referred to. Resistance is denoted by R, self-induction by L, capacity by C; and equivalent resistance, self-induction, and capacity by BR’, L’, and C’ respectively. The currents (maximum value) in the main line and branches are denoted by I,,, and by I,, 1,, &e. Gis the angle of advance or lag according to whether it is positive or negative. w is the angular velocity, 2.e. 2m x frequency, and H the maximum value of the impressed electromotive force. * Communicated by the Authors. 242 Drs. Bedell and Crehore on the Figs. 1 and 2 are triangles of E.M.F.s for simple circuits with resistance and self-induction, and resistance and capacity Fig. 1. WZ KE aan E p/p Gu™ respectively. Rotation is in all cases counter-clockwise, and the impressed E.M.F. is considered as the sum of that neces- sary to overcome the resistance and the self-induction or Fig. 3. capacity. Fig. 3 is the H.M.F. triangle for a single circuit with resistance, self-induction, and capacity, the current being Equivalent Resistance &c. of Parallel Circutts. 2738 in advance of the impressed E.M.F. in this particular case, in which + is taken as greater than Lo. Cw Fig. 4, In fig. 5, OBA, OCA, ODA are the E.M.F. triangles for the several branches of a divided circuit (see fig. 4), with an impressed E.M.F. equal to OA. The main current, I, is the vector sum of the currents I,, I,, I; in the branches. By taking the projections of the currents upon the line OA, and then upon a line perpendicular te OA, we obtain the equations I,. cos 9=I, cos 0, +1, cos 6,+...=2Icos6.. . (1) I, sin =I, sin 0,4 I,sin 6,+...=ZIsin@. . . (2) In the right triangle OF A, K a L,= a = sa ea (3) Rn +( on —Lie) I@W 274 Drs. Bedell and Crehore on the . cos = SSS ae a= ~ Ue ) : / a, — le >: reer .. «en Substituting in (1) and (2) these values, and similar values for I,, cos 6,, sin @, ; and I,, cos @, sin @, &e.; we have a Leas - Rk’ R i 7 Glia (pega R Ha Le) R'+(— Le) Lt ; : ij Py eee wae = ql iH ‘ Rize ey R+(Go—-Le) C’w Cw C—C*Lo’ = 2 iets Clef 1 The letters A and B are introduced to simplify the results. Dividing (7) by (6), B 3 tan 0= =. bok i is, ee or Comparing (4) and (6), we obtain 2 2 A= a or R= “7 oe (9) Comparing (5) and (7), we obtain sin? @ 1 in? 6 Bo= =———, or Gg —Lo="F—. . (10) eo For cos? @ and sin? @ we may substitute the values cos? G= Bees | > 2) foveetstan’ oo 1 Bw? A?+ Be” + Xe in? eee ee ~ I+ cot? Oo, ie Hquivalent Resistance &e. of Parallel Circuits. — 2°75 With these substitutions equations (9) and (10) become A R= ay Ro? ° * ° ° (11) 1 ; Bw eae eo” eee Here A and Bo each stand for a summatien, as expressed in (6) and (7), and are calculated from the particular values of the resistance, self-induction, and capacity of each branch. This gives a definite value to the equivalent resistance, R’, aecord- 1a . Wa —L’w, according to (12). There may be an indefinite number of values assigned to L’ or C’ according to values assigned to the other. If the right-hand member of (12) is positive, we may consider that the equivalent circuit has no self-induction, 7. e. L’=0, and calculate the equivalent capacity. If this member is negative, we may consider that the equivalent circuit has no condenser, 7. ¢. C’/=0, and_ calculate accordingly the equi- valent self-induction. The angle of advance or lag of the main current is obtained from equation (8). If there is no condenser in any branch, the expressions for A and Bo are obtained by substituting C=o0 in the summa- fions in (6) and (7), thus A= > ing to {11), and a definite value to R?+ L?o” and ay R?+ L?w? Equations (11) and (12) with these values for A and Be give the equivalent resistance and self-induction of parallel circuits containing resistance and self-induction only. These expres- sions are the same as those obtained by Lord Rayleigh. If there is no self-induction in any branch, the substitution of L=0 in (6) and (7) gives Bo = > ree se ae oe : il and i Caw Co Dee eee sae 12 4. D CO? Rew? + 1 276 Dr. F. T. Trouton on the Theory of Use of a With these values, equations (11) and (12) give the equi- valent resistance and capacity of parallel circuits containing resistance and capacity only. Physical Laboratory of Cornell University, July 1892. XXIX. On the Theory of the Use of a permanently Magne Core in the Telephone. By Frup.T. Trouton, M.A., D.Sc.* N the development of the Bell Receiver it was found in practice that a louder effect could be obtained by employ- ing a magnetized core instead of a plain soft-iron core. This appears to be at variance with ordinary experience, it being generally supposed that for a maximum effect the core in electromagnets should be of soft iron. An explanation of this which seems to be commonly received attributes the improved results to the diaphragm of the receiver being kept permanently in a taut condition, it being supposed to be thus for some reason in a hetter condition for taking up the vibrations than when left*relaxed. However, no very elear ideas on the subject seem to be current. For some time past I have been experimenting with the object of clearing up this point; the result of which is to almost entirely refer the improved effect, obtainable by per- manently magnetizing the core of the telephone, to the fact that the mechanical force on the diaphragm, or armature of the electromagnet, is proportional to the square of the mag- netic force. Suppose, for example, H to be the magnetic force at the diaphragm due to a permanent magnetization in the core, we may take the mechanical force on the diaphragm to be a constant multiplied by H?; and should H vary by an amount oH through the passage of a current of ordinary telephonic dimensions, we have the corresponding variation (8F) in the mechanical force, which for loudness in the telephone is the important factor, dependent on the amount of the permanent magnetization ; thus )F=C.H6dH. In this way it is seen that the variations in the force on the diaphragm, and conse- quently its amplitude of vibration, will depend on the per- manent magnetization as well as on its variation produced by the currents passed through the telephone. Or, to look at it another way, it is important that initially the armature have strong magnetic poles ready to be acted on by this variation T. * Communicated by the Author. + This would suggest magnetized steel diaphragms for louder effects. permanently Magnetized Core in the Telephone. 277 Another advantageous result arising from permanently magnetizing a soft-iron core by the addition of a steel magnet (as in a common arrangement of the telephone) probably is that the permeability of the soft iron is increased thereby; so that our 6H itself (that is to say, the magnetic force at the armature arising from the current) is greater than were the current to act on an initially unmagnetized core. This effect probably is always unimportant compared with the last. The effect here alluded to may perhaps be better understood by reference to the well-known curves of magnetization (fig. 1), Fig. 1. where it will be at once seen that equal small variations in the magnetic force (horizontal direction) produce effects in the magnetization (vertical direction) dependent in amount on the initial position on the curve. An apparatus found convenient in these experiments is shown in fig. 2. It consisted of an ordinary tambourine, LLLLLLLA inn armed at the centre with a little piece of iron, and thus could be made to vibrate by an alternating current passed through an electromagnet with a soft-iron core placed telephonewise close up to the little piece of iron. The core could be perma- nently magnetized to the extent desired, either by a current passed through an additional coil wound on for the purpose, or by bringing up in line with the core a permanent magnet to a variable distance. 278 Dr. F. T. Trouton on the Theory of Use of a With such an apparatus it is easy to arrange for showing the gain in sound brought about by permanently magnetizing the core of the electromagnet. For this purpose an alter- nating current is employed of insufficient intensity to affect the diaphragm much; one then brings up the permanent magnet to the position shown in the diagram, on which the sound is found to be immensely improved. A similar result of course can be ebtained by a permanent current sent round the additional ceil. To distinguish between the parts of this improved effect due respectively to the two causes mentioned above, one can either connect up the additional coil wound on the electro- magnet with an electro-dynamometer or with a telephone. if the improved action on bringing up the steel magnet were entirely due to the second cause (namely the fact that then the same current is able to produce more lines of force in the iron than before), the deflexion of the dynamometer or the sound in the telephone would increase in like proportion with the loudness of the tambourine. The experiment proves that though there is undoubtedly a slight increase in the number of lines of force, shown by aslight increase in the defiexion of the dynamometer, the chief improvement is to be referred to the other cause. A similar experiment was made with the additional coil wound en the armature instead of on the core. This was effected by taking a rather thicker piece of iron than that which had been previously employed for the arming of the diaphragm of the tambourine. On connecting up a telephone with the coil, it could be arranged so as to have the sound from the telephone and from tke tambourine, with the steel magnet removed, of equal intensity ; then, on bringing up the magnet as before, the sound of the tambourine is found to be far and away louder than the telephone. Lest it might be thought that the increased effect could be due to the diaphragm coming nearer to the electromagnet in consequence of the attraction arising from the permanent magnet, it may be mentioned that similar effects are pro- duced on placing the magnet on the opposite side to that as shown in the diagram. In this position the diaphragm lies between the magnet and the electromagnet, and consequently tends to be drawn further away from the electromagnet. A somewhat curious effect is to be observed with this apparatus on employing an intermittent current instead of an alternating one. With an alternating current a continuous increase in sound is of course produced indifferently by either - pole of the permanent magnet as it is brought up from a dis- permanently Magnetized Core in the Telephone. 279 tance to the position of contact with the core. Not so with an intermittent current. One pole effects, as before, a con- tinuous increase in the intensity of the sound as it is brought up from a distance to the position of contact with the core ; but the other pole at first, as it is brought up, steadily d¢mz- nishes what small amount of sound the current alone may be capable of producing; reaching a minimum, the sound again begins to increase, and continues doing so until the position of contact is arrived at. In this case the permanent magnet is opposed (that is, lies-in the opposite direction) to the mag- netization produced in the soft iron by the intermittent cur- rent. The position of minimum sound evidently is that in which the average ‘permanent magnetization resulting from the intermittent current is just neutralized by the permanent magnet. Tor, as we have seen already, a small change in the magnetizing force applied to an electromagnet should pro- duce least mechanical force on the armature under such circumstances. A few rough experiments bearing directly on this subject were also made, an account of which may perhaps not be out of place here. A long thin cylinder of soft iron was attached to one arm of a balance, and hung down with its lower end dipping into a coaxial solenoid consisting of two separate layers of wire, through each of which independent currents could be sent. One coil (A) was used to produce various intensities of field by the passage of various currents. The other (B) was for effecting a small increment in this magnetization by the passage of a small current of constant amount through it. In each case equilibrium was restored to the balance by the addition of the necessary weights. It required ‘03 gram to balance the small current in B when acting alone. It required in all *15 g. for equilibrium when a current was running in A, which acting alone was balanced by ‘09 g. Again, when this last was increased so as to require ‘52 g., the two coils acting together required *72 g. Here the same provocation produced an effect of 6 and of 20 when under these two circumstances. The numbers themselves agree with theory as well as could be expected in the case of such a crude experiment. We deduce from the considerations given pre- viously, SF /8F/ = V B/v F’. Putting in the numbers, we have respectively 20/6 or 3°3, and V52/V9 or 2°4. [ 280 ] XXX. On the Application of Interference Methods to Spec- troscopic Measurements.—II. By AtBert A. MicHELson*. [| Plates V.—VIII.] f piaa theoretical investigation of the relation between the distribution of light in a source, as a function of the wave-length, and the resulting “ visibility curve” has been given in a paper bearing the same title as the present one in the ‘ Philosophical Magazine’ for April 1891. The physical definition of “ visibility ” there adopted is I,—I, ana in which I, is the intensity at the centre of a bright inter- ference-band, and I, the intensity at the centre of the ad- joining dark band. In order to interpret the actual curves obtained by observation of interference-fringes, it is first necessary to reduce the results of the eye-estimates of visi- bility, which may be designated by V,, to their absolute values as above defined. For this purpose two quartz lenses, one concave and the other convex, and of equal curvatures, were mounted with their crystalline axes at right angles to each other between two Nicols. Under these conditions a series of concentric interference-rings appeared. If a be the angle between the principal section of the polarizer and the axis of the first quartz, and » the angle between the axis and the analyser, the intensity of the light transmitted will be 3 : : K(ty—t I=cos’(m—«a) —sin 2e sin 2w sin 8) where ¢, is the thickness through the first quartz and ¢, that through the second. If the analyser and polarizer are parallel, a=, and J=1—sin?2« sinter 4) : whence Lei, and I,=1—sin?2«, I,—I, | 1—cos?2e ams 4 1,41, 1+ cos?2«° * Communicated by the Author. f I take this opportunity of presenting my acknowledgments and thanks to the Smithsonian Institution for the funds necessary to carry out this research ; to the Clark University for the facilities it has placed at my disposal; and especially to Mr. F. L.O. Wadsworth, Assistant in Physics of Clark University, for the valuable services he has rendered and his unflagging zeal in furthering this investigation. On Spectroscopic Measurements. 281 This curve, together with the mean of a number of eye- estimates, is given in fig. 2, Woodeut. From these the following table of corrections may be obtained :— Vie Cor. Vv. Cor. ‘00 "00 "05 —'12 "05 +:03 60 —'14 "10 +04 "65 —'15 -hd +°03 “70 —'16 20) +°02 ‘to —'16 “ZO "00 *80 —'14 “30 —'03 °85 —'13 °35 —'05 ‘90 —'ll “40 — 07 "95 —'08 “45 — 08 1:00 "00 “50 — 10 The curves show a general tendency to estimate the visi- bility too high when the interference-bands are clear, and too low when they are indistinct. This tendency may be modi- fied by a number of circumstances: thus, it increases with the refrangibility of the light used; it is greater when the field contains a Jarge number of bands than when there are but few; it is greater while the visibility-curve is falling than when it is rising ; it does not seem to be greatly affected by the intensity of the light; finally, it varies on different occasions and with different observers. Notwithstanding these disturbing causes, the result, after applying the correc- tion, will rarely be in error by more than one tenth of its value, and ordinarily the approximation is much closer than this*. * The formula for visibility deduced in the preceding paper is in which C =\$(x) cos kxd2, S =\9(2) sin kedx, P=|9(w)de, k=2rD, D= Difference in path, and (x) represents the distribution of light in the source. In this expression no account was taken of the effect of extraneous light, and it was assumed that the two interfering pencils were of equal intensities. It can be shown that the error due to both these causes 282 Mr. A. A. Michelson on the Application of As stated in Part I. of this paper, the observations neces- sary to construct the visibility-curves, from which the distri- bution of light in any approximately homogeneous source is to be deduced, may be made with any form of interference apparatus which allows a considerable alteration in the difference of path between the two interfering streams of light. The apparatus actually employed for this purpose was de- signed for the comparison of wave-lengths, and while admi- rably adapted for the observation of visibility-curves, it con- tains many parts not necessary for this use. Fig. 1, Woodcut, presents the plan of an arrangement which, while showing all the essential parts, is much less complicated. Starting from V,a vacuum-tube containing the substance whose radia- tions are to be examined (and which is usually enclosed in a metal box in order that it may be raised to any required temperature), the light is analysed by one or more prisms forming a spectrum from which any required radiation mat be separated from the rest by passing through the slit S*. 5 The light from §$ is rendered nearly parallel by a colli- mating lens, and then falls on a transparent film of silver, on tends to lower the visibility; but in either case the correct values may be obtained by multiplying by a constant factor. In the first case, let e be the intensity of the extraneous light, and V’ the resulting visibility ; then, by definition, Moe (I,+e)— (+e) the Lees Stare 3 2e Sect Geert ees L-L V= (40+, 40 ~ hth 7 7° eee whence V=(1+7)V’. In the second case, let p be the ratio of intensities of the interfering pencils ; then it can readily be shown that the resulting intensity is I=(1+ p’)P+2(C cos$—S sin 9), and hence the visibility is yr 20 MOPS? zi _ + iW) oD “ 1 2 whence V= = SY 2 If the interfering pencils differ by 25 per cent., the factor in differs from unity by about 4 per cent.; so that, in most cases, this cause of error may be neglected. 5 * In the case of close groups of lines, the image of the source is first thrown on a slit; otherwise the lines at S would overlap. Interference Methods to Speetroscopie Measurements. = Fig , 20° 30° Bquotion Dovd Carve — Ball Curve - V — pte wt 20 Bye Bistimabes = We ca LS wtta Se ees Se Ee ES 283 fi Ce 284 Mr. A. A. Michelson on the Application of the surface of the plane parallel plate G,*. Here it divides, part being transmitted to the fixed plane mirror M, and part reflected to the movable mirror M,. These mirrors return the light to the silvered surface, where the first part is re- flected and the second transmitted; so that both pencils coincide on entering the observing-telescopef . A little consideration will show that this arrangement is, in all respects, equivalent to a film or plate of air between two plane surfaces. The interference phenomena are, there- fore, the same as for such an air-plate. The theory of these interference-bands has been given in an article entitled “‘ Interference Phenomena in a new form of Refractometer,’ Philosophical Magazine for April 1882. As is there shown, the projections of the bands are, in general, conic sections, the position of maximum distinctness being given by the formula = $ tan 7 cos’@, in which ft, is the thickness of the equivalent air-plate, where it is cut by the axis of the telescope, ¢, the inclination of the two surfaces, 6 and i, the components of the angle of inci- dence parallel and perpendicular respectively to the inter- section of the surfaces, and P, the distance of the plane of maximum distinctness from the surfaces. If @ be small the variations of P with @ may be neglected, and we have then cdo ~ tand or with sufficient accuracy, tan 2, From this it will be seen that the focal plane varies very * The light entering the telescope is a maximum when the thickness of the silver film is such that the intensity of the transmitted light is equal to that of the reflected light. The silvering has another important advantage in diminishing the relative intensity of the light reflected from the other surface. Indeed, for this purpose it is advisable to make the film heavier ; even so thick that the reflected light is twice as bright as the transmitted. This does not affect the ultimate ratio of intensities of the interfering pencils—for what is lost by transmission on entering the plate G, is made up by reflexion on leaving it, the effect being simply to diminish somewhat the whole intensity. Another advantage of the thicker film is that it can be made uniform with far less difficulty than the thin film. It may be mentioned that with this form of instrument the interference-fringes in white light present a purity and gorgeousness of coloration that are surpassed only by the colours of the polariscope. + The second plane parallel plate G, is made of the same thickness as the first, and is required to equalize the optical paths of the two pencils. Interference Methods to Spectroscopic Measurements. 285 rapidly with 2, so that, unless ¢=0, it is impessible to see all parts of the interference-bands in focus with equal distinct- ness. If, however, ¢=0, that is,if the two surfaces are strictly parallel, then P=, and if the observing-telescope is focused for parallel rays, all parts of the bands are equally distinct. Under these circumstances the interference-fringes are concentric circles, whose angular diameter is given by A s=—. cos Diy If for A we put 2%—nd, and for cos 3 its approximate 92 value 1— —, we have De nr nu to * In order to obtain an idea of the order of accuracy required ‘in this adjustment, suppose the angle 3 to be so small that its influence on the distinctness may be neglected. The intensity at the focus of the observing-telescope will be 2ar I ==(( cos’$xAdaedy, where x= > ° If the aperture be a rectangle whose height is 2b, and width 2G, - ji +a ce 26 cos*4«Adz. —a But A=2(to+ $2), whence sin 2«pa ae a ie sale aa 7, | a+ cos 2Kty Sach ] The maximum value of I is sin =] oe E z 2np 4” and the minimum value is sin 2xg@a 2b | a— ge |; whence ae sin 2npa fale Qicpa In attempting to verify this formula by actual observation, one is met by the difficulty that all parts of the bands are not in focus at the same time, the right and left bands being more distinct than the central one, to which attention ought to be directed. Notwithstanding the rather rough character of the Phil. Mag. 8. 5. Vol. 34. No. 208. Sept. 1892. x 286 Mr. A. A. Michelson on the Application of observations, the results agree fairly well with theory. If do is the ratio of the wave-length to the width of the rectangular aperture, the above formula becomes _ sin 277o/do ar 27 $/Po 4 from which the second column in the following Table was calculated. b/Py: V (cale.). V (obs.). 0-0 1:00 1:00 lk "94 “94 2 "79 73 3 "00 “40 “4 "24 ae "OD ‘00 ‘09 “6 "15 ZN at 22 ‘09 8 “19 07 Oy pare eas ‘05 1g (ce 00 "04 From this table it appears that if the visibility is to be estimated by observations with a telescope of 12 millim. aper- ture (or with a circular aperture about one fourth greater), an error in the adjustment of the surfaces of a second of arc would produce a diminution of 4 or 5 per cent. in the visibility. Accordingly, if the ways on which the mirror-carriage moves are not true to this degree, it is necessary to make the adjust- ment for every observation. This can be done with very great accuracy by moving the beam of light from side to side and adjusting the mirror until there is no perceptible alteration in the size of the rings. Since the admissible error in adjustment is inversely propor- tional to the aperture, the observations may be facilitated by making this as small as possible if there be light to spare. This is all the more necessary for the same reasons, if the surfaces be not true. However, the error due to this source may be easily corrected (since all the observations are affected alike) by multiplying by a constant factor. In order that the visibility-curve may extend as far as pos- sible, it is necessary that the vapour should be very rare. Accordingly, in all but a few cases to be mentioned later, the substance to be investigated was enclosed in a vacuum-tube, which was previously heated to drive off any moisture or occluded gases. The vapour was rendered luminous by the discharge from Interference Methods to Spectroscopic Measurements. 287 the secondary of a large induction-coil, whose primary cur- rent was interrupted by a rotary break attached to the arma- ture of an electric motor, making about 20 to 30 breaks per second. The steadiness of the light thus obtained was far greater than with the ordinary Foucault interrupter. Probably it would have been still more satisfactory to use an alternating dynamo properly wound to give a strong current with compa- ratively few alternations. The box surrounding the vacuum-tube was heated just suffi- ciently to give a steady bright light, and the temperature then kept as nearly uniform as possible. This temperature was usually taken to represent that of the vapour within the tube. This is, of course, only a rough approximation to the truth ; and in some cases the estimate was much too low. As it was not intended to include in the present work an elaborate study of the effect of temperature, this matter was not of great consequence. It may be suggested, however, that a very much closer approximation to the real temperature could be obtained by winding a platinum wire about the capillary portion of the tube, and deducing the temperature from the variation of its resistance. A preliminary experiment in which a platinum wire passing through the tube and heated by a current until the platinum spiral outside the tube was raised - to fixed temperatures, would give a means of deducing, from the indications of the spiral, the true temperature within the tube. These adjustments being effected, the screw of the “ wave- comparer ” was turned to zero; that is, till there was no dif- ference of path between the interfering pencils. At this point the visibility should be as great as possible, and was accordingly marked 100. The screw (of 1 millim. pitch) was then turned through one turn, thus giving a difference of path of 2 millim., and the visibility again estimated, and so on. The curve was then drawn, giving the estimated visibility for each 2 millim. difference of path ; and this was corrected for the personal equation, as before described. Hydrogen. The full curve in fig. 30, Pl. V., represents such a curve for the red hydrogen-line* at a pressure of about 1 millim. and a temperature of about 50° C. * The hydrogen was prepared by dropping distilled water upon sodium amalgam, and allowing the gas to pass through sulphuric acid into the vacuum-tube, which was repeatedly exhausted until the spectrum of hydrogen was nearly pure. _ 2 288 Mr. A. A. Michelson‘on the Application of The dotted curve represents Van cos *7/30 *. It follows that the visibility-curve is practically the same as that due to a double source, whose components have the intensity ratio 7 : 10, and in each of which the light is distri- buted according to the exponential law, expressed by the first term. The formula for a double source, where the components are similar, is = xX 1+7°+ 2r cos 27 = ye D v2 1+7+2r x in which D, the period of the curve, is inversely proportional to the distance between the components. But D=NdA,=(N+1)A,, whence r2 a=rA,;—Ao= i Hence, in the present instance we have for the distance between the components of the red hydrogen-line 1/30 x (6°56 x 107)?=1-4 x 107° millim. or 0°14 division of Rowland’s scale. Again, if 6 be the “half-width” of the spectral line (the value of « when $(2) =4), then 32K 252 tees O(a) = 2 SF antiV Sem If A be the value of X for V=3i, then 6= = = or, with sufficient accuracy, 6= ae Substituting the value of § in the equation for V, we have ae V=2". The value of A in the hydrogen curve is 19. Accordingly, after reducing to the same units as above, we have 6==0°049. * As frequent use is to be made of the function 1+7?+2r cos ont 1+7?+427 : it will be abbreviated to the form cos r/D. Interference Methods to Spectroscopic Measurements. 289 From these data fig. 3a, Plate V., was constructed, the full curve showing the distribution of light in the source. Fig. 4b, Plate V., gives, in the full curve, the corrected values of the visibility of the blue hydrogen-line, at the same temperature and pressure as before. The dotted curve repre- sents a double exponential, as before. The formula for this curve is V=2 PP cos *7/28, thus giving «=0°08 for the distance between the components, and 6=0°057 for the “ half-width” of each. These values give for the distribution of light in the blue hydrogen-line, the full curve in fig. 4a. Oxygen. Fig. 5, Plate V., represents the results obtained from oxygen prepared by heating a tube containing mercuric oxide, drying the gas by sulphuric acid, and exhausting and filling repeatedly till the spectrum was nearly pure. ‘The lines are much less bright than those of hydrogen ; and in order to obtain satisfactory results, the current had to be increased so far that the tube was frequently broken. Notwithstanding the somewhat uncertain character of the observations, it will be seen from fig. 5a that the curve for the orange-red line cor- responds very well with that given by the formula V=2 ©" -36 + °32 cos 27X/2°69 +16 cos 2rX/4°85 +16 cos 27X/1°73]* The agreement between the coefficient 275" and the general curve drawn through the maxima is also shown in fig. 5d, Plate V. The interpretation of these results is that the orange-red oxygen line is a triple, whose components have intensities in the ratios 1: 1:1/2, and whose distances apart are 1°51 and 0:84 respectively, and whose “ half-width ” is 0027. This is shown in fig. 5e. Sodium. The results obtained from metallic sodium in the vacuum- tube are so varied, the character of the lines being so consi- derably altered by temperature and pressure, that a complete study is at present impossible. This is especially true of the yellow lines.; and the difficulty is considerably increased on account of the insufficiency of the dispersion used, which does not permit the separate examination of the lines. Some reference to the changes mentioned will be given at the close 290 Mr. A.A. Michelson on the Application of of this paper. At present it will suffice to take a particular case—the pressure being very low, and the temperature about DOW ets The full curve in fig. 6 6, Plate V., gives the experimental result for the visibility at the maxima for yellow sodium, cor- rected for the personal equation. The dotted curve corre- sponds to the formula V=2 ee cos 7/50 cos “1/140. The complete equation, assuming that the two lines are alike, is V=2 4% cos +8/0°58 cos *7/50 cos “1/140. The interpretation of these results is that each of the sodium- lines is a close double, as shown in fig. 6 a. The yellow-green sodium-line at X=5687 isadouble whose components are about the same distance apart as the yellow ir. It was found to be far less variable than the yellow; and the full visibility-curve, neglecting slight irregularities, gives the experimental results corrected for personal equation. Fig. 7b, Plate V., shows that its components are single, and correspond in distribution of light fairly well with the expo- nential curve, fig. 7a. The same may be said of the orange-red double at 6156 also, except that this seems to have a companion of feeble intensity. The doubles at 5150 and at 4982 were also examined, the curves showing nearly the same results as the red. Zine. The temperature at which the radiations from metallic zine could be conveniently observed was in the neighbourhood of the melting-point of the glass of which the vacuum-tubes were made. But few observations were recorded, though these were quite consistent. The results of the observations, cor- rected for personal equation, are given in figs. 8 and 9, Plate V. The former is the record obtained from the red line near 6360, and shows that this line is single, the distribution of light agreeing very well with a simple ‘exponential curve, the “‘ half-width ” being 0°013. The latter shows the results of observation on the blue line near 4811. The dotted curve is the visibility-curve due to a distribution represented in fio. 19 2. * The curve given above was obtained a year ago; and since then it has been impossible to reproduce it exactly. Interference Methods to Spectroscopic Measurements. 291 Cadmium. : Metallic cadmium in the vacuum-tube at a temperature of about 280° givesa number of very bright lines, widely sepa- rated, and varying very slightly with temperature or pressure. Fig. 10 6, Plate VI., shows the experimental visibility-curve of the red line near 6439, corrected for the personal equation, together with the simple exponential curve V=2-*/™. _The remarkably close agreement leaves no doubt that the distri- bution of light in the source follows very nearly the expo- nential law giving the curve in fig. 10a, in which the “ halt- width ”’ of the source is 0:0065. The result of a single set of observations on the green line at 5086 is given in fig. 116, Plate VI., the approximate agreement between the full line and the dotted curve (which corresponds to the equation V=27*"™ cos +2/115) showing that the source is a close double, the intensity of whose com- ponents is in the ratio 5:1, and whose distance apart is ‘022, the “half-width” of each component being 0°0048. The curve for the blue radiation at 4800 is given in fig. 126, Plate VI., and shows that the results may be approximately represented by V=2~*!* cos-1/32, which corresponds to the distribution of intensity given in fig. 12 a. Thallium. The metal is not sufficiently volatile at the temperatures attainable, but the chloride answers admirably, giving a brilliant green light, the visibility-curve varying but little with temperature. This curve is given in fig. 13 0, Plate VI., together with the dotted curve representing the equation =1¢0s°2/160 V4V,?+ V.7+4V,V_. cos 27 X/25°3, injwhich.V,=2* ("and V,=2° <7. This is the visibility-curve due to a double source, each of whose components is a close double, as shown in fig. 13 a. Mercury. Mercury in a vacuum-tube gives two yellow lines 5790 and 5770, a very brilliant green line at 5461, and a violet line at 4358. The yellow lines are not very bright, and are so close together that it is somewhat difficult with the dispersion employed to prevent the light from overlapping. Notwith- standing these difficulties, the close agreement of a number of observations shows that the curve for the lower line, given 292 Mr. A. A. Michelson on the Application of in fig. 140, Plate VI., is a close approximation to the truth. Neglecting the effect of a line of feeble intensity at a distance of about -24 from the principal line, the distribution of light in the source is represented in fig. 14a, which gives for the visibility-curve V=2 V3Vi4+ V2+6ViV_ cos 27 X/28, in which V,=27~”™ and V,=2> ~*~? cos °5/280. Fig. 155, Plate VI., represents the results of observa- tions on the upper yellow line, omitting some peculiarities due to the presence of one or more lines of feeble intensity. The curve agrees closely with the formula V=t V8V/24+ V2+ 6V1V_2 cos 27-X/70, in which V,=2>*7"" and V,=2-*7", which represents the visibility-curve produced by two lines of intensities 1:3 and separated by 0°019 divisions as shown in fig. 15 a. The green mercury-line is one of the most complex yet examined. The constituent lines are nevertheless so fine that the interference-bands are frequently visible when the difference of path is over four tenths of a metre. The full curve in fig. 16, Plate VI., gives the results of observa- tions corrected for personal equation, while the dotted curve represents the equation VHD COV 2 + 03 V0? + 28V, V_ cos 27 X/31°4, in which V,='62 +°38 cos 27 X/360 and V,='77+°23 cos 277 X/110. This is the visibility-curve corresponding to the distribu- tion represented in fig. 16a. The components of the line, for simplicity, have been assumed to be symmetrical, as figured ; but the observations are not sufficiently accurate to determine whether, for instance, each component is a double or a triple line. In this case also, as in the preceding ones, it is impossible from the data given to determine whether the smaller component is to the right or left of the principal line. A direct observation with the grating showed, however, that the smaller component is towards the red end ef the spectrum. The full curve shows that there is at least one other line— probably more than one—whose intensity is roughly one twentieth of the principal line, and whose distance from it is about three times that of the chief components. The violet mercury-line is much more difficult to observe than the others. The results obtained by observation, cor- rected for persunal equations, are given by the full curve fig. 176, Plate VI. The formula for the dotted curve is Interference Methods to Spectroscopic Measurements. 293 V= V'88V,?+°12V,V, cos 277 X/238, in which V,=27*"""[-62+°38 cos 27X/200] and Vy the resulting distribution of light shown in fig. 17 a. The results of the preceeding work are collected for com- parison in fig. 18, Plate VIII., together with the D group in the solar spectrum. From these, as well as from the curves, it will be seen that it is easy by this method to separate lines whose distance apart is only a thousandth of that between D, and D,, and even to determine the distribution of light in the separate components. The conditions most favourable to high values of the visibility are low density and lew tempera- ture, and these conditions were complied with as far as possible. Still, in many cases, the range of visibility due to slight variations of the conditions show that the behaviour of each substance must be carefully studied. under all possible circumstances of temperature, pressure, strength of current, size and shape of the electrodes, diameter of the vacuum- tube, &e. The effect of temperature and of pressure on the visibility may be readily accounted for on the kinetic theory. In fact, there is but little doubt that these are the chief if not the sole causes of the broadening of the spectral lines and the consequent diminution of visibility ; the latter cause acting by altering the period of the source by frequent collisions, and the former, by the change in the wave-length of the light due to the motion of the source in the line of sight. If, now, the density of the vapour is very low, the second cause may be ignored, and it will be shown that in the case of hydrogen this is the case when the pressure is one or two millimetres. In most of the cases investigated the pressure was so low that the discharge passed with difficulty. Supposing, then, the effect of collisions to be insignificant, let it be proposed to find the effect due to the motion of the molecule in the line of sight. If v be the mean velocity of the molecule and V that of light, then the formula for the resulting visibility- curve, as given by Lord Rayleigh * is h=(/—a")/(l+a"). If the definition of visibility as given above be taken, however, this becomes V=a'=exp |-" (+) | * “On the Limit to Interference when Light is Radiated from Moving Molecules,” Phil. Mag. April 1889, 294 Mr. A. A. Michelson on the Application of If A be the difference of path at which the visibility is reduced to half its value at X =0, then pee @ Lk, T T () Be a bo. or approximately, If we take for hydrogen »=2000 metres per second, thee = 22500. Again, if we ignore the difference in the temperature (about which there is considerable uncertainty), at which the other substances were examined, the velocities v would vary inversely as the square root of the atomic weight, and the number of waves in the difference of path at which the visibility is 0°5 is therefore 22500 /m. Considering the difficulties and uncertainties of the problem, the following Table shows a remarkable agreement between the values actually found and the calculated results *. = r Substance. At. Wt. d. A. N= <. N. (Calc.). 1: pe pee oe eP me or 1 656 19:0 30000 22500 IE bs ds agile morsdacnpaae ae 1 486 8:5 18000 22500 Oe sete apie czas ese 16 616 | 340 55000 80000 2 te Pipe are boner a4 he 23 616 66:0 | 107000 108000 Wie 1 Os8. cededcekee eae 23 589 80:0 | 1383000 108000 ING oe av sicaswp opeeteeie bbe 23 567 62:0 | 109000 108000 IN uss wuss caters cantare 23 515 44-0 85000 108000 1 i ih Rin Mo se AE 23 498 55°0 | 110000 108000 Pies tgs A Se 65°5 636 66:0 | 104000 182000 A EPO ee 65°5 481 470 98000 182000 OOF ris ceans» axmeneatcente 112-0 644 1380 | 215000 238000 aa cement oh 112-0 509 1200 | 236000 238000 Gas (is. bs eee 1120 480 64:0 | 134000 238000 Hee. ssccntramemeemetee= 200°0 579 230°0 | 400000 317000 ET oo naan ce emcee 200°0 577 154:0 | 270000 317000 8 ean yl a8 200°0 546 230°0 | 420000 317000 Wie, th. Ss lectehe aero 200:0 436 100°0 | 230000 317000 Mle cases. ase seen eaeeeene 203°6 535 220°0 | 400000 322000 * It should be stated that the value of A for the yellow sodium-line, if taken from the curve, would be much larger than that given. The latter was the mean of a number of observations taken within the past month. As has been stated before, this particular curve has not been obtained since last year. A few other substances, very difficult to examine, either because the lines are too feeble, or because the spectrum is so unstable, have given results not quite so consistent as the above, though all are of the same order of magnitude as that required by theory. Interference Methods to Spectroscopic Measurements. 295 In order to show conclusively that the effect of density may be neglected in the foregoing observations, as well as to ascertain the law governing the broadening of spectral lines by pressure or density, a series of observations was made on the red hydrogen-line at varying pressures, with the results shown in fig. 19a, Plate VII.* From these curves the following Table was calculated :— Pressure in millim. é. 90 "128 71 "116 AT “O95 23 “OCF 13 "056 9 °053 a "050 9) "048 In fig. 19 the curved line gives the relation between 6 and = , and shows clearly that when p is less than 5 millim. the effect of collisions has almost entirely ceased. If we take as variables 6 and p, the results agree very closely with the straight line 6—6)=kp, in which 6)='047 (the “ half-width ”’ of the line at zero pressure in the units adopted), £=°00093, and p is the pressure in millimetres f. The same results were found for the blue hydrogen-line, though, as might be expected, these were not so consistent. It thus appears that in the case of hydrogen—and probably in all other cases—the width of the spectral line diminishes towards a limit as the pressure diminishes, which depends upon the substance and its temperature; and that the excess of width over this limit is simply proportional to the pressure. In general, it may be said that, under considerable ranges of temperature and pressure, the character of the visibility- curve remains the same ; but it may be important to note that there are a number of exceptions to this rule, among which the green mercury-line and the yellow sodium-line may be especially mentioned. Thus, fig. 20a, Plate VII., represents the visibility-curve usually’ observed for the green mercury-line, and fig. 20 represents that obtained when the vacuum is so high that the discharge passes with difficulty, while fig. 20 represents the * The numbers against the curves denote pressure in millimetres. + In the figure, the numbers representing values of the abscissz for this line should be multiplied by 100. 296 Mr. A. A. Michelson on the Application of intermediate stage. This last observation was obtained by placing the mercury in an atmosphere of hydrogen whose pressure could be measured by a McLeod gauge. It might be objected that the presence of a foreign sub- stance might of itself affect the distribution of light in the source, and therefore the form of the curve. In order to test this point, a series of observations of the red hydrogen-line was taken, while the tube contained liquid mercury, which was heated until the mercury-spectrum was at least ten times as bright as that of the hydrogen. The character of the visi- bility-curve was not perceptibly altered. In the same series of experiments it was found that, pro- vided the pressure of the hydrogen remained constant, the effect of a change in temperature from 75° to 140° had no appreciable effect on the result. In this connexion it may be mentioned that the character of the curve for the green mer- cury-line was not essentially altered when, in place of metallic mercury, the nitrate, iodide, or the chloride was substituted, the only important effect being a diminution in the visibility in the order named. In the case of yellow sodium light, it has already been mentioned that the character of the curve is more variable than that of any other line thus far examined. This is illus- trated by the curves in figs. 21a and 21), Plate VII. It has not been possible thus far to devote the attention which a systematic investigation demands. These changes are very puzzling to trace, but undoubtedly much of the difficulty is due to the fact that the dispersion employed was not sufficient to permit the separate examination of the components. Still, there can be no doubt that the width of the lines, their dis- tances apart, and their relative intensities vary rapidly with changes in temperature and pressure. In addition to the preceding investigations of visibility- curves for light emanating from a rare yas or vapour in a vacuum-tube, the curves for sodium, thallium, and lithium in the flame of a Bunsen-burner have been observed ; and the results are given in fig. 22, Plate VII. The thallium- and lithium-lines are clearly double ; the distance between the components of the former agreeing very well with the results obtained with the vacuum-tube. These substances were brought into the flame in the ordi- nary way, and the results obtained were at least as good as when a finely divided solution was used according to the method of Gouy. It appears from these curves that the width of the line is about ten times as great as when the vacuum- tube is used. Butif the temperature of the flame be taken Interference Methods to Spectroscopic Measurements. 297 at 1500° C. and that in the vacuum-tubes at 350° C., the lines should be only twice as broad in the former case as in the latter. It appears, then, that notwithstanding the small quantity of substance present (barely enough to colour the flame), the real density must be comparable to that of the vapour of the substance boiling under atmospheric pressure. The principal object of the foregoing work is to illustrate the advantages which may be expected from a study of the variations of clearness of interference-fringes with increase in difference of path. The fundamental principle by which the “structure” of a line or group of lines is determined by this method is not essentially different from that of spectrum- analysis by the grating, both depending, in fact, on interference phenomena; but in consequence of the almost complete freedom from errors arising from defects in optical or mecha- nical parts, the method has extraordinary advantages for this special work. A glance at fig. 18, Plate VIII, will give a fair idea of the “ resolving-power ” of the method as compared with that of the grating. In order that the comparison be quite fair, however, it would be necessary to take for a com- parison-spectrum that of the substances here used, and under the same conditions. With the best instrumental appliances now in use, it is difficult to “resolve” lines as close together as the components of either of the yellow sodium-lines. It is evident, however, that by Light-wave Analysis (if I may venture so to call the foregoing method) a tenth of this dis- tance is obviously within the limit ; indeed, if the width of the lines themselves be less than their distance apart, there can be no limit. SUPPLEMENT. I. It has already been pointed out that in many cases it is difficult or impossible to decide between two or more distri- butions of lines which give very nearly the same visibility- curve ; and when there are many lines in the source, the combinations of intensities and arrangement of these from which a type may be selected is enormously great. Indeed, even when the number of lines is greater than three, except- ing perhaps the cases where the lines may be in pairs (as in the case of yellow sodium-light), the resulting visibility-curve becomes so complex that it is very difficult to analyse. Doubt- less in many cases where the components are not too close, the grating will give the information necessary for the investi- gator to select the proper combination. It may readily be shown that the formula ye tS? —aps 298 On Spectroscopic Measurements. for the vzszbilzty-curve due to a distribution of light, y=¢(2), is identical with that of the cntensity-curve at the focus of a telescope provided with apertures which produce this distri- bution in the light passing through. Accordingly, if a telescope be provided with apertures adjustable in width (or length) and distance apart, the diffraction-image of a distant illuminated slit will give, at once, a representation of the whole visibility-curve ; and by adjustment of intensities and dis- tances any particular visibility-curve may be more or less accurately copied, thus furnishing a means of studying the relations between V and (wv), which, while giving perhaps only a rough approximation to the truth, may prove more convenient than analytical or graphical methods. II. One of the purposes which led to these investigations was the search for a radiation of sufficient homogeneity to serve as an ultimate standard of length. It will appear from the curves of cadmium that there are three lines which may be used for this purpose. The red cadmium-line is almost ideally homogeneous, and will readily permit the estimation of a change of phase in the interference-fringes of one hundredth of a fringe in a total distance of 200 millimetres, or over 300,000 waves. Both the green and the blue lines are fairly well adapted for the purpose, and will prove very valuable as checks. Hach of these, however, has a small companion, and it is necessary to know the effect of this in altering the phase of the inter- ference-bands. If @ be the fraction of a wave by which the position of a minimum is shifted on account of the presence of the com- panion, a the number of “ periods” in the difference of path, and r the ratio of the intensities, then : rsin 27a x ema +r cos 27a Thus, if 7 =1/4, @ isa maximum when « is about 1/3 ; and for this we have, approximately, d= —°04. This is the largest correction to be applied, and is negative if the brighter line has the greater wave-length. It is theo- retically possible, by this means, to determine, in case of an unequal double, or a line unsymmetrically broadened, whether the brighter side is toward the blue or the red end of the spectrum. III. It has been argued that, even if all practical diffi- culties in making large gratings could be removed, nothing * See Phil. Mag. April 1891, page 345. (The value of 7 is the reci- procal of that here used.) Level of No Strain in a Cooling Homogeneous Sphere. 299 further could be gained in resolution of groups of spectral lines, on account of the real width of the lines themselves, caused by the lack of homogeneity in the radiations which produce them. The results of the preceding investigations show that, while this is very far from being true with present gratings, such a limit undoubtedly exists. ‘The accordance between the measured widths of eighteen lines shows, further, that this broadening of lines in a rare gas can be fully accounted for by the application of Doppler’s principle to the motion of the vibrating atoms in the line of sight, and indeed furnishes what may be considered one of the most direct proofs of the kinetic theory of gases. The form of the ultimate components of all the groups of lines thus far examined is found to agree fairly well with an exponential curve, ¢(z)=e—®”, which shows that the distri- bution of velocities cannot vary widely from that demanded by Maxwell’s theory. If the limit above mentioned were due solely to the motion of the molecule, and the radiating substance could be rendered luminous while its temperature was very low, it might be possible to observe interference-phenomena with difference of path of many metres. But it must be considered that, since every vibrating molecule is communicating its energy to the eether in the form of light-waves, its vibrations must diminish in amplitude; consequently the train of waves is no longer homogeneous even though the vibrations remain absolutely isochronous, and the result is a broadening of the line and limitation of the difference of path at which interference is visible. XXXI. Note on the Level of No Strain in a Cooling Homoge- neous Sphere. By M. M. P. Rupsx1, Privat Docent in the University of Odessa * [This note relates to the geological theory of the formation of mountains which is based on the fact that, when a sphere cools and contracts, the stratum which contracts most rapidly is situated below the surface, and accordingly the superficial layers are com- pressed and crumpled. This theory is developed in a paper by Mr. Davison and in a subsequent note by Professor Darwin in Phil. Trans. Roy. Soc. vol. clxxviii. (1887) A. pp. 231-249. It has been urged against this theory that the level of no strain lies too near the surface to explain the formation of mountains. But M. Rudski here shows that if the initial temperature be not uni- form, the level of no strain will lie deeper. Such an hypothesis * Communicated by Professor G. H. Darwin. 800 Level of No Strain in a Cooling Homogeneous Sphere. appears justifiable, when we reflect that previous to consolidation some cooling of the molten material must have taken place by con- vective currents. Ihave ventured to slightly expand the beginning of M. Rudski’s note, since it seemed to me somewhat too brief.— GH. Da i v denotes the temperature in the cooling sphere, the equation of cooling is, in Thomson’s notation, dv On: 2 ey va =e == e Py e e ° e dt die * > dr) (1) Now it may be shown thatif K be the modulus of stretching in any stratum, then dK € d?v ai => 5 |r APE dr, = e - ° (2) where ¢ is a certain constant, and where the inteyral is taken from the stratum 7 down to such a depth that there is no change of temperature™. If (2) be integrated by parts, then, on the assumption that at the centre of the sphere there is no change of temperature, we have from (1) ge alee eee dt dt r adr hide sa Fe =ne(= 2-5). = - 4 ° (3) Then the strata are compressed, unstrained, or stretched according as dv > 3x dv dt S+Pa2x id 14 pee b. i acl ae Water and Mercury at nearly Perpendicular Incidence. 311 A few calculations from ths original expressions will serve to indicate the field of these approximations. “ir ; p= yp CSI Op 1 20 sa A x 10467, T= 7p x “9541, 1 G2 Bae. =e : + T 2x 75 x 10004 From (5) we get as the last factor 100050. H= 5, B=20°,. Og= 14°: 51':8; 1 1 7) ae . a e S?=7, x 12021, T= 7 x ‘8158, 1 2 pa es ra . : ae ?=2 x a9 * 1:0090. By (5) the last factor is 1:0080. Again, u=s, G=307 Op 22-14, ge] 15189, T= x-5866 AY : AY : 1 z 2 anes . 87+ TP?=2 x Zo * 1:0527. According to (5) the last factor is here 1°0405. It appears that in the case of water the aggregate reflexion scarcely begins to vary sensibly from its value for @=0 until 6=20°, a property of some importance for our present purpose, as it absolves us from the necessity of striving after very small angles of incidence. I will now describe the actual arrangement adopted for the experiments. The source of light at A, (fig. 1, Pl. X.), isa small incandescent lamp, the current through which is con- trolled with the aid of a galvanometer. It is so mounted that its equatorial plane coincides with the (vertical) plane of the diagram. Underneath, upon the floor, is placed the liquid (B) whose reflecting power is to be examined. At QC, just under the roof, the direct ray AC and the reflected ray BC are turned into the same horizontal direction by two mirrors silvered in front and meeting one another at C under a small Z2 312 Lord Rayleigh on the Intensity of Light reflected from angle. The eye situated opposite to the edge © and looking into the double mirror thus sees the direct and reflected images superposed, so far as the different apparent magnitudes allow. D represents a diaphragm and E a photographic portrait-lens of about 3 inches aperture which forms an image of A and A’ on or near the plane F. At F is placed a screen perforated with a hole sufficiently large to make sure of including all the rays from A, A’ which pass D. To determine this point an eye-piece is focused upon F, so that the images of A, A’ are seen nearly in foeus. Some margin is necessary because the images of A, A’ cannot (both) be accurately in focus at F. These adjustments being made, an eye placed behind F and focused upon C sees the upper mirror illuminated by the direct light (from A), and the lower illuminated by the reflected light (from A’). And if the aperture at F is less than that of the pupil of the eye, the apparent brightnesses of the two parts of the field are in the same proportion as would be the illuminations on a diffusing screen at C due to the two sources. The advantage of the present arrangement, as compared for example with the double-shadow method, lies in the immense saving of light. In the case of water there is a great disproportion (of about 50 to 1) in the illumina- tions as seen from F. In order to reduce the direct light to at least approximate equality with the reflected, Talbot’s device* of a revolving disk was employed. This is shown in section at I, and in plan at I’. The angular opening may be chosen so as to allow for the loss in reflexion, and for the farther disadvantage under which the reflected hght acts in respect of distance. The disk finally employed was of zinc, stiffened with wood, and covered on both faces with black velvet. It was at first proposed to work as above described by eye estimations ; but the necessity for a ready adjustment capable of introducing small relative changes of brightness leads to further complications. Moreover, the large disk which it is advisable to use for the sake of accurate measurement of the angular opening, cannot well be rotated at the necessary speed of 20 or 25 revolutions per second. Jor this reason, and also for the sake of obtaining a record capable of being examined at leisure, it was decided to work by photography. This in- volves no change of principle. The photographic plate H simply takes the place of the retina of the eye. But now the integration of the effect over a somewhat prolorved ex- posure (of several minutes) dispenses with the necessity for a rapid rotation of the Talbot disk, and allows us to obtain at * Phil. Mag. vol. v. p. 327 (1884). Water and Mercury at nearly Perpendicular Incidence. 313 will a fine adjustment by screening one or the other light from the plate for a measured interval of time. In practice the direct light was thus partially cut off, a mechanically held sereen being advanced a little above the plane of the revolving disk. The reader will not fail to observe that the incomplete coincidence of the times ef exposure has the disadvantage of rendering the calculation dependent upon the assumption that the light is uniform over the duration of an experiment. Error that might otherwise enter is, however, in great degree obviated by the precaution of choosing the middle of the total period of exposure as the time for screening. The above is a sufficient explanation of the general scheme, but there are many points of importance still to be described. With respect to the source of light, it was at first supposed that even if the radiation upwards and downwards could not be assumed to be equal, at any rate a reversal by rotation of the lamp through 180° in the plane of the diagram would suffice to eliminate error. On examination, however, it appeared that owing to veins in the glass bulb the radiation in various directions was very irregular, so much so that it was feared that mere reversal might prove an insufficient precaution. The difficulty thus arising was met by covering the bulb, or at least an equatorial belt of sufficient width, with thin tissue-paper, by which anything like sudden variations of radiation with direction would be prevented, and by causing the lamp to revolve slowly about its axis during the whole time of exposure. The diameter of the bulb was about 14 inch, and the illuminating-power rather less than that of one candle. Another point of great importance is to secure that the light regularly reflected from the upper surface of the liquid, which we wish to measure, shall be free from admixture. It must be remembered that by far the greater part of the light inci- dent upon the liquid penetrates into the interior, and must be annulled or at any rate diverted into a harmless direction. To this end it is necessary that the liquid be free from tur- bidity and that proper provision be made for the disposal of the light after its passage. It is not sufficient merely to blacken the bottom of the dish in which the water is con- tained. But the desired object is attained by the insertion into the water of a piece of opaque glass, held at such a slight inclination to the horizon that the light from the lamp regularly reflected at its upper surface is thrown to one side. As additional precautions the disk and its mountings were blackened, as were also the walls and ceiling of the room in which the experiments were made. 814 Lord Rayleigh on the Intensity of Light reflected from The surface of water must be large enough to avoid curva- ture due to capillarity. Shortly before an experiment it is cleansed with the aid of a hoop of thin sheet-brass about 2 inches wide. The hoop is deposited upon the water so doubled up that it includes but an insensible area, and is then opened out into a circle. In this way not only is the greasy film usually present upon the surface greatly attenuated, but also dust is swept away. The avoidance of dust, especially of a fibrous character, is important. Otherwise the resulting deformation of the surface causes the field of the reflected light to become patchy and irregular. We come now to the silvered glass reflectors, which are assumed to reflect the direct and reflected lights equally well. It seems safe to suppose that no appreciable error can enter depending upon the slightly differing angles at which the reflexion takes place in the two cases. But the mirrors are liable to tarnish, and, indeed, in the earlier experiments soon showed signs of being affected. The influence of this tarnish would be much greater in photographs done upon ordinary plates, sensitive principally to blue light, than in the estimation of the eye ; and it was thought desirable to eliminate once for all any question of the effect of differential tarnishing by interchanging the mirrors in the middle of each exposure. For this purpose a somewhat elaborate mounting had to be contrived. It was executed by Mr. Gordon and answered its purpose extremely well. The mirrors are carried by a brass tube B (fig. 2), which revolves in an external tube AA rigidly attached to the stand of the apparatus. A lateral arm OU, some inches in length, projects from B, and near its extremity bears against one or other of two screw-stops D. The lower end of B carries perpendicular to itself a brass plate HE (fig. 3). The mirrors GG are of plate-glass and are fixed by cement to two brass plates FF. The latter plates are attached by friction only to EH, being on the one hand pushed away by adjusting- screws HH, and on the other held up by four steel springs I. The edges of the reflecting surfaces meet accurately in a line passing through the axis of rotation, and the stops D are so adjusted that the transition from the one bearing to the other corresponds to a rotation through precisely 180°, so that on reversal the common edge of the reflectors recovers its position. The two mirrors were originally silvered in one piece, and the common edge corresponds to the division made by a diamond-cut at the back. These arrangements were so successful that in spite of the reversal between the two parts of the exposure the division line appears sharp in the photo- graphs and exhibits no appearance of duplicity. Water and Mercury at nearly Perpendicular Incidence. 315 When not in use the reflecting-surfaces are protected by a sort of cap of tin-plate, which fits loosely over them. The improvement thus obtained was very remarkable, the mirrors not suffering so much ina month as they formerly did in a day before the protection was provided. The following are the measures of distances required for the calculation. From the division-line C to the axis of rotation of the lamp meio. b), AC = 82:21 inches ; AB [-28.°BC=93'15, so that AB + BC=104°43. The factor expressing the ratio of the squares of the distances is thus 1°6137. The angle of incidence is best obtained from a measurement of the horizontal distance between C and A. This proved to be 114 inches ; so that ; 1 sin ?= 104-3 = HL and 6=634°. This applies to all the experiments referred to in the present paper. The estimation of the angular opening in the disk used for the water experiments depended upon measurements of corre- sponding chord and diameter. The chord, measured by means of the screw of a travelling-microscope, was*7574 inch. The radius, expressed in terms of the same unit, was found to be 7°79. Hence, if «be the angular opening, Crew 4 = 167. The ratio in which the direct light is reduced is thus 167 «167 180 x60 ~ 10800 It will now be necessary to give some details with respect to the actual matches as determined photographically. At first the intention was to employ ordinary plates (Ilford), which worked very satisfactorily. But when the attempt was made to compare the result with theory, the comparison was found =*01546. 316 Lord Rayleigh on the Intensity of Light reflected from to be embarrassed by uncertainty as to the effective wave- length of the light in operation. Moreover, as these plates are scarcely sensitive to yellow and green light, the effective wave-length is liable to considerable variation with the current used to ignite the lamp. Photographs were indeed taken of the spectrum of the lamp as actually employed, but the un- symmetrical character of the falling off at the two ends made it difficult to fix upon the centre of activity. Recourse was then had to Edwards’ “ isochromatic ” plates. The spectrum of the lamp, as photegraphed upon these plates after passing through a pale yellow glass, was very well defined, lying with almost perfect symmetry between the sodium and the thallium lines. It was, therefore, determined to use these plates and the same yellow glass in the actual experiments, so that A=4(5892 + 5349) = 5620 could be taken as the representative wave-length. The only disadvantage arising from this change was in the necessary prolongation of the exposure, which became some- what tedious. Although no dense image is required or indeed desirable, the exposure should be such that the development does not need to be forced. Two photographs, with different times of screening, were usually taken upon the same plate, the object being to obtain a reversal of relative intensity, so that in one image the semicircle representing the direct light should be more intense and in the other image the semicircle representing the reflected light. The best way of examining the pictures depended somewhat upon circumstances. When the exposure and development had been suitable, the most effective view for the detection of a feeble difference was obtained by placing the dry picture, film downwards, upon a piece of opal glass. The light returned to the eye had then for the most part traversed the film twice, with the effect of doubling any feeble difference which would occur on simple transmission. Under favourable circumstances it was possible to detect a reversal between the two images when the difference amounted to 83 per cent. A few such experiments might therefore be expected to give the required result accurate to less than one per cent. With the Edwards’ plates an exposure of 12 minutes was found to be necessary. This was divided into two parts of 6 minutes each, with an interval of one minute during which the mirrors were reversed. About the middle of each period of 6 minutes the direct light was screened off for a time which varied from picture to picture. For example, on June 6, the time of screening for one picture was 71 seconds, and for the Water and Mercury at nearly Perpendicular Incidence. 317 second picture 48 seconds. This means that while in both pictures the exposure for the reflected light was 12 minutes or 720 seconds, the exposures for the direct light were re- spectively 720—2 x 71=9578 seconds, and 720—2 x 48=624 seconds. The water was distilled, and its temperature was 17°77 C. The examination of the finished pictures showed that the contrast was reversed, so that the total exposure (to the direct light) required for a balance was intermediate be- tween 578 and 624, and, further, that the first mentioned was the nearer to the mark. The general conclusion derived from a large number of photographs was that the balance corresponded to a total screening of 121 seconds, viz., to an exposure of 720—121 =599 seconds. This is for the direct light, the exposure to the reflected light being always 720 seconds. The ratio of exposures required for a balance is thus DI 20 = and this may be considered to correspond to a temperature of 18°C. We can now calculate the observed reflexion for 64° inci- dence, reckoned as a fraction of the incident ight. We have 599 167 (104-43)? 720 ° 10800 \ 82-21 The above relates to the impression upon Edwards’ plates after the light had been transmitted through a yellow glass. When Ilford plates were substituted and the yellow glass omitted, the reflexion appeared decidedly more powerful, and the ratio of exposures necessary for a balance was about 425 : 480, or 637: 720. It appears, therefore, that the re- flexion of the light operative in this case is some 6 per cent. more than before, or about ‘0220 of the incident light. As to a large increase of reflexion there was no doubt; but, owing perhaps to variations in the quality of the light, the agree- ment between individual results was not so good as before. It now remains to calculate the reflexion as given by Fresnel’s formule ; and it appears from the discussion at the commencement of this paper that we may ignore the small angle of incidence (63°) and take the formula in the simple form given by Young, viz. :— R= (u—1)4/(u+1)2, As to the value of w for water, Wiillner* gives #, =1°326067 —-000099 ¢ + °380531A-?, * Poge. Ann. Bd. cxxxlii. =°02076. 318 Lord Rayleigh on the Intensity of Light reflected from ¢ denoting the temperature in Centigrade degrees. Applied to 18° and to A=5620, this gives = 1:333951, whence (u—1)*/(w+1)?=-02047. The reflexion actually found is accordingly about 14 per cent. greater than that given by Fresnel’s formule. In order to estimate the effect, according to the formula, of a change in index, we may use dR 40u ei or, in the case of water, d5R/R=56y nearly. To cause a variation of 14 per cent. in the reflexion, du would have to be 003, and to cause 6 per cent. 64 would have to be "012. The latter exceeds the variation of mw in passing be- tween the lines D and H. The agreement with Fresnel’s formule is thus pretty good, but the question arises whether it ought not to be better. Apart from a priori ideas as to the result to be expected, I should have estimated the errors of experiment as not likely to exceed one half per cent., and certainly no straining of judgment in respect of the photometric pictures would bring about agreement. On the other hand, it must be remembered that one per cent. is not a large error in photometry, and that in the present case a one per cent. error in the reflexion is but one in 5000 reckoned as a fraction of the incident light. While, therefore, the disagreement may be real, it is too small a foundation upon which to build with any confidence. It only remains to record the results of some observations upon the reflexion from mercury. In these experiments the revolving disk was dispensed with, and the photographs were taken upon Edwards’ plates through yellow glass. The angle of incidence and all the other arrangements remained as before. In order to obtain a balance it appeared that the direct light required to be screened for 64 seconds out of 120 seconds. ‘The reflexion is accordingly 56 /104°43\2 120 ( SST moe 2 The mercury was of good quality, and was filtered into a glass vessel just before use. The level was adjusted to be the same as that adopted for the observations upon water. A aes Water and Mercury at nearly Perpendicular Incidence. 319 surface thus obtained would not be free from a greasy layer, but it is not probable that this would sensibly influence the reflexion. APPENDIX. The calculation of the reflexion depends upon the assump- tion that the reflecting surface is plane ; and a very moderate concavity would suflice to explain the small excess in the observed number for water over that calculated from Fresnel’s formule. It is thus of importance to assure ourselves that the concavity due to capillarity is really small enough to be neglected. For this purpose an estimate founded upon the capillary surface applicable in two dimensions will suftice. If @ be the inclination to the horizon at any point, « the horizontal and y the vertical coordinate, the equations to the surface are :— z=2a cos }0+alogcoti@, y=2asin 30, where a’=T/gp. At a great distance from the edge, d=0, y=0, r=0. At the vertical edge of a wetted vessel, 0=47. The origin of w corresponds to ete Oe In the case of water T=74, p=1, and g=981 C.G.S. ; so that a='274 centim. In the experiments upon reflexion the part of the surface in action was about 11 centim. away from the boundary, so that x/a=40, and @ is very small. For the curvature 1/p=y/a?=2 sin $0. /a ; or for our present purpose. 1/p=0@/a. To find @ we have approximately, cot LO=e*, or 0=4e-®., 320 Professor Ewing on Joints Accordingly 1 z pp. ‘274xe8" This may be multiplied by 4 to represent the increase of effect in the actual circumstances as compared with what is supposed in the two-dimensional problem; but it remains absolutely insensible in comparison with the other curvatures involved. — XXXVI. On Joints in Magnetic Circuits. By Professor Ewine, 1.A., F.R.S.* N the ‘ Philosophical Magazine’ for September 1888 an account was given by Mr. William Low and myself of experiments we had made to examine the influence of a plane of transverse section in an iron bar which formed part of a magnetic circuit. It was found that a transverse cut intro- duced what is now frequently spoken of as magnetic resist- ance, even when the faces at the cut were scraped up to be as nearly as possible true planes, and were brought into ex- cellent mechanical contact. Only when a very considerable force was applied to press them together did the resistance of the joint appear to vanish ; except in this case it required a stronger magnetomotive force—a greater number of ampere- turns in the magnetizing coil—to bring the magnetic induc- tion up to any assigned value when the bar was cut than it had required when uncut. In this respect the joint is equi- valent to a narrow crevasse of air or other non-magnetic material. We proceeded to calculate the width of the air- gap which should be equivalent, in magnetic resistance, to the joint, and obtained values which made it appear that the width of this equivalent air-gap became reduced when the magnetization was forced up to high values. Having occasion recently to revise these results I found that an incorrect procedure had been followed in calculating, from the experimental data, the width of the equivalent air- gap. When the necessary correction is made it appears that this width remains constant, or nearly constant, whether the magnetization be weak or strong. The object of this note is to point out the error in the former calculation, and to state the corrected values. Suppose we have to deal with the magnetization of a com- plete iron circuit, of uniform cross section, and of length J. Let the permeability of the iron be yu, for any induction 8. The * Communicated by the Author. in Magnetic Circuits. 321 line-integral of magnetic force producing this induction is = , when there is no joint. Next suppose a joint to be introduced and to be equivalent in magnetic resistance to an air-gap of width x. The line- integral of magnetic force is then at +Bz. Mb Distinguish by dashes the values of 8 and yw in this case from their values when there was no joint. Then, taking the same number of ampere-turns in the magnetizing coil in both cases, we have / Bi_ BE , fee 2p from. which ees Bere ou (aie) ae aA (s z) In the paper referred to we omitted to take account of the variation which the permeability of the iron undergoes when $ is changed to B’. It is, however, essential to distinguish between yw’ and yw if the plan is followed of finding « by com- paring the inductions reached in the two cases under equal values of the magnetomotive force. A better way, however, is to draw the curve of magneti- zation in relation to magnetomotive force for each of the two cases, and then find, by measurement from the curves, what is the difference in magnetomotive force for one and the same value of the induction. This simplifies matters by securing that yw shall be the same when there is a joint as when there is none. If we write § for the magnetic force when there is no joint, and §! for the mean magnetic force required to produce the same induction 8 when there is a joint, we have ee! [ gi= 2, and Hil= = + Bz ; ! =] 2) a ( = Thus, let a curve giving 8 and § be drawn, and another curve, with the same axes, giving $8 and §’, then the distance from which Magnetic Induction %. 322 Professor Ewing on Joints between the curves measured parallel to the axis of , when divided by the corresponding value of %, is proportional to the width of the equivalent gap. When this method is ap- plied to the experimental results which were published in the paper referred to it will be found that, for any given joint, the values of )’—¥ are nearly proportional to 8. In other words, x remains nearly constant instead of becoming diminished as the magnetism is pushed towards saturation. One or two instances may be quoted. 0 2 4 6 8 10 12 14 16 «18 20 22 24 26 28 Magnetizing Force due to Solenoid. In Table I. experimentally found values of the magnetic force due to the magnetizing solenoid are given, along with the inductions observed when there was no joint and when there was a joint. The joint in this case was between care- fully finished surfaces, approaching the condition of true in Magnetic Circuits. 323 planes. No externally applied mechanical force was used to force the surfaces together. TABLE I. Influence of a smooth joint in an iron bar. Magnetic Induction %. Magnetizing Force due to solenoid. Without joint. With joint. 4 3950 3000 6 6900 5300 8 9250 7400 10 10900 9150 15 13250 12000 20 14300 13500 30 15200 ~ 14900 In fig. 1 these results are shown by curves, and a third curve (the dotted line) is added to show the values of 5'—H in terms of $. It will be seen that this curve is nearly a straight line ; it should be straight if the magnetic resistance of the joint were strictly constant. Table II. gives values of the width of air-gap equivalent in magnetic resistance to the joint, as determined from this experiment. TABLE I]. 8 Width of air-gap equivalent ; to the joint, in centimetres. 4000 0:0026 6000 0-:0030 8000 0:0031 10000 0:0031 12000 0°0035 14000 0:0037 Data relating to a second experiment in which a cut with true-plane surfaces was introduced in another iron bar will be found in the former paper. Fig. 2 shows the graphic treatment of this example, and it will be seen that the dotted curve which exhibits §’— is in this case slightly concave towards the axis of 8, whereas in the former example it was 324 Professor Ewing on Joints slightly convex. The deviations from straightness are in neither case more than may well be set down to uncer- tainty in the experimental data ; and taking the two examples together it seems very probable that the joint is equivalent to an air-gap of sensibly constant magnetic resistance. In Fig. 2. eM >] xt 1 Magnetic Induction ¥, Me Me fen Se se es Magnetizing Force due to Solenoid. the first example the width of the equivalent gap was about 0-0033 cm.: in this case its mean value is about 0:0036 cm. Small as these gaps are, it is difficult to believe that the metallic surfaces were actually separated so far, and it seems more probable that the magnetic resistance of a joint is due in part to a reduction of permeability in the metal itself at and close to each surface, as a result of the influence of surface conditions in affecting the grouping of the molecular magnets. Since the joint is equivalent to an air-gap of sensibly con- stant width it is easy to apply a graphic construction, like that used by Lord Rayleigh in the case of ellipsoids*, to de- termine the influence which the joint must exert in any cyclic or other magnetizing process ; or, conversely, to determine from experiments made on a circuit in which there is a joint the form which the magnetization curve would assume if the joint were not there. The effect of a gap or of a joint is to make the true magnetic force less than the apparent mag- netic force (namely, the force due to the magnetizing sole- noid) by an amount which is proportional to the magnetiza- tion. We have H=5’— oe relation of S to the magnetizing force in a ring without a * Phil. Mag. August 1886, p. 180. Thus a curve giving the in Magnetic Circuits. 325 joint will serve to show the relation of 8 to the externally applied magnetic force on a ring with a joint if we re-draw it, merely shearing it over by inclining the axis along which 8 is represented through an angle the tangent of which (inter- preted upon the proper scale of 8 and of §) is equal to - To take a practical instance, suppose we have a ring of soft iron forming a magnetic circuit say 30 cm. long. Sup- pose the ring, when tested in the solid state, to give the eurves of 8 and § shown in fig. 3, which are actual curves Ss DERI R30 «ons Bena | | | | } | 10 0.00 a oar inca ‘goes | | 8 lO | & & Be force ete ia for a soft wrought-iron ring. Let it be required to deduce from these curves the relation which the magnetism would have to the externally applied magnetizing force if the ring, instead of being solid, were cut into two half rings, with carefully faced ends placed in the best possible contact. Each cut, as we have seen, is equivalent to a gap 34, cm. wide. The two together make x equal to z21,5 of the length of the ring. Hence, if we draw a line OPQ in the figure inclined so that is —1 when % is 4500, the intercepts between it Phil, Mag. 8. 5. Vol. 34. No. 209. Oct. 1892. 2A 326 Profs. Dewar and Fleming on Electrical Resistance and the axis of 8 represent the self-demagnetizing forces dae to the cuts at all stages of the magnetizing process. Curves. giving the applied force §' in relation to § for the cut ring may then be drawn by using as abscisse the distances of points on the original curves from the line OPQ, measured parallel to the axis of §. The reverse of this construction will of course serve to deduce true magnetization curves (for uncut metal) from experiments made with a magnetic circuit which contains a joint or a gap. _ The figure is drawn to scale for an actual case, and serves to show that the division of a ring 30 cm. long into two half rings abutting against each other with the smoothest possible joints has the effect of reducing the residual magnetism from 9000 to 6000, which is the height of the point P. A similar shearing over of the curves, with consequent reduction of the residual magnetism occurs, to a greater or less extent, in magnetic tests of bars when these are made to form part of a eee magnetic circuit by the addition of a massive iron yoke. 3 XXXVI. On the Electrical Resistance of Pure Metals, Alloys, and Non-metals at the Boiling-point of Oxygen. By JAMES Dewar, LL.D., F.BR.S., Professor of Chemistry in the Royal Institution, §c., and J. A. Fuemine, IA., D.Se., PRS., Professor of Electrical Technology in University College, London, &c.* §1. QYEVERAL observers have studied the behaviour of metals as regards electrical conductivity at low temperatures. In particular MM. Cailletet and Bouty (Journ. de Physique, July 1885) have made observations of | the resistance and resistance-change with temperature of various metals at —100° C. by the employment of liquid ethylene as a cooling agent. Wroblewski (Comptes Rendus, 1885, vol. ci. p. 161) measured the electrical resistances of wires of electrolytic copper at various temperatures, 100° C., 20° C., 0° C., —100° C., and gives also figures for the com-. parative resistance of the same wires at the critical point of nitrogen, the boiling-point of nitrogen, and the temperature of the solidification of nitrogen. ‘The possession of means for producing very considerable quantities of liquid oxygen as well as liquid ethylene has placed at our disposal an opportu- nity of carrying out some investigations on the comparative electrical resistance of a number of pure metals, alloys, and. + Communicated by the Authors. es of Pure Metals §c. at the Boiling-point of Oxygen. 327 non-metals at the low temperatures obtainable by the evapora- tion of liquid oxygen at ordinary barometric pressures and the ebullition of liquid oxygen under reduced pressures of about 25 or 30 millim. Since liquid oxygen, as already pointed out by one of us, is a very perfect insulating fluid, it is quite a simple matter, if once sufficient of the liquid gas is ob- tained, to measure the electrical resistance of a wire or small rod of the metal when immersed in liquid oxygen, and thus entirely at the same temperature as the evaporating liquid. A series of observations has accordingly been made by us on the specific electrical resistances of various metals, non-metals, and alloys over a range of temperature varying from +100° C. to nearly —200° C., and although these in- vestigations are as .yet incomplete, the observations already made seem to be of sufficient interest to render it worth while to place them on record. § 2. A number of small resistance-coils were prepared of wires of different metals and alloys in the following way. thin rectangular sheet of mica, about 5 centim. long and 1 or 2 centim. in width, had a number of nicks cut in the edges, and round this was wound loosely the wire whose resistance was to be determined. The ends of the wire were brought out through two holes in the mica and soldered to two stout copper terminal wires formed of high-conductivity copper wire well insulated with indiarubber. The ends of these terminal wires were bent over so as to dip into mercury cups (see fig. 1). The small resistance-coil so formed could then be lowered into a test-tube full of the liquid gas or other fluid, by means of which the temperature of the wire was deter- mined. ‘The majority of the wires used had a length of 50 or 100 centim. and a diameter of ‘003 of an inch (3 mils). The electrical resistance of these small coils of wire was measured by a Wheatstone’s bridge, kindly lent for the purpose by Messrs. Elliott Bros. The coils of this bridge were of platinum-silver, and adjusted to read in B.A. units at 15°°5 Cent. The balance was determined by the use of a highly sensitive mirror-galvanometer, using the current from a single Helsen’s dry cell. The experiments were carried out in the Royal Institution laboratories, in a room which remained approximately at about 20° C. during the whole time. A series of wires of pure metals was obtained, and also others of known alloys. Mr. J. 8. Sellon and Mr. G. Matthey, of Messrs. Johnson and Matthey of Hatton Garden, kindly provided for us carefully drawn wires of absolutely pure annealed platinum, pure gold (999°9 degrees of fineness), pure silver, aluminium, and tin. Also wires of alloys of 2A 2 328 Profs. Dewar and Fleming on Electrical Resistance platinum-silver, iridium-platinum, rhodium-platinum, palla- | dium-silver, as well as wires of palladium, and of nickel. igo. Resistance-Coil used in taking resistance of Wires in Liquid Gases. (a) Mica rectangle wound round with the wire to be measured. (6) Stout insulated Copper-rod Connexions, From the London Electric Wire Company was procured some electrolytic copper wire of the highest conductivity obtainable, and also from Messrs. Griffin & Co. some pure annealed iron wire. Other wires of commercial materials, e. g., German- silver, platinoid, tinned copper, tinned iron wire, and com- mercial tin, were also mounted. The mean diameter of all these wires was measured with the microscope-micrometer to the nearest ten-thousandth of an inch, and the length also carefully ascertained. The elec- trical resistance of each of these small resistance-coils was then taken at six or seven fixed temperatures, as follows :— _ (1) At about 100° C. when immersed in a paraffin-oil or glycerine bath heated with boiling water. of Pure Metals &c. at the Boiling-point of Oxygen. 329 (2) At about 20° C. when immersed ina paraffin or alcohol bath at normal temperatures. (3) At about 0° C. when in a paraffin-oil or alcohol bath . cooled by melting ice. (4) At —80° C. when immersed in a bath of ether and solid carbonic acid. (5) At —100° C. when immersed in a test-tube full of liquid ethylene boiling freely under atmospheric pressure. (6) At —182° C. when immersed in a test-tube full of liquid oxygen boiling freely under atmospheric pressure. (7) At —197° C. when immersed in a closed tube containing liquid oxygen boiling under a reduced pressure of about 25 or 30 mm. of mercury. § 3. In the case of the measurements in liquid oxygen and ethylene, the liquid gases were contained in double test-tubes holding about 200 cub. centim., which were kept filled up as fast as the liquid boiled away, in order that the resistance- coil might always be fully covered. Hach measurement of resistance was repeated several times. We thus obtained a mean observed resistance, which is the mean resistance as measured on the bridge. This had then to be corrected by deducting the resistance of the leads up to and including that of the mercury cups in which the resistance-coil terminals were dipped. This correction was found to be equal to 025 B.A.U. at 20° O. The corrected resistance is the resistance of the coil after this deduction. A further correction was then applied for the temperature of the Bridge coils, which were used at a temperature of about 19°°5 C., or 4° C. above the temperature at which they were correct ; and also a reduction to true ohms, effected by the use of the factor -9866 recom- mended by the Board of Trade Electrical Committee (1 B.A. unit=*9866 true ohm). | If R is the corrected resistance in B.A. units of any coil of wire, the length of which is / centimetres and the mean dia- meter d mils (1 mil=-001 inch), and if p is the resistance of one cubic centimetre of the metal between opposed faces, in absolute electromagnetic units, then it was found that p and R were related by the equation 2 p = 5000 5 R; in which the numerical constant embodies all the above cor- rections. In this reduction no correction has been applied for the change in volume which the wire undergoes when raised or lowered in temperature. Hence the values ofp thus obtained for various temperatures are not the true equi-volume specific resistances, or resistances across opposed faces of one true centimetre of the metal whatever its temperature; but the 330 Profs. Dewar and Fleming on Electrical Resistance values of p are the relative resistances at the different tempe- ratures of one cubic centimetre of the metal taken at the atmospheric temperature and allowed to shrink or expand with the temperature. They are therefore the relative specific resistances for constant mass and not for constant volume. These latter cannot be obtained until we know the true mean coeflicient of cubieal or linear expansion of the various metals and alloys between 0° C, and —100° ©. and —200° C.; but in any case, even for the most expansible metal, the correction would probably not amount to one-half of a per cent., or to less than the error created in the measurement of the resist- ance by an uncertainty in the temperature to the extent of one degree Centigrade. § 4. The following Tables give these specific resistances for constant mass (p) at the various temperatures (¢°) of the different pure metals, alloys, and impure metals employed. The mean value of the various results is given, and the temperature in degrees Centigrade placed over the number de- noting the specific resistance in absolute electromagnetic units. If these specific resistances are plotted out in a series of curves, taking the absolute temperature as abscissee, we find that all the lines of resistance are more or less curved lines which tend downwards in such a way as to show that if pro- longed beyond —200° C. they would probably pass through or near the origin or absolute zero. These curves of resist- ance can be divided into three classes :—(i.) those of metals such as iron, nickel, tin, and perhaps copper. whieh are con- cave upwards; (ii.) those of metals such as gold, platinum, and palladium, and probably silver, which are concave downwards towards the axis of temperature ; and (iil.) those of metals such as aluminium, which are apparently nearly straight lines. In the case of a metal of the first class, such as iron, the resistance changes with the temperature in such a way that the rate of change of resistance with temperature inereases as the temperature increases. In other words, the second dif- ferential of resistance with respect to temperature is posztzve. In the case of a metal of the second class, such as platinum, the second differential of resistance with temperature is nega- t2ve ; that is, as the temperature increases the rate of change of resistance with temperature decreases. _ This distinetion between such metals as platinum and nickel, in respect of their variation of resistance with temperature, has been noted by Professor Cargill G. Knott (Proc. Roy. Soc. Edin. vol. xxxiii. 1888, p. 187), in a memoir on the Hlectrical Resistance of Nickel at High Temperatures. Prof. Knott’s observations on the resistance of nickel were made between 0° and 300° C. Our own were between + 100° and ddl < Ss = SS) os Platinum. 8 S | Gold. i = ™ Silver. S RQ a Copper. > 8 A Iron. oy w "S| Aluminium. = » S Nickel ickel. a > Tin. $1 Q Ho} tl Th °C, = pP = god Or QP a=) tC. —_———— -—— ————_—_—_| —_——. at p ll TABLE I. F Temp. of Temp. of Temp. of Temp. of Temp. of eruie Temp. of | Temp. of deveen boiling ! melting : boiling boiling boili myataneeoline ao ice carbonic | ethylene. | ox gen Ne gapaale ; ‘ acid. y ; y . m vacuo. 100°-2 18° 1° _g9° | —100° | —182° | —197° 10912 8698 8248 6133 5295 2821 9290 ~ 9690°5 | 2993 0°-6 89° | —100° |: —182° E 2639 2096 1952 1400 1207 604 - 100° | ~~ 19 0°-5 —go0° | —100° | —182° cs 2139 1643 1561 1138 962 472 - 9302 | 18°95 | 07 = 100° | —182° | —197° 1881 1447 1353 rai 272 178 9694 | 18°15 1° % —100° | =182° | —197° 13777 9455 8659 si 4010 1067 608 Scions eticeo, ic —100° | —182° 2 4658 3503 3185 1928 894 94°-5 90° 192 =g0° «| 100°: | —182° . 18913. 13494. 12350 7470 6110 1900 ak 99°-3 90° 0°-8 = 2R0@ |, SadG0P: I> 2.182° Bs 13837 10473 | 9609" 6681 5671 - Q5BB «kk wt : Specific Resistance (p) in Blectromagnetic Units of various Pure Metals, at different Temperatures. Remarks. Pure annealed soft pla- tinum wire. Purest soft gold wire. Pure silver wire. Pure electrolytic copper, annealed. Pure soft iron wire, an- nealed. Hard drawn pure alu- minium wire. Pure nickel, prepared by Mr.. Ludwig Mond’s process from compound of nickel and carbonic oxide. Pure tin wire. ee 332 Profs. Dewar and Fleming on Electrical Resistance —200° C. Accordingly the distinction between these two metals, in this respect, extends over a very large range of temperature. The most interesting fact which these experiments have brought out is the enormous decrease in specific resistance experienced by the perfectly pure metals when cooled to these low temperatures. Thus the electrical resistance of a given pure iron wire at —197° C. is only one twenty-third part of that which it is at +100° C. In the case of pure copper the ratio of resistance is about one to eleven for the same change of temperature. The very smallest impurity greatly affects this decrease. In the case of some nickel wire sup- posed te be pure we found that the specific resistance at zero Centigrade was 13387 electromagnetic units, and that the specific resistance at —182° C. was 6737. On repeating the measurement, however, with some absolutely pure nickel, obtained by Mr. Ludwig Mond’s process of depositing the metal on glass by heating the gaseous compound of nickel and carbonic oxide, we found a very different result. In this last case, although the specific resistance at zero Centi- grade was not very different, viz. about 12000 units, the specific resistance at —182° C. was only 1900, showimg a far greater decrease. For the perfectly pure metals, there- fore, it seems probable that as the temperature is lowered towards the absolute zero the specific electrical resistance decreases so that it either vanishes at the absolute zero or reaches a very small residual value. Clausius made the sug- gestion in 1858 (Pogg. Ann. vol. civ. p. 650) that the elec- trical resistance of all pure metals is proportional to the absolute temperature. Owing tothe marked curvature of the resistance-temperature lines this statement is only very ap- proximately true for a few metals, and not at all for others, but it yet remains not improbable that the electrical resist- ance of all pure metals would at the absolute zero be either null or exceedingly small. § 5. In the course of our experiments we found that the trend of the curve of specific resistance drawn with absolute temperatures as abscissze seemed to give a very good indication of the chemical purity of the metal. If that curve tended downward so as to indicate that if prolonged it would pro- bably pass through the absolute zero, the metal was indicated as pure. If, however, as in-the case of the palladium wire and the first nickel wire used, it did not so tend, then impurity was probably in some way present. We next directed our attention to alloys, and the following is a Table of the results obtained for them, treated in the same manner as regards temperature :— 333 ing-point of Oxygen. of Pure Metals &c. at the Boil TABLE II. Specific Resistance (p) of various Alloys in Electromagnetic units at different Temperatures. Temp. of : Temp. of Temp. of Temp. of | Temp. of boiling ay of melting eee boiling boiling Remarks, water. aes ice. ‘acid ethylene oxygen. Platinum-Silver. ge Ob: oho 18°°35 1? —80° — 100° —182° | The alloy commonly used for Ag=66 °/,, Pt=33 °/,.| o = 27400 26905 26824 26311 26108 25537 resistance-coils. Be Bese ee | 2 0,— | 99°38 | 18°45 0°:8 —go° | —100° | —192° oF pct nan Salyer. 0 =| se712 | 34688 | 34534 | 83684 | sa2g0 | gy51g | Commercial wire. tC 100° 18°'45 0°°8 —80° —100° —182° | Martino’s Platinoid; German- Platinoid. Silver with 1 or 2 °/, of 0 = 44590 43806 43610 43022 42385 41454 Tungsten. Palladium-Silver. CC: Ee 99°°8 20° 0°°8 —80° —100° —182° | From Messrs, Johnson and Pd=20°/,, Ag=80°/,.| ¢ = 15409 14984. 14965 14482 14256 13797 Matthey. Be ace] toot aes 028 a Sia SiS ed Phosphor-Bronze. 0 we 9071 8581 8483 8054. 7883 7371 Commercial wire, Platinum-Iridium. (20 a= 100° 18°'8 0°'6 —80° —100° —182° !From Messrs. Johnson and Pt=80)°/,, dr=20°°,. ine = 31848 29870 29390 27504 26712 24440 Matthey. Platinum-Rhodium. t° Cue 100° 18°°8 0°°8 —80° —100° —182° | From Messrs, Johnson and 18417 14532 13719 10778 9834 7134 Matthey. 334. Profs. Dewar and Fleming on Electrical Resistance On charting the above figures for the alloys, it is found that the resistance-lines are very nearly straight lines with but little slope, not one-tenth that of the pure metals, when the constituent metals of the alloy are chemically very different. This is the case with the platinum-silver, platinoid, and German-silver alloys. When, however, the constituents of the alloy are chemically similar, as in the case of the platinum-iridium and platinum-rhodium alloys, the resistance- lines plotted in terms of the absolute temperature slope down at a much steeper angle, but never, as in the case of the pure metals, in such manner as to indicate that if prolonged they would pass through the absolute zero. In a similar manner _the impure metals also behave. We have laid down on sucha resistance-temperature diagram the lines representing the change of specific resistance with temperature for palladium not known to be pure, for nickel not pure, for commercial iron wire, for ordinary bismuth, for gold not pure, and other commercial metals, and found that the resistance-lines of these metals do not slope down at such an angle as to indi- cate that if prolonged they would pass through or near the origin or absolute zero. The lines for these impure metals are in position more or less like the lines for alloys formed of similar metals. § 6. Another fact of considerable interest has been ascer- tained. We know that for temperatures above zero Centi- grade carbon behaves like an electrolyte as regards change of resistance with temperature, that is, its specific resistance decreases as temperature increases. Hence it was a matter of importance to examine the behaviour of carbon as regards electric resistance when cooled to —182° in liquid oxygen. For this purpose we employed the carbon filaments of in- candescence lamps, taking the treated filaments of Hdison- Swan lamps and the dense adamantine carbon employed in the Woodhouse and Rawson incandescence lamp. In both cases we found that when cooled to this low temperature the specific resistance of the carbon continuously increased instead of decreasing, as do the metals. The following Table III. gives the specific resistance (p) in electromagnetic units of these carbons at various temperatures. | § 7. We have not yet completed the examination at similar low temperatures which we propose to make of the behaviour in regard to electrical resistance of such non-metals as selenium and sulphur, and such metals as arsenic and antimony, with marked chemical affinities with non-metals. Also the be- haviour of such insulators as mica, glass, guttapercha, india- rubber remains to be examined. It is known that the electrical resistance of these bodies decreases with rise of Qe eee eee ee 339 TasB.E ITI. Specific Resistance (p) of Carbon at various Temperatures. lamp. S > S ‘> es “> S < Se Carbon from an Edison- eo C= xs Swan incandescence 5 lamp, No. 1. p= RY ~~ =~ e = | Carbon from an Edison- ?? C= ae Swan incandescence cD lamp, No. 2. p= me 8 ~— ~ e a Carbon (Adamantine) C= s from a Woodhouse and a = Rawson incandescence = RY Cae ‘> boiling water. On 3835 x 103 100° 6168 x 10° | Temperature of | Temperature Temperature of of air. melting ice. 18°°9 I 4049 x 10° 4090 x 10° 18°°9. 1° 3911 x 10° 3953 x 10° 18°79 0°38 6303 x 10° 6360 x 10° Temperature of melting carbonic acid, — 80° 4189 x 10° — 80° 4054 10° -- 80° 6495 x 10° Temperature of boiling ethy- lene. — 100° 4218 x 10° —100° 4079 x 10° —100° 6533 x 10° Temperature of boiling oxygen. — 182° 4321 x 10° — 182° 4180 x 10° 336 On the Electrical Resistance of Pure Metals &c. temperature. It is not improbable that we may find for such bodies a maximum electrical resistance at the lowest attainable temperatures, and that it may prove to be the case that pure non-metals approach a maximum specific electrical resistance and pure metals a minimum specific electrical resistance in proportion as the absolute zero of temperature is approached. In any case it is a matter of considerable interest to complete the examination of the change of conductivity with diminished temperature for all the metals in a state of the greatest chemical purity. MM. Cailletet and Bouty expressed the results of their experiments on the electrical resistance of metals between zero Centigrade and —100° C. by giving the value of the mean coefficient of resistance-change as fol- lows :—If R, is the electrical resistance of the metal at ¢° C., and Rp its resistance at 0° C., then we may write i R,(1 SEIS): When ¢ is in the neighbourhood of —100° C., @ has the following values, stated below, for the different metals. We give in one column the results of the experiments of MM. Cailletet and Bouty, and in the other coefficients for the same temperature range calculated from a part of our results in Table I. TaBLeE IV. a = mean coefficient of resistance-change between 0° and —100° O. Metal. Cailletet and Dewar and Bouty. Fleming. STLVOE: csp cectogeerep renee tetas ‘00385 00384 Aluminiuiny, S505 onansenstm=san5's 00388 00390 (PO DPEL jcc nccebhaesteeeeneaaeh ces. 00423 00410 HVOW. Gost sesscecentancesseerstees ‘00490 00531 Platimamyy Sete. cetoesase nag: : "00340 00354 PENTA cows decease nese ceiaar chase 00424 ‘00509 Masnestim "tetecnsr-7-ssaees-.<- OOSOQ yale, Tee aaa WIERCUTY © K the force in the fluid near the solid is increased, while within the solid it is decreased, as compared with the forces in the absence of the solid. The total energy is, however, increased by the presence of the solid, and if the field is variable the solid is acted upon by a resultant force which urges it toward the stronger parts of the field. If, however, K;=E-(P,—P,), . Parte. and pe a,—P) [= Sree e e ° . ° ° . (2) Differentiating equation (1) and combining the result with equation (2), we obtain di d Bae ie, 138 Ge at Gorm” H d or G2 dee LCT The integral of equation (3) for the case that R, L, and C are constants is Qe Sarl Saga 1 R? 1 Re? Bad Sete : @=e 2L { Asin La 7 a2: tt Boos LG 412°" A and B are integration-constants, and e the base of the natural system of logarithms. For t=0 we have z=0, and hence B=0, and equation (4) reduces to ot A i= Ae co Sete AC) (5) . (4) 394 Dr. W. L. Robb on Osecllations Substituting this value of i in equation (1), we have — 2L es - sin yh ae a2"! R2 ot 1 ee +LAe 2L taf ae B® ia 413°! = H—(P,—P,). For t=0 we have P;— P,=0, and consequently from equation (6) we obtain and from equation (5), € “iat sing fe ‘ zs R2 LO: Ae ae i Substituting this value of 7 in equation (1) and reducing, we obtain Bea! ae le P,—P,=E- ne pS jy Bain Loi" eae oY ER +2LA/to- ip mew ro ~ ae} rata) a 1 ee A Le. f pf ae Gp mz ——_---— IN or P,—P,=E- aa sre Lo ae"! LG air Le pi + are tan wou}: . i He? 1 ati 7 re let ee If in equation .(7) we le LO > ae = 2mn, and (6) in the Charging of a Condenser. 395 d re tan 21, \/ L Rt ] final i = arc tan —. —— — —_.. we have as our final expression , R LG 41” sal for the difference of potential of the two surfaces of the con- denser, Heo at’ L ae. bh nh (Qrnt+). . . (8) 4 Equation (8) is only exact on the assumption that the capacity of a condenser is independent of the time. The capacity being a function of the time*, the equation is simply valuable as indicating the general character of the oscillations, and how their amplitude should be influenced by changes in the values of any of the various quantities that enter into the expression for the amplitude. From equation (8) we have the amplitude of the oscilla- R Ee 2° RC /1 aL an increase in the amplitude of the oscillations in the differ- ence of potential of the two surfaces of the condenser corre- sponds to a decrease in the time or of the resistance, and to an increase in either the electromotive force, the coefficient of self-induction, or the capacity. The following determinations, which are to be considered simply as qualitative in their nature, were made to see in how far the observed variations in the charge followed laws similar to those given by equation (8) for variations in the difference of potential of the two surfaces of the condenser. Two con- densers were used in these experiments. One was a standard microfarad mica condenser, manufactured by Carpentier of Paris, and subdivided into 0°5, 0:2, 0:2, and 0-1 microfarad. The other was a microfarad paraffin condenser, manufactured by the Societe d’exploitation des cables électriques of Cor- taillod, Switzerland. tions equal to From this it would follow that * Willner, Svtzungsb. kénigl. bayer. Akad, 1877, p. 1. 396 Dr. W. L. Robb on Oscillations 1. Effect of Diminishing the Time. A series of determinations was made to find the effect ot diminishing the time of contact of the spheres (see fig. 1) upon the variations observed in the charge of the condenser. The simplest way of varying with certainty the time of con- tact was io use, as the smaller sphere, spheres of different sizes. An increase in the diameter of the sphere lengthens the time of contact. Three different spheres were used, and in Tables Il. and III. the results of twenty consecutive obser- vations with each sphere are given. In the actual work, however, at least fifty observations were always taken to ensure that the observations given in the following tables were representative in their character and included the extreme variations. Tassie II. (See also diagram 2.) Mica Condenser (normal charge * = 2°010 microcoulombs). | Diameter of Sphere, | Diameter of Sphere, || Diameter of Sphere, S,=0°49 centim. || S,=0°79 centim. S,=2°16 centim. Observation} —____}/____ number. | Charge of | Deviation | Charge of | Deviation) Charge of | Deviation) Condenser.) from the || Condenser.| from the | Condenser. | from the Micro- normal Micro- normal || Micro- normal coulombs. | charge. |; coulombs. | charge. | coulombs. | charge. I 2074 +0°064 1-987 —0°023 1-983 —0:027 2 2072 +0062 2011 +0°001 1-982 — 0-028 3 L971 --0:039 1-988 —0:022 1-988 —0°022 4 2-120 +0°110 1994 —0°016 1-925 —0-025 +) 2°136 +0°126 1-967 —0°043 1-991 —0:019 6 27115 +0:105 2°011 +0°001 1-983 —0:027 7 2-071 +0°061 1-996 —0-014 1-980 —0-030 8 2044 +0:034 1-987 — 0023 1-983 —0-027 9 2 061 +0:051 2-021 +0°011 1-982 —0°028 10 2°038 +0:028 1-985 —0°025 1-980 —0-030 1] 2-065 +0°055 | 1:986 —0:024 1-982 —0-028 12 2150 +} +0:140 1-987 —0023 1-988 — 0-022 13 2-093 +0°083 || 1976 — 0-034 1-985 —0°025 14 2-030 +0:020 2-024 +0014 1-981 — 0-029 15 1-987 —0°023 1971 —0:039 1:988 —0-022 16 1-966 —0:044 1-969 —0-041 1-983 —0°027 Ws 2-081 +0071 1-984 — 0-026 1-985 --0:025 18 2107 +0:097 2-007 —0-003 1983 —0 027 19 2-060 +0:050 2-009 —0-001 1-980 —0-030 20 2065 +0°055 2-016 +0°006 1-988 —0-022 * By the term “normal charge” is meant the constant charge which the condenser eventually assumes when charged for a considerable time. rge. Deviation from the normal cha: Microcoulombs. —0'1 —0°2 | Observation number. Be : in the Charging of a Condenser. TABLE III. Diameter of Sphere, | S,=0°49 centim. | Micro- 2-020 1:808 1-871 1-969 1-890 1:977 2-098 1-887 1-890 1-987 1-829 1-942 1-813 2-094 1839 1-843 2-096 2-041 1°832 1925 Charge of | Condenser. | coulombs. | Deviation| from the normal | | charge. Diameter of Sphere, S,=0°79 centim. (See also diagram 2.) 397 Paraffin Condenser (normal charge =2°044 microcoulombs). Diameter of Sphere, S,=2°16 centim. Charge of Condenser. Micro- coulombs. 1-921 1:877 1°892 1°845 1-890 1-859 1-868 1-930 1-868 1:886 1-863 1-885 | 1-905 1-895 1-900 1-843 1-902 1-914 1-934 1:853 Deviation from the normal charge. | Charge of | Condenser. | Micro- coulombs. 1-915 1922 1-914 1-915 1-906 1-915 1915 1-912 1916 1-915 1:916 bia O28 1-915 1924 1-914 1915 1917 1-906 1-915 1914 Deviation from the normal charge. —0°129 —0°122 —0'130 —0-129 —0'158 —0°129 —0°129 —0:132 —0°128 —0:129 —0128 —0°116 —0-129 —0°120 —0:130 —0°129 —0°127 —0°138 —0129 | —0-130 Diagram 2. Mica Condenser. Twenty consecutive determinations. | se Mia ape ak o—.—© >) 99 ” aE (See Tables IT. .... + ‘Lime of charging least. and [TI.) Br hese ae ee Paraffin Condenser. Twenty consecutive determinations. intermediate. greatest, An inspection of Tables IL. and III. and of the graphical representation of these observations given in diagram 2, shows that when the diameter of the sphere 8,, and consequently the time during which the condenser receives a charge, 1s increased, the variations in the charge rapidly diminish. 398 Dr. W. L. Robb on Oscillations. 2. Liffect of varying the Lesistance. A series of determinations was made to find the effect of varying the resistance of the charging circuit upon the varia- tions observed in the charge of the condenser. The resistance was varied by changing the suspension of the sphere S, (see fig. 1). In one set the suspension was a single silver wire, 0:05 millim. in diameter and 6 centim. long. In a second set a band made up of ten such silver wires, and consequently having only one tenth the resistance, was used. In the third set a single platinum wire 0°05 millim. in diameter and 6 centim. long was substituted for the usual suspension of silver wire. The results obtained with these various resistances are given in Tables IV. and V. Taste IV. (See also diagram 3.) Mica Condenser (normal charge = 1°978 microcoulombs). | R=0:04 ohm. R=0°41 ohm. R=3'72 ohms. | ie : Charge of Deviation || Charge of | Deviation || Charge of | Deviation Condenser.) from the ||\Condenser. | from the || Condenser.) from the Micro- normal Micro- normal Micro- normal coulombs. | charge. || coulombs. | charge. || coulombs, | charge. 1 2-092 +0:114 1:969 —0-009 1-916 — 0-062 2 1:985 +0007 1:949 —0°029 1-908 —0:070 3 1°863 —0°115 2:015 +0:037 1:910 — 0:068 4 1:932 —0°046 1:928 —0°050 1:931 —0:047 5 2:091 +0118 1:964 —0°014 1:908 —0:070 6 1813 —0°165 1:987 +0:009 1:908 —0-070 7 2°015 +0°037 1-961 —0:017 1910 —0-068 8 2°180 +0°202 1:978 +0-000 1919 —0°059 2 1:936 — 0042 1:934 — 0-044 1:908 —0:070 10 2116 +0°138 2021 +0:043 1:926 —0°052 1 2-004 +0:026 1-902 —0:076 1-939 —0-039 12 2081 +0°053 1:963 —0-015 1-916 —0:062 13 2:045 +0:067 1-984 +0-006 1:909 — 0-069 14 2°136 +0°158 1961 —0:017 1:916 —0:062 15 2:054 +0°076 1:979 +0°001 1917 —0:061 16 2:133 +0°155 1:924 — 0-054 1:924 —0:054 17 2-184 +0°206 1:957 —0:021 1:918 —0-:060 18 1°922 — 0-056 1°885 —0:093 1:926 —0:052 19 2:080 | +0°102 1:922 | —0:056 1:908 — 0-070 20 | 2084 | +0°106 1:976 | — 0-002 1°926 — 0:052 in the Charging of a Condenser. 399 TaBLE V. (See also diagram 3.) Paraffin Condenser (normal charge= 1°996 microcoulombs). | | | | R=0-04 ohm. R=0-41 ohin. | R=38°72 ohms. ce ee Charge of | Deviation | Charge of | Deviation, Charge of _ Deviation Condenser. | from the | Condenser.) from the | Condenser.) from the Micro- normal || Micro- normal | Micro- | normal coulombs. | charge. | coulombs. | charge. | coulombs.| charge. | | 1 1-987. | -0009 || 1-870 | —0-196 1-764 | —0-232 2 2012 | +0:016 1875 | —0-121 L778 f= Ole | 3 1605 | —0-391 1851 | —0-145 1:787 | —0-209. | | 4 1934 | —0062 || 1:896 —O' 100 1-778") eC Se | a 1:950 | —0°046 || 1:829 —0'167 W771 =| —0°225 6 1916 | —0O080 | 1:865 —O 131 1 C776) | =O 220 7 1964 | —0:032 | {°830 —0O°166 |} 1746 | —0°250 8 1-980 | —0016 | 1820 —0'176 1778 —0:218 y 1670 | —0326 || 1:862 —0°134 1778 —0°218 10 1822 | —0-174 | 1°803 —(0-193 1°764: —()'232 1] 1829 | —0:167 | ESTs —0°118 1'770 — 0°226 1Z 2047 «=| +0051 || -1°897 —0-099 1°758 —0°238 13 1976 | —0:020 || 1-°850 —0O146 | 1778 | —0218 14 1:710 —0'286 || 1:889 —=(O107 |) ~ 1-760: |.—0236 | 15 1923 | —0-073 || 1840 | —0156 | 1-770 | —0-226 16 2005 | +0009 | 1874 | —0122 | 1-768 | —0-228 17 1-835 —O161 | 1°800 —0196 | 1:774 — 0-222 18 1970 | —0026 || 1:895 —O101 | 1:°774 —0°222 19 1°886 —0110 | 1875 =(O121 || . i764 | —0:232 20 1-960 —0:036 | 1°896 —0°100 1756 | —0:240 | | Diagram 3. (See Tables IV. and V.) Mica Condenser. Paratfin Condenser. Twenty consecutive determinations. Twenty consecutive determinations. rge. Microcoulombs. Deviation from the normal cha +....+ R=0°04 w. x — x R=041 w. @— —© R=3°72 4. An inspection of Tables IV. and V., and of the graphical representation of these observations given in diagram 3, shows that the variations in the charge diminish rapidly as the resistance of the charging circuit is increased. 400 Dr. W. L. Robb on Oscillations 3. Lffect of varying the Electromotive Force. In order to determine the effect of varying the electromo- tive force of the battery used in charging upon the variations in the charge, two sets of observations were made. In one set the battery used consisted of five accumulators joined in multiple are. In the second set the battery consisted of fifteen accumulators joined in multiple arc in groups of five, and the resultant groups joined in series; in all other respects the conditions under which the two sets of observations were made were identical. Tables VI. and VII. give the results of these observations. : TasLe VI. (See also diagram 4.) Mica Condenser. 20 2016 —0059 | 6°152 —0:073 pet ee | E=6-225 volts. | | | Normal charge = 2'075 || Normal charge = 6:225 | microcoulombs. | microcoulombs. | Observation SMG Gwen GREE | number. | Charge of | Deviation | Charge of Deviation | | Condenser. from the || Condenser. from the | Micro- normal | Micro- normal | coulombs. charge. | coulombs, charge. | 1 / 2°135 +0060 | 6210 —0-015 2 27100 +0°025 || 6°240 +0°015 3 | 2053 —0:022 | 6°400 +0175 4 2:057 —0018 | 6°482 +0°257 5) 2075 +0000 | 6°453 +0°228 6 | 2°057 —0018 | 6°352 +0°127 7 2049 — 0-026 6 181 —0 044 8 | 2.058 —0-017 6°487 +0°262 9 2°051 — 0-024 6148 —0-077 10 2°094 +0-019 6°605 +0°380 11 2°102 +0:027 67148 —0-077 12 2058 —0:017 6643 +0°418 13 | 2.094 +0019 || 6-170 —0-055 14 / 2°103 +0028 6170 —0:055 15 | 2-007 —0:068 6°222 —0°003 16 2:053 —0°022 6°281 +0056 17 2°136 +0061 6153 —0-072 18 / 2°035 —0:040 | 6°562 +0°337 19 | 2061 —0-014 6-409 +0°184 in the Charging of a Condenser. 401 TaBLE VII. (See also diagram 4.) Paraffin Condenser. E=6 225 volts. | | E=2-075 volts. | Normal charge = 2:093 || Normal charge = 6°279 | microcoulombs. microcoulombs. | Observation |——M—@§-_——______ | — : | number. | Chargeof | Deviation Charge of Deviation | : Condenser. from the || Condenser. from the Micro- normal | Micro- normal / coulombs. charge. || coulombs. | charge. 1 1970 —0123 || 5-993 | —0:286 z 1-955 —O'138 || 5817 | —0'462 | 3 1:973 —0':120 || 6°152 —0°127 | 4 1:993 —0:100 6°135 | —O144 | a) 1:970 —0'123 || 5-954 | =—Ogas © | | 6 1962 | —0O131 || 5985 | —0-294 | 7 1955 —0'1388 | 5-922 | —0°357 8 1°95d —O1358 | 6°183 | — 0096 9 1-932 —O0'161 6°24] —0:038 10 1:996 —0:097 || 6°220 | —0-059 | 11 1915 —O178 || 6°220 | —0:059 12 1972 —O'121 || 5-902 —0:°377 i 1-939 —0°154 6°240 | —0'039 14 1°921 —0:172 G17 | —0°162 15 1-961 —0°132 6:076 | —Q:203 16 1-992 —O101 | 6°243 | —0:036 iv 1:977 —0'116 6°188 | —0-091 18 1972 —0-121 6°254 —0 025 19 1-965 —-128 |} 6-251 | —0028 20 I-95 —0°138 6°195 — 0084 Diagram 4, (See Tables VI. and VII.) Mica Condenser. Paraffin Condenser. Twenty consecutive determinations. Twenty consecutive determinations. ge. Deviation from the normal char Microcoulombs +... + H=2 voits @©——@® E= volts. 402 Dr. W. L. Robb on Oscillations An inspection of Tables VI. and VII., and of the graphical representation of these observations given in diagram 4, shows that the variations in the charge are increased when the electromotive force of the battery used in charging the con- denser is increased. 4, Effect of varying the Self-induction. In order to determine the effect of varying the self-induction in the charging circuit two sets of observations were made. In the first set, No. 7 B.W.G. copper wire was used in making the circuit, and the total length of copper wire used was 50 centim. In the second set an additional piece of copper wire, of the same diameter and 120 centim. in length, was included in the circuit, making the total length of copper wire 170 centim. In this way the self-induction was increased without appreciably altering the resistance. Tables VIII. and IX. give the results of these observations. Taste VIII. (See also diagram 5.) Mica Condenser (normal charge = 1°971 microcoulombs). | Smaller Self-induction. | Larger Self-induction. | | = i Bedi ny Charge of | Deviation Charge of | Deviation | ‘ Condenser. | from the Condenser. | from the | Micro- | normal || Micro- | normal coulonibs. charge. | coulombs. charge. 1 1-980 +0009 | 1-754 —O0'217 2 1-968 —0003 | 2 010 +0-069 3 1 982 +0011 | 1:964 — 0007 4 1-969 —0002 | 1-782 —O0'189 5 1-958 —O0018 | 1-745 —0°226 6 1-958 07013: | 1853 —0118 7 1956 —0015 || 1-880 —0091 8 2-008 +0037 = || 1-855 —O'116 9 1978 +0007 | 1°655 —0316 10 1-950 —0-0z1 | 1884 —0:137 11 1-952 —O0019 | 1829 —0°142 12 1-968 —0:003 || 1-731 —0°240 13 1-974 =O: oi 1-723 —0'248 14 1958 —O013 | 1-971 +0000 15 1-950 —0 021 1-920 —0:051 16 1967 —0:004 1813 —0-158 ‘le 2:010 +0:039 1871 —0100 18 2-008 +0:-037 1900 —O-O71 . 19 1940 00380 tt | | ea 20 1972 +0001 1943 —CO28 in the Charging of a Condenser. 403 Tarte 1X. (See also diagram 5.) Paraffin Condenser (normal charge = 2°093 micro- coulombs. ) Smaller Self-induction. || Larger Self-induction. | eg Charge of Deviation || Charge of | Deviation | Condenser. | fromthe || Condenser. from the | Micro- normal || Micro- normal coulombs. | charge. || coulombs. charge. | 1 19a2 | == 0161 ., || 1-885 —0:208 2 1-926 | —O167 || 1-817 —0-276 3 1929 | —0O164 | 1 873 — 0220 4 1-936 —O157 || 1°735 —0°358 5 1:936 —0:157 | 1818 —0:275 6 1:932 —O0-°161 1-920 —0'173 7 1:932 OF AG liens | 1-780 — (313 8 | 1-984 —O-159 | 1-832 —0:261 9 1:914 —0'179 | 1°854 — 0-239 10 1:932 ==()"161, | 1°829 —0:264 | 1B | 1:942 —O151 \| 1°735 —0:358 | 12 | 1:934 —O159 || 1-891 — 0-202 13 | 1-940 —0°153 | 1°833 —0:260 14 | 1903 —0:190 1:883 —0-210 | 15 1:924 —0:169 1°868 —0°225 | 16 1-932 —0:161 1°785 — 0-308 17 _ 1:935 —0°158 | 1812 —0°281 18 _ 1-942 —O°151 | 1°830 — (0-263 19 1:931 —0°162 | 1°880 —0°213 20 1:936 —0:157 1:744 —0°349 Diagram 5. (See Tables VIII. and IX.) Mica Condenser. Paraffin Condenser. Twenty consecutive determinations. Twenty consecutive determinations. ge. Microcoulombs. Deviation from the normal char +....+ Smaller self-induction. ©— —© Larger ¥3 9 An inspection of Tables VIII. and LX. and of the graphical representation of these observations given in diagram 5, shows that the variations in the charge are greatly increased by increasing the self-induction in the charging circuit. 404 Dr. W. L. Robb on Oscillations 5. Hffect of varying the Capacity. The paraffin condenser was constructed so that a part of its capacity could not be used. Consequently the effect of varying the capacity upon the variations in the charge was only determined with the mica condenser. The results of these observations are given in Table X. TABLE X. (See also diagram 6.) Mica Condenser. Peace ee = | Capacity = 1 microfarad. || Capacity = 0'5 microfarad. | Normal charge = 1-976 || Normal charge = U-988 | | microcoulombs. | microvoulomb. | Observation | = ; a number. | Charge of Deviation || Charge of Deviation | | Condenser. from the Condenser. from the Micro- normal | Miero- normal conlombs. charge. | coulombs. charge. 1 } 2010 +0-034 0-993 +0:005 | 4 1°925 —0:051 0-983 —0005 | 3 | 1°858 —O118” | 0-979 —0-009 | 4 1-972 —0:004 | 0-969 —O019 | / 5 1-908 —0068 | 0-974 —O014 | | 6 | 1-945 —0-031 | 0979 —0-009 | 7 | 2-015 +0:039 | 0-991 +0003 | | 8 | 1:965 —0-011 || 0-974 —0014 | | 9 1917 —0:059 | 0-975 - 0013 | 10 1-975 —0001° | 0-991 +0°003 | 11 1-947 —0:029 0978 —0-010 | ot RH el) ERP —0-013 0-959 —0-019 13 1:955 —0:021 0-989 +0°0UL | } da a eg +0:003 || 0-988 —0:005 | | 1D | 1925 —O0051 | (974 —O0-014 - | | 16 1:957 —O0019 | 0-970 —0-018 17 | 1-965 —O01L | 0973 —0015 4 18 1-957 —OO019 0-991 +0003 19 1-948 —():028 0-983 | —0:005 | 20 / 1-900 —0:076 | 0-983 — 0-005 Diagram 6. (See Table X.) Mica Condenser. Twenty consecutive observations. oe | z 2 oe a — v3 Sas | 3 a +....+ C=1 microfarad. ©— —© C=0'5 microfarad. An inspection of Table X. and of the graphical representation in the Charging of a Condenser. 405 of these observations given in diagram 6, shows that the variations of the charge are increased when the capacity of the condenser is increased. Although the preceding work is simply qualitative in its character, one is justified, as a result of these observations, in concluding that oscillations in the charge of a condenser occur during the charging, and that the amplitude of these oscillations diminishes rapidly as the time increases, and may be increased either by diminishing the resistance, or by increasing either the electromotive force, the self-induction, or the capacity. Since completing the above experiments I have begun a set of determinations, quantitative in their character, for the purpose of deducing the relation between the charge of a condenser and the time of charging. In order to show more clearly the character of the oscillations that occur in the charge, the results of a preliminary determination with a mica condenser are given in Table XI., and the corresponding curve is shown in diagram 7. Taste XI. (See also diagram 7.) Mica Condenser (normal charge = 2 050 microcoulombs). Time of Charging. sea : : Fiandecdtleicendihs pine of rea | Bee toeatT: icrocoulombs. | 1°26 3°78 2°52 0-71 3°78 312 5:04 1:33 6°30 2°75 7°56 1°65 8°82 2°41 10'1 1°72 LES 2-26 126 1:83 13°9 2°15 151 1°89 16°4 2-12 176 1°93 18°9 2:09 20°2 1:96 21:4 2-06 22:7 2-00 27°1 2°02 36'1 2°02 el. Mag. . 5. Vol. d4. No. 210. Nov. 1892. 2F 406 On Oscillations in the Charging of a Condenser. e of Condenser. crocoulombs Ghatge Hurdred-thousandths of a second. ‘ume of charging. The resistance of the circuit in this determination was about 0:03 ohm, and the condenser was charged by means of the well-known von Helmholtz pendulum*, manufactured by L. Zimmermann, of Heidelberg. The usual manner of making the connexions was, however, slightly modified, as indicated in fig. 4. Fig. 4. Condenser Yl Bure IE 4 / ee ee ee ee ee re ee eee ee The sections of the two steel bars attached to the lower part of the pendulum proper are denoted by A and B. The battery was always in connexion with A and insulated from B. * Helmholtz, Monatsberichte der Berliner Akad. May 25, 1871, p. 294. Lord Rayleigh on Interference Bands. 407 The binding-posts denoted by P and the metal parts attached to them were insulated from each other by being fastened to hard rubber bases. As arranged, the circuit was completed by A coming in contact with the lever C, and broken by B coming in contact with the lever D. In order to insure good contacts the surfaces at the points where the circuit was made and broken were plated with gold. By means of the micrometer-screw attached to one of the hard rubber bases, the time between the contacts that made and broke the circuit could be varied. One complete turn of the drum, which was subdivided into one hundred divisions, corre- sponded to a difference in time of contacts, and consequently to a duration of charge of 0 00018 second. The result of this determination accords fully with the earlier results ob- tained with the mica condensers. In the determinations with a paraffin condenser, described in the first part of this article, the observed charges were usually considerably smaller than the normal charge, and than the charges given to the mica condenser under the same conditions. It is therefore to be expected that, when the determination just described is repeated with a paraffin condenser, a curve will be obtained differing somewhat from that obtained for the mica condenser. The above work was carried out in the laboratory of Prof. H. F. Weber, at Ziirich, and I take this opportunity of thanking him for his kindness in permitting me to use the laboratory at all times, and for his numerous valuable sug- gestions. August 18th, 1892, XLVII. On the Interference Bands of Approximately Homo- geneous Light; in a Letter to Prof. A. Micuetson. By Lord RayueieH, Sec. R.S.* Dear Prof. MIcHELson, AY BEN we were discussing together the results of your interesting work upon high interference, you asked my opinion upon one or two questions connected therewith. I have delayed answering until I had the opportunity of seeing your paper in print (Phil. Mag. Sept. 1892), but now I may as well send you what I have to say. First, as to the definiteness with which the character of * Communicated by the Author. 2 2 408 Lord Rayleigh on the Interference Bands the spectral line $() can be deduced from the “ visibility- curve.” By Fourier’s theorem, tae ti ; 7 ro) 2 ee \ du { COS UE \ cos uv d(v) dv w No L ie o y +0 | + sin uz ( sin uv h(v) aw} : io 3] or in your notation, if we identify w with 27D, Ge je LY du 4c cosue+S sin ue f Hence, if C and S are both given as functions of yw, (2) is absolutely, and uniquely, determined. However, the visibility-curve by itself gives, not both C and 8, but only C?+8*; so that we must conclude that in general an inde- finite variety of structures is consistent with a visibility- curve given in all its parts. But if we may assume that the structure is symmetrical, S=0; and ¢ is then determined by means of C. And, since V?=C?/P?*, the visibility-curve determines C, or at least C?. In practice, considerations of continuity would always fix the choice of the square root. Thus, in the case of a spectral band of uniform brightness, where V? = sin 2arn/an?, we are to take C = sin mn/7n, and not C= + (sin 2an/7?n?). In order to determine both C and 8, observations would have to be made not only upon the visibility, but also upon the situation of the bands. You remark that ‘it is theoreti- cally possible by this means to determine, in case of an unequal double, ora line unsymmetrically broadened, whether the brighter side is towards the blue or the red end of the spectrum.” But I suppose that a complete determination of both C and 8, though theoretically possible, would be an extremely difficult task. If the spectral line has a given total width, the “visibility” begins to fall away from the maximum (unity) most rapidly when the brightness of the line is all concentrated at the edges, so as to constitute a double line. It is interesting to note that in several simple cases the bands seen with ever increasing retardation represent the of Approximately Homogeneous Light. 409 character of the luminous vibration itself. In the case of a mathematical spectral line, the waves are regular to infinity, and the bands are formed without limit and with maximum visibility throughout. Again, in the case of a double line (with equal components) the waves divide themselves into groups with intermediate evanescences, and this is also the character of the interference bands. Thirdly, if the spectral line be a band of uniform brightness, and if the waves at the origin be supposed to be all in one phase, the actual compound vibration will be accurately represented by the corresponding interference bands. But this law is not general for the reason that in one case we have to deal with amplitudes and in the other with zntensities. The accuracy of correspondence thus requires that the finite amplitudes in- volved be all of one magnitude. A partial exception to this statement occurs in the case of a spectral line in which the distribution of brightness is exponential. Another question related to the effect of the gradual loss of energy from communication to the ether upon the homo- geneity of the light radiated from freely vibrating molecules. In illustration of this we may consider the analysis by Fourier’s theorem of a vibration in which the amplitude follows the exponential law, rising from zero to a maximum, and after- wards falling again to zero. It is easily proved that mr cos re = sa) du cos ux {e—@—7P Aa? 4. gut r)? ha? & in which the second member expresses an aggregate of trains of waves, each individual train being absolutely homogeneous. If a be small in comparison with 7, as will happen when the amplitude on the left varies but slow ly;) e Geir ey may De neglected, and e——”/42? is sensible only when wu is very nearly equal to r. As an example in which the departure from regularity con- sists only in an abrupt change of phase, let us suppose that ar(z) = +s1n (2774/1), the sign being reversed at every interval of ml, so that the + sign applies from 0 to ml, 2 ml to 3 ml, 4ml to 5 ml, &e., and the negative sign from a to 2ml, 3 wal to 4 ml, &e. As the analysis into simple waves we eae _ «2 cos (27n2a/2ml) wes mt (1—n?/4m?) ” the summation extending to all odd values 1, 3, 5,... of n. The fundamental component cos (27«/2ml) and every odd 410 Lord Rayleigh on Interference Bands. harmonic occur, but not to the same extent, When x is nearly equal to 2m, the terms rise to great relative magnitude. The most important are thus 27x Qe cos "(1 5-) hy cos “7 (1+ 5), Xe. ; and it is especially to be remarked that what might at first sight be regarded as the principal, if not the solitary, wave- length, viz. /, does not occur at all. Besides communication of energy to the ether, and dis- turbance during encounters with neighbours, the motion of the molecule itself has to be considered as hostile to homo- geneity of radiation. The effect, according to Doppler’s principle, of motion in the line of sight was calculated by me on a former occasion and is fully regarded in your paper. But there is another, and perhaps more important, consequence of molecular motion, which does not appear to have been remarked. Besides the motion of translation there is the motion of rotation to be reckoned with. The effect of the latter will depend upon the law of radiation in various directions from a stationary molecule. As to this we do not know much, but enough to exclude the case of radiation alike in all directions, as from an ideal source of sound. Sucha symmetry is indeed inconsistent with the law of transverse vibrations. The simplest supposition is that the radiation is like that. generated in an elastic solid, at one point of which there acts a periodic force in a given direction. In this case the amplitude in any direction varies as the sine of the angle between the ray and the force, and the direction of (trans- verse) vibration lies in the plane containing these two lines. A complete investigation of the radiation from such molecules vibrating and rotating about all possible axes would be rather complicated, but from one or two particular cases it is easy to recognize the general character of the effect produced. Suppose, for example, that the axis of rotation is perpen- dicular to the axis of vibration, and consider the radiation in a direction perpendicular to the former axis. If w be the angular velocity, the amplitude varies as coswt, and the vibration may be be ae by 2 cos wt. cos nt=cos (n+ w)t+cos (n—o)t. The spectrum would thus show a double line, whose compo- nents are separated by a distance proportional to a. Again, if the ray be parallel to the axis of rotation, the amplitude is indeed constant in magnitude, but its direction Raouit’s Law of the Lowering of Vapour-Pressure. 411 rotates. The plane-polarized rays into which the vibration may be resolved are represented as before by cos wt . cos nt. There is of course one case in which these complications fail to occur, 7.e. when the axis of rotation coincides with the axis of vibration; but with axes distributed at random we must expect vibrations (n+) to be almost as important as the vibration n. The law of distribution of brightness in the spectral line would probably be exponential, as when the widening is due to motion of molecules as wholes in the line of sight. It will be of interest to compare the magnitudes of the two effects. If v be the linear velocity of a molecule and V that of light, the comparison is between w and nv/V, or between mand v/A. If r be the radius of a molecule, the circum- ferential velocity of rotation is wr, and we may compare or with vr/A. Now, according to Boltzmann’s theorem, rw would be of the same order of magnitude as v, so that the import- ance of the rotatory and linear effects would be somewhat as X:7. There is every reason to suppose that > is much greater than 7, and thus (if Boltzmann’s relation held good) to expect that the disturbance of homogeneity due to rotation would largely outweigh that due to translation. Your results seem already to interpose serious obstacles in the way of accepting such a conclusion ; and the fact that light may thus be thrown upon a much controverted question in molecular physics is only another proof of the importance of the research upon which you are engaged. am, Yours very truly, RAYLEIGH. September 23, 1892. XLVI. An Attempt to give a Simple Theoretical Expla- nation of Raoult’s Law of the Lowering of Vapour-Pressure. By ¥. G. Donnan *. | AOULT’S Law, as is well known, applies to the lowering of the vapour-pressure of a liquid, due to the solution in it of a non-volatile substance. It asserts that the relative lowering of vapour-pressure is proportional to the ratio of the number of molecules of the dissolved substance to the total number of molecules in the liquid. Expressed in sym- bolical language this is ‘bom AS ae ja ae a * Communicated by the Author. 412 Mr. F. G. Donnan on Raoult’s Law of where /=original vapour-pressure, /’=reduced pressure, n=number of molecules of dissolved substance, and N= number of molecules of solvent. The factor k never deviates far from unity, and for solutions of medium concentration it is sensibly equal to unity, so that for such solutions we have ia as Ne This is the usual form in which Raoult’s Law is applied. This equation has been theoretically deduced from the known laws of osmotic pressure by Van’t Hoff and by Arrhenius. The following 1 is an attempt to deduce it from the kinetic molecular theory. Suppose we have a liquid in equilibrium with its vapour at a certain temperature ¢ Let the corresponding vapour- pressure be f. We shall suppose the liquid and its vapour to be contained in a closed vessel, and we shall also suppose the temperature to remain constant, so that no permanent disturbance of equilibrium due to a permanent change of temperature need be considered. For this purpose we may imagine the walls of the containing vessel to be perfect conductors of heat, and suppose the vessel surrounded by a very large reservoir at constant temperature ¢. This equilibrium of the liquid with its vapour is regarded as conditioned by a mutual and equal exchange of molecules. A certain definite number, x, of molecules escape per second from the superficial film of the liquid and pass into the vapour, while the same number, x, pass per second from the vapour into the liquid. Although the escape of any one particular molecule from the liquid into the vapour, and vice versa, 1s in itself a fortuitous occurrence, yet, owing to the very great number of molecules, a constant average is main- tained and equilibrium preserved. This is the usual view of liquid-vapour equilibrium, on the kinetic hypothesis. Considering the vapour, «x will, ceteris paribus, be propor- tional to the number of molecules per unit volume—that is, ‘proportional to the density, and therefore to the pressure. Thus we may write e=cf,t being constant. ‘The number of molecules in the liquid remains constant. Call this nnmber N. Now suppose we dissolve in the liquid n molecules of a substance which exerts no appreciable vapour-pressure at the temperature ¢, 7. e. whose molecules, for some reason or other, cannot escape at this temperature from the superficial film into the outer space. We have now a modified state of affairs within the liquid, It is allowable to suppose, as the Lowering of, Vapour-Pressure. 413 before, that x molecules per second get an opportunity of escaping from the liquid, since there is nothing to alter this condition of things, as the molecules of the dissolved sub- stance move freely about like those of the solvent. But, assuming a homogeneous composition of the solution, the of the x theory of chances shows that only the fraction So molecules which get suitable opportunities will be molecules of the solvent, and therefore able to escape. Thus we have N N-+n now only x molecules escaping every second from the liquid. Be a diiialy the previous mobile equilibrium is disturbed, for the liquid will now be gaining molecules, and the vapour losing them. This will continue until the rates at which the liquid and vapour gain molecules again become equal. ‘The value of the . N e . « fraction Rin? however, changes during this process, inas- much as N becomes N+v, where v is increasing and repre- ‘sents at any moment the number of molecules gained by the liquid (we must suppose that the film of pure solvent thus formed is dispersed and homogeneity secured by agitation of the liquid). Thus when equilibrium is again attained we N+v shall have Nin+oe the vapour per second, and vice versa. If fis the new vapour- pressure, we have now N+v Meee a? « molecules escaping from the liquid into where v has now its final value, 7. e. the total number of molecules gained by the liquid. If we divide the members of this equation by the members of the previous equation, we obtain :-— N+v By" Ninao f? whence CP Oe ae a Nt+n+tv f° n will be a small fraction and Now if the solution is dilute, + = 414 Raoult’s Law of the Lowering of Vapour-Pressure. v will be very small in comparison with N+n. So that we may write, in accordance with Raoult’s Law :— , ee olf" Neen ff For concentrated solutions, v will be greater and ee. will not be a small fraction, so that in this case it is just possible v will have an appreciable effect, in which case we should have n pee N+an ale where & is less than unity. But as v will evidently never become of much importance in comparison with N+n, & will never deviate far from unity, as Raoult observed. In the above investigation, v stands for the number of molecules gained by the liquid. But it is evident v is con- nected with the change of vapour-pressure. At first, number of molecules per unit volume of vapour=A/. Denoting time by t, the rate at which the liquid gains molecules= a Hence the number of molecules gained by the liquid from volume V of vapour in time ¢ is expressed by vg di Tape t=0 Ut Thus we have :— n _f—f Nat Vag OF This expression assumes that the whole volume V remains saturated. Assuming that the fundamental equation of the kinetic theory of gases is applicable to the space occupied by the vapour (and the liquid surface acts towards the vaporous space just as if it were an immovable wall), it is easily seen that 2 eee m where m = mass of a molecule, and v = velocity of mean square of the molecules (the pressures / and 7" being measured in absolute units). [ 415 ] XLIX. On the Appreciation of Ultra-visible Quantities, and ona Gauge to help us to appreciate them. By G. JoHN- STONE STONEY, J/.A., D.Sc., PRS., Vice-President, Royal Dublin Society *. I. Description of the Gauge. Poo a quadrant of the earth’s meridian to be straightened out, and used as the base-line of a wedge- shaped gauge. Set a metre upright at one end of this base, and from the top of it draw the inclined plane to the other end. This completes the gauge. It is, in fact, a wedge with a slope of one in ten millions. We shall only require the last ten metres of this gauge, next its apex ; and it is this portion which I propose as a standard for the measurement of small quantities. Small quantities are to be measured by the ordi- nates of the gauge, that is by the little perpendicular distances from its base-line up to its sloping top. Another and perhaps a better way of conceiving the gauge is to take a base-line that is only ten metres long, to erect a micron} at one end, and from the top of this to draw the incline to the other end. This will give the same slope as before—a gradient of one in 10,000,000. II. Illustrations of the very Acute Angle of this Gauge. 1. A wedge with an angle of 1” would furnish a slope of one in 206,265. Ours has a slope of only one in 10,000,000. It is, accordingly, between 48 and 49 times more acute: in other words, its angle is less than the forty-eighth of 1”, which is a much smaller angle than can be measured by any astronomical instrument. 2. Prolong the gauge beyond the ten metres. Then the slightly differing diameters of the red corpuscles in human blood are equal to the ordinates of the gauge at from 70 to 80 metres from its apex—about as far as street-lamps are from one another. 3. At 10 kilometres distance (over six miles) the ordinate is exactly one millimetre. * From the ‘Scientific Proceedings’ of the Royal Dublin Society, vol. vii. p. 530. Communicated by the Author. + The micron is a measure that has come of late years into general use among microscopists. It is the thousandth of a millimetre, which is the same as 1/25400th of an inch. The micron is between the seventh and the eighth part of the diameters of the little red corpuscles in human blood, which are tolerably uniform in size and are familiar objects to all workers with the microscope. 416 Dr. G. J. Stoney on the 4. And to reach an ordinate which is as long as an inch, we should have to go to a distance of 254 kilometres from the apex—about 158 miles, or across Ireland. However, in the study of molecular physics, we are dealing with measures that are fractions of a micron; so that the ten metres of our gauge that are next its apex are enough for us to retain. III. Relation of the Gauge to Angstrém’s Map. The wave-lengths of light are the longest of the small quantities with which we need concern ourselves; and the gradient of the gauge has been specially chosen to be con- venient in measuring them. For this purpose, lay the gauge on Angstrém’s map of the “Spectre normal du Soleil,” making the points on the base-line at 4, 5, and 6 metres from the apex of the gauge coincide with the positions 4000, 5000, and 6000 on his map. This can be done, since Ang- strém’s scale is a scale of millimetres. Then the actual length of X (the wave-length in air) for each ray represented in his map, is the ordinate of the gauge (i. e. the vertical distance from the horizontal base of the gauge up to its sloping top) immediately over the line of the map representing the ray*. TV. The * Minimum visibile.” The minimum visibile (2. e. the smallest separation at which two points must stand to admit of their being seen as two, by the help of such coarse waves as the waves of light) is about half the wave-length of the light admitted to our microscope; that is, it 1s the ordinate of our standard gauge at some point between two and three metres from its apex. All smaller magnitudes are ultra-visible. V. The Larger Ultra-visible Magnitudes. 1. Ponderable matter is in the gaseous state when its molecules are so little crowded that they have room to dart * Rowland’s great photograph of the solar spectrum is on a scale which is about three times larger than that of Angstrom’s map; and, from the exigencies of the case, the lengths of the degrees upon it differ slightly from strip to strip. To adapt a gauge to it, begin by extending both ways the scale of the strip under examination till it reaches zero in one direction and 10,000 in the other. Over the 10,000 mark erect a micron, and from the top of it draw the inclined plane to the zero mark. This is the gauge whose ordinates will be the wave-lengths of the rays represented in that strip of the map. Lach strip will require its own gauge; but none of them will be far from 30 metres long, so that they are about three times more acute than the proposed standard gauge. Appreciation of Ultra-visible Quantities. 417 to a certain distance along a free path, in the intervals between their encounters with one another ; and information as to the average length of these little journeys can be deduced from experiments on the viscosity of gases. If the gas is a tolerably “ perfect” one; at the ordinary tempera- ture ; and exposed to the pressure of one atmosphere, the average length of the “free path” of the molecules is small. In fact the observed amount of the viscosity assigns to it in air a length equal to the ordinate of our gauge at a distance vf something like three quarters of a metre (30 inches) from its apex ; and although the mean length of the free path differs from one gas to another, it is in all a magnitude of this order*. Note that this is a good deal smaller than what we have found to be the ‘‘ minimum visibile.” Within the receiver of an air-pump the free path becomes longer, until at the excessive attenuations that Mr. Crookes obtains by working his compound Sprengel-pump for a long time, its average length may even reach to several centi- metres, which would be the ordinate of our gauge at a distance from its apex of some hundreds of miles. Pon- derable matter is then in what Mr. Crookes calls the radzant state. 2. The average spacing of the molecules in a gas (1. e. their average distance asunder at any one instant of time) may be obtained in various wayst; e. g. it may be deduced from the ~ * See Philosophical Magazine for August 1868, p. 188. + Calling the average length of the free path A, the average interval between the molecules o, and the average ‘‘ diameter of a molecule” 6; we can obtain A from experiments on viscosity, we get 5/o from observing the condensation which the gas undergoes when liquetied, and one other equation between A, o, and 6 would enable us to obtain all three. Now it is evident that A (the average length of the journeys of the molecules) will, ceteris paribus, increase if o (the space between the molecules) is increased, which may be effected by expanding the gas, and will decrease if 6 (the distance within which the molecules sensibly act on one another) is increased, which may be effected by exchanging one eas for another. It is, in fact, a function of these two quantities and of others, viz. of the velocities of the molecules (the mean of the squares of which is known from the pressure and density of the gas), of the events that occur in the struggle of two molecules with one another during their brief encounters, and of the time occupied by these struggles. The events that occur during the encounters and the time they last are not sufficiently known for the actual equation to be set down: but hypo- theses can be framed in regard to them—as, for instance, that the mole- cules when they encounter simply rebound like hard elastic globes —which enable us to ascertain what function o/A would be of 5/o if the hypothesis were true, and thus enable us to judge what kind of magnitudes 6 and o are. The quantities \ and 6 vary within wide limits from gas to gas; but it is one of the elementary propositions in the kinetic theory of gases that « 418 Dr. G. J. Stoney on the last measure by taking into account the degree in which the gas falls short of being “ perfect,” 2. e. of accurately fulfilling the law poe LT pe ays qv? where p is pressure, v volume, and T absolute temperature. Judged in this and other ways, it appears that the average interval between the molecules of any of the more perfect gases, when at atmospheric pressures and temperatures, is something like the ordinate of our gauge at the distance of one centametre from its apex*. If the vacuum in a Sprengel- pump be carried so far as to reduce the pressure to one millionth of an atmosphere (which is not very far from the greatest exhaustion that can be attained), the average spacing (which may be called the average spacing of the molecules) is nearly the same in all nearly “‘ perfect” gases when compared at the same pressure and temperature. This is, in fact, the truth that underlies and gives its value to Avogadro’s erroneous hypothesis that at the same temperature and pressure the size of the gaseous molecnles of all substances is the same. In the present state of science it is desirable that every practicable effort should be made to determine with more exactness the value of this important physical quantity. * Phil. Mag. for August 1868, p.140. If we assume, in conformity with the estimate in the text, that the molecules of a gas at, say, 21° C. and 760 millim. pressure, are as numerous within a given space as would be a number of points cubically disposed at intervals of a ninethet-metre asunder (this being the ordinate of our gauge at the distance of one centi- metre from its apex); then the number of molecules of the gas in every cubic millimetre of its volume is a uno-eighteen—the number repre- sented by 1 with eighteen 0's after it. Hence, in a litre of the gas there will be a million times more, 2. e. a uno-twentyfour of molecules, Now at the above-mentioned temperature and pressure a litre of hydrogen weighs just one twelfth of a gramme (-083’). Hence the mass of each molecule is the twentyfourthet of this (¢, e. the fraction represented by 1 in the numerator, and 1 followed by twenty-four 0’s in the denominator), 2. e. itis =8"3 xxvic* of a gramme; and according to this computation the chemical atom of hydrogen, being the semi-molecule, has as its mass 4'-16 xxvi** ofa gramme. This is probably somewhere in the neigh- bourhood of the true value; so that we may regard the mass of a chemical atom of hydrogen as a mass probably not more than a few times more or a few times less than the twenty-fifthet or twenty-sixthet of a gramme. This seems the best approach that can at present be made to estimating the mass of a chemical atom. The determination depends upon the average spacing of the centres of the molecules of a gas at standard temperature and pressure (see last footnote); and if this very important physical magnitude, which is common to all perfect gases, can be ascertained with more accuracy, we shall get a proportionally better estimate of the mass of a chemical atom, Of course if the mass of the atom of any one element, e. g. hydrogen, be determined, the masses of all the others become known by the chemical tables of atomic weights. Appreciation of Ultra-visible Quantities. 419 of the molecules will have increased up to being equal to the ordinate of our gauge at a distance of about a metre from the apex. It is instructive to observe that even this enlarged interval is ultra-visible, and that in this so-called extreme vacuum there remain something like a million millions of mole- cules of the gas in every cubic millimetre of the space within the receiver—. e. about a thousand in every cubic micron. 3. The magnitude to be next considered is the diameter of a molecule. By this is to be understood the distance within which the centres of two molecules must come, if they are sensibly to deflect each other’s path. This size of the gaseous molecule, as it may be called, is intimately related to the ratio of the last two measures to one another, and may be deduced from that ratio. Or it may be obtained by observing the condensation which a gas or vapour undergoes when passing into the state of a liquid or solid. Estimated in either of these ways, it appears to be usually about the 8th, 10th, or 12th of the last measure—that is, it is something like the ordinate of our gauge at a distance of one millimetre from its apex. VI. Smaller Magnitudes. The diameter of a gaseous molecule, as above defined, is the smallest measurement for which the present gauge is suggested as convenient, as it is also the smallest magnitude of the actual size of which any approximate estimate has been made. But we have, through the spectroscope, indications of important events in nature that are perpetually going on within each gaseous molecule, and probably on a very much smaller scale. For example, an easy calculation will show that the motion within the molecules of sodium to which the principal double line in its spectrum is due*—a motion which is repeated 508,911,000,000,000, ¢. e. more than five hundred millions of millions of times, every second within each mole- cule—would need to have a velocity several times greater than that of the earth in its orbit (which is a velocity of 30 kilo- metres, 19 miles, per second) if the range of these motions is the whole diameter we have attributed to the molecule. This consideration, though not decisive, is nevertheless quite sufficient foundation on which to base the expectation that, if ever we are able to ascertain the actual range of this motion and others of a like kind, they will turn out to be much smaller than the ordinate of our gauge at a distance of a millimetre from its apex; so that if ever we discover any * See Stoney on Double Lines, Scientific Transactions of the Royal Dublin Society, vol. iv. p. C03. 420 Dr. G. J. Stoney on the way of quantitatively estimating such events, we shall require another and more acute-angled gauge to aid us in appre- ciating them. | VIL. Ofthe Borderland of the Visible. Meanwhile the gauge now proposed will, it is hoped, help the scientific student to obtain a more connected view of nature, by placing before him in somewhat clear evidence the relation in which some of the larger molecular events stand to the dimensions of the smallest objects he can see with his microscope. He should never forget that even the most minute of these microscopic objects is an immense army of molecules*, or semi-molecules, crowded together, more numerous indeed than all the inhabitants of Hurope. The individuals that constitute the battalions are not seen, nor is there the least glimpse of the active motions that are without intermission going on among or within the individuals : nay more, waves of light are too coarse to supply our microscopes with infor- mation about the evolutions of the companies, regiments, and brigades of this great army. It is only when the entire army shifts its position that anything can be seen; and my object will be attained if the contrivance I have proposed helps in any degree to bring about a better balance of thought rela- tively to the cosmos in which we find ourselves: it is so difficult to avoid making the small range of our senses a universal scale with which to measure all nature. Where, for instance, is the justification for our alleging that any visible speck of protoplasm is undifferentiated ? And, in fact, are not subsequent events perpetually rebuking this rashness ? A convenient object to help in connecting visible with ultra-visible magnitudes, is the marking on the frustule of the Pleurosigma angulatum (or, Gyrosigma angulatum), one of the commonest of test-objects. The little brown specks are easily seen with the higher powers of a microscope if a good condenser and the proper stop be used, and their distance asunder from centre to centre is somewhere between ‘64 and ‘65 of a micron, according to the best determinations I can make. This is a trifle more than the spacing deduced from Professor Smith’s measurement of the interval between the * That is, of molecules such as are present in the gaseous state of the ultimate chemical constituents of the speck of matter under examination. These in a highly organized substance like protoplasm are associated into much larger organic groups, that may be called mega-molecules, and may be likened to the companies or regiments of the brigades and corps that make up the army. Appreciation of Ultra-visible Quantities. 421 rows. It is the ordinate of our gauge at about six and a half metres from its end, and is the wave-length of a ray of red light not far from the red hydrogen-line, the line C of the solar spectrum; so that the brown dots succeeding one another in a row mark off in the field of the microscope the successive waves of this particular ray of light. The dots are arranged in rows parallel to the sides of an equilateral triangle, and with oblique light com'ng at right angles to any one of these sets of rows, the dots will elongate and almost run into one another in a way that makes the rows look like a ruling of parallel lines. These parallel lines are at shorter distances asunder than the dots in the ratio of /3 to 2, and accordingly present to the eye intervals equal to the wave-length of a green ray less refrangible than the line Hi of the solar spectrum. The interval in this case is the ordinate of our gauge at a distance of about five metres and a third from its apex. Furthermore, what we have found above to be the minimum visibile is a little more than one third of the interval from centre to centre of the dots, or a little less than half the interval of the rows. It is well illustrated by the Pleuro- sigma markings. In fact, judging from similar markings on other scales, the round dots would be seen as rings were it not for their small size, which prevents the opposite sides of the ring from being seen as two objects. They accordingly look like disks*. VIII. Of the Nomenclature of Small Measures. It will often be found convenient to connect the proposed standard gauge with another useful way of describing small magnitudes. Let us understand by a sixthet a unit in the sixth place of decimals, 2. ¢. the fraction of 1/10°, and let us use the phrase sixthet-metre, or metre-sixthet, to mean the sixthet of a metre, in the same sense in which we say half- inch or quarter-inch to mean the half or quarter of an inch. We can then conveniently express the following table of equivalents. The ordinate of the standard gauge, at a distance :-— Of ten metres from the apex = a sixthet-metre. Of one metre % » =a seventhet-metre. Of one decimetre ,, 5 = an eighthet-metre. Of one centimetre 5 =a ninthet-metre. Of one millimetre 5 =a tenthet-metre. * The white pearl-like specks which take the place of the dots when they are a little out of focus, must not be mistaken for their being seen as rings. They are an optical effect, and of larger size Plul, Mag. 8. 5. Vol. 34. No. 210. Nov. 1892. 2G 422 Dr. G. J. Stoney on the The sixthet-metre is identical with the micron spoken of above. Many writers represent it by the symbol p. The ninthet-metre is the thousandth part of a micron. It has sometimes been called the micro-millimetre, and is by some writers represented by tue symbol py. The tenthet-metre is the same measure as is usually called the tenth-metre. It has also sometimes been called the tenth-metret. IX. Of the Smallest Magnitudes that have been Measured. The most minute magnitudes that have been actually measured are differences of wave-length. These can be de- termined with truly astonishing precision by observations with the diffraction-grating spectroscope, so much so that they carry us down to magnitudes that are fractions of the diameter of a gaseous molecule. The observation is most easily made in the case of close double lines. In these the interval be- tween the two constituents is due to the difference of their wave-lengths, and by measuring the former the latter can be ascertained. Thus, an interval of one degree on Angstroém’s or Rowland’s map indicates a difference of wave-length amounting to a tenthet-metre, which, as we know, is the ordinate of our standard gauge at a distance of a millimetre from its end. But lines have been seen to be double with Prof. Rowland’s gratings, in which the separation of the two constituents is not more than from 1/30th to 1/100th part of a degree. In the latter case the difference of wave-length is only one twelfthet-metre. This is the ordinate of our gauge at a distance from its apex which is little more than the diameter of a single blood-corpuscle, and may be taken to be the smallest measurement that can as yet be directly effected with certainty. The following is a list of close double lines which I have myself seen in the solar spectrum with a small Rowland’s grating * :— The solar line 03, and a multitude of other close doubles, shown as such on Rowland’s map (2nd Series, 1888). The less refrangible of the two E lines. The line in the E group at \=5264'4 of Rowland’s scale. * The grating is flat, nearly an inch and three-quarters long, contains about 25,000 lines, and the observations were made in the fifth spectrum. In this spectrum the image is formed by bringing together every fifth wave of light out ofa series of 125,000 consecutive waves, and the ruling must be sufficiently accurate to effect this. Appreciation of Ultra-visible Quantities. 423 The nicke! line which is nearly midway between the two D lines ; and the line at X=5892°6. This is the second from the nickel line towards D,, of the eleven lines which are at all tinses visible between the two D lines. The least refrangible constituent of the triple line at A=5328°7. This last is about the closest double that my spectroscope will resolve. There is no micrometer on my instrument, so that | cannot give measures, but I estimate the coarsest of these, those first mentioned, to have a difference of wave- length under two eleventhet-metres—and in the closest, that last mentioned, it cannot be more than a very few twelfthet- metres. Most of them could be measured with a good micrometer. This can be accomplished with one of the smaller of Professor Rowland’s splendid gratings ; and he himself and other observers have carried matters farther, by taking pho- tographs with the best of his great six-inch concave gratings. This may give some idea of the marvellous precision of this, the latest and most searching appliance for exploring Nature. By it a brilliant series of discoveries have already been made in stellar astronomy; and we may anticipate still greater achievements from the distance to which it can throw its plumb-line into the obscure depths of molecular events. X. Time Relations. The fragments of time that can be appreciated with accuracy in this way are even more wonderful in their minuteness than are the differences of length. Time relations, however, lie somewhat outside the scope of the present essay ; but they, too, should be carefully pondered by anyone who wants to know what Nature really is. And after thus taking the best survey that he can, he should bear in mind that all he can do is to gauge the little that man has been fortunate enough to detect ; and that far more may lie beyond the ken of any human being than the immense range which now lies within it. He should also reflect that the few molecular events that are already known succeed one another with such astonishing rapidity that the swiftest visible motions are, in relation to them, as sluggish and as gradual in their progress as are the changes in the configurations of the constellations owing to the proper motions of the fixed stars, in their relation to us and to the events we can see occurring about us on the earth. In fact, the thousandth of one second of time is, in relation 2G 2 424 Dr. G. J. Stoney on the to them, comparable with some such period as twenty or thirty thousand years in its relation to man’s slow thoughts, or the driftings about of those accumulations of molecules which are the only kind of objects he can perceive even with the highest powers of his microscope. These wistble objects, these arnues of molecules massed together, seem to him sometimes at rest and sometimes in motion; but in either case strenuous activity within and between the mole- cules themselves never ceases, nor the perpetual response between them and the ether through which they keep up a communication with one another at a distance. The magni- tude of the consequences throughout all Nature of this unflagging intercourse between molecules cannot be ap- proached by the utmost thought we can give to it. It is quite impossible for us to appreciate it adequately. The human eye placed anywhere intercepts a small fragment of the messages in their transit, and is thus a detector of their presence. But it does so roughly. It jumbles up the immense detail which even our spectroscopes can show to be included within this fragment. Yet even so, how much our eyes show us wherever we turn them, and with what seems to us such marvellous promptness! The spectroscope in some respects penetrates farther as a detector. Even it, however, fails to reach much detail that we know to be present, e.g. it cannot tell us the innumerable interruptions or the various orientations or the phases of the actual motions. And, at the best, both these detectors together can give us but a very slender notion of the real activity that is going on, and of the precision and fulness with which the molecules everywhere about us are energetically exchanging many millions of different messages with one another every second. Such is Nature as it really is. XI. On the Bearing of these Determinations on other Branches of Study. Determinations such as those dealt with in this paper have a bearing upon almost every study that is occupied either in the interpretation of material Nature, or investigating the relation between the thoughts of animals and the operations that go on in their brains ; inasmuch as the whole of material Nature is found, on careful analysis, to rest on molecules— on their mutual relations and motions, on the events going on within the molecules, and on those which they excite in the medium in which they move. 1. One example of this influence upon other studies is the general limitation which molecular determinations impose Appreciation of Ultra-visible Quantities. 425 upon the methods employed in dynamical inquiries, as pointed out in a Paper by the present author on “ Texture in Media,” [Scientific Proceedings of the Royal Dublin Society, vol. vi. p. 392; Phil. Mag. for June 1890, p. 467. ] 2. The direct bearing of the inquiry upon chemistry is obvious. It is briefly referred to in the Scientific Transactions of the R. D. Society, vol. iv. p. 608. In fact the record of the chemist is not unlike what one often sees upon a tomb- stone—“ Born in such a year; Died in such another ;”— while the intervening /ife is passed over in silence. So the chemist submits two or more substances to their mutual influence, and finds that such and such substances emerge ; but he takes little note of the eventful time during which all the protracted contests of the reaction have taken place, which, if it has lasted for only the five-hundred-thousandth of one second, has been as long in reference to the activities of the molecules as a long life of 60 years would be in reference to all the thoughts and actions of a man, 3. The minemum visibile, as defined above, is between the fourth and fifth of a micron, and a speck whose volume is tue cube of this may be regarded as the smallest organic speck that the biologist can distinguish from other specks by the highest powers of his microscope. Its volume is accordingly about one-hundredth of a cubic micron—about the 1/7000th part of the volume of one blood-corpuscle. Now, liquid or solid material, if resolved into its chemical elements, and if these be brought into the gaseous state, will, at the tempera- ture and pressure of the atmosphere, expand about 1000 times. Hence the foregoing speck, if thus resolved into gas, would occupy about ten cubic microns. But this volume of gas at that temperature and pressure contains about a uno-ten (10,000,000,000) of molecules, which for the most part will consist each of two chemical atoms. Hence the number of chemical atoms in our speck may be taken to be about two uno-tens. Our speck, perhaps, consists of very complex organic molecules ; but however complex each of these may be their number must nevertheless be very great. For, let us make the liberal allowance of 2000 chemical atoms for each organic molecule, and the number of these very complex molecules will be about ten millions. This is an army quite large enough to admit of an immense amount of differentiation within its ranks—of very active operations within and among the complex molecules or between brigades of them—all of which are ultra-visible events. These are facts which every biologist should keep constantly before his mind when carry- ing out his investigations and interpreting them, and especially 426 Dr. G. J. Stoney on the when he is tempted either to speak or think of “ undifferen- tiated protoplasm.” 4, A still more striking instance is presented when we consider the operations of the human mind. Here I will make the usual assumption, that every perception or other thought in the mind is accompanied by a physical event occurring in the brain, which is connected with it in such a way that neither presents itself without the other. Of this event we know that it is of a kind that arises only in living: brains and in them only while the man is either awake or dreaming. We also know that it is of a kind that lasts for a considerable time when it does occur, viz. throughout the duration of the perception or other thought in the mind. This last consideration is very significant, The event in the brain with which human perception or any other human thought is associated must be one which can last while the thought lasts, ¢. e. for a time immensely long when compared with the or iinal molecular events that are going on. The event may, for example, be such an event as a strain conse- quent on a stress, whether dynamical or electromagnetic, acting on some part of the brain; or it may be of the nature of a forced vibration or current. These are events which would continue in existence so long as the stress is applied, and will cease when the stress is removed: they fulfil the requisite time conditions. Another event which would fulfil the time conditions is an undulation—dynamical, electro- magnetic, or of any other kind. The waves that make up an undulation may continue in it but a short time, some passing off while others come on, and the motions or stresses of which each wave consists may be such as succeed each other with extreme rapidity, w hile all the time the undulation viewed as a whole continues as much unchanged as a human thought does while it lasts. Hence an event of this kind may, so ‘far as its relation to time is concerned, be that event in the brain which is intimately associated with human thought. Possibly the event we are in search of may be found among the pro- cesses of metabolism whereby nutrient matter brought by the blood becomes part of the brain; or more probably among those processes in which matter that had formed part of the brain separates and is swept away either by the blood or lymphatic vessels. Events of this kind, including every interference with or modification of those here specified, and the many other events which like them may be described as stream effects, are marked by the peculiarity that a vast number of molecules are concerned in them in such a way that different molecules successively take up the running. Appreciation of Ultra-visible Quantities. 427 All such events fulfil the necessary condition of continuing temporarily in existence, as each of our thoughts does, for a time which may be immensely long compared with the funda- mental events within or between the molecules, or in the interfused wether. It must, however, be borne in mind that it is upon these fundamental events that the whole super- structure rests; and that stream effects are in relation to them of the nature of very small outstanding residual events which remain over, when the rest—the great bulk of the events that are actually happening—are such as balance one another *. It is evident that it is among protracted events such as those spoken of in the last paragraph that we must search for the physical event in the brain with which human thought is associated. It is also evident that it is with but a very small selection out of the vast number of such events occurring in the brain during life, that the thoughts of which man. is conscious can be directly associated. All the rest of the innumerable stream events, and all the underlying funda- mental events within and between the individual molecules, go on besides. Now, what happens in the brain is an index to us of what is going on in that portion of the Autic Universe f which is most closely connected with human thought. For what goes on in the phenomenal world (to which the brain belongs) is an index to us of what is going on in the Autic Universe (to which our thoughts belong), in the same sense in which a weather-cock is an index to us of the direction of the wind ; since what occurs in the phenomenal world is dependent upon and determined by what occurs in the Autic Universe (see passim a paper by the Author, “On the Relation between Natural Science and Ontology,” Scientific Proceedings of the Royal Dublin Society, vol. vi. p. 475). Hence in the Autic Universe there are events as closely related to the thoughts which exist in my mind at any time, as are the other physical events going on in my brain at that time to those few which are directly associated with my thoughts. The autic events here spoken of probably more or less distantly resemble the events within my own mind of which I am conscious, inas- much as all the physical events in the brain in a certain * Wind is such a residual event. It arises whenever there is a small preponderance in one direction of the very much swi«ter velocities with which the individual molecules of air are at all times darting about in all directions. So also the current in a river is a similar residual event. t+ Auta, actual existences (of which the thoughts that are in my mind are a sample directly known to me). The Autic Universe, the totality of all really existing things, of all auta. 428 On the Appreciation of Ulira-visible Quantities. degree resemble one another; and, in particular, the time relations between them and between them and my thoughts are the same as the time relations of the physical events that, as a consequence, go on in my brain. This body of events in the Autic Universe and the thoughts or other auta between which they occur, may suitably be spoken of as a synergos (cuvepyos, a fellow-worker) which is ever on the alert to work along with my mind, and on which the thoughts that are ny mind as much depend (in the autic sense of that word), as do those few physical events in my brain which are associated with my thoughts depend (in the physical sense of the word) upon the vast multitude of other physical events also going 7 in my brain. The one cannot even exist without the other. The lesson to be learned from all this is that psychology and the other branches of metaphysics, as presented by the ablest men who were unaware of the existence of this synergos and of the large degree in which it intervenes in all that happens in the mind, will now have to be rewritten. In memory ; in the association of ideas ; and in the other mis- called “faculties and operations of the human. mind,” it is little we do: it is much that is done for us*. Man’s mind— * Take, for example, some particular instance of memory. I remember where I sat at breakfast this morning, where my companions sat, also several particulars of what was said during breakfast, of the gestures of my companions, of my own motions, of what I ate, of the equipage on the table, and so on. None of these things have occupied my thoughts since breakfast, till I sat down to write these lines; since when they have all come into my mind. - Now what does all this mean? It means that the group of thoughts which I call myself—my mind—and which at each instant is a group the several parts of which are connected and interacting in that way that we call being within one consciousness, is a group of thoughts that has undergone change; that one instance of this change has been the dis- continuance of the above-recited perceptions that formed part of the group at breakfast-time, and that another instance has been the occurrence now within this group of what is more or less an imperfect and much modified repetition of some of those perceptions, accompanied by the additional thought which we call being aware that they had at breakfast- time occupied a place within the group in the fuller form of complete perceptions. During the intervening hours none of these occupied any place in the group either in their fuller or in their modified form: never- theless, there must have been, somewhere in the autic universe, a chain of causation connecting the original perceptions and the memory of them. Of this chain the first link was a part of my mind, the last link 7s a part of my mind, but the connecting links have been in the synergos. This becomes clearer when we turn to the ebjective or phenomenal world, which is a kind of shadow thrown in a special way by the suc- cession of events that occur in the autic universe (see Stoney “ On the Relation. between Science and Ontology,” Scientific Proceedings of the Royal Dublin Society, vol. vi. pp. 502,503, 504). My thoughts, which On the Law of Error and Correlated Averages. 429 the little changeful group of interwoven thoughts ‘that is himself—is a very small part of the great Autic Universe. We must shift our centre, and exchange the metaphysician’s narrow Ptolemaic for a broad Copernican view of existence. L. The Law of Error and Correlated Averages. By Professor F. Y. Epcewortu, M.A., D.C.L.* NHIS is a contribution to the investigation of the most general conditions under which the exponential law of error is fulfilled ; together with some applications to the theory of “ correlated averages” f. I. The simplest case in which a group of measurable objects renge in conformity with the law of error is where every member of the group is a sum of a certain number of items, or elements, each of which has or has not a certain quality, e. g. the colour white, or, as it may be expressed, assumes the value 1 or 0, with a certain average frequency (e. g. 1 just as often— or half as often—as 0, in the long run) and at random, or in are a part of the autic universe, are shadowed by certain objective changes in my brain; and the term synergos means that other portion of the autic universe which is shadowed by all the other events that happen objectively in my brain. It appears from physical considerations that the particular stream effects or other changes in the brain that were the shadows of the perceptions I had at breakfast-time, cannot have occurred alone, but were accompanied by more subtile motions or changes in the brain, which were the shadow of, and thus betokened, certain closely associated events then going on in my synergos. These again were suc- ceeded by mctions, changes, or states of strain in the brain during the intervening hours, all of which were a part of the varying shadow of the synergos as it underwent whatever changes took place in it during that interval. Moreover, these intervening events in the brain were of such a kind, as the result has proved, that they have been now followed up by motions or changes in the brain which resemble those that were the shadow of my thoughts at breakfast-time, and which are a part of the group of events now going on in my brain that are the shadow of those thoughts that constitute my mind as it exists at present. Softening of the brain is the shadow cast within the objective world when very unfortunate events have happened in the autic universe— events which have included a weakening of the power which the synergos and the mind previously had of mutually acting on one another, or else which have prevented the full formation within the synergos of some of the intermediate links of causation spoken of above. Either of these would involve a partial loss of memory. * Communicated by the Author. + See Galton (Proc. Roy. Soc. 1888), “Co-relations and their Mea- surements;” and Weldon (Proc. Roy. Soc. 1892), “Certain Correlated Variations in Crangon vulgaris;” also “Correlated Averages,” by the present writer in the Philosophical Magazine for August 1892, 430 Prof. F. Y. Edgeworth on the such wise that the value assumed by one item does not affect that ofanother. This is a case fully discussed by the classical writers on Probabilities; under the guise of problems relating to games of chance. Thus if we take several batches of balls, every batch containing x balls, each of which is either black or white, the probability of a white being p*; then the fre- quency of white balls is approximately represented by a curve of error, of which the centre corresponds to np, the most pro- bable number of white balls ina batch, and the modulus is V2np(1—p)}. I venture to refer to my reproduction, in a former number of this Journalt, of Poisson’s reasoning on this problem, in order to recall a proposition which will be required here : namely, that the limits, on either side of np, up to which the approximation holds good are of the order V nt. | II. An easy transition brings us to a more general case in whicheach item has any finite limits a, and a, (a greater than a). These limits need not be identical for each item; ; provided that the range of any one item, say @,—-@,, is small in comparison with the sum of the ranges S(a,—a,). Nor are we confined to the supposition that each item should assume one or other of two values a, and «,; it may assume any one of an indefinite Peed of values indicated by the curve of distribution =/,(z), representing the frequency with which the 7th pe assumes each value w ; where /, may have any form whatever, continuous or not, prov ided that it does not extend beyond @, and a,, and that the integral between those limits is unity. Moreover, it is allowable to affect each item with a different factor or “ weight” 3; provided that no weight is pre- ponderant—large in comparison with the sum of all the other weights. When we have thus substituted, for a “sum,” a linear junction of independently varying items, we have reached the extent of generalization to which Laplace thought it necessary to carry ~ the investigation for the purpose cae the theory of errors §. III. We enter on a less trodden path when, following the lead of Mr. Glaisher ||, we pass from a “ linear” function of items to any function whatever. To make the transition less violent, let us break it up into two steps; and first consider * n being a large number, and p not a very small fraction. T Pai Mag. 1887, xxiv. p. 330. { There is some ap roximation outside these limits; but not of the degree usually assigne § There is a eood account of Laplace’s analysis in Todhunter’s ‘ History of Probabilities, art. 1001 e¢ seqq. || Memoirs of the Astronomical Society, vol. xl. p. 105. Law of Error and Correlated: Averages. 431 the case in which each member of a group is—not a linear funetion—but some function of a linear function of numerous elements oscillating in the manner defined above: say I'(3); where B= pe, + pote t Ke. + Darn 3 D1, P2, Ke. are of the same order of magnitude; 2, #2, Ke. oscillate respectively between limits which we may write without loss of generality 0—a,, 0—a,, &e.; while Sa may be taken as the unit. Now by the usual theory & fluctuates about its average value—say X—in conformity with a * Probability-curve of which the modulus is of the order —- 1 (Sa being =1). Put 3=F(X+ &) and, expanding, write 2 F(B) =F(X) +£9"(X) +5 P(X +68) (where @ is a proper fraction). This expression is approxi- mately equal to its first two terms F(X)+€&F’(X) ; provided that, for the values of F(&) with which we are concerned— at most from 0 to 1 (if Sa=1)—the function is free from singularity and continuous in senses defined by the con- dition that SP X+ 0&) is small in relation to F’(X), for values of & of the order n That condition holding, we may reason thus:—Of the group formed by the varying values of & the greater part is, by the usual theory, arranged according to a probability- curve with centre X (the average value of &), up to a dis- tance from that centre, +£&, where £ is a small fraction. But every value of F(X+&)=F(X)+é8F’(X) nearly. Therefore the greater part of the group F(&) ranges in conformity with a probability-curve whose centre is F(X), and whose modulus is that of & multiplied by F’(X). For example, let F(&)=5?; where & is the sum of m * Above, p. 430, and the article there referred to. + This is a kind of assumption continually made, I think, by mathe- maticians. To take an instance cognate to the present subject, Laplace, when introducing the Method of Least Squares (Theorie analytique, Book IT. ch. iv. art. 20) supposes the datum of observation to be a func- tion of the “element” which it is sought to determine. Anapproximate value for this value having been obtained; and this value, plus a correc- tion z having been substituted for the element; “ expanding in ascending powers of z [en reduisant en série par rapport a2z| and neglecting the square of z, this function will take the form 4+pz.” It will take that form only upon the condition above stated. 432 Prof. F. Y. Edgeworth on the independent elements, each element in random fashion assu- ming either the value 0 or +7 with equal frequency in the long run. Then, according to the magnitude of m, and the degree of accuracy required, the group formed by the varying values of & may be regarded as conforming to a probability- curve whose modulus is \/ a i, up to a distance from the central value (F i) amounting to the quartile, octile, decile, &c., as the case may be. Thus of a group of values assumed by & about a quarter occurs between the limits 3 2 and oa —% My) — ; where q is the “quartile” for modulus unity, ='476..,. Another quarter occurs between zit an/ = i Similarly the octiles, deciles, &. are — m . i. alee mM » 2 2 2 2 where 7, s, &c. are coefficients obtained from the tables ; either ‘below, or not much above unity. And so on, up to the largest percentile up to which the approximation is accurate. Now let this group be deformed by squaring each of the observations. The new median, quartile, octile, &c. will of course be the squares of the respective old ones. The new 2 9 median will be 3 the new quartiles (B+q4/ ey es m2 ik ms Rae, OP m* ye ee L_) 2 See (1- = 2 75 4( Ty ie, TE ae are Now ihe last-written two expressions differ from the ex- pression outside the brackets and from each other by small quautities ; so that but a small proportion of the group occurs between the. limits m? me 2 m? me )e gil io. (F + 759) and G ero eg ts and in the corresponding interval in the neighbourhood of the quartile below the Mean. Thus the new quartiles are approxi- distant respectively from the new median, 2, by 2 3 mately ices 732) 27, In like manner it may be shown at — Law of Error and Correlated Averages. 433 that the new octiles, deciles, &c. are aries al m? me t , ar) (7+ 4+ 5s v?, de. (r,s, &e. — : meaning assigned in the last paragraph). Thus the translated curve Na oars is such ee its quar- mi tiles are eG m ag 5 mye , its octiles +r—= Ji ; ; m= its deciles ( mn +s—= Ji Q peproximately a probability-curve whose centre is 7*, and mye that is, the new distribution is modulus % i In the preceding example the conformity of & to a proba- bility-curve is known a priort by the usual theory, referred to in our second section*. Let us take another example in which this conformity is known «a posteriort by actually observing the measurements of a group : for instance, heights of men. The annexed numerals give the number of men per thousand of each particular height in inches; as ascertained by Mr. Elliott from the measurement of some 25,000 Ame- rican recruits (International Statistical Congress, 1863). Thus there are 121 men (per thousand) of the height 70 inches, that is, as I understand, between the heights 69°5 and 70°5 inches. 1 0 OO, 1 2 20 48 7 i117 134 gees. of, 60. 61 62 65.64. 65.-. 66 ... 67 moran 6121 6800 CU CG 13 oF ne 1 cee (0. 412, 22 odd 1k TD 16. 11 2f8 Now let the curve or locus of distribution thus constituted be translated by squaring each of the measurements ; while each compartment thus dislocated carries with it, so to speak, the men appertaining thereto. . g. the 121 men who were originally found between 69°5 and 70°5 are now distributed between 69°52 and 70°5?. It will be found that the trans- lated observations fulfil the law of error in the same sense as the original ones ; that is, in the sense in which a planet is proved a posterior: to move in an ellipse. The parameters being calculated from some of the observations, other obser- vations are found to tally with the curve thus determined. Iiet us adopt a uniform method of calculating the para- * The case might be brought under our first section by a mere change of unit. 484 Prof. F. Y. Edgeworth on the: meters (the centre and the modulus) for the primary and for the dislocated curves ; namely, from the Arithmetic mean, and from the distance between certain percentiles which are given by observation. In the primary group, above the point 69-5 occur 3805 observations, °305 of the total thousand ; and below the point 66°5 occur 264 observations, '264 of the total. Whence between the Mean and the point 69°5 there should occur *9—"305="195 of the total; and between the Mean and the point 66°5, °236 of the total. Now it is found from the tables that °195 of the total number corresponds to *361 of the Modulus, :236 of the total to °447 of the Modulus. Accordingly we have ‘808 (=°361+4°447) modulus, equated to the interval between 66°5 and 69°5 inches = 3 inches. Whence the modulus = 3+°808=3°'71 inches. Let us now apply this result to predict the number of observations at particular heights (not too near the ex- tremity, where the fulfilment of the Probability-curve is not to be looked for). To predict the number above 70°5 (which gives the number between 69°5 and 70°5, the number above 69°5 being already taken for granted) we are to employ the Arithmetic mean, which is 68°20. 70°5—68'2=2°3 inches ='62 modulus. Now, according to the Tables, the proportion outside the point which is at a distance of 62 modulus from the Mean=4$(1—'6194) of the total =$ x ‘380 x 1000=190. The real number is 184. To find the number above 71:5 we have 71:°5—68:2=3°3 inches =*89 modulus; corresponding to $(1—*7918) of the total= 43-2082 x 1000=104. ‘The real number is also 104. Proceeding similarly with the lower limb, to predict the number below 65°5, we have 68°2—65°5=2-7=nearly °728 modulus ; corrresponds to $(1—°6971) total =4:302 x 1000 =151. Whereas the real number is 147. By parity we find below 64:5, 79 by calculation, against 72 observed. The question is now whether we shall fare equally well when we apply the same method to the group which is formed by squaring each observation in the manner explained. By parity of calculation the modulus of the new curve x°808 = (69°5?—66°5?) inches =408. Whence the new modulus =505 inches. Also the Arithmetic mean of the new group is 4657'38. Accordingly, to predict the number of observations above (70°5)? we have 4970°25 —4657°38 = 312:87 = 62. modulus. Whence the calculated number is 190, exactly the same result Law of Error and Correlated Averages. 435 as before. By parity the number above (71°5)? is calculated to be 102. With regard to the lower limb, in order to predict the number below 65°5?,we have 4657°38 —4290°25 =367:13='727 modulus, which corresponds to 4(1-(6914+°7 x 0067) ) x 1000 =4(1—°6961) x 1000=152 ; a result which is wider than the former result from the true figure 147 by only an unit. For the number below (64°53)? there is found by parity a result not much worse than that which was obtained from the primary observations. I have applied a similar test to the group which is formed by. cubing the original observations. The results of both verifications are embodied in the annexed Table * :— 7 , Below Below | Above Above | 64°5 in. 65:5 in. 70°5 in. T1sd: in: so) ee 72 147 184 | 104 From the original } 79 151 | 190 104 3 observations ...... | 3 | 3 From the cage a 89 152 190 02 3 GSC Bie 2 | @) From the cubes ......... 86 151 191 98 This verification might, I think, have been predicted from the circumstance that the Arithmetic means of the squares and cubes differ by very little from the respective medians, 68-2? and 68:28 (the square root of the one A1ithmetic mean being 68-2, the cube root of the other 68°3, each correct to the first decimal). Now the distortion to be apprehended is the unsymmetrical extension of the upper, and shrinking of the lower limb; but this cannot be considerable, while the Median and Arithmetic meau are nearly coincident. Accordingly we may add to the verifications above re- corded other instances in which the consilience between the Median and Arithmetic mean is preserved. Thus in the case of observations on the height of adult males recorded by the Anthropometric Committee of the British Association (Report of the Brit. Assoc. 1883), the Arithmetic mean of the primary observations (expressed as per-milles, and upon the under- * T have to thank Mrs. Bryant, D.Sc., of the North London Collegiate School, for having worked the greater nart of the arithmetical examples in this and the preceding paper (Phil. Mag. Aug. 1892). 436 Prof. F. Y. Edgeworth on the standing * that each number of men,e. g. 155, entered against a certain height, e. g. 67 inches, means that there were 155 men between 66°5 and 67:5 inches) is 67:023. The Arithmetic mean of the squares is 4498°671, very nearly the square of the primary Arithmetic mean, viz. 4492°08. Also the Arith- metic mean of the cubes is not far from the cube of the Arithmetic mean. Similar verifications would no doubt be obtained, if we employed, for F, any other simple function, e.g. c log & or ce-®, The limits within which the rale may be expected to hold will appear, if we consider an exception. Let E=(#—X), and F(X) =(#—X)?=€£? (employing the same notation as on p. 431). If the varying values of € range under a probability-curve, then the squares of these measurements will noét range under such a curve. The operation of squaring will cause the negative limb of the Fig. 1. R 1 s J 1 1 ' ‘ a D - a original curve to be screwed round to the positive side ; and, in addition to this displacement, there will be the distortion —72 . caused by substituting, for the error-function Fig 2, the 1 pad 1 —_——_——_¢€¢ ‘ function DP A/ BA re 7 * A misunderstanding: for as I have learnt, since the above was printed, the entry 155 men against 66 inches means that there were 155 men above 66 inches ; but our argument is not affected by pushing up the whole set of measurements en ddoc half an inch higher. t In general substituting for 2, in the error-function, d—-1(2), and multiplying by = p—1 (2); where d(x) is the new value of an observa- tion which originally measured x«— or rather something between 2 and w+dx, . FRE Law of Error and Correlated Averages. 437 This dislocation is exhibited in the annexed diagram (fig. 1); where BAB’ is the original curve, «8 what it” becomes when each of the observations (measured from the centre 0) is squared. If for F(E) be put the cube or any odd power of («—X), then the original probability-curve under which the values of w (and #«—X) ranged will be transformed to a curve sym- metrical on both sides of X, but not a probability-curve. These exceptions arise when FE is a function of the deviation of an observation from its average value. The rule is fulfilled when F is a simple continuous function of the observations themselves measured, as is usual with concrete quantities, from an origin below the least possible value—as we measure human heights, or death-rates, or other statistics, from zero ; and not from 674 inches, or ‘2 per cent., or whatever the average may be. If any such function of an observation is substituted for the observation itself in a group obeying the law of error, we may expect that the transformed group will also obey that law. We have here the explanation of incidents which must have puzzled many students of Probabilities: why Mr. Galton should have found the Arithmetic and Geometric means of observations to give sensibly identical results (Proc. Roy. Soc. 1886); how Quetelet could be justified in affirming that both weights and heights of men obey the law of error (Anthropométrie) ; supposing, as is plausible, that the weight of a man is apt to be proportioned to the square of his height. In fine we have here an answer to the objection which has been made to Quetelet’s doctrine of the Mean Man by Cournot and other high authorities on Statistics. The objection is thus stated by the eminent Prof. Westergaard (‘L’heorte der Statistik, p. 189) :— ‘Suppose we had measured for a number of men three lines of the body which make a right-angled triangle, and we wished to determine the corresponding triangle of the Average-man, Then it may be shown [es zeigt sich dann so fort| that the three averages do not make a right-angled triangle. Call the sides , 5, ¢, a,, 6,, ¢, &e., ¢,, ¢, &c. being the hypothenuses; then the aa of the average-triangle are Dye, meets ands Se = Sa/ gak Bs Vv nr i) Vv One should have accordingly (3a) + (26) =v a? + 0) But this can only occur—except by accident—[in der Legel nur Phil. Mag. 8. 5. Vol. 34. No. 210. Nov. 1892. 2 H 438 Onthe Law of Error and Correlated Averages. dann zutreffen wurde] when the quantities a,, b,,¢, are proportional to a,, b,, ¢, &c.; which in general is not the case” *. Suppose that the two sets of measurements which we have above adduced as examples had been made not on American recruits and English adult males, but upon two organs a and b, so related to a third ¢ that c= /q2+62. The values of a’, as we have shown, vary according to a law of error ; and so do those of 6?. Accordingly, by universal admission the sum a?+Jl? will vary according to the typical law ; and we have shown that in general, if a quantity varies according to this law, so also will its square root. Thus the Arithmetic mean of the observed e’s will fit the Arithmetic means of the observed a’s and 0’s. The gist of the reasoning, it will be remarked, is that the greater part of a group conforming to the law of error is apt to be packed within limits which are narrow relatively to the largest possible member of the group, and even the average member ; in the symbols above used § (the deviation) is small relatively to X (the average); for the greater part of the group, at least, up to some percentile near the extremities. ‘his is true by the Laplace-Poisson theory above adverted to, even in the case most unfavourable for the argument where X is measured from a point just below the least possible ; as in the example given at p. 432. Even then it would be safe to treat (X+&)%, or 4/X+&, as equal to the linear function X + 2&, or 4/X +14 oe £, respectively. But for natural groups the origin should perhaps be placed at some distance below the smallest possible observation. The smallest possible dwarf must be well above zero. Upon this view the smallness of the modulus in comparison with the dis- tance of the centre from the origin becomes more decided. As a matter of fact, the ratio of the modulus to the mean value (the order of our +X) is found upon an average of several instances, taken from Mr. Galton’s men* and Mr. Weldon’s shrimpst, to be from 5 to 55. Mr. Galton in some authoritative observations on this topict{ assigns for the ratio in question (in the case of human stature) +. (To be continued. | * Cf. Cournot, ‘Théorie des Chances, Ch. X. Art. 123. Morselli follows Cournot in attempting to demonstrate a@ priori the impossibility of constructing a type involving numerous means of different organs (Metodo an Antropologia, 1880, p. 26). + Proc. Roy. Soc. 1888 and 1892. Phil. Mag. vol. xlix, p. 44. He gives for the ratio of the mean to the probable error, “about 30.” + eo ee LI. Lurther Data on Colour-Blindness.—No. IL. By Dr. Witu1AM Pots, £.2.S.* if the July number of the Phil. Mag., seeing the importance that had been attached to the subject of Colour-blindness, 1 contributed some data as to certain early investigations of my own. Since that time the interest in the matter has been much increased by the prominence given to it at the late meeting of the British Association in Hdin- burgh, and by criticisms made on some of the views still held on itin this country. It may therefore be useful to invite more attention than has yet been given, to opinions expressed on it by foreign writers of eminence and authority who have made it their special study ; and I propose to add some notes with this object. I will begin with a writer whose reputation as physicist, physiologist, and practical oculist was second to none,— the late Professor F. C. Donders of Utrecht. He took great interest in Colour-blindness, made many yaluable investigations, and wrote much thereupon. The position he took was peculiarly independent. He embraced and strongly advocated Young’s trichromic theory, but entirely dissented from the application of it so commonly made to dichromic vision. At the same time, though he agreed with some of Hering’s fundamental principles, he opposed his colour theory. And, consistently enough, having objected to the current explanations, he brought forward a hypothesis of his own, remarkable for its originality and its consonance with modern biological science. Professor Donders came over to the meeting of the British Medical Association at Cambridge in August 1880, and gave there an able lecture on Colour; but his views are still but little known here. They are contained chiefly in two elaborate articles, namely :— “ Ueber Farbensysteme,” in Graete’s Archiv fir Ophthal- mologie, vol. xxvil. part i. 1881. ‘Noch einmal die Farbensysteme,” /bid. vol. xxx. part i. 1884. In giving his ideas about Normal Vision, he considers that our natural impressions point to four “ simple’? colours— red, yellow, green, and blue (which is one of the starting- points of Hering’s theory) ; but he believes these sensations are caused by the combinations of three more-highly satu- * Communicated by the Anthor, at 2 440 Dr. W. Pole on Colour-Blindness. rated ‘‘fundamental” energies, corresponding to red, green, and violet, which is essentially the Youn o-Helmholt doctrine. He describes the phenomena of dichromic vision, according to the testimony of patients (laying some stress on my own case), confirmed by his own experiments and observations. But he prefers to denote the contrasting colours as “‘ warm ’ and “ cold”’ respectively. He then discusses at some length (what is the most im- portant part of his work for our purpose here, namely), “The connexion between the Normal and the Abnormal Systems,” and he puts the question thus :— Is one of the energies belonging to the normal system wanting in the dichromic system? Or,in other words, Does every di- chromic system consist of two of the energies of the trichromic system ? Maxwell and Helmholtz assumed this ;—and they believed they found it so. After discussing at some length the reasonings and caleula- tions of Maxwell and Helmholtz, the author continues :— Now do these facts prove that the colour wanting in colour- blindness is generally one of the fundamental colours ? We call those colours fundamental which do not spring from others, but are necessary to form others. We assume that every fundamental colour shall base its specific process in definite retina- elements, and in order to characterize it more closely, we determine its subjective luminosity as a function of the wave-length, inasmuch as we presuppose that this will coincide with the intensity of the retina-process. Without regard to colour-blindness we come to the result that the terminal colours of the spectrum, red and violet, and the central, green, are the fundamental colours, as Young had already shown. Now it happens that in Red-blindness the wanting colour is not the spectral red, but a red which approaches to carmine, i. e. a red which does not appear in the spectrum, and is only to be obtained by a mixture of two spectral colours, red and violet. With Green-blind patients there is reason enough to look for the wanting colour in their neutral spot; but this by no means decides that the bluish green which corresponds to this neutral must be one of the fundamental colours of the normal system. The final result of this, therefore, is that we have no right to consider the colours wanting in the various forms of colour-blind- ness as the fundamental ones of the normal system. He adds elsewhere :— In a theoretical point of view the simple falling away of one of the energies is inadmissible. It does not harmonize with our idea of the origin of things, that, of three activities which develop themselves in a reciprocal relation as an organic whole, one should Dr. W. Pole on Colour-Blindness. 441 be absent without the defect having an influence on the other two. Edmund Rose was the first to point out this difficulty in the application of Young’s theory, but he was, in my opinion, quite wrong in inferring that this was fatal to the theory. The retina is not a thing formed by human hands. It is not an instrument with three strings, one of which is broken for the colour-blind. It is a living instrument—genitum non factum—whose three differ- ently-tuned strings have developed themselves in combination with each other. If one is absent, then the tuning of both the others is certainly not what it would have been under a regular develop- ment of the whole. We have also to consider the subjective sensations of the colour- blind. | I presuppose that they see the ordinary daylight as the normal eyes see it, 2. e. neutral, colourless. Herschel, in his remarkable letter to Dalton, says, ‘‘ when your two colours are in equilibrium, they form your white ;” and in the case of Mr. Pole also he says he “is strongly disposed to believe that he (Mr. Pole) sees white as we do.” The reason why, according to my view, the white of the colour- blind should correspond with that of the normal-eyed, lies deeper: What the total light brings forth must necessarily be the manifes- tation of the total process, and according to the nature of things this is, in opposition to the partial process, neutral, or it would become so, if it were not so already. Colour-blind people who have due regard to their sensations, see in white no third colour, but only the negation of the other two, something neutral. Mixed with either of these, it leaves the colour unchanged, and only re- duces the saturation. Nobody can believe that the white to the Green-blind should be a purple, such as for the normal eye is formed out of red and violet, and no one has ever believed it. According to the assumption that the white light of the colour- blind is neutral, colourless, I consider their fundamental cclours as complementary. In common with the declarations of intelligent colour-blind persons, I have assumed that the Red-blind must have for their warm colour a yellow, leaning towards green, and the Green-blind a yellow leaning towards red; and that blue or violet, as cold colours, will correspond with these. And I consider these assump- tions as already, to a certain extent, proved by the revelations of one-eyed colour-blindness. ? In the lecture given at Cambridge, 1881, Donders repeats this opinion : he says :— The warm and cold must be considered as complementary colours ; to which of our sensations they correspond cannot well be told, probably the cold colour is blue or violet, and the warm one is yellow, approaching to red or green. 449 Dr. W. Pole on Colour-Blindness. From. these extracts it is clear that he agrees in the view, drawn from observation, that the colours seen in dichromic vision are, generally speaking, yellow and blue, and he does not consider that there is any such connexion between the dichromic and the normal vision as should require them to be otherwise. But he goes farther, and puts forward a view of his own, which he believes may explain Colour-blindness without interfering with the Young-Helmholtz theory. He considers dichromie vision may be a step in the evolution of the colour- sense, anterior and introductory to the present normal vision. He appears to have been led to this by the analogy of the remarkable variation of colour-perception in the different parts of the human retina. He describes this at some length, shows how practically to examine and test it, and notices the remarkable arrangement of full normal colour-perception in the central portion, dichromatism in a surrounding ring, and total colour-blindness in the outer periphery. The Young-Helmholtz theorists had tried to explain this, as they did dichromatism, by the falling away of one or two of their fundamental sensations ; but Fick had shown the unsatisfactory nature of this explanation, in which opinion Donders entirely concurs. His view is that this structure of the eye points to a gradual development of colour-seeing power. In the first instance, the whole eye was in the state that its exterior ring is now, namely, having a power of vision of light and shade without colour. That afterwards an improved state set in, with two colour- sensations, which, beginning in the centre, gradually extended to a certain diameter over the retina. This was the state corresponding to the present colour-blind or dichromic vision. That, thirdly and lastly, a still further improvement set in, extending to a smaller circle, and giving the present normal vision. And all these states, be it observed, still remain in the human eye. Now if we call in the well-known phenomenon of atavism, an exceptional return in certain individuals to a former inferior state of development, the whole explanation lies open before us. In some few cases the inner circle of the three-colour sensation is absent, and the dichromic ring extends, as in former ages, to the centre. These are the cases of dichromic vision. In some cases, rarer still, the structure reverts to the still earlier type, where both the trichromie and dichromic On Graphic Solution of Dynamical Problems. 443 states are absent, and these are the very rare people with vision of light and shade only. Donders even thinks that he can trace, within the ocular area, a vestige of a difference of the kind existing between red and green blindness, the former having a ‘shortened spectrum, and the latter being a stage nearer perfect vision, which, if it were established, would be an additional element in the analogy. The idea of a complete system of evolution for colours might then be sketched out somewhat as follows :— 1. Achromic vision (light and shade only). 2. Dichromic imperfect vision (called ‘ Red-blindness ” short spectrum, low sensitiveness to the long-waved rays). 3. Dichromic perfect vision (called “ Grreen-blindness ” See spectrum, full sensitiveness to the long-waved rays). Trichromic imperfect vision (as pointed out by Lord Rayleigh), with low sensitiveness to certain colours. 5. Trichromic perfect vision. These classes would be subject to intermediate gradations, as in other evolutionary development. Looked upon in this way, colour-blindness would be only an imperfect development of normal vision, not springing out of it, as the Young-Helmholtz explanation would suggest, but antecedent to it. It would be a system whose two energies resulted independently from the decomposition of white light, and, therefore, would be complementary to each other. Donders also cites, as favouring this view, the peculiar mode of hereditary transmission of the defect, according to the unanimous testimony of experts. A patient transmits it, not to his sons, but to his grandsons through a daughter, who is free from it herself: thus causing it to skip over one generation. Athenzeum Club, S.W, October, 1892. LIL. On Graphic Solution of Dynamical Problems. By Lord Kervin*. | Mie method of drawing meridianal curves of capillary surfaces of revolution, described in ‘ Popular Lectures and Addresses,’ vol. i., 2nd edition, pp. 31-42, and illustrated by woodcuts made from large scale curves, worked out ac- cording to it with great care and success by Professor Perry when a student in the Natural Philosophy Class of Glasgow * Communicated by the Author. 444 Lord Kelvin on Graphic Solution University, suggests a corresponding method for the solution of dynamical problems. In dynamical problems regarding the motion of a single particle in a plane, it gives the following plan for drawing any possible path under the influence of a force of which the potential is given for every point of the plane. Suppose, for example, it is required to find the path of a particle projected, with any given velocity, in any given direction through any given point P, (fig. 1). Caleulate the normal component force at this point; and divide the square of the Fig. 1. velocity by this value, to find the radius of curvature of the path at that point. Taking this radius on the com- passes, find the centre of curvature, Cp, in the line, P)K, perpendicular to the given direction through P», and describe a small arc, PyP,Q;, making P,Q, equal to about half the length intended for the second are. Calculate the altered velocity for the position Q,, according to the potential law ; and, as before for Po, calculate a fresh radius of curvature for Q, by finding the normal component force for the altered direction of normal and for the velocity corresponding to the position of Q;. With this radius, find the position of the centre of curvature, C,, in P,C,L, the line of the radius through P,. With this centre of curvature, and the fresh radius of curvature, describe an are P,; P,Q, making P,Q, equal’ to about half the length intended for the third arc ; calculate radius of curvature for position Q, ; draw anare P,P;Q;; and continue the procedure. This process is well adapted for finding orbits by the ‘trial and error’ method described in my article ““On Some ‘Test Cases’ of the Maxwell- Boltzmann Doctrine regarding Distribution of Energy,’ sect. 13; Proc. Royal Soc., June 11, 1891. of Dynamical Problems. 445 The accompanying curve (fig. 2) has been drawn with great care, and with very interesting success, In the ‘ trial and error’ method of finding the first and simplest orbil, hy my secretary, Mr. Thomas Carver, for the case of motion defined by the equations ax 3 = — yn". dt? The initial point Py was taken on one of the lines cutting the axes of « and y at 45°, and at first at a random distance from the origin. A trial curve was worked according to the method described above, and was found to cut the axis of « at an oblique angle. Other trial curves, with unchanged energy-constant, were worked from initial points at greater or less distances from the origin, until a curve was found to cut the axis of w perpendicularly. This curve is one-eighth part of the orbit; and is shown in fig. 2 repeated eight times in order to complete the orbit, which is symmetrical on the two sides of the axes of w and y,. As an interesting case of motion related to the Lunar Theory, suppose the mass of the moon be infinitely small in comparison with the mass of the earth; and the earth and sun to have uniform motions in circles round their centre of 446 Lord Kelvin on Graphic Solution gravity. Let (z, y) be coordinates of the moon relative to OX in line with the sun, outwards, and OY perpendicular to it in the direction of the earth’s orbital motion. The well- known equation of motion relatively to revolving coordinates gives, for the equations of the moon’s motion, if a denote the distance from O (the earth) of the centre of gravity of the sun and earth, a as ie. Lg ee ; de & OE (20 2)= gee ° ° ° (1) d*y dat Fas —. ay co a er where V is the potential of the attractions of the sun and earth on the moon, and @ the angular velocity of the earth’s radius-vector. From this we find, for the relative-energy equation hee. day? (qe + ae }=Etiwt@ty)—V, . . @) where E denotes a constant ; and for the relative-curvature equation we find dxd’y —dyd*x _ dt Nd? (dx? + dy”): Mita eS, + itd? ° (4) where N denotes the component perpendicular to the path, of the resultant of (X, Y) with dV XA=o{(e+a)— =-, «ws «os da’ dV banal 7s ° e ° ° . e ° (6). Hence if g denote moon’s velocity and p the radius of curva- ture of her path, relatively to the revolving plane XOY, we have 29 ~H+$e0' (2 y")— Vi 7". ae and Li) eo Se Tg Bigeee or: Soot Se Calling 8 the sun’s mass, and a his distance from the earth, and supposing the earth’s mass infinitely small in com- parison with the sun’s, we have B ae Fie eo and therefore wa? m [(ata) yt te? ie ee where m denotes the earth’s mass, and r= (x? +y’). of Dynamical Problems. Hence —V=}o? (2a*—2ax+ 2u7—y?) + 2 b one Pt VES OO he —X=2(3 =) Ct eae dy dt? dt big hen a a a gi and Sig PSN —2q From equations (12) and (13), G. W. Hill has, with four different values of E, found x and y explicitly in terms ¢, for the particular solution in each case which gives the simplest orbit (relatively to the revolving plane XOY); of which the one which presents the greatest deviation from the well-known ‘variational’ oval of the ele- mentary lunar theory is a symmetrical curve with two outwardly projecting cusps cor- responding to the moon in quadratures. He supposed this to be the most extreme deviation from the variational oval possible for an orbit sur- rounding the earth. Poincare, in his MJeéthodes Nouvelles de la Mécanique Ceéleste, p. 109 (1892), admiring justly the manner in which Hill has thus ‘si magistralement ’ studied the subject of finite closed lunar orbits, points out that there are solutions corresponding to looped orbits, transcending Hiil’s, wrongly supposed ex- treme, cusped orbit. Mr. Hill 3 g?=2E + 3x? + a | eee rs | Zee m 3 447 (11). With this, and with a=1 and m=1', for simplicity in the numerical work which follows, we have (12) (13) (14) (15). 448 Notices respecting New Books. tells me that he accepts this criticism. The labour of working out a fairly accurate analytical solution for any of Poincaré’s looped orbits, by Hill’s method, would probably be very great. I have therefore thought it might interest others besides ourselves to apply my graphic method to the drawing of at Jeast one of Poincaré’s looped orbits, in our Physical (and Arithmetical) Laboratory in the University of Glasgow. Figure 3 represents a looped orbit, which has been worked out accordingly by. Mr. Magnus Maclean, Chief Official Assistant of the Professor of Natural Philosophy, from the equations (14) (15) above. The initial values used for obtaining the curve, were c=2; y=0; b=10; 2H=—130 ; and .*. gp’?=882 and pyp=4'8. LIIL. Notices respecting New Books. Organic Dyestuffs. Chemistry of the Organic Dyestuffs. By R. Nierzxi1, PAD, Translated, with additions, by A. Coutin, Ph.D., and W. RicHarp- son. (London: Gurney & Jackson, 1892.) HE German editions of this little volume are so well known to ail chemists who interest themselves in the tar colouring-matters that Messrs. Collin & Richardson have done good service by pre- senting Dr. Nietzk’s work in an English form. The author, it is perhaps needless to state, is Professor m the University of Basle, and is best known in the chemical world as one of the most suc- cessful investigators into the constitution of the complex organic colouring-matters which science furnishes to technology. Coming from the pen of such a recognized authority as Dr. Nietzki, no special commendation is necessary to assure English students and technologists that they have received a most important and valuable contribution to their literature. The translators have done their part of the work also with commendable skill, and have fairly well expressed the author’s meaning throughout. Une special feature of the present work is its purely scientific treatment of a subject which is necessarily intimately connected with manufacturing processes. There are already in Germany several exhaustive works on the technology of coal-tar colouring- matters, notably those of Schultz and Mihlhiuser, but while these are replete with manufacturing details and reprints of bulky patent specifications, Dr. Nietzki concerns himself more especially with the classification and constitution of the compounds, and his work appeals therefore to the purely scientific chemist as well as to the technologist. Only sufficient technology is introduced to make the scientific discussion coherent and intelligible. Many of the Notices respecting New Books. 449 compounds described also have at present no technical value, but are introduced in illustration of general methods of formation or on account of their historical interest. The volume is in its way a model of conciseness, extending to little more than 300 pages and comprising fifteen chapters, an introduction, an appendix, and a remarkably complete list of references occupying about fifteen pages. ‘The reader who masters the contents of the work will, in spite of the conciseness referred to, find that he has obtained an accurate and comprehensive grasp of the chemistry of the coal-tar colouring-matters, and this is perhaps the very best tribute that can be paid to the merits of Dr. Nietzki’s production. A brief analysis of the contents will suffice to show what extent of ground is covered. The introductory chapter deals with the general question of colour and the relation between chemical constitution and tinctorial power. In this connexion Witt’s theory of the Chromophor and Chromogen is explained. It is of interest to note also in this chapter the acceptance of the view as to the probable existence of trivalent phenyl, C,H,", in aromatic compounds as first suggested in the pages of this Magazine by Meldola in 1887 (vol. xxiii. p. 513). Among other subjects touched upon is that of the theory of dyeing, the action of mordants, the formation of lakes, the testing of dyestuffs for tinctorial value; and the chapter concludes with a brief historical summary which is fair on the whole, although open to correction on one or two points. In fact, this particular section is hardly as complete to date as the author might have made it for this English edition of his work. There is an error on p. 7 which confuses the meaning of the whole paragraph in which it occurs, viz. “a valent chromophor” for a “ polyvalent chromophor ” as it stands in the original. Also we object to the adoption in English of the literal translation “methane rest ” on p. 9. The fifteen chapters following the introduction are devoted to the various groups of colouring-matters arranged according to a very convenient system of classification introduced originally by Dr. Nietzki and now generally adopted. The first chapter deals with nitro-compounds and comprises such well-known dyestuffs as picric acid, dinitronaphthol, acid naphthol yellow, &c. The large and important class of azo-dyes form the subject of the second chapter. With respect to the much-discussed question of the constitution of the amido and oxyazo-compounds the author prefers to retain the old formule, because, as he says, there is just as much evidence for the view which regards these compounds as containing NH, and HO as for the later theory of Zincke, Goldschmidt, &e. according to which the groups C=O and C=NH are present. The probability is that both views are correct and that the com- pounds can behave in one way or the other according to the nature of the reagent which acts upon them. In this chapter there is a slight error, present also in the German edition. One of the 450 Notices respecting New Books. alternative formule on p. 29 ascribed to Zincke is not due to that author, as will be found on reference to the original paper. Oxyquinones, such as the important anthraquinone dyestuffs and the quinoneoximes, formerly known as nitrosophenols, form the subject of the third chapter. The author here unreservedly accepts the isonitroso-formula of Goldschmidt. The fourth chapter deals with Ketoneimides and Hydrazides, the sole representative of the former class of compounds being auramine, and the latter class being represented by tartrazine. The derivatives of Triphenylmethane form the subject of the fifth chapter, and are dealt with under the three headings of Rosaniline Dyestuffs, Rosolic Acid Dyestuffs, and Phthaleins. Owing to the large number of colouring-matters coinprised under this group this is necessarily one of the longest chapters in the book, extending over about 40 pages. The sixth chapter is devoted to the Quinoneimides, viz. Indamines, Indophenols, and Thiazines, the latter comprising the well-known Methylene Blue of Caro. In this same chapter the author includes the oxyindamines or oxazines, typified by the first known member of the series, Naphtkol Blue or Meldola’s Blue. In accordance with his own researches Dr. Nietzki also comprises the dichroines or Liebermann’s Phenol dyestuffs in this chapter. The Azines, treated of as Eurhodines, Eurhodols, Saffrauines, Magdala red, and Mauveine, constitute the seventh chapter, which is also of considerable length. A short chapter, the eighth, is devoted to Aniline Black, a subject on which the author is well known to be an authority. Chapter 1X. treats of Indulines and Nigrosines ; Chapter X. of the Quinoline and Acridine dyestuffs, and it is of interest to note that the author includes the natural dyestuff berberine in this group, to which it is evidently related, since it gives pyridine-tricarbonic acid on oxidation. Indigo dye- stuffs are discussed at some length in Chapter XI., and an excellent summary of Prof. von Baeyer’s results is given. In Chapter XIL., devoted to Euxanthic Acid and Galloflavine, we notice the objection- able Teutonism :—“ Euxanthic acid contains as chromogen the rest (sic) of diphenyleneketone-oxide.” Canarine is dealt with in a short page constituting Chapter XIII., and a similar Chapter (XIV.) suffices for Murexide. The dyestuffs of unknown con- stitution forming the subject of Chapter XV. are Hematoxylin, Hematein, Brazilin, Morin, Quercitrin, Archil and Litmus, Cochineal, Catechu, and several other well-known and largely-used vegetable colouring-matters. Any of these may, with the advance- meut of chemical science, have their constitution explained or even ecome transferred to the true tar-products before the appearance of the next edition of the work. In an industry of such importance, and we may add of such sreat scientific interest as that of the coal-tar colouring-matters, the value of a work like that forming the subject of the present Notices respecting New Books. 451 notice cannot be too highly appraised. From our own estimate of its soundness, and our knowledge of the author’s capability of speaking with the highest authority, viz. that of the original investigator, we can only state that Dr. Nietzki’s book is alike indispensable to chemist and technologist. Lightning Conductors and Lightung Guards, By Oxiver J. Loven, D.Sc., F.RS., LL.D. Crown 8vo. London: Whittaker & Co. 1892. Dr. Lover’s work on this subject is the result of bis having agreed in 1888 to deliver two lectures at the Society of Arts. It is not for the first time that the preparation of such lectures has led to great developments in the practical applications of scientific principles. In this case, the mathematician and physicist owe their thanks to the Society of Arts as wellas the mere engineer or person who applies scientific principles to useful purposes. There can be no doubt that Dr. Lodge has effected a complete revolution in our notions as to protection from lightning-discharge. Five years ago, all engineers and nearly all physicists, in thinking of this subject, applied to it Ohm’s law for steady currents. A few telegraph-engineers knew that something more than Ohin’s law ought to be applied, and interesting experiments had been made by Prof. Hughes and others. Dr. Lodge has threshed out the whole subject, experimentally and mathematically ; he has encountered much opposition of a kind that was to have been expected; he has overcome this opposition, and at the present time his views are held to be correct. There is no such thing as absolute protection of a building or instrument from evil effects due to lightning-discharges any more than there is absolute protection of a building from earthquakes. Protection is merely a question of degree; and although Dr. Lodge assumes with Clerk-Maxwell that a powder-magazine “ can, if desired, be absolutely protected from internal sparking by enclosing it in a metallic cage or sheath,” there can be no doubt that the statement is too absolute. A summary of Dr. Lodge’s views, which may now be called the accepted views, is given in Chap. 21 in nine pages, which no practical man will find any difficulty in understanding. The rest of this volume of over 500 pages is mainly devoted to a description of the experiments and other evidence and mathe- matical reasoning which have led Dr. Lodge to hold these views, with the controversial matters which were brought forward by his opponents. The experiments are very interesting, and so is the mathematical reasoning. In both, very simple cases are taken to give and illus- trate rather vague general rules for studying the exceedingly com- plicated cases which occur in practice. Until the experiments of Professor Hughes, six years ago, very little attention was paid to 452 Notices respecting New Books. _ the self-induction of short pieces of wire or rod. It is now generally known that the impedance in even large copper rods or stmps may be enormous, and that when there is a condensing action and rapid discharge, great surging of electricity may take place and the potential differences which may exist between difterent parts of such rods may be enormous. The currents on the outsides of conductors may fuse the metal when there is no current and no heating inside. Weare led to the use of round rods of iron because their strength and endurance are of more importance than any other qualities. The reader of this book sees clearly how Dr. Lodge was led to the study of those electric oscillations which Prof. Hertz began to consider a year or so previously. It is good that these lectures and papers and extracts from discussions before learned societies should be published in a book form. Now that they are published, it would be good to have a much smailer book published with those parts left out which are merely historically interesting. Volcanoes, Past and Present. By Epwarp Hur, W.A., LL.D., F.RS. 270 pages, with 41 illustrations and 4 plates of rock- sections. 8vo. 1892. Walter Scott, London. Tus handy little volume is one of the ‘Contemporary Science Series”; and, with its many nice illustrations, is-intended as a pup ular account of volcanic phenomena, their characteristics and probable causes. A bibliographic sketch of some of the chief works on volcanoes and earthquakes is first given; and is sup- plemented by numerous foot-notes throughout the book. Then follow some notes—on the form, structure, and composition of volcanic mountains,—on the lines and groups of active volcanic vents,—and on mid-oceanic voleanic islands. A condensed account of the active, dormant, and extinct volcanoes of Europe forms Part II.; and those of other parts of the world are briefly described in Part III. The districts in the British Islands in which volcanic energy was active in the geological Tertiary period are succinctly treated in Part [V.; and in Part V. the author notices older vol- canic rocks, both in other parts of the globe (India, Abyssinia, and Cape Colony), and in the British Isles. The eruption of Krakatoa and some earthquakes, as well as the volcanic aspect of the moon, together with theoretical considerations, are referred to in Parts VI. and VIT. An Appendix offers “a brief account of the principal varieties of volcanic rocks,” with the four plates of ‘‘ magnified scclae s showing their structure. This little book on a large subject has been niet by the pen of a ready writer, who, appreciating the picturesque and the wonderful, but not working up to the strict scientific standard of the investigators whom he mentions at pages vill and 260, has produced a somewhat dilettante sketch of the results-of volcanic Notices respecting New Books. 453 action, interesting to some of the reading public, but not to be taken as a complete or safe guide for students. They had better have recourse to fuller and more complete works, which embody recent researches on the mineralogy, petrology, and physics of volcanoes and earthquakes. Principles of the Alyebra of Physics. By A. Macrartane, M.4A., DSe., LL.D., F.RS.EL. Salem, Mass., U.S.A., 1891; 8vo, pp. 52. Tis brochure is a reprint of a paper read before Sections A and B of the American Association for the Advancement of Science on the occasion of its Meeting in August 1891 at Washington. The author, an alumnus of the University of Edinburgh, now Professor of Physics in the University of Texas, published some few years back a volume with the title ‘“‘ Physical Arithmetic,” the object of which was to substitute direct calculations from first principles for the use of formule, in elementary concrete examples and exercises. In the present paper Dr. Macfarlane criticises some of the critics of the bases of the Calculus of Quaternions as laid down by Hanitton and of GRASSMANN’S system as expounded in the Ausdehnungslehre, and proposes such modifications as have commended themselves to his experience. The objections made to the square of a vector being a negative quantity,—or, more generally, to the sign of the scalar part of the product of two vectors—; the “want of harmony between the notation of Quaternions and that of Determinants,” and the dis- cordance of sign between the square of the Hamiltonian VY and Laplace’s operator are noticed in particular and commented on ; as well as the limitation of the Quaternion to tridimensional space in contrast to the general-dimensional character of Grassmann’s method, insisted on by Profs. Gibbs (Letters to ‘ Nature’) and Hyde (‘ Directional Calculus’). After these preliminaries, occupying some sixteen pages, Prof. Macfarlane addresses himself systematically to the development of a basis for the Calculus of Quaternions, beginning with ‘ Definitions and Notation,’ then pro- ceeding successively to the ‘ Addition and Subtraction of Vectors,’ ‘ Products of two Vectors’ (introducing generalized Cos. and Sin. to replace Sa/3 and Va{3), of ‘Three,’ and of ‘ Four Vectors.’ The interpretation in the last two cases involving the consideration of space of four dimensions. And here it may be remarked that the author, in general full and clear in his demonstrations, in a critical ease at the foot of p. 88 is satisfied to tell his readers that “it can be shown” that a certain expression is scalar in a space of three, but directed in one of four dimensions. Justice could not be done to Prof. Macfarlane’s views within the space at command by any attempt to reproduce them here; but under the head of “ Quaternions ” his definition may be quoted :— ‘“‘ By a Quaternion proper (a) is meant an arithmetical ratio (a) Phil. Mag. 8. 5. Vol. 34. No. 210. Nov. 1892. 2 I 454 Notices respecting New Books. combined with an amount of turning” (A, axis a), viz. a= da, index A. It “may be expressed as the sum of two components one of which has an indefinite axis and the other the same axis as the Quaternion”: viz. if A is less than a quadrant, a=a (cos.A.a, index 0-+sin. A.a, index 7/2). Here it may be of interest to recall Hamilton’s original definition as given ~ in the Phil. Mag. for July 1844, viz.: Q=w+iv+jy+kz, where ?v=—1...y=—jzi=k...and no linear relation connects 7, 7, k The later definition (adopted i in the ‘ Lectures’ and ‘ Elements ) of a Quaternion as the operator which changes one vector into another, is not the same thing exactly, Prof. Mactarlnvic explains, as the quotient of two vectors. Sums and Products of Quaternions are then dealt with and ‘‘ Quaternion Exponentials.” “¢ Scalar Differentiation,” “‘ Matrices,” “ Vector Differentiation,” and “* Gene- ralized Addition ” occupy the concluding fifteen pages of what is thus a tolerably complete monograph on the bases of the Calculus written from the point of view which commends itself to the author. That all his readers will be prepared to adopt these views can scarcely be hoped—quot homines, tot sententie being apparently the rule here too. But Prof. Macfarlane has con- tributed an interesting and well thought-out essay to the small stock of Quaternion literature. J.J. W. Watts’ Dictionary of Chemistry. By H. Forstmur Morury, /.4A., DSc., and M. M. Patrison Murr, M.A. Vol. III. London: Longmans, Green, & Co., 1892. Tuis third volume of the new edition of Watts’ Dictionary of Chemistry, by Dr. Morley and Mr. Muir, shows no falling off in quality as compared with the preceding volumes, which have already been noticed in the pages of this Magazine. The work extends to 856 pages, and comprises references and articles under the letters Ito P inclusive. Among the longer articles, that on Tron, by Mr. Muir, occupies 14 pages, and is followed by a useful article on the Iron Group of Elements by the same author, who summarizes the chief properties and relationships of Lron, Man- ganese, Cobalt, and Nickel in the course of 3 pages. The subject of Isomerism is treated of by Dr. Armstrong in about 9 pages, an amount of space which hardly enables the author to do full justice to this all-important topic. It is to be regretted that under this restriction so much space is devoted to the historical treatment. The author has, however, managed to compress a very large amount of information into the space allotted tohim. He criticises adversely the recent stereochemical develop- ments of the theory of isomerism by Wislicenus and others. None of the objections urged against these views appear to us, however, to be fatal, and whatever decision may be arrived at as the result of future investigation it cannot be denied that the extension of the Le Bel-Van’t Hoff hypothesis, by Wislicenus, Victor Meyer, &c., Notices respecting New Books. 455 has been one of the most prolific suggestions in organic chemistry since the “benzene theory” of Kekulé. The article concludes with some suggested changes in the terminology of the subject which chemists will do well to consider carefully whether they adopt them or not. The next article demanding notice is that on Isomorphism by Dr. Hutchinson, which extends to nearly 8 pages. The author has found it necessary, in view of the development of the subject since the time of Mitscherlich, to modify the definition of the term. Dr. Hutchinson considers isomorphism to be “a part of that branch of physical chemistry which studies the relations between the chemical composition and crystalline form of bodies, and which from a knowledge of the constitution and chemical properties of a substance seeks to predict its system, form, and crystallographic constants.” The author justly points out that this final aim has rarely been achieved as yet, and he then goes on to discuss the relations between the chemical composition and crystalline form under the three headings of Polymorphism, Morphotropy (includ- ing Isomorphism), and Isogonism (“ bodies not chemically related possess the same form’’). The article.as a whole is a decidedly valuable summary of existing knowledge with respect to a subject which has hardly as yet received its proper share of attention. The article on Ketones by Dr. Japp displays this author's well- known special knowledge of these interesting compounds. A somewhat lengthy article on Lead is contributed by Mr. Muir, and the same author writes on Magnesium, the Magnesium Group of Elements, Manganese, and Mercury. The article on Metal- lurgical Chemistry by Dr. Huntingdon is much too short, extending over only 4 pages. Mr. Crookes discusses the Rare Metals in some 8 or 9 pages profusely illustrated with maps of spectra, This article, as might have been anticipated from its authorship, is not only a masterly summary of our present knowledge—to which Mr. Crookes has himself made such splendid contributions —but it is equally valuable from its suggestiveness. The state at which we have arrived appears to be this :— “ We have, therefore, some thirty bodies of which the so-called rare metals are composed, or, at least which they contain; and a variety of facts point to the conclusion that we have by no means come to the end. Several even of the new bodies give signs of a capability of further splitting up, if they are examined with sufficient nicety and persistence. It is far from unlikely that when the various methods of research known as fractionation have been more generally applied we may have to deal, not with thirty, but with nearer sixty, unknown bodies.” In answer to the question as to what these bodies into which the rare earths have been resolved are the author goes on to say :— “Pending, therefore, the completion of a series of investigations, chemical and optical, which will probably occupy several generations of chemists, it may be safest to call these recently observed bodies not, as 456 Notices respecting New Books. yet, elements, but quasi-~ or meta-elements. Our notions of a chemical element have been enlarged; hitherto the elemental molecule has been regarded as an aggregate of two or more atoms, and no account has been taken of the manner in which these atoms have been agglomerated. The structure of a chemical element is certainly more complicated than has hitherto heen supposed. We may reasonably suspect that between the molecules which we are accustomed to deal with in chemical reactions, and the component or ultimate atoms, there may intervene sub-molecules, sub-ageregates of atoms, or meta-elements, differing from each other according to the positions which they occupy in the very complex structures commonly known as didymium, yttrium, and the like.” A careful consideration of this important utterance by Mr. Crookes will lead readers to the conclusion that, after all, the re- solution of the rare earths by the methods of fractionation and phosphorescent spectra may simply be a separation into modifica- tion of the nature of isomerides or polymerides having different optical properties. The basic element or metal may be the same form of matter throughout whole groups, and if so we should expect that compounds giving different “ radiant” spectra would give identical emission spectra under the influence of the more violent disruptive discharge. The problem appears to require attack from this side also before we can hope to arrive at any finality. The article on Mineralogical Chemistry is short and is con- tributed by Mr. L, Fletcher, of the British Museum; this is sufficient guarantee of its soundness. About 7 pages are devoted to an important discussion of the Molecular Constitution of Bodies by Prof. J.J.Thomson. In the first part of this article the author gives a brief account of the researches which have led to the con- clusion that matter possesses a molecular structure, and he then proceeds to consider the theories of such structure. Although Prof. Thomson’s article is essentially physical it is well in place in the present volume; the points of contact between chemistry and physics are in fact becoming more and more numerous with the progress of research and speculation. To give one example: in discussing Von Helmholtz’s theory of the ‘“ electric charge on the atom,” the author says:—‘‘ On this view of molecular structure the ‘bonds of affinity’ of chemists have a distinct physical mean- ing, as they are the tubes of electrostatic force connecting the atoms.” It is impossible to do more than give this brief reference to Prof. Thomson’s article ; it must be read to be appreciated at its full value, which in our judgment is certainly the highest among the contributions to the volume. The article on Molecular Weights by Mr. Muir is a kind of appendix to the article on Atomic and Molecular Weights in the first volume necessitated by the later researches of Raoult, and the application of this investigator’s methods to the determination of the molecular weights of chemical compounds. Considerable space is devoted to Molybdenum and its compounds by Mr. Muir; and his co-editor, Dr. Morley, treats of Naphthalene ina lengthy contribution of over 40 pages. This article is very Intelligence and Miscellaneous Articles. 457 complete and necessarily overlaps to some extent the article on this same subject in the companion ‘Dictionary of Applied Chemistry,’ edited by Prof. Thorpe. It is interesting to notice as an illustration of the inseparable union between theoretical and applied chemistry that the theory of the constitution of the naphtha- lene derivatives is more completely discussed by Mr. Wynne in Thorpe’s Dictionary than by Dr. Morley in the present article. Among other lengthy contributions under this letter are two by Mr. Muir on Nitrogen and the Nitrogen Group of Elements, which together occupy about 20 pages. The article on the Periodic Law by Mr. Douglas Carnegie is also noteworthy as an admirable statement of this subject. Enough has been said of the contents of the present volume to show that it is as full of interest to chemists as its predecessors, and the editors are again to be congratulated on the results of their labours. We have tested the work over and over again by referring to those topics with which we happen to be personally familiar, and have found it in no way behind its companion volumes in completeness and accuracy. Among the contributors whose articles we have not hitherto mentioned are Dr. Halliburton, who writes on Milk and Muscle; Dr. Rideal, who contributes the articles on Paraffin and Petroleum ; Mr. Shenstone, who treats of Ozcne ; and Dr. Tilden, who writes on Pentinene. Some of the present contributors have not before appeared in the pages of the new Dictionary, but their names indicate that the editors have known where to look for authoritative treatment of the respective subjects. LIV. Intelligence and Miscellaneous Articles. ON THE COLOUR OF IONS. BY W. OSTWALD. if follows from the dissociation hypothesis of Arrhenius that the absorption-spectrum of a very dilute solution of a salt is the sum of the spectra of the ions. If one of the ions be so chosen that it exerts no absorption in the region investigated and is therefore colourless, the colour of all the salts formed of one coloured ion and several colourless ions must be identically the same. This conclusion has been tested and confirmed by the author on a series of salts, both by actual observation and by photographs. The photographs were so arranged that the spectra of one series of salts were on a plate under each other, so that they could be con- veniently compared. The following salts were investigated :— 1. Permanganates.—The solutions investigated were prepared by mixing a decinormal solution of barium permanganate with an equal volume of an equivalent solution of the desired metal, and then diluting with 50 times as much water. ‘The final solutions 458 Intelligence and Miscellaneous Articles. contained therefore a gramme-equivalent of the ion MnO, in 500 litres. The first four absorption-bands measured from red had the same position in 15 different.salts of permanganic acid. 2. Fluoresceine and its derivatives.—The solutions to be compared were prepared in the same way as the permanganates, by decom- posing solutions of barium salts of known strength with the sulphates. (a) Fluoresceine-—The salts showed the same position of the absorption-spectra with a strength of one gramme-equivalent in 4080 and the same position of the absorption-bands. (b) Hosine.—-In opposition to the observations of G. Kriiss (Zeitsch. fur phys. Chem. ui. p. 320, 1888) and v. Knoblauch ( Wied. Ann. xlii. p. 767, 1891), all the salts investigated showed the same absorption-spectrum. The author attributes this difference to the different preparation of the specimens used. ‘The salts which the above-named experimenters had used were prepared by double decomposition and purified by long continued washing. With eosine-aluminium the author observed in the wash-waters on continued washing a displacement of the absorption-bands towards the side of the longer waves of light. Hence there are probably two very similar compounds ; for example, two eosines isomeric as regards the positions of the bromine atoms, the salts of which are of different solubility. In the mode of preparation adopted by -Kriiss the more easily soluble part was removed, while in the ‘method which the author adopted the same mixture (if such there was) in the same proportion was used. (c) Jodeosine.— There was the same absorption for the different salts. If dilute solutions of aluminium sulphate and eosine-barium or potassium are mixed the absorptiou-band is very feeble, and disappears with any excess of aluminium sulphate. Hence the aluminium salt of iodeosine is not probably present in the solution as a regularly dissolved salt, but is probably present as a colloid body. This conclusion is supported by the fact that the electrical conductivity of a solution of 4000 litres is about 20 per cent. less than that calculated for a mean value if iodeosine-potassium and aluminium sulphate are brought together in equivalent quantities ; whereas, for instance, with a mixture of eosine-potassium and cadmium sulphate the value calculated is nearly obtained. (d) Dinitrofluoresceine and (e) the tetrabrom- derivative of orcin- phthaleine also confirm the conclusion drawn from the dissociation hy pothesis. 3. Ltosolic acid.—Solutions were prepared which contained an equivalent of the neutral salt in 2600 litres. Both by direct observation and by photographs they showed considerable variations in their spectra. The author ascribes this to an hydrolysis. When solutions were prepared by mixing the neutral solutions of barium rosolate with barium hydrate equivalent to the barium they contained already; and when then quantities of the various sulphates were added equivalent to the whole barium present, Intelligence and Miscellaneous Articles. 459 and the dilution extended to 2650 litres, the rosolates were in the presence of an equivalent of free base, and the hydrolysis was sufficiently avoided. Unlike the previous great deviations these solutions showed now only small differences. The salts of the following groups showed finally equal spectra : (4) Diazoresorcine, (5) Diazoresorufine, (6) Chromoxalates. The absorption is relatively small; the investigation was therefore made by the photographic method. The chromoxalates all exert an absorption in the violet and ultraviolet. The boundaries had all the same position with the exception of the solutions of copper and aluminium salts, which exceptions the author ascribes to a hydrolytic decomposition. (7) Safrosine.—Here again the aluminium and copper salts show a deviation which is distinct with the former and just perceptible with the latter. (8) p-Rosaniline, (9) Aniline Violet—The different intensity of the absorption-spectra which the salts of some acids show is explained by their insolubility. (10) Chrysaniline, (11) Chrysoidine—The salts of this last group which were prepared with various acids showed a develop- ment like that observed with rosolic acid. They have different spectra, which, however, assume the same appearance, if the solution contains an equivalent of the acid in the free state along with the neutral salt. The deviations originally observed may be referred to an hydrolysis.—Abhandl. der Kgl. Stichs Gesellschaft, vol. xviii. p- 281, 1892. Bulletin der Physik, No. 8, p. 534, 1892. —————_— ON THE PHYSICAL SIGNIFICATION OF } IN VAN DER WAALS’ EQUATION. BY EK. HEILBORN. What is called the covolume of gases, b, is proportional, as gene- rally accepted, to the space w occupied by the whole of the molecules contained in the unit volume. It may therefore be written b=Au, where A is a constant for all gases. Vander Waals finds (“ On the Continuity of the Liquid and Gaseous States of Matter,” p. 384 of the English translation) from considerations based on the theory of probabilities the number 4, while O. E. Meyer (Kinetische Theorie der Gase, p. 298), also from theoretical considerations, deduces 4 V2. | To decide this question we may utilise the equation deduced by Dorn and Exner, v—1 Uu= ca 2? 460 Intelligence and Miscellaneous Articles. where x is the refractive index of gases for infinitely long waves. In this way we get for Hydrogen at 4°-4..b6=0-0,5244 (Waals); w=0-0,938 ; A=5-69, Ethylene at 101°. .6==0-0,254 (Waals); w=0-0,467 ; A=5-62. From this we have as the mean A=5°655=4 V2. Exner’s [ep. vol. xxvii. p. 369, 1891; from Beiblitter der Physik, No. 8,,1892, p. 503. ON THE THEORY OF THE USE OF A PERMANENTLY MAGNETIZED CORE IN THE TELEPHONE. In No. 208 of this Journal Dr. Trouton has explained the meaning of the permanent magnet in telephonic receivers. Although his explanation is undoubtedly true, there is a second side to the question that he has wholly left alone. According to my investigations, the permanent magnet in the telephone plays two roles, quite different from each other :— 1. It serves to give the sounds the required loudness. 2. It serves to give the sounds the true pitch. My researches have shown me that when a non-polarized telephone is joined to the secondary of an induction-coil, in the primary of which a microphone is placed, the sounds given out by the telephone are an octave higher than those produced before the microphone. Of course it follows from this, that such a telephone will be unable to speak intelligibly, as all the vowels lose their character by this change of pitch *. The experimental proof of this fact is very difficult, owing to the inevitable remanent magnetism of every iron core and also to the feebleness of the sounds given by a non-polarized telephone ; but is very easily proved when a speaking condenser is used instead of a telephone. For my experiments on this subject, I refer to my papers “La Polarisation des Récepteurs Téléphoniques ” and ‘Emploi de la Pile auxiliaire dans la Téléphonie,” which are to be found in the Archives Néerlandaises, t. xix. and t. xx., 1884 and 1885. J. W. Ginvay. Delft, October 2, 1892. * The vowel O thus becomes A, pronounced as in Dutch or in French, but the other vowels are quite lost. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIES.] DECEMBER 1892. LV. An Electrolytic Theory of Dielectrics. By A. P. CHATtTock *. [’ is well known that a displacement of electricity along the axes of certain crystals is capable of accounting for the phenomena of both pyro- and piezo-electricity observed in them, the fact having been pointed out by Lord Kelvin in 1878. Such a displacement may be conceived of as taking place (1) by the separation of the two electricities within previously unelectrified conducting atoms or molecules, or (2) by the relative motion of two sets of initially and oppo- sitely charged molecules, distributed throughout the crystal. There seems to be no third way for the displacement to occur if electricity cannot exist apart from matter. To explain the displacement by the first of these hypo- theses amounts, in the present state of our knowledge, to little more than a confession of ignorance. The mechanism for such a process has yet to be invented. But with the second hypothesis the case is different. Given a certain arrangement of unalterably and oppositely charged mole- cules, and both pyro- and piezo-electric phenomena follow at once from mechanical principles. Moreover, it is possible to calculate an approximate value for the molecular charges assumed ; and the fact that this comes out of the same order of magnitude as that of the charges carried by ions in electrolysis, seems to me to lend no little support to the view that such charges exist. * Communicated by the Author. Phil. Mag. 8S. 5. Vol. 34. No. 211. Dec. 1892. 2K 462 Mr. A. P. Chattock on an Arrange two sets of equally, oppositely, and permanently charged spheres, A, B, C,..... (fig. 1) in a frictionless, A Beran he horizontal, non-conducting tube which just fits them; so that they stand in line, alternately + and — along its length. Their mutual attractions will at once bring them into contact with each other; and the resulting chain of spheres will be electrically polarized—a sort of electrostatic magnet; the amount of free + or — electricity at either end being equal, roughly speaking, to half that possessed by each of the spheres. Now endow the spheres with a mutually repulsive tendency, such as is possessed by the molecules of a body due to their mechanical heat-motion. This will open out the chain length- wise, but it will not do so uniformly. There will be a pair- ing tendency among the spheres—A tending to pair with B, C with D, and so on. This may result in the complete dis- integration of the chain—the spheres flying off in pairs; but if the repulsive action is not powerful enough to effect this, it will still give rise to a partial pairing; with the result that A will be nearer to B than B to C, and so on; A not having separated from B quite so faras B from C. This means, of course, that of the whole number of lines of force emanating from A, a greater proportion goes to B than before, and therefore fewer remain to pass out into space and give rise to external effects. In other words, the chain possesses less free electricity at its ends than before ; and when the pairing is complete it possesses none. The application of the above to pyro-electricity is obvious. It is only necessary to suppose that, in the building-up of the crystal, chains of alternately positively and negatively electrified molecules occur parallel to its pyro-electric axis ; and, provided no alteration in the formation takes place during the process, a change of temperature from absolute zero (where the molecules are in contact and the free charge is a maximum) to evaporation-point (where they fly off in pairs and the free charge has vanished) will give rise to an alteration in the free charge at the end of each chain equal Electrolytic Theory of Dielectrics. 463 in amount to about half the charge possessed by each mole- cule. But there is no necessity for the chains to be opened out by heat in order to cause pairing of the molecules. Tension along their lengths will give rise to the same effect, while compression will correspond with cooling; the result of either being what is known as piezo-electricity. Such an agreement of heating with tension and of cooling with com- pression appears to exist*. It is true that it is not invari- able; but there seems to be no great difficulty in supposing that in those cases where the rule is reversed, disturbing effects may exist due to some action on the chains of the molecules on either side of them; for while both heating and tension affect the chains similarly, they produce opposite effects in planes at right angles to them—heating decreasing, while tension in- creases the proximity of the molecules therein. The theory does not therefore necessitate the above agreement—it only renders it likely; but the calculations which follow can only be made when the agreement occurs. The point at which the free piezo-electric charge is a maximum, and which corresponds with the absolute zero of temperature in the case of pyro-electricity, occurs at a pres- sure which is sufficient to bring the molecules into contact. As this pressure is reduced the pairing tendency comes more and more into play, until, after changing from compression to tension, the breaking-point is reached. Now the break occurs when the cohesion of the weakest place in the test- piece is overcome by the pull; and, on the above view, the weakest place will be found between the molecules which are farthest apart in the chains (flaws being supposed absent), i, e. between B and C, D and H, &e. (fig. 1). On the other hand, from the fact that when a crystal breaks it does so under a definite tension, it follows that the pairing is not com- plete at the point of rupture. Hence all that can be said is that the pairing is on the way towards completion, and, in the light of what follows, that an appreciable amount of it has been already accomplished when this point is reached. A change of stress, then, along the piezo-electric axis of a crystal, from a compression which brings the molecules into contact to a tension which breaks the crystal, will alter the free charge at the end of each molecular chain by an amount which is less than, but probably of the same order of magni- tude as, half the charge possessed by each molecule. * Wiedemann’s Llectricitat, ii. p. 342. 2K2 464 Mr. A. P. Chattock on an Molecular Charge. If, then, the number (n) of molecular chains per sq. em. of cross section of a crystal were known, and if it were possible to measure the alteration (p) in the density of the free charges on its two electrified surfaces, due to an alteration (either pyro- or piezo-electrically) from the state in which the mole- cules are in absolute contact to that in which they are com- pletely paired,it would be easy to calculate the value (g) of the charge possessed by each molecule. For,by what has been said, ud 2 Taian’. This it is not possible, with any accuracy, to do; but neverthe- less, both from pyro- and piezo-electric data, a rough estimate of the value of g may be formed. J.and P. Curie * have measured the piezo-electric constant for tourmaline and quartz: for the former they found it to be D°4 x 10-8, and for the latter 6°4 x 10-° E.S. units of electri- city per dyne. onsider a cubic centimetre of one of these crystals, cut with two faces perpendicular to the piezo-electric axis. It is very probable that the molecules of a solid body occupy from about 4 to 4 the space occupied by the body itself f. vm 1—/ i is therefore about the length by which a crystal 1 centimetre long must be shortened in order to bring its molecules into contact. Let M be the average value of Young’s Modulus for this compression, and the force in dynes per sq. cm. which will effect the compression is (1 -a/ i.) = 04M 4°5 i ; This quantity added to the tenacity (T) represents the whole range of stress to which the crystal can be subjected, and this multiplied by the average value (k) of the piezo-electric constant over the whole range of stress = (04 M+T)&, a quantity which is less than p in the above equation for g. In this expression T can be known accurately, but M and & are only measurable under normal conditions; and it cannot be doubted, of course, that even if the crystal could be kept from crushing, the values of both M and k& would undergo very * Comptes Rerdus, xcili. p. 204 (1881). + Riicker, Journ. Chem. Soc. liii. p. 222 (1888). Electrolytic Theory of Dielectrics. 465 great alterations long before the molecules were brought into contact. Nevertheless, the following considerations seem to me to render it probable that if M and & are represented by their normal measured values in the above expression, the resulting number will not be hopelessly different from what it ought to be. T is small compared with 0-4 M, as M is almost certainly larger than the measured value of M. The expression (0-4 M+T)& therefore reduces itself practically to the inte- gral of the product M& with alterations in length of the crystal between the limits 0 and 0'4 cm. Now M and £ are, from their nature, closely connected (if the chain hypothesis be true). Whatever alteration one undergoes, the other must undergo a similar one and in the opposite sense. If, for in- stance, at high pressures M increases, it will be due to the fact that the molecules resist being brought together more than they do at low pressures ; but this is only another way of saying that they are more reluctant to unpair at high than at low pressures: in other words, that the value of & has di- minished, and to an extent which is roughly comparable with the increase of M. The variations of M and & thus tend to compensate each other, and it therefore follows that the varia- tions in the value of their product will be small compared with those of either factor separately, and that the assumption that Mk is constant over a wide range is probably not very far from the truth. Mallock* gives the value of M for tourmaline parallel to the axis as 1:°3x 10" C.G.S. Hence we may write (0-4M+T)k = 2°8x10* 7= n where T is the range of temperature within which the mole- cules pass from the state of absolute contact with each other to the state of complete pairing (7. e. the range from the absolute zero of temperature to the evaporation-point of the crystal), and « is the mean value of « throughout that range. Now « is necessarily unknown, and in all probability very * Wied. Ann. xl. p. 264. + It is perhaps worth mentioning that an exceedingly rough attempt of my own to integrate the current from a crystal of tourmaline with time during a known alteration of temperature corroborates Riecke’s result so far as the order of magnitude is concerned. It indicates a value of x greater than any of his, but the mazimum galvanometer reading was only half a millimetre. Electrolytic Theory of Dielectrics. 467 different from « measured at ordinary temperatures; and in the expression «TI there is no compensating action, such as occurred in the corresponding expression kM for piezo-elec- tricity. Hence it is inadmissible to substitute the measured value of « for x. If, however, instead of T we write a quan- tity corresponding to Young’s Modulus—if, in other words, we make pyro-electricity a question of change of length of the crystal, like piezo-electricity, rather than of change of temperature—the difficulty of not knowing the true value of « vanishes to a great extent, as it did in the case of kh. Instead, therefore, of taking for T a range of temperature equal to the absolute temperature of evaporation, calculate the alteration of temperature required to bring the molecules into contact (7.e. to shorten the crystal by 0°4 of its length) in terms of the mean coefficient of expansion («) during such an alteration; add to this the temperature of evaporation (¢) reckoned from 0° C., and the result is T. g thus becomes 2K ve +t) a Saameean fie Now by precisely similar reasoning to that employed in the case of the product &M for piezo-electricity, it may be shown that as « alters in one direction so 1/a alters in the other; their product thus tending to constancy. ue may therefore be ‘ a represented by = («and « being for normal temperature) with the same sort of accuracy that £M is represented by kM. Unfortunately ¢ cannot be considered negligible compared with sa as this is by definition equal to 273. If, however, a we strike out t from the equation, we shall be calculating g from the amount of electricity set free during an alteration of the crystal from the condition in which its molecules are in contact to the condition of normal temperature and pressure ; that is to say, for precisely the same change that was used for the piezo-electric calculation. The values of g obtained by these two methods should therefore agree with one another, at the same time that both are too small. According to Fizeau, « for tourmaline =0°000009. Hence poms i. iia 0-000009 Mere <4~x 1016 >2x13~x >3x 10-, 468 Mr. A. P. Chattock on an a value which is in striking agreement with that obtained from the stress data. It may be well to refer here to an argument brought for- ward by Wiedemann in his Electricitdét * against Lord Kelvin’s theory, by which he seems to consider that the latter is rendered untenable. According to him a pyro-electric crystal when carefully broken at right angles to its axis does not exhibit opposite charges on its new surfaces, as it should do if polarized throughout. This argument would be, of course, unanswerable if it were applied to magnetic polarization, and it holds also in the corresponding electrical case provided the separated electricities both occur within the dimensions of a single atom. But if they occur in different parts of a complex molecule (and under this head may be included the pairs of oppositely charged molecules already referred to) they may be separated by dissociation of the latter, and Wiedemann’s result seems to me to be just what should be expected. At the instant of breaking, the molecules at the point of rupture will be violently agitated. Some of them, in the conflict, will almost necessarily become dissociated, and if their parts are oppositely charged it follows that the infinitesimal gap between the newly formed surfaces will, for the moment, be playing the part of a nearly perfect conductor. Instantly, therefore, a rush of negatively-charged particles will take place to the positively- charged surface and vice versd; and the surfaces will be ren- dered more or less neutral, as Wiedemann found them to be. Hven without this action, the field in the gap would probably be strong enough to start discharge in air. Cohesion. The agreement of the above estimations of the value of g both with each other and with the ionic charge leads naturally to the search for a possible connexion between the hypo- thetical molecular charges and other physical properties of the crystals. Imagine a mixture of two equal numbers of particles, equally and oppositely charged, and uniformly spaced. The mass will cohere in virtue of the lines of force which pass in all directions from positive to negative particles. Draw an imaginary surface through the mass, in such a way that it cuts none of the particles, but passes midway between those on either side of it, Provided the mixture of the opposite charges is thorough, and their distance apart not too small, this surface will everywhere cut lines of force, roughly speaking, * Vol. u. pp. 3388, 346. Electrolytic Theory of Dielectrics. 469 at right angles ; the number of lines passing across it from a particle on either side being, on the average, one sixth of the whole number emanating from that particle. If g be the charge per particle, and d the distance from centre to centre of adjacent particles, the average number of lines crossing the Anr i : surface per sq. cm. = | ; and the mechanical tension across it per sq. cm. is therefore approximately j 2 2 nS je +—8r2 i. This quantity represents the cohesive force of the mass, and is consequently equal to the internal pressure per sq. cm. of the particles due to their heat-motions. Pairing of the particles will, however, reduce the cohesion ; for it will strengthen some bonds at the expense of others, and the cohesion depends on the strength of the weakest part. On the other hand, the above formula is only accurate if the closeness of the lines of force cutting the imaginary surface is constant all over it. This will not be strictly the case ; and want of uniformity will increase the cohesion, as the latter is proportional to the mean square of the field strength, while the calculated value is in terms of the square of the mean field. These two causes of error thus tend to compensate—to what extent it is impossible to say, but sufficiently, in all pro- bability, to allow of a determination by the formula of the order of magnitude of g. Applying it to the crystals already discussed, we have therefore 2 = ~ cohesion per sq. cm. = internal pressure per sq. cm. Now, of the value of the internal pressure in a solid it is only possible to obtain the very roughest idea, by applying Boyle’s law and assuming that the unoccupied space ina solid is three fourths its apparent volume. The volume of a substance in the form of vapour is, on this assumption, about 1700 times as great as the unoccupied space within it when in the solid form ; from which the internal pressure comes out by Boyle’s law to be about 2x 10° dynes per sq. cm. Of course this is the merest approximation to the true value, but it serves as a guide to a better determination in the following manner :— When tension is applied to a rigid body, the internal pressure along the line of pull is reduced, the sum of these two forces (tension and pressure) being now balanced by the cohesion. The maximum possible tension (.e. the tenacity otf the sub- stance) must therefore be less than the internal pressure in the 470 Mr. A. P. Chattock on an unstrained condition ; assuming that the cohesion does not increase on the application of stress, which seems unlikely. Now the tenacity of quartz in fibres of ordinary thickness is about 7x 10°*. Hence the internal pressure must be at least equal to this; and as its calculated value is so much less, its true value will probably not greatly exceed 7 x 10° (as it might have done if its calculated value had been larger). For the purpose, therefore, of obtaining the order of mag- nitude of g from the above equation (which is all that can be hoped for from such a rough calculation), it will probably be allowable to write tenacity for internal pressure in the formula, bearing in mind, however, that the value of g so obtained will be too small. Putting for d the value 4 x 10-8 cm., we have 2 eer alts whence q >5x10-%, a number which is not to be distinguished from the ionic charge of hydrogen. Of course the extreme closeness of the agreement is purely accidental ; but I cannot think that the agreement of the values of g calculated by the above three independent methods, both with each other and with the ionic charge, can be due to mere chance. On the other hand, it does not follow immediately that cohesion is due to electrical attraction because a good value can be calculated from ionic charges. As Dr. Lodge has shown f, the work done by the attraction of two oppositely charged ions is of the same order of magnitude as that done by their chemical affinity in bringing them together ; and consequently, if cohesion can be shown to be numerically equal to either of these forces, it may just as well depend on one as on the other. It is only when considered in con- nexion with the electrical properties of crystals (for which, if the view here put forward be accepted, initially charged molecules are required on electrical grounds) that the above result suggests that cohesion may turn out to be an electrical and not a chemical phenomenon. ‘The idea is at any rate in accordance with the most striking feature of cohesive attrac- tion—its variation, namely, in inverse proportion to a high power of the distance between the cohering particles. For a mass of oppositely charged particles this power could not be less than the cube, and would probably be nearer the fourth power. * Boys, ‘ Nature,’ xl. p. 249. His value is 7:8X10°=50 tons to the inch. t “Seat of E.M.F.” Electrolytic Theory of Dielectrics. 471 Of course a theory which is to account for the cohesion of pyro-electric crystals must also account for that of all dielectrics. And in fact, according to the present view, pyro-electric crystals only differ from other crystals in the peculiar chain-like arrangement of their molecules, whereby the cohesive force, as it were, shows itself externally ; whereas, in the case of other substances, it is kept entirely within their bounding surfaces. As regards the variations of tenacity in different substances, they are, I think, to be easily accounted for by the different sizes, masses, and perhaps shapes of the molecules as compared with the charges they carry. Varia- tions in these quautities will give rise to a greater or less degree of pairing, as well as to alterations in d and in internal pressure, sufficient to account for a greater range of variation than is observed in the tenacity of homogeneous bodies. (Porous substances like cement do not fall under this head, of course. They may be regarded as containing an infinite number of flaws, and their tenacity will therefore be too small.) The chief difficulty is, however, met with in the case of elementary substances. It is not easy to see where a want of symmetry occurs in the combination of two like atoms, to account for their being or becoming oppositely charged. The question is a large one, and one which is intimately connected with the manner of discharge of electricity through simple gases. It is, for Instance, uncertain whether the two atoms of an oxygen molecule are oppositely charged to start with, or become so by induction on entering «a strong electrostatic field. It is, however, extremely probable that after separation they possess their ionic charge ; so that, unless there is a lémit to the capacity of an atom for having electricity induced on it, the former view is perhaps the more likely. I merely mention this to show that the existence of opposite charges in like atoms is not @ priorz impossible. Specific Inductive Capacity and Dielectric Strength. Two other physical constants with which initial molecular charges should be closely connected are those of Specific Inductive Capacity and Dielectric Strength. From both these quantities it is possible to obtain a value of g. Place the mixture of oppositely charged particles, discussed under cohesion, in an electrostatic field. If the field is not too strong there will be a slight movement of all the + par- ticles in one direction and of the — particles in the opposite direction. This will be equivalent to coating the surfaces of the mixture at which the lines of force enter and leave it with 4792 Mr. A. P. Chattock on an — and + electricity respectively. These coatings, which are really due to a slight protruding of all the — particles in one surface and all the + in the other, correspond exactly with the “fictive layers” of electricity, which for mathematical purposes are assumed to appear on the bounding surfaces of a dielectric under similar conditions. Arrange that the lines of force of the field enter and leave the mixture normally, and consider a particle in the interior. It is urged with a force of fg dynes in the direction of the field (f being numerically equal to the slope of potential at the point considered). If fg be not too great, the particle will be displaced a distance 6, which is small compared with the distance d between it and its nearest neighbours. In other words, d has been shortened by 8 on one side and increased by 6 on the other in the direction of #7. Let M be Young’s Modulus for the mixture. Then, if there are n par- ; 2M o., ticles per sq. cm. of a surface at right angles to /, > +738 d the mechanical force required to produce the displacement 6 in the particle considered. Hence Now if it were possible to continue the process of displace- ment by increasing / until the + particles had moved a distance d relatively to the — particles, the fictive layer at either end would contain 5 particles per sq. cm., and its density would therefore be L" =p. Consequently, if the 2 displacement is only 6, the value of p will be = ' 3 But ee sn a Ss where K is the specific inductive capacity of the mixture. Hence . qn 8 (K~—1)f pg aero sl sete 1 Combining (1) with (2), and putting z for n, Electrolytic Theory of Dielectrics. 473 The measured value of K for quartz is 4°6, and of M 7°5 x 10™. Hence, for quartz, = oe LO", If K be calculated from the index of refraction (w) of quartz its value is 2°4, which makes g= 14x 107", (As, however, M was measured by a “slow ’”’ method, the first value of qg is likely to be the more correct.) For tourmaline K (obtained from w) =2°7, M=1°3 x 10”. Hence ga x 10s For glass K=7 (about) for slow electrification, and Bread x 10", GOO ie 2 These values are about double the ionic charge of oxygen, so far as it is known; and the method of calculation is at the same time far more reliable than those that have hitherto been discussed. ‘There are no very uncertain assumptions in it ; and the only quantity not accurately known is d, which enters equally into all the methods. It is therefore satis- factory that the results obtained by this particular method should be so good. Applying now these considerations to liquids and gases, the results are at first sight less satisfactory. Taking water as an example of a liquid dielectric, K=1°8 (calculated from uw); M*=2 x 10", — ire es For bisulphide of carbon K = 1-8 (measured), M=1°6 x 10”, = 16x 10-”. These values are small; and if we take gaseous oxygenj the * M is now the coefficient of volume elasticity, of course. For solids it makes very little difference, theoretically or practically, whether this or Young’s Modulus is used. + In using the formula for a gas there are one or two modifications to be made. ‘he number of molecules in the protruded (fictive) layer of electrified particles at either end of the gas is d/d,? per sq. cm., where d is the thickness of the layer (=3 10-8) as before, and d, is the distance apart of the gas-molecules, say 5°3x10—8 (=dx 4/1200). Hence the equation for g comes to be (K—1)Mdd,° in 2a 7 It is worth noticing that the direct variation of (K—1) with the density of the gas (1/d,*) follows from this as it should do. A474 Mr. A. P. Chattock on an discrepancy is far greater: K=1-:00054, M=1°1 x 10°, and d=raxit, g=Scl0=: The following considerations, however, seem to me to show that these results are just what they should be, on the view of initially charged particles. According to that view, the oppositely charged particles tend to pair more and more as the temperature rises. When the pairing has gone on to a certain extent the body becomes liquid (provided, of course, the critical pressure is not overstepped), the bonds holding the paired particles together having been strengthened at the expense of the others to such an extent that the latter are no longer capable of holding the particles in relatively fixed positions. Ata still higher temperature they have become so weakened that the paired particles fly off in the form of vapour. Now Young’s Modulus expresses the difficulty experienced in bringing the particles of a body nearer together. Whena body is compressed it yields chiefly at those places where the particles are furthest apart. Hence, in the case of a body composed of partially paired particles, the yielding will chiefly occur where the pairing is least, z. e. between the particles B and UC, D and £, &e. in fig. 1. The same is of course true of tension. When, however, the body is strained electrically the case is different. The electrostatic field takes hold, as it were, of all the + charges throughout the dielectric, and urges them past all the — charges, which it pulls in the opposite direction. The resistance to this straining will obviously depend prin- cipally on the firmness with which the. paired particles hold together. It will be determined, therefore, by the forces between the particles A and B, C and D, &c., and will consequently be greater than M. From this it follows that as the particles become more and more paired (2. e., as they pass from the solid to the gaseous condition) M will get less and less, as it is known to do, while the resistance to electrical strain will probably not differ much from the value it possessed in the solid state. And, in fact, if for M in the above three cases of fluids we substitute the value 2x10”, which is fairly representative of M for the most elastic solids (it is necessary to take a high value, since, by what has just been said, low elasticity only applies to mechanical strain in bodies where pairing is pronounced), we obtain for g :— Electrolytic Theory of Dielectrics. 475 For water and bisulphide of carbon . 1°8x107", For oxygen Lyin COS se el abd numbers which are in satisfactory agreement with the ex- pected result. It would be necessary to divide the M for paired particles by 2, as it acts, as it were, on one side only of each particle ; but the increased nearness of the particles due to pairing probably increases M sufficiently to render this unnecessary. Even in solids there must be some difference between the electrical and mechanical elasticities. Pairing must have taken place to a certain extent, or there would be no pyro- or piezo-electricity. If, however, we may take the values of q obtained from these two phenomena as an indication of the amount of pairing actually achieved at ordinary temperatures in unstrained solids, that amount is probably about a tenth of what it might be, and may therefore be neglected for the present purpose. It only remains now to consider the question of dielectric strength. Divide the tenacity (T) of a substance, in any direction, by the number of molecules in each sq. cm. of a plane at right angles to that direction. The result is the greatest stress which a single molecule can bear without being torn from its neighbours. Hence, if M represent the mean value of Young’s Modulus up to this point, and A the corresponding displace- ment of the molecule, we have, by (1) :— 2M AT If now force be applied to each molecule by an electro- static field, the disruptive discharge should occur at the point where the molecules are subjected to this maximum possible stress. In other words, by what has been said above, the density of the fictive layer at the point of discharge will be given by the equation oa as a = Prax. Nowp —4 ; = where - represents the slope of poten- tial in the direction of the applied field (the surfaces at which 476 Mr. A. P. Chattock on an the lines of force enter and leave the substance being, as before, normal to the lines). Hence, by combining these equations, _dV (K—1)dM tds * rl ae zt being here the maximum slope of potential which the substance will stand dielectrically. M is unknown; but, judging by what is known of the behaviour of solids under stress, it is probably considerably less than the value of M measured within the limits of elasticity. To test this formula I made a few rough experiments on the dielectric strength of glass, in the form of the thin plates used for the covers of microscope-slides. The plates, which were 13 inch diam., were held at their centres between platinum knobs of 24 mm. diam. Parallel with these knobs was a micrometer spark-gauge, the length of which was gradually increased until the glass was pierced. The following table gives the results obtained :— Thickness (s) of | Max. value of V dV max. glass in cm. in E.S. units. ds, 0-023 70 3000 0:023 77 38300 0:025 78 3100 3130 Taking the value of K for glass as 7, T=6°5 x 10°, and M=5"'5 x10", we have p< Mee el Now, in addition to the fact that M is probably less than M, T is also likely to be too small on account of flaws in the material. These would not affect the dielectric strength nearly so much as they affect the tenacity, because the stress in the latter case has to be transmitted from point to point in the material, and may therefore have to be shared by more or fewer molecules according as the material is truly homo- geneous or not, whereas in the dielectric the stress is applied Electrolytic Theory of Dielectrics. 477 more directly to each molecule, without depending so much on the positions of its neighbours, and is therefore more constant in amount. Hence, owing to both these causes, the above value of q is too large. The source of error discussed above under specific inductive capacity, which is due to the resistance of a body to electrical stress being greater than that to mechanical stress on account of pairing, is as negligible here for solids as it probably was there. Indeed, even in the case of liquids it is not nearly so important a factor in the calculation as before ; for the true value of T (for electrical stress) is also affected by the same cause, and it is the ratio of M to T which enters into the expression for g ; and, in fact, if instead of the ratio =; we write the ratio of the volume of a liquid to the change of its volume when strained to the breaking-point under tension, not only does this source of error become negligible, but the one which depends on the fact that M is less than M disappears altogether. For the above ratio and M are by definition the ab same. ‘There are, however, two minor sources of error still remaining in the case of liquids. One arises from the experi- ~ mental difficulty of ascertaining the maximum extension that a pure liquid will bear without breaking; the measured values being almost certainly too low in consequence of foreign matter in solution. ‘The other is due to the fact that the molecules of a liquid are able to move about among them- selves. This property will enable them, when the field of force is very strong (as it is just before disruption occurs) to fall into line with it and form chains—a tendency which must also exist in solids, but is probably almost unsatisfied on account of their rigidity. Now a chain of alternately oppo- sitely electrified molecules is a line of high specific inductive capacity, and the formation of a large number of such lines through the liquid may easily raise the value of K by an appreciable though unknown amount. Hence, as these two remaining sources of uncertainty tend to compensate, and it is impossible to say which will preponderate, it will be fairer to substitute the sign ~ for < in the equation of g for liquids. A few experiments on sparks in liquids, due to a very sudden rise of potential between platinum electrodes, are in fair-agreement with the formula when viewed in the light of the above. Phil. Mag. 8. 5. Vol. 84. No. 211. Dec. 1892. 21, A78 Mr. A. P. Chattock on an Vv in E.S. dV. Substance. sin cm. ate aE Water) (5.14.22 0:015 53 3500 eat ee 0-0165 56 3400 Oils eich cee 0-018 56 3100 The potential was measured by comparing the spark in the liquid with a spark in air arranged parallel with it; the assumption being made that the same potential was necessary to produce the air-spark whether the potential was applied suddenly or slowly. M for water is 2x10'°. Berthelot * has succeeded in stretching water by 74,5 of its whole volume. K for water (calculated from 4)=1°8. We have therefore g=220x10~: In the accompanying table the values of g calculated by the various methods discussed in this paper are collected for convenience of reference. Their agreement with each other and with the ionic charge seems to me to be quite beyond the range of chance. Hxcept in one case theory has been able to anticipate any marked divergence from the ionic value, as is shown by the sign > or <. On the other hand, it is well to emphasize what has already been pointed out, that even granting the reality of the above relations, the existence of initially charged molecules is not in every case necessitated by the success of the method of calcu- lation. It has been shown above that Method III. affords equally good evidence that cohesion is due to chemical affinity, to ionic charges, or perhaps to both. The order of magnitude of the resulting value of g will in each case be the same. So, too, Dielectric Capacity has been shown by Clausius to be explicable on the hypothesis of induced charges in conducting molecules ; and the ratio of the volume of the molecules to the space they occupy, when determined by this theory, is in close agreement with the same ratio calculated from the behaviour of gases under pressure. Hither this agreement, therefore, or that of the results of Method IV. with the other five determinations of g, must be accidental; for neither theory of dielectric capacity implies the other in any way, so far as * B.A. Report for 1888, p. 583, 479 Electrolytic Theory of Dielectrics. Data of Calculation. I. Pyro-electric Constant and Coefficient of Expansion....,. } TI. Piezo-electrie constant and Young’s Modulus......,.......- III. Tenacity eer eoesrereoseee ester eoeres eee IV. Specific Inductive Capacity et POLASUICIEY? sh oad aasaiys ewe os he V. Dielectric Strength, Specific Inductive Capacity, Elas- ticity, and Tenacity............ VI. Electrolysis ............ I Quartz. Tourmaline, Oxygen. Glass. Water. ae +3107)? SS isclur se 14107!” 5.2100" Teles Ba >1'5x10-? | >08x10~” 2:3x1077! 21x107" 1:2x107 26x 1071 183x107! 5 nee Caen > <—1:2x107!° 2:6x10- Lee sie eee Vaio ee SESS EE ee 2L2 480 Electrolytic Theory of Dielectrics. I am able to judge. With Method V., however, acceptance of the formula seems to necessitate acceptance of initial charges ; and this is still more the case with Methods I. and II., where electrification is produced without the application of any apparent electrostatic field. So far, at any rate, as the results of the table are concerned, I think it will be admitted that the hypothesis of initial ionic charges is the simplest and most consistent method of ex- plaining them. Indeed, if it were not for the difficulties met with in the consideration of elementary substances, and espe- cially of metals, one would be almost tempted to imagine that chemical affinity and ionic attraction were one and the same thing. The difficulty, however, of accounting for opposite charges in similar particles—the fact that the pairing ten- dency is so small in metals that monatomic vapours are the rule—the low compressibility of liquid mercury, which also points to incomplete pairing—and especially the. fact that metals conduct without any apparent convection of their par- ticles—all these points are clearly opposed to the idea that the particles of metals are oppositely and permanently charged. It is true that the phenomena of thermo-electricity, like all other cases where electricity is produced by apparently non- electrical means, are likely to find an easy explanation if the previous existence of electricity, as such, in the substance is postulated. So, too, magnetic permeability mzght conceivably be due to the spinning of permanently charged atoms. But these are the only indications of internal charges in metals that I know of, and they are very slight. On some grounds it almost seems as if the force of chemical affinity in some way passes over into that of ionic attraction either during, or after, or even just before the act of combina- tion of wnlike particles, while, from symmetry, it remains unaltered in the case of lzke particles. The possible equality, mentioned above, of the two forces, lends some colour to the idea. There is, however, plenty to be said against it. If more were known of the manner in which monatomic vapours conduct, there would be a better chance of coming to a con- clusion on this question. As it is, it must be left for the present as a serious, but perhaps not insurmountable difficulty from the point of view of this paper. One other point I may refer to. For convenience, the word pairing has been used to indicate the combining tendency of oppositely electrified particles, or, rather, the reluctance of the two electricities to part company while the substance-expands and disintegrates under the action of heat. Of course, it is not necessary that this process should be Influence of Obstacles on the Properties of a Medium. 481 liinited to two particles only. Given the proper temperature with corresponding conditions of mass, shape, and distribu- tion of charge on the particles, and, as it seems to me, almost any amount of molecular complexity is possible. That I have not taken this possibility into account does not, however, vitiate the results here brought forward, as they do not pre- tend to greater accuracy than that of their order of magnitude. It is the cumulative value of these results which will, I hope, be regarded as sufficient reason for the publication of what is at best an incomplete piece of theory. Univ. Coll. Bristol. LVL. On the Influence of Obstacles arranged in Rectangular Order upon the Properties ofa Medium. By Lord Ray.eicuH, Secoh.S.* HE remarkable formula, arrived at almost simultaneously by L. Lorenz f and H. A. Lorentz{, and expressing the relation between refractive index and density, is well known ; but the demonstrations are rather difficult to follow, and the limits of application are far from obvious. Indeed, in some discussions the necessity for any limitation at all is ignored. I have thought that it might be worth while to consider the problem in the more definite form which it assumes when the obstacles are supposed to be arranged in rectangular or square order, and to show how the approximation may be pursued when the dimensions of the obstacles are no longer very small in comparison with the distances between them. Taking, first, the case of two dimensions, let us investigate the conductivity for heat, or electricity, of an otherwise uniform medium interrupted by cylindrical obstacles which are ar- ranged in rectangular order. The sides of the rectangle will be denoted by a, 8, and the radius of the cylinders by a. The simplest cases would be obtained by supposing the material composing the cylinders to be either non-conducting or per- fectly conducting ; but it will be sufficient to suppose that it has a definite conductivity different from that of the remainder of the medium. By the principle of superposition the conductivity of the interrupted medium for a current in any direction can be de- duced from its conductivities in the three principal directions. * Communicated by the Author. + Wied. Ann. xi. p. 70 (1880). { Wied. Ann. ix. p. 641 (1880). 482 Lord Rayleigh on the Influence of Obstacles Since conduction parallel to the axes of the cylinders pre- sents nothing special for our consideration, we may limit our attention to conduction parallel to one of the sides (a) of the rectangular structure. In this case lines parallel to @, Fig. 1. symmetrically situated between the cylinders, such as AD, BC, are lines of flow, and the perpendicular lines AB, CD are equipotential. Ir we take the centre of one of the cylinders P as origin of polar coordinates, the potential external to the cylinder may be expanded in the series V=Ao+ (Ayr + Br") cos 0+ (Agr? + By) cos 80+..., (1) and at points within the cylinder in the series V'=C)+ Cyr cos 0+C,r* cos 80+..., . . (2) @ being measured from the direction of a The sines of 6 and its multiples are excluded by the symmetry with respect to =, and the cosines of the even multiples by the sym- metry with respect to 0=}7. At the bounding surface, where r=a, we have the conditions V=V’, vdV'/dr=dV/dr, v denoting the conductivity of the material composing the cylinders in terms of that of the remainder reckoned as unity. The application of these conditions to the term in cosn@ gives | Lis, : 3,= Ley ne. ° . ° s e ° (3) In the case where the cylinders are perfectly conducting, y=oo. If they are non-conducting, v=0. in Rectangular Order upon a Medium. 483 The values of the coefficients A,,B,, A3,B;... are neces- sarily the same for all the cylinders, and each may be regarded as a similar multiple source of potential. ‘The first term Ag, however, varies from cylinder to cylinder, as we pass up or down the stream. Let us now apply Green’s theorem, {(oZ-vs,) a=0 i — deena to the contour of the region between the rectanyle ABCD and the cylinder P. Within this region V satisfies Laplace’s equation, as also will U, if we assume CO te. 3) sie deo) Over the sides BC, AD, dU/dn, dV/dn both vanish. On CD, dV jdn ds represents the total current across the rectangle, which we may denote by C. The value of this part of the integral over CD, ABis thuseC. The value of the remainder of the integral over the same lines is — V,@, where V, is the fall in potential corresponding to one rectangle, as between CD and AB. On the circular part of the contour, U=acos@, dU/dn=—dU/dr=—cos 0; and thus the only terms in (1) which will contribute to the result are those in cos@. Thus we may write V=(A,a+ Ba!) cos 6, dV /dn= — (A,— Bya-?) cos 0; so that this part of the integral is 27B,. The final result from the application of (4) is thus aC—BV,+27B,=0..... . (6) If B,=0, we fall back upon the uninterrupted medium of which the conductivity is unity. For the case of the actual medium we require a further relation between B, and V,. The potential V at any point may be regarded as due to external sources at infinity (by which the flow is caused) and to multiple sources situated on the axes of the cylinders. The first part may be denoted by Ha. In considering the second it will conduce to clearness if we imagine the (infinite) region occupied by the cylinders to have a rectangular boundary parallel to «and 8. Even then the manner in which the infinite system of sources is to be taken into account will depend upon 484 Lord Rayleigh on the Influence of Obstacles the shape of the rectangle. The simplest case, which suffices for our purpose, is when we suppose the rectangular boundary to be extended infinitely more parallel to a than parallel to @. It is then evident that the periodic difference V, may be reckoned as due entirely to Hx, and equated to He. For the difference due to the sources upon the axes will be equivalent to the addition of one extra column at +x, and the removal of one at —% , and in the case supposed such a transference is immaterial*, Thus V,=—He .°. os {ee simply, and it remains to connect H with B,. This we may do by equating two forms of the expression for the potential at a point z,y near P. The part of the potential due to Hz and to the multiple sources Q (P not included) is Ajp+A,7 cos 6+ Az 7* cos 804+....3 or, if we subtract Hz, we may say that the potential at 2, y due to the multiple sources at Q is the real part of Ao+ (A, —H)(#@+ ty) + A3(a+iy)?+A;(e+iy)?+... . (8) But if w', y' are the coordinates of the same point when re- ferred to the centre of one of the Q’s, the same potential may be expressed by D{B, (2! + iy’) +Ba(a' +2y’) PF +....3, 2. (9) the summation being extended over all the Q’s. If &,7 be the coordinates of a Q referred to P, w=a—E, y'=y—73 so that B (a +ty') "=B, (a+ty—E—in)™. Since (8) is the expansion of (9) in rising powers of (x+iy), we obtain, equating term to term, H—A,=B, Do + 3B; >.+5B; Set. ee —l 2.3 Agel. 2.3)B, 2,493.24 oR oe. =£1.2.3.4.5 Ag=152.3.4°55' Bs, +3.45 56.7 Bee and so on, where Sn Det ig isyush of .=B,(v'a-?+%,), e = ° (15) and the conductivity is seat (14) 27a? 1 roe aB(v + a2.) ° . 2 2 e ° (1 6) The second approximation gives | 2 “= =v +a? >.— SaSe, aes eS (17) and the series may be continued as far as desired. The problem is thus reduced to the evaluation of the quan- tities ,, 24,..- We will consider first the important: parti- cular case which arises when the cylinders are in sguare order, that is when B=a. €and 7 in (11) are then both multiples of a, and we may write Saori ee. » Bs, oon een where ee Ce em 2 the summation being extended to all integral values of m, m’, positive or negative, except the pair m=0, m'=0. The quantities S are thus purely numerical, and real. The next thing to be remarked is that, since m, m’ are as 486 Lord Rayleigh on the Influence of Obstacles much positive as negative, 8, vanishes for every odd value of n. This holds even when e@ and £ are unequal. Again, = > (m! + im) —2n — 7-22 5 (—im! +m)-*" = (—1)"3(—zm!+m)—* =(—1)"8 Whenever n is odd, Sen = — Son, or Son vanishes. Thus for square order, 2n° SS oS ha oa Mie Suet ae (20) This argument does not, without reservation, apply to S.. In that case the sum is not convergent; and the symmetry between m and m’, essential to the proof of evanescence, only holds under the restriction that the infinite region over which the summation takes place is symmetrical with respect to the two directions « and 8—is, for example, square or circular. On the contrary, we have supposed, and must of course continue to suppose, that the region in question is infinitely elongated in the direction of «. The question of convergency may be tested by replacing the parts of the sum relating to a great distance by the corre- sponding integral. This is didy eo cos 2n8 r dr do ery = \) La a= p—2nt2/( In + 2) : so that if n> 1 there is convergency, but if n=1 the integral contains an infinite logarithm. We have now to investigate the value of S, appropriate to our purpose; that is, when the summation extends over the region bounded by e= +u, y= +v, where u and v are both infinite, but so that v/u=0. If we suppose that the region of summation is that bounded by z=+v, y= +v, the sum and herein Fig. 2. vanishes by symmetry. We may therefore regard the sum- mation as extending over the region bounded externally by w= +e, y= +v,and internally by v= +0 (fig. 2). When v is very great, the sum may be replaced by the corresponding in Rectangular Order upon a Medium. 487 dx dy , =2 ({ ae, a ibe: Soares Reaulyy the limits for y being +v, and those for 2 being v and w. Ultimately v is to be “made infinite. We have integral. Hence eee OU. 2 i aa _, @tiyye etiv a“2—w x+y?’ and ie a = 2 tan™' «© —2 tan7™’ 1=47. Accordingly Sf = Ce ee nr 2) In the case of square order, equations (10) (12) give 2 He yas, Sosy ery... ma 3a Ta} eos Tg S82 - a eae (23) and by (14) era haa Conductivity = 1-—,- re ee) We. Ae ee) If p denote the proportional space occupied by the cylinders, ‘ PTE Ies -AeMe yp 35 Baek: At Ge an Conductivity = | aera eae ee ape ee 1 J Sp" ao ip" TPT yar a ate Of the double summation indicated in (19) one part can be effected without difficulty. Consider the roots of sin (E—im7)=0. They are all included in the form E = m’a+imr, where m is s any integer, positive, negative, or zero. Hence we see that sin ae may be written in the form A Ce mr mmr +17 wnt — 17 wm + 2a 488 Lord Rayleigh on the Influence of Obstacles in which Thus A = —sin imm. log {cos —cotzmr sin 1 EY = les G-= —— im E gue (1 e a) a If we change the sign of m, and add the two equations, we get 3 | : sin”2m7 log 1 aa ‘i logy 1- casa} AGE {i- cae ea i whence, on expansion of the logarithms, sin’é sint& sin®é sinzmmr = 2sinttm@r 3 sin*2mt apf ole ae | (imm)* ~~ (tma+m)* (imm—T)y 1 1 | (2ma)* a sae Ne ze (imma —a7 )* one a +36 1 al ae Ea pies igh i By expanding the sines on the left and equating the corre- sponding powers of £, we find pee. 4 ae ag et dk Le a A wings O 2 {im)? © (tm+1)? © (t¢m—1)? © (em+2)? a: cemrnp sr | ‘) 1 1 2ar* a {im)* Ls (tm+1)* ee ae ee eps i 1 Q7° 1 Ginyt* (¢m+ it 2 Es amy aati * Ta tmar Aaa In the summation with respect to m required i in (19:) we are to take all positive and negative integral values. But in the case of m=0 we are to leave out the first term, corre- sponding to m’/=0. When m=0, 7 1 Yl ——————. = — sin*ima (tm)?—— 33” in Rectangular Order upon a Medium. 489 which, as is well known, is the value of Pee FP L =n): 92 faa Hence Se=2r $3 sin—imar+im; . . . «| (30) m=1 and in like manner 4 m= y= - +20* > {—2sin—imrt+sin-mmt,. . (31) m=1 9,-6 m= Ss = aii + 27° & S75 sin-imar 27 ° 35 m=1 ‘= —sin-“imar+sin—&mm}. . . (32) We have seen already that 8,=0, and that S,=a. The comparison of the latter with (30) gives oS epam by: 1 eae | —2 STS Sees eee y es sin™imm= 5 — ee te (33) We will now apply (31) to the numerical calculation of S,. We find :. 2 t m. | — sin” “7m. Bin *237. ] 00749767 -0000562150 2 : 395 2 3 : 3 Sum 00751165 "0000562152 so that =m X705235020. . ee Ole Ba) In the same way we may verify (33), and that (82) =0. | If we introduce this value into (26), taking for example the case where the cylinders are non-conductive (v'=1), we get / ih me 1+p—°3058p*" ° 6 e ° e (35) From the above example it appears that in the summation with respect to m there is a high degree of convergency. 490 Lord Rayleigh on the Influence of Obstacles The reason for this will appear more clearly if we consider the nature of the first summation (with respect to m’). In (19) we have to deal with the sum of (w+ y)—, where y is for the moment regarded as constant, while x receives the values =m’. If, instead of being concentrated at equidistant points, the values of « were uniformly distributed, the sum would become ‘ae ax ey Ce) is Now, n being greater than 1, the value of this integral is zero. We see, then, that the finite value of the sum depends entirely upon the discontinuity of its formation, and thus a high degree of convergency when y increases may be ex- pected. The same mode of calculation may be applied without difficulty to any particular case of a rectangular arrangement. For example, in (11) | Ly==(mla+imB)—? =a (m! +imB/a)—?. If m be given, the summation with respect to m’ leads, as before, to 7 ~ sin? (immB/a)? >(m! +imB/a)-? and thus aS =2n? S sin-?(immB/a) +40. . . (36) m=1 The numerical calculation would now proceed as before, and the final approximate result for the conductivity is given by (16). Since (36) is not symmetrical with respect to « and £, the conductivity of the medium is different in the two principal directions. When B=a, we know that «-?2,=7. And since this does not differ much from 37’, it follows that the series on the right of (36) contributes but little to the total. The same will be true, even though £ be not equal to «, provided the ratio of the two quantities be moderate. We may then identify «>, with 7, or with 37’, if we are content with a very rough approximation. The question of the values of the sums denoted by 22, is intimately connected with the theory of the 6-functions*, inasmuch as the roots of H(u), or 6,(7u/2K), are of the form 2m K+ 2m'tK’. * Cayley’s ‘ Elliptic Functions,’ p. 300. The notation is that of Jacobi. in Rectangular Order upon a Medium. 491 The analytical question is accordingly that of the expansion of log 6,(z) in ascending powers of «. Now, Jacobi* has himself investigated the expansion in powers of « of 6 (a) =24g'4 sin e—g%4 sin 8x + g?/4 sin 5a—...$, (87) where ger SAiOc is | eT) ORR) So far as the cube of x the result is a 2-1-5 { 3KE— @—K)K | fs D being a constant which it is not necessary ae to specify. K and E are the elliptic functions of # usually so denoted. By what has been stated above the roots of 6,(x) are of the form ay (ite? ING IR 2 apes A OY so that S4m+in'K'/K}—?=4{3KE—(2—#?)K*}, . (41) the summation on the left being extended to all integral values of m and m’, except m=O, m'=0. This is the general solution for 2. If K’=K, #?=4, and y4{m-+im'}-?=2 {2KE—K?} = 7, since in general f, HK'+ E’K—KK’'=37. In proceeding further it is convenient to use the form in which an exponential factor is removed from the series. This is ee Ab Aba Atyt 0, (a) = Aze—24Be 4 Aas 3r + 53 BT 78 ae +... ih in which (42) meee ee a igo ee) Te 7 Tie. =, =a4B, s2=B(a’—2B'), s,=aB(a'—66%), the law of formation of s being S41 == 2in(2m + 1) B4sp_1 +48 dsm/dB—B8B*dsm/da, (44) * ‘Crelle,’ Bd. 54, p. 82. + Cayley’s ‘ Elliptic Functions,’ p. 49. 492 Lord Rayleigh on the Influence of Obstacles while a=k?—k?, B= Vv (kk'). + ape ee I have thought it worth while to quote these expres- sions, as they do not seem to be easily accessible; but I propose to apply them only to the case of square order, K’=K, ce= f=. ieee MB l/r, a=0,. BHl/s/2 sac eee S=PR, 3 =0, s=—28’, s3=0, s,=—368?, and A? At at ie 0, (2) = Teer 4 1 S ae + ote LH. - (47) Hence Cig Ne ee Aba? log yg Qn 2.5? — 16.35.51. If +2,, +A, ... are the roots of 0,(x)/z=0, we have —-2— z —4__ AS —6— —§ AS EN? = =, yr == po Wage! Seer =a Now by (40) the roots in question are 7 (m-+2m’), and thus or? ar? A8 S,=7, Se= go 4 Ss= 797 5T? (49) in which 2 12 1, 12.881, 12.82.52 1 A= —K=ltogt eat oreo gt =1°18034. Leaving the two-dimensional problem, I will now pass on to the case of a medium interrupted by spherical obstacles arranged.in. rectangular order. As before, we may suppose that the side of the rectangle in the direction of flow is a, the two others being B and y. The radius of the sphere is a. The course of the investigation runs so nearly parallel to that already given, that it will suffice to indicate some of the steps with brevity. In place of (1) and (2) we have the expansions V=Aot+ (Ayr + Byr-*?) Yy+ cee + (A,r + Boyt DY, eee ee V'=C0,4+ 0, Yyr+ eee + CnYnr”™ + oe ey Te (51) in Rectangular Order upon a Medium. 493 Y,, denoting the spherical surface harmonic of order n. And from the surface conditions V=V', vdV'/dn=dV/dn, we find a= a4 enti 6s (52) We must now consider the limitations to be imposed upon Y,. In general, B Mae = @;,(H,cos s6+ K,sinsd), . . (58) s=0 where 5 — cin’ gs G — (n—s) (n—s—1) N—S— Ee ©),=sin 0 (cos 7 ED (o7 iy COS 29 + ee i: (54) @ being supposed to be measured from the axis of x (parallel to «),and ¢ from the plane of xz. In the present application symmetry requires that s should be even, and that Y,, (except when n=0) should be reversed when (7—6@) is written for 0. Hence even values of n are to be excluded altogether. Further, no sines of sp are admissible. Thus we may take Bee ee ae Hn he it nw we (8D) Y,=cos? @— 3 cos 6+ H, sin?@ cos A cos 2h, . (56) Y;=cos°?9—10 cos? 0+ 3% cos 0 + L, sin? 4 (cos’ @—+ cos 6) cos 26 + Lijsin 7 cos O'eos' 2g e2 4) BY LID In the case where 8=y symmetry further requires that Ee es ee, eos ee CO) In applying Green’s theorem (4) the only difference is that we must now understand by s the area of the surface bounding the region of integration. If C denote the total current flowing across the faces By, V, the periodic difference of potential, the analogue of (6) is aC—ByV,+47B,=0. .... . (89) We suppose, as before, that the system of obstacles, ex- tended without limit in every direction, is yet infinitely more extended in the direction of « than in the directions of Band y. Phil. Mag. 8. 5. Vol. 84. No. 211. Dec. 1892. 2M 494 Lord Rayleigh on the Influence of Obstacles Then, if Hw be the potential due to the sources at infinity other than the spheres, V;= He, and C.. By ,— 47B ; aca aByH J? so that the specific conductivity of the compound medium parallel to « is AqrB, 2 1 ae aByH e rn) e e- e ° ° (60) We will now show how the ratio B,/H is to be calculated approximately, limiting ourselves, however, for the sake of simplicity to the case of cubic order, where 2=fB=y. The potential round P, viz. Act An Ae Ye may be regarded as due to Hx and to the other spheres Q acting as sources of potential. Thus, if we revert to rectan- gular coordinates and denote the coordinates of a point rela-~ tively to P by 2, y, z, and relatively to one of the Q’s by w', y', 2, we have Ao-+ (A, —H)x#+A3(a?—Bar?) +... gob gpl? 207 . v 2 ve =B,> 3 + Bs = —_ yy AT ja te ta . (61) -in which Si (yas > Vgc —> v= Z = ys z=z—6, if £,, € be the coordinates of Q referred to P. The left side of (61) is thus the expansion of the right in ascending powers of 2, y, z Accordingly, A,—H is found by taking d/dz of the right-hand member and then making 2, y, z vanish. In like manner 6A; will be found from the third differential coefficient. Now, at the origin, Bu ds d —& 7? dé vr’? dé p° pre ad dz in which pr2=E2 + 92 + 62. It will be observed that we start with a harmonic of order 1 and that the differentiation raises the order to 2. The law that each differentiation raises the order by unity is general ; and, so far as we shall proceed, the harmonics are all zonal, in Rectangular Order upon a Medium. 495. and may be expressed in the usual way as functions P,,(u) of p, where w=&/p. Thus 3 =—2p-* P, (u). In like manner, dw —3e'r? _ d B 3p? __ 8 17 Ha. Z THE p' ie p~° P(r), and a a’ a ad? E ir Peed a eee +P): The comparison of terms in (61) thus gives A,—H= —2B, xp? P,—8 Bs > Ome P,t+ olatis A,= —4B, 2p-* Py+.... or aie (62) Tn each of the quantities, such as Sp °P,, the summation is to be extended to all the points whose coordinates are of the form le, mea, Ne, where J, m, n are any set of integers, positive or negative, except 0,0,0. Ifwe take «=1, and denote the corresponding sums by Sz, Sy,...., these quantities will be purely nume- rical, and . Ropes: Soy lee aa BO jens: «: agp (OS) From (52) (62) we now obtain Ha 2+y7 eo ypoe l—y 7 a = a eo which with (60) gives the desired result for the conductivity of the medium. We now proceed to the calculation of 8. We have Lpitish i gMPot yy d (EY) 0 By the symmetry of a cubical arrangement, it follows that = (2/p?) = = (a°/p’) = &(8"/p") s so that if S were calculated for a space bounded by a cube, it would necessarily vanish. But for our purpose 8, is to be calculated over the space bounded by =+0, n=xv, 2M 2 496 Lord Rayleigh on the Influence of Obstacles ¢=-+v, where v is finally to be made infinite; and, as we have. just seen, we may exclude the space bounded by E=+v, n=+v, S=+0; so that 48, will be obtained from the space bounded by i, £-@, Ls te se Now when p is sufficiently great, the summation may be replaced by an integration ; thus s.=— {|| ( \aely ) a dnat. In this, ec’ ae v Ee are Crem ae cdg ye 2v? -o WAP HEE) +L) (27+ 7)” and finally a vdé eae 2d0 _» (P+ EL) (2? +E7)? Jo V (2+ tan’d) TF UNS ee uz / (2—87) Ate Thus owt 4 2 se If we neglect a'°/«", and write p for the ratio of volumes, viz. : Aaa 3 gage ee Ge ee ge eet ane (66) we have by (60) for the conductivity (2+¥)/(l—v)—2p (2+yv)/(lL—v) +p’ or in the particular case of non-conducting obstacles (v=0) 1—p oe In order to carry on the approximation we must calculate S, &. Not seeing any general analytical method, such as was available in the former problem, I have calculated an (67)* (68) * Compare Maxwell’s ‘ Electricity,’ § 314. in Rectangular Order upon a Medium. 497 approximate value of S, by direct summation from the formula 3d&* — 308? + 3% 8p* q We may limit ourselves to the consideration of positive and zero values of &,,¢€. Every term for which &, y, € are finite is repeated in each octant, that is 8 times. If one of the three coordinates vanish, the repetition is fourfold, and if two vanish, twofold. The following table contains the result for all points which lie within p?=18. This repetition in the case, for example, of p?=9 represents two kinds of composition. In the first p?=2? +274 2=9, Sp and in the second ps Ory 9. The success of the approximation is favoured by the fact that P vanishes when integrated over the complete sphere, so that the sum required is only a kind of residue depending upon the discontinuity of the summation. The result is poms eee ete ee MO) | | | | p> p° eet Ys ee ee ee 0, 0, 1 1 + 35000 | 0, 0, 38 9 | +4 -0144 or Od Ist 2 eS OS toy |) 22-0948 "da oe a 3 Eye }O9G0T || TAR SN |) Tae We s-0075 poo 2) 4 i 1094 2, 22) 12 | = 0062 | owt, 2. ''5 aol Gotan) is eS. Cols at e2° | 6 SO Osor eb Seon l4 Ve -009 22 | 8 =a 607 0.0, ae |) 16. be 4-004 192 | 19 = 0277 «||. 2, 2.3. |. 17. | = 0061 | WenOrple4: 19/17, lye 0085 The results of our investigation have been expressed for the sake of simplicity in electrical language as the con- ductivity of a compound medium, but they may now be applied to certain problems of vibration. The simplest of these is the problem of wave-motion in a gaseous medium obstructed by rigid and fixed cylinders or spheres. It is assumed that the wave-length is very great in comparison with the period (a, 8, y) of the structure. Under these cir- 498 Lord Rayleigh on the Influence of Obstacles cumstances the flow of gas round the obstacles follows the same law as that of electricity, and the kinetic energy of the motion is at once given by the expressions already obtained. In fact the kinetic energy corresponding to a given total flow is increased by the obstacles in the same proportion as the electrical resistances of the original problem, so that the influence of the obstacles is taken into account if we suppose that the density of the gas is increased in the above ratio of resistances. In the case of cylinders in square order (35), the ratio is approximately (1+ p)/(1—p), and in the case of spheres in cubic order by (68) it is approximately (1+3p)/(1—p). But this is not the only effect of the obstacles which we must take into account in considering the velocity of pro- pagation. The potential energy also undergoes a change. The space available for compression or rarefaction is now (1—p) only instead of 1; and in this proportion is increased the potential energy corresponding to a given accumulation of gas*. For cylindrical obstruction the square of the velocity of propagation is thus altered in the ratio 1 —p sf Sepp ede so that if ~ denote the refractive index, referred to that of the unobstructed medium as unity, we find pe=1+p, (w’—1)/p=constant, . .< u- |=)» eee which shows that a medium thus constituted would follow Newton’s law as to the relation between refraction and density of obstructing matter. The same law (70) obtains also in the case of spherical obstacles ; but reckoned abso- lutely the effect of spheres is only that of cylinders of halyed — density. It must be remembered, however, that while the velocity in the last case is the same in all directions, in the case of cylinders it is otherwise. For waves propagated parallel to the cylinders the velocity is uninfluenced by their presence. The medium containing the cylinders has there- fore some of the properties which we are accustomed to associate with double refraction, although here the refraction is necessarily single. To this point we shall presently return, but in the meantime it may be well to apply the formule to the more general case where the obstacles have the pro- perties of fluid, with finite density and compressibility. * ‘Theory of Sound,’ § 303. or in Rectangular Order upon a Medium. 499 To deduce the formula for the kinetic energy we have only to bear in mind that density corresponds to electrical resist- ance. Hence, by (26), if o denote the density of the cyiin- drical obstacle, that of the remainder of the medium being unity, the kinetic energy is altered by the obstacles in the approximate ratio (c+1)/(co—1) +p ePiienh—s sca pee The effect of this is the same as if the density of the whole medium were increased in the like ratio. The change in the potential energy depends upon the “compressibility” of the obstacles. If the material com- posing them resists compression m times as much as the re- mainder of the medium, the volume p counts only as p/m, and the whole space available may be reckcned as 1—p + p/m instead of 1. In this proportion is the potential energy of a given accumulation reduced. Accordingly, if u be the refrac- tive index as altered by the obstacles, Be wt Ly oC Dime Se ew alo) The compressibilities of all actual gases are nearly the same, so that if we suppose ourselves to be thus limited, we may set m=1, and | »_ (c41(o-1) +p. a GC) Cees ae at or, as it may also be written, a] me os es eonaiant..- a, «Clap +1 p ; In the case of spherical obstacles of density o we obtain in like manner (m=1), >. e+1)/(o—1)+p pm Geer 8 wi pete p or = constant, es. ee Ue In the general case, where m is arbitrary, the equation ex- pressing p in terms of pw? is a quadratic, and there are no simple formule analogous to (74) and (76). lt must not be forgotten that the application of these formulze is limited to moderately small values of p. If it be 500 Lord Rayleigh on the Influence of Obstacles desired to push the application as far as possible, we must employ closer approximations to (26) &c. It may be re- marked that however far we may go in this direction, the final formula will always give yp? explicitly as a function of p. For example, in the case of rigid cylindrical obstacles, we have from (85) : bale 1+p—'3058p*+ .. =P) {0g It will be evident that results such as these afford no foundation for a theory by which the refractive properties of a mixture are to be deduced by addition from the corre- sponding properties of the components. Such theories re- quire formule in which p occurs in the first power only, as m (76). If the obstacles are themselves elongated, or, even though their form be spherical, if they are disposed in a rectangular order which is not cubic, the velocity of wave-propagation becomes a function of the direction of the wave-normal. As in Optics, we may regard the character of the refraction as determined by the form of the wave-surface. The eolotropy of the structure will not introduce any cor- responding property into the potential energy, which depends only upon the volumes and compressibilities concerned. The present question, therefore, reduces itself to the consideration of the kinetic energy as influenced by the direction of wave- propagation.. And this, as we have seen, is a matter of the electrical resistance of certain compound conductors, on the supposition, which we continue to make, that the wave- length is very large in comparison with the periods of the structure. The theory of electrical conduction in general has been treated by Maxwell (‘Hlectricity” § 297). A parallel treatment of the present question shows that in all eases it is possible to assign a system of principal axes, having the property that if the wave-normal coincide with any one of them the direction of flow will also le in the same direction, whereas in general there would be a diver- gence. To each principal axis corresponds an efficient “ den- sity,” and the equations of motion, applicable to the medium in the gross, take the form f 3 are dd d’n dé ae dé ei? eet, dP. a hey? ia ee where &, 7, ¢ are the displacements parallel to the axes, m, is the compressibility, and olde dap de" in Rectangular Order upon a Medium. 5OL If X, w, v are the direction-cosines of the displacement, l, m, n of the wave-normal, we may take E=20, n=pw0, C= v8, é= eet my+nz—Ve) where Thus 7 dd/dz = — 1l0(iXN+mp-+nyv), &e., and the equations become o AV? = ml(iXN+mpt+nv), ouV* = mm(lA+mpt+nyv), ovV? = mn(Ir+mp+nv), from which, on elimination of \ : w: v, we get ee mt ey mal + Bint en? (78 pea BGs =a ym’ +n, . ) if a, b, ¢ denote the velocities in the principal directions eee The wave-surface after unit time is accordingly the ellip- soid whose axes are a, b, e. 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HOOD HO ae | siete | 100 199 ie ®) os H 0) HnO09 HO |loqssorg nS 1) | "100 of: 0 en (0) *HOOO0*HO OH || "lous jo a a 2 "HO ~—s ||: *H®0000°HO ‘soinqeroduray, poarosq() pur poyrpno[ry uoomyoq soouoroyiqT “TT Tey I, Determination of Low Temperatures. 515 It is obvious, however, that by taking a smaller range of pressure—say from the lowest to 2000 millim., or from 4000 millim. to the highest pressure, and by altering the constants, a very much better agreement would be obtained. Hxtra- polation beyond the limits of pressure chosen would, how- ever, introduce very large errors indeed. In conclusion it may, I think, be stated that in a great number of cases, including pairs of widely different bodies, the formula of Ramsay and Young gives better results than that of M. Colot; the latter, which is very convenient, may, however, be employed, even for very wide ranges of pressure, when the constant B is small, and for small ranges of pressure when B is large. LX. Note on the Determination of Low Temperatures by Platinum-Thermometers. By HK. H. Grirrirus, .A., Assistant Lecturer at Sidney College, and G. M. CuarK, B.A., Sidney College”. N connexion with Profs. Dewar and Fleming’s recent work on the resistance of certain metals and alloys at very low temperatures, and the suggestion that they have made—viz. that the resistance of certain pure metals (amongst which is platinum) vanishes at absolute zero,—the following facts may be of interest. Being in the possession of the constants of several platinum- _ thermometers, whose accuracy has been exposed to severe tests, we have calculated the temperature at which R=0,—by assuming the possibility of applying Callendar and Griffith’s method—from the formulsze _ R-R and gel: ete i—pt=34 my ay where R,, Ry are the resistances in steam under normal pressure, and melting ice, respectively, and R is the resistance at temperature ¢; the value of 6 tor each thermometer is determined by observations of the resistance in boiling sulphur (¢=444°°53). | The following table gives the constants and results :— ‘* Communicated by R. T. Glazebrook, M.A., having been read at the meeting of the Cambridge Philosophical Society, October 31, 1892. 516 Messrs. Griffiths and Clark on the Determination of ae | Bou. Thermo: Reference. R,- R,/Ry | 6 pt. LAN ges meter. Oiilcadar’s ae E nies 50845 | 1:3460 | 1-46 |—289-04|—274-12 Phil. Trans. A. 1891, t . 4. Y é Nats. |{ op ISL, 188. \ 9-8558 | 1:3484 1-638 |—287-03 |—270°60 Na. i - — 59865 | 13482 1-648 | — 287-19 | —269°60 Nz. : i 88749 | 13480 | 1-639 |—287-34/ 27086. M,. eee ae 42967 | 13381 | 157 |—295-81 |—279-15 M,. ; : 4:1732 | 1:3383 | 1:57 |—295-58|—278-97 H. ies mk ae aa 185216 | 13463 | 1-474 |—288-77!-278-71 The actual numbers used in these calculations are those taken from the papers referred to in column 2. All the other thermometers mentioned by Griffiths (Phil. Trans. A. 1891, pp. 43-72) were made with single electrodes for rapid observations ; and thus every observation includes their stem-resistance, which evidently cannot become zero, so that the above investigation with their constants would be of no use. The above table seems to corroborate the conclusions arrived at by Profs. Dewar and Fleming, and at the same time is a valuable testimony to the accuracy of the method adopted by Callendar and Griffiths in the papers referred to, for it did not appear probable that their formule could bear the strain of extrapolating over a range of nearly 300°; this is more especially the case when it is remembered that the wires used have different resistance coefficients and different values for 8; the origin of these differences is probably to be found in slight impurities in the platinum. We have reason to believe, from some recent experiments conducted by us, that the rise in temperature of the wire, caused by the current necessary to determine its resistance, is greater than is usually supposed. If the difference of potential at the ends of the wire is kept constant, the effect of the error thus introduced is to make the absolute zero too low; * Cf. with the value given by Joule and Thomson, —273°°7, Proc. Roy, Soe. vol. x. p. 502. Low Temperatures by Platinum-Thermometers. 517 if, however, the current is kept constant, the absolute zero works out too high. In the determination of the con- stants of thermometers N, a rough attempt was made to keep the current constant; but in the remaining thermo- meters no alteration was made in the electromotive force as the resistance of the coil, and therefore of the bridge, increased. It appears probable that some of the discrepancies in the above table are due to this cause. A simple method of graduating platinum-thermometers is thus suggested. Assuming, as we are fully entitled to do, that the curve t—pé is a parabola, three points only are necessary for its complete determination : the points hitherto adopted have been 0°, 100°, and 444°°5. The necessity of guarding against loss by radiation, gain by superheating, &c., when determining the resistance in sulphur, renders it a somewhat troublesome operation to those observers who are not provided with the necessary apparatus. In cases where a high order of accuracy is not a sine qué non, and where the platinum is known to be fairly pure, we may assume that when ¢=—273°7 then R=O: therefore, if R, and Ry are determined by direct observations in steam (760 millim.) and melting ice, the instrument may be considered as graduated, since — —273°7— pt 273-7 , 273-7) 100 100 =o tame es re As an example of the order of accuracy which could be thus obtained, we append the following table, the first column giving the true temperature, the others the error introduced by graduating the various thermometers by the suggested method. where x 100. | | temp, (Callendar’s| Natz.| Na | No. ee |e, ema ——_ | |] — |] —_— | —_—_ i lati) 0 0 0 0 0 0 50 OL | +08 | +08 | +08) 0 | —14 | —14 100 || 0 0 0 ( 7 B 0 | 4-44 | 4-43 150 +03 | —25 | ~-25 | —23 | 518 ~ Prof. F. Y. Edgeworth on the Law Although the above discrepancies may in some cases appear large, it must be remembered that we ave here the whole of the error in each ease, and that such errors as non-uniform bore, zero chan ges (both temporary and permanent), unequal graduation, changes of condition, sticking, &c. are totally absent, while the experimental difficulties of the air-thermo- meter are avoided, Thus we conclude, thata platinum-thermometer—especially of the H pattern *—would be a convenient instrument for the determination of very low temperatures where an error of O°'5 is immaterial; and that further light on platinum- thermometry only strengthens our belief in its correctness over the range of —273° to + 700°. Sidney College, Cambridge. October 1892; _ ——_—___— - . ~ —s. blinds =. er te * tetas LXI. The Law of Error and Correlated Averages. By Professor F. Y, Epgeworru, W.A., D.C.L. (Continued from p, 4388. | IV. ¢ VONTINUING the investigation of the conditions under which the law of error is fulfilled, we come now to the general case in which every. member of a group is any function of a number of elements, each, us before, oscillating, according to any independent lajw of frequency, through a range which is relatively small. Let the function be F(a, a, Ge. &,). Put, for v,, x, +&, 3 for wa, xo t&+ &e.f, where x, Xo, &e. are the average values of the corresponding variables. By d ad expansion—putting FY’, I',', &e. for ap KF, 7 I’, &e.—we have ¢ vy Ws F (vy, ve, Ke.) = F(x, xo, We.) + Ey Fy! (x), xo, dee.) + Ea Fy'( x) ,X0,e.) + Ke. ; forming a linear function of &), &, &e.; if the terms containing higher powers of &,, & may be neglected. This may, in general, be done, provided that F(x, v, &e.) is free from * A full description of the construction of H will be found in the British Association Report on Electrical Standards, 1890, reprinted in the ‘ Electrical Review, No. 670, p. 863; and a short account on p. 153, Phil. Trans. A. vol. elxxxii. 1891. There is some inelegance in using € to denote here a deviation which 8 8 is of the order L and ina former passage a deviation which is of the 1 re : order —= (ante, p. 431). But as the subject involves three orders of n magnitude, it is not very easy to distinguish them by differences of type. of Error and Correlated Averages. 519 singularity and continuous in the sense before defined *, in the neighbourhood with which we are concerned: throughout the set of values formed by assigning to each of the variables yy Xe, Ke. all the values of which it is capable. {xamples of this theory may be seen in all the important cases in which the law of error is fulfilled : namely, errors of observation (proper), gunnery, and natural groups generally; in all which departments the fulfilment of the law is doubtless due to the independent action of numerous small agencies. The apparent number of exceptions is much reduced by the consideration that the rule still holds good if we substitute for &, in the above statement, any function of §&,, say &, which does not become infinite for any of the values assumed by &; and make similar substitutions ft for &, &, &e. Hor example, the law of error will be obeyed by the varying values of B(x, + 01, Xo +82, Xc.), where CE, be’, So Eaty; wrovided that Fy!(x,), Iy'(x.), &e. neither vanish, nor become 1 B79. = 9\%8/) ’ infinite ; and Jy (ay), bg9/’(v,), &e. are not very large §, Nor, even thus construed, are all the cases in which I*\/(x;), I,/(xg), Ke. all vanish genuine exceptions. When the first term of the expansion vanishes, the function in general reduces to the second term, which is of the form Why’ (Xq Xb Ke.) x O° + Waa!" (X1y XQ) Ke.) x Co" + Ke, } 2h 19 (X41, Xo, Ke. ) x 183 -+ Xa, ‘ Which in general may be reduced toa sum of n squares of the form 2+ 92+ &e. ; where each » is a linear function of the &s. ‘The systems of the »’sand the @s being thus related, it is allowable to regard the y’s as the independent, and the €’s as the dependent, variables. Accordingly the original function reduces to a sum of numerous elements independently oscillating. Which is the essential condition for the fulfilment of the law of error. ‘lor exaniple, if for Il’ we have (ay —X,)* + (@g— XQ)? + Ke, the first term of the expansion vanishes for the point O, = xj, We xs, ue.: * Ante, p. 481. { By a proper change in the law of frequency for the elements, this case may be reduced to the preceding, { The elemental laws of frequency and the functions ¢ may both be unsymmetrical, § Ante, p. 431. 520 Prof. F. Y. Edgeworth on the Law but the second forms a sum of squares ; the particular values of which sum are grouped around its average value according to the law of error. The genuine exceptions arise when the compound function cannot be reduced to a number of independently fluctuating elements : for example, F(a, X25 &c.) = (2, —X;) x (&g—Xe) Xx Xe. x (%n—Xn), a continued product of residuals ; or a continued fraction such as ]+1 + 1+(x,;—2,) + j—+1 + (x; —22) + Ke. The simplest class of exceptions seems to be that which has been already instanced *; where the compound function involves the elements only implicitly, through a linear function of those variables ; and the first differential becomes zero or infinite for the average value of the linear function. The function is thus of the form F(X-+€), where & fluctuates in conformity with a probability-curve about the average value zero, and F’(X) is either zero or infinite. This class may be subdivided according as one or other of the last-named conditions is fulfilled. If F’(X) is zero, X is a point of maximum, or at least infinite, frequency ; if F’(X) is infinite, X is a point of zero frequency. This is seen by puting HEX 4H, ESE — Then the law of frequency for w is : Error-function [F_,(u)] x > F_,(u). But d e 1 dat) = weK se)’ Whence the proposition. Simple examples are (a—X)’ and 2—X; which take the forms & and V respectively, when for « we put X + &, The law of frequency for w=§? is T, as already stated, bud athlete / Jae 27 y ~ The case of u=& is analogous. Fig. 2 is designed to re- present this type. When w= ¥€, it should seem that only half the original curve is reproduced in the transformation; the other half becoming impossible. The transformed curve is Dak apes SS sce eo. Y / 1 3 * Ante, p. 436. + Ibid, of Error and Correlated Averages. 521 having two branches ; between which the values of v +é should be equally distributed. The curve has two maximum Fig. 2. ¥ 0 x . Cc . . points w= rw 9? and is concave from zero up to a distance from the centre, on either side, 2 times as great as that of the maximum-point ; after which the curve becomes convex. As the maximum-point is near the Median of the corresponding limb, which is /°476...c (corresponding to £=+°476...c, which is the quartile of the original curve), and the curve is concave about that central region, its general appearance will not differ very sensibly from that of the Probability-curve. The result will not be very different if for the square root be Fig. 3. substituted the tth root. Fig. 3 is designed to illustrate this type. The probable errors, both of the original and the 522 Prof. F. Y. Edgeworth on the Law transformed curve, are in the neighbourhood of the maximum- points Q;, Qe. ae a _ The question may now be raised: whether one of these types—especially the more distinctive one, where F’(X) =0— is likely to be realized in the (exceptional) case of the action of numerous independent agencies not resulting in the pro- bability-curve. The following problem may have wide ana- logies. Let a body be moved from rest in a right line by the simultaneous action of numerous small impulses, each as likely to be in the positive as in the negative direction. The law of frequency for the momentum of the body will be the proba- bility-curve ; for the energy of the body is the curve represented in fig. 2. V. So far we have supposed the “elements” to assume different values in virtue each of a single variable x, or 2». Let us now generalize this supposition by supposing that each element fluctuates according to a law of frequency which in- volves two (or more) variables. Thus let the rth element, / , vary _ according to the law of frequency z=/,(a, y); it is required to determine approximately the law of frequency for F((,, /;, &e.), where F is a function which fulfils the conditions above - prescribed. Now the law of frequency for J,, given in terms of a and y, is capable of being expressed in terms of w only. For the (proportionate) number of the values of /,-which pertain to a particular value of 2 (which occur between 2 and «+ Az) are obtained by integrating between extreme limits with regard to y, while w is constant. Thus the law of frequency for lL, may be written :— = {Mewar if we put b,, 8,, for the limits of y in the rth element. But, by the preceding section, if 1,, l, &. each vary ac- cording to a law of frequency involving 2 only, F(d,, d,, &c.) is approximately an error-function in «#: that is, of the form He-*”. Now the function in «2 of which this form is an approximation should be obtained by integrating F with regard to y between extreme limits, 2 being treated as constant. And by parity a similar conclusion is obtained by integrating with respect to z,—y being treated as constant— on the one hand the elemental functions and on the other hand the compound function F. Thus F is approximately such a function that if it is integrated with regard to y, between ex- treme limits, « being treated as constant, the result is approxi- mately of the form He-*” ; and if itis integrated with regard of Error and Correlated Averages. 523 to x between extreme limits, y being treated as constant, the . . a , = nD 2 rn . e,e . result is approximately of the form Ke~*”, This condition is fulfilled by functions of the form Je—®, where R is a quadratic function of « and y*. By parity it may be presumed that, if the law of frequency for each element involve three or more variables wv, y, 2, the law of frequency for the compound is Je—®, where R is a quantic of the second order. Assuming the appropriateness of this form, I have in a former articlet shown how to calculate the coefficients from the measurements of correlated organs. I now justify that assumption. , The general theorem, besides its direct applications to the measurement of organisms, has other interesting consequences. We have here a theoretical ground for the practical use of the * The @ priort demonstration that this form should prevail is con- firmed, in the case of correlated organs, subject to a number of independ- ent influences, by @ posterior? verifications, such as that which was given in a former paper (Phil. Mag. Aug. 1892, p. 197); and the following. If the assumed form prevail, we should expect that p,,, the coefficient of correlation obtained by assigning values of one organ « and observing the corresponding values of another organ y (op. et. p. 191) should be equal to p, obtained by assigning values of y and observing the corresponding values of vw. ‘That this condition is very nearly fulfilled by the measure- ments of three hundred men supplied to me by Mr. Galton appears from the following results calculated by Mrs. Bryant, D.Sc. ; p,., p.1, &c., having the meaning just explained ; and the suffixes 1, 2, 3 re‘erring respectively to stature, cubit-length, and knee-height. Pi= "79 Po, = "81 Pri='93 Porss'83 Pxo=e'Ol Bug iui The correspondence is as close as, in view of the probable error (incident to the limited number of observations), was to be expected. The results obtained from the positive and negative deviations from the average, results which should be equal, in theory, are in fact as follows :— From aoae pi AS, Deviation. Deyiation. | suche el ee Sie heals ges ai ee, ee ee The analogous verifications obtained by Mr. Galton (Proc. Roy. Soe. 1886 and 1888) and Prof. Weldon (7did. 1892) are to be added. + “On Correlated Averages,” Phil. Mag. Aug. 1892. 524 Prof. F. Y. Edgeworth on the Law method of least squares in the reduction of observations re- lating to several variables. For let it be granted that all errors-of-observation are in general effects of numerous small agencies—a proposition both & priort probable and verified in the case of observations relating to one quantity by copious experience; then it follows from the reasoning of this section that the law of frequency for the errors of two (or more) quantities is of the form Je~*, where R is a quantic of the second degree, which can be put in the form of a sum of two (or, as the case may be, more) squares, such as A(#—x)?+By—y)’s and then it follows, as Merriman* and others have reasoned, that the method of least squares is the solution of a maximum- problem: to determine the origin x, y from which the observed system of values have most probably emanated. Again, the extended law of frequency presents on a larger scale the spectacle of order emerging from chaos which is © formed by the evolution of one universal law in the com- pound from any particular law in the elements. There is a difference, too, in quality as well as in scale. When we con- sider elements involving one variable only, it is only with regard to laws of frequency that the melting of discord into rhythm is observed ; but in the case of elements involving two or more variables, that phenomenon is observed with respect to statistical laws generally. Whatever, in the ele- ment, is the function connecting « with the y which on an average corresponds thereto ; in the compound, generally, the relation between zw and y will be linear. For example, let the law of frequency for each element form such a cubic surface that the ordinate which on an average corresponds to any assigned w is given by a parabola y=Aa*; yet in the com- pound the correlation between x and y is linear, of the form y—Y=r (e—X); where X and Y are the average values of x and y (in the compound), and 7 is the coefficient of correla- tion which Mr. Galton has made familiar t. By parity in the case of several variables, however complicated the laws connecting the variables in the elements, the correlation be- tween the composite organs is of the form a(a—X)+h(y—Y) +9(e—Z) ; where a, h, g, &. are the coefficients investigated by the present writer in a former number of this Journal {, It is not to be supposed that this conclusion amounts to no _* Method of Least Squares,’ arts. 41, 47. + Proc. Roy. Soc. vol. xlv. p. 140, et seg. ¢ August 1892, of Error and Correlated Averages. 525 more than what is deducible from Taylor’s theorem, because for short distances from the maximum of F(z, y) (the surface of frequency for x and y) the value of y most likely to attend any assigned value of # may, in general, be regarded as a linear function of w One might as well attribute to Taylor’s theorem the fulfilment of the law of error in the case of a single variable, on the ground tuat any continuous curve symmetrical about a certain point may be identified for short distances from that point with a probability-curve. That consideration alone would not have enabled us to predict the fulfilment of the law up to a sensible distance from the centre—that exact consilience between the quartiles, octiles, deciles, &c. of theory and of fact which has heen observed by Mr. Galton and others. Those verifications allow us to pre- sume the existence zn rerum natura of elementary agencies in such numbers and of such magnitude as to generate a proba- bility-curve of sensible dimensions. From that presumption we may reason down to the genesis of a probability-surface also of sensible dimensions. And this conclusion is confirmed by specific experience: the sort of correlation which is dedu- cible from that hypothesis having been, in fact, observed by Mr. Galton and Prof. Weldon. ‘There is required here not only the use of Taylor’s theorem, but also the presence of the essential condition of the law of error; the action of numerous independently variable elements. In fine, the theorem of this section completes the answer which was given in a former section to the objection brought by Cournot and Prof. Westergaard against Quetelet’s prin- ciple of the Mean Man; in substance, that the mean organs obtained by averaging a number of specimens might not fit each other. It is now seen that the objection in the form in which it has been stated * implies a very rudimentary con- ception of “ correlation,” viz. that to assigned values of some organs the corresponding value of another organ is i each particular instance and not merely on an average a function of those assigned values. This sort of exact correlation probably exists in natural history only with regard to those geometrical relations between lines and angles which the objectors indi- cate ft. The objection in this form has been answered in the * Ante, p. 437. | + Their instance, it will be remembered, was: two sides of a right- angled triangle and the base. As right angles are not very prevalent in organic nature, a happier instance might be: for three “organs,” two sides a and 6 and the contained angle C; and, for the correlated organ, the base ¢; which = ¥ a?+0?—2abcosC. A concrete example of such correlation is the triangle formed by the breasts and the throat (Quetelet, Anthropométrie, p. 224). Phil. Mag. 8. 5. Vol, 34. No. 211. Dee, 1892. 20 526 Law of Error and Correlated Averages. last section. If the objection is understood in a less narrow sense as suggesting that antinomies arise when correlations other than linear are supposed, the answer furnished by this section is that correlations other than linear are not to be supposed (in the case of composite organs). VI. In the investigation of the most general conditions of the law of error, one more step has still to be taken. We have so far supposed every member of a group to be a definite function of a number of elements, each of which varies according to some definite law of frequency. But now it is to be added that the conclusion is not seriously affected if we. suppose these functions to be themselves variable ; provided that they preserve certain characteristics approximately constant, namely, the parameters of the probability-curve. These are its centre and its dispersion :. or P1X1 t+ poXo + Ke. +p,x, (=X), prey" + proce” + Ke. + p,,76,7(=C’), where py, 2, &e. are “ weights ” (identical with the Fy? (x1, X2, &e.), Fy (x1, X2, &e.), employed before) ; x,, X:, &c. are the average values of the elements as before ; ¢”, ¢”, &c. are each the modulus squared for the corresponding elements. Now these parameters will not be seriously affected if we suppose each of the quantities x and c’ to be vilea ia about an average value. For the centre will then become changed from VPkS SpAx )\. X to X(1 Fine which differs from X by a percentage of the order S supposing the relative fluctuation of each x to be the same as that which according to the usual theory is ascribed to X, namely, of the order —., For, if Aw is of the order x+ Wn, SpAew (by the usual theory) will be of the order /n x (x+ Vn) or x; and x is of the order X~n. By parity of reasoning, if C4", C9’, &e., each fluctuate over a relatively small range, the value of CO? will not be seriously affected. In fine, the result will still remain unaffected if we attri- bute to each p a relatively small variation Ap; corresponding to a variation of 2Ap on p’. So loose are the conditions which suffice for the fulfilment of the law of error. [ 527 7 LXIL. Notices respecting New Books. A Treatise on Analytical Statics, with numerous ewamples. By E. J. Rouru, F.R.S. Vol. I. (Cambridge University Press, 1892: pp. xii+ 224.) RADERS of Dr. Routh’s first volume were told in the preface to that work thet he “felt that such subjects as Attractions, Astatics, and the Bending of Rods could not be adequately treated at the end of a treatise without making the volume too bulky or requiring the other parts to be unduly curtailed. These remaining portions will appear in a second volume.” No long delay has intervened, and the book before us treats the above-named subjects in the author's usual clear and graphic manner and completes, we presume, what Dr. Routh proposes to publish on the macter of Statics. The main portion of the work (126 pages) is devoied to the interesting branch of Attractions. As the arthor points out, “The reasons for believing the truth of the general law,” which is the fundamental one in the subject, “cannot be properly ex- plained until the reader has advanced some way in the study of Dynamics. At the same time a large number of theorems which are independent of all dynamical considerations follow from this law. Experience has shown that it is important for the student to acquire an early acquaintance with these results, as he cannot prosecute his studies in the higher Dynamics without their assist- ance. It has therefore been found advantageous to study the attractions of bodies as a part of Statics. For this purpose we assume the truth of the law of attraction as a working hypothesis and postpone its verification as a law of nature uutil the studeut has read Dynamics.” It is superfluous to examine in any detail such an exhaustive account as this is, seeing that every student of the subject, if he has not enjoyed a viva voce exposition of it from the author, should strive to master its convenis. All previous writers seem to have been laid uader contribution and, with scrupulous care, each important result is assigned to its fixst discoverer. This citation of original authorities we have, in previous notices of Dr. Routh’s works, had occasion to commend. A further excellent feature in the present case is the full Index at the end in addition to a fairly exhaustive table of contents. The object of the chapter on the Bending of Rods is “to discass the stretching, bending, and torsion of a thin rod or wire.” It closes with a section on Rods in three dimensions. The final chapter, on Astatics, opens with an historical note which states that the subject seems to have been first studied by Moebius, in his Lehrbuch der Statik, 1837. Indexes are furnished to the last two chapters, and the whole of the text bears evidence to the extreme care with which it has been drawnup. This is to be accounted for on the ground that both volumes are the outcome of many years’ lectures (see preface to vol. i.). [528 J LXIII. Intelligence and Miscellaneous Articles. THE EFFECTS OF SELF-INDUCTION AND DISTRIBUTED STATIC CAPACITY IN A CONDUCTOR. BY FREDERICK BEDELL, PH.D., AND ALBERT C. CREHORE, PH.D. HE solution obtained by Sir Wm. Thomson for the variation of the current and the potential at different points in a conductor possessing static capacity is given by Mascart and Joubert, LP’ Hlectricité et Le Magnétisme, vol. i. § 233, and is treated at length by Mr. T. H. Blakesley in his book on Alternating Currents. The object of the present communicavion is to give the solution for the case of a conductor possessing self-induction as well as distributed capacity, and to note the effects produced by the intro- duction of the self-induction. The rate of change of the charge on an element of the cable is equal to the difference of the currents flowing into and out from it ; and so, writing g for charge and 7 for current at any time, and w for the distance of any point from the origin—positive direction being that of current-flow—we have dq di ay. = — ae dx. If ¢ is the potential of an element, its charge is g=Ceda, where C denotes the capacity per unit length of the cable, and the first equation may be written oe epee e e ° e ° e ° . (1) By Ohm’s law the current in an element is equal to the total E.M.F. (the sum of the impressed and that of self-induction) divided by the resistance; and, if R is the resistance per unit length and we assume the back E.M.F. of self-induction per unit length to be equal to the rate of change of the current multiplied by a constant L, we may write Rae os ce In some cases this assumption may approximately represent the true effect of self-induction, and the results obtained from this particular assumption may show the nature of the effect of self- induction even in cases where the assumption is not justifiable. The differential equation for potential is obtained by eliminating 2 from (1) and (2), and is We de de qe tC za RCT =0. The current equation, obtained in the same way, is similar when 7 is written instead of e. Intelligence and Miscellaneous Articles. 529 The general solution of these equations is R t— Ckz)s e= Dhke Ct h, k __ 2 __¢—Ckz) i= Dhe tL h, k where ¢ is the Naperian base, and / and k are constants to be determined. If the impressed E.M.F. is harmonic, and at the origin e=Esin wt, where w denotes angular velocity, the solution for the potential at any point of the conductor at any time becomes e=He*”’ sin (wt-av). a non The solution for the current at any time across any section of the conductor is ye . —EV Cw a eed = ane -P) ars sin (attaw+ tan Py aa (4) In these equations Im. denotes the impedance, (R?-+ L’w)’ ; The solutions in equations (3) and (4) show that the potential and current are propagated in harmonic waves whose amplitudes decrease with the distance from the origin according to a loga- rithmic decrement. At any point of the conductor the potential and current vary as simple harmonic functions of the time with constant amplitudes which are different for every point of the conductor. The current wave is propagated in advance of the potential wave by an angle @ such that tan 0= = This phase- difference diminishes with increase of frequency when there 1s self- induction, but becomes a constant angle of 45° when L=0. The wave- me length is = and the rate of propagation is =. The wave-length and rate of propagation each become less as the self-induction increases. The wave of higher frequency will have the shorter length and be propagated the faster. This difference in rate of propagation of waves ot different frequencies is most marked when there is no self-induction. 1 The distance at which the amplitude decreases to —th of its r exe i ? value is 5 Tapani! the time for the decrease is alta The rate of decay is most rapid when there is no self-induction. The waves of higher frequency decay more rapidly than those of lower frequency ; when there is no self-induction this difference in the rate of decay is the greatest. 530 Intelligence and Miscellaneous Articles. The difference in the rates of propagation and decay of waves of high and low frequency doubtless constitutes the limitations to the use of the telephone. As the several harmonic components of a complex tone advance along a conducior, they keep shifting their relative phases according to the difference in their rates of propa- gation and also change their relative intensities according to the difference in their rates of decay, thus changing the resultant combination tone ai d materially altering its quality. These effects are always present in circuits containing distributed static capacity but are not so marked when there is no self-induction.—Szlliman’s Journal, November 1892, Physical Laboratory of Cornell University, July 1892. EXPERIMENTS ON THE CONDUCTIVITY OF INSULATING BODIES. BY M. EDOUARD BRANLY, M.D. Insulating bodies in a very thin layer seem apt to become conducting, and their conductivity presents some peculiar characters, an idea of which is given by the following experiment. Form a circuit containing a Daniell cell, an ordinary galvano- meter, and fine metallic filings contained in a- tube of glass or ebonile, between two metallic rods acting as electrodes. The file-dust opposes to the current of the Daniell cell an enormous resistance, often over ten million ohms, so that the needle of the galvanometer remains in equilibrium at zero. The thing being thus, if one produces in that circuit a sudden displacement of electricity at high potential, whether by exciting at a distance a spark of a Leyden jar, or simply by putting for a few moments a point of the circuit in contact with one of the poles of a battery of one or two hundred cells, the file-dust will easily convey the current of the Daniell cell; its resistance is lowered to some hundred ohms or even less, and such conductivity persists during many hours, or even many days. It is immediately and completely suppressed by a very slight shock on the tube containing the filings. In order to show that the shock has not suppressed the con- ductivity by a modification in the relative positions of the molecules, one may take instead of the tube with metallic filmgs another tube containing a carefully made mixture of resin and metallic dust, in proper proportions, amalgamated at the temperature of fusion of the resin. Such a mixture, pasty when hot, becomes solid and extremely hard when cold. Such a tube, which presents an infinice resistance to the current of a single Daniell cell, becomes conducting under the same circumstances as the metallic dust with intervals of air ; and its conductivity disappears also undera slight shock. You may thus reproduce and suppress it alternately as often as you wish. These experiments may be diversified in a great number of ways.—Comptes rendus de l Académie des Sciences, 24 Nov. 1890 and 12 Jan. 1891; Bulletin dela Socété internationale d électriciens, no. 78, May 1891. Communicated by the Author. Intelligence and Miscellaneous Articles. 531 ON THE THEORY OF MAGNETIZATION. BY PROF. DR. WASSMUTH. The lecturer first of all develops two new formule for Poisson’s theory of magnetization by means of an equation of Green’s. If k is the magnetizing number, V the inducing, Q the induced, and V+Q=¢ the total potential, he obtains the equations 1 ° ad _ (4b Qat [Qe itn | or dnj and a Q; + 4 thd; =", os bol ds, dn, which formule may in some cases be advantageously employed along with the usual one, It is thus shown how simply all the known methods of approxima- tion for the problem of magnetization (Beer, C. Neamann, Riecke) may be deduced from these equations, and therefore also the corresponding series. It is next investigated in how far the introduction of the idea magnetic resistance,” at any rate in a special case, is justifiable. With this view is treated the magnetization of an incomplete ring, one, that is to say, which has an air-break, and is wound uni- formly and completely with wire. By the aid of a strongly con- verging series, the following expression is obtained for the number N of lines of force with a close approximation, . AnSz i 2rk l\ X where ¢ is the strength of the current, s the number of all the windings, / the mean length and R’z the section of the ring, A the length and R’z the section of the air-break, and Kh =1+47k the coefficient of magnetic permeability. In the denominator there is actually along with the magnetic resistance of iron a sum which is proportional to the ratio ‘length: section of the air-break,” and which can be taken as magnetic resistance of the air-break. But this calculation shows us also that the factor with which this ratio pepappoars multiplied differs from unity, by which what is T called *‘ the dispersion of magnetic lines of force” finds an exp!ana- tion.— Abstract of a Lecture given to the Natur. med. Verein in Innsbruck. Communicated by the Author. - | fi. os2° | INDEX to VOL. XXXIV. AIR-THERMOMETRY, apparatus for, 2. Alloys, on the electrical resistance of, at the boiling-point of oxygen, 326. Alum, on the absorption of radiant heat by, 141. Andesites of Devonshire, on the, 385. Arons (L.) on electrolytic polari- zation, 140. Auroras observed at Godthaab, on, 148. Averages, on correlated, 190, 429, 518. Barus (C.) on the measurement of high temperature, 1; on the thermoelectrics of platinum-iri- dium and platinum-rhodium, 376, Basalts of Devonshire, on the, 385. Basset (A. B.) on the theory of the collapse of boiler-flues, 221. Bedell (Dr. F.) on the equivalent resistance, self-induction, and capa- city of parallel circuits with har- monic impressed electromotive force, 271; on the effects of self- induction and distributed static capacity in a conductor, 528. Boiler-flues, on the theory of the collapse of, 221. Boiling-points of different liquids at equal pressures, on the, 510. Bonney (Prof. T.G.) on the so-called Gneiss of Carboniferous age at Guttannen, 138. Books, new :—Hunt’s Systematic Mineralogy, 210; Basset’s Phy- sical Optics, 213, 386; Houston’s Dictionary of Electrical Words, Terms, and Phrases, 215; Flet- cher’s Optical Indicatrix, 217 ; Mendeleetf’s Principles of Chemis- try, 301; Violle’s Cours de Phy- sique, 381; Nietzki’s Chemistry of the Organic Dyestuffs, 448 ; Lodge’s Lightning Conductors and Lightning Guards, 451; Hull’s Volcanoes, Past and Present, 452; Macfarlane’s Principles of the Al- gebra of Physics, 453; Watts’s Dic- tionary of Chemistry, 454; Routh’s Analytical Statics, 527. Bosanquet (R. H. M.) on the illu- niinating-power of hydrocarbons, 120, 355. Bragg (Prof. W. H.) on the ‘elastic medium ” method of treating elec- trostatic theorems, 18. Branly (Dr. E.) on the conductivity of insulating bodies, 530. Breath figures, on, 180. Browne (R. G. M.) on the precipi- tation of sea-borne sediment, 136. Butler (G. W.) on the lithophyses in the obsidian of Lipari, 138. Capillary force, on the instability of a cylinder of viscous liquid under, 145, Cells, on the measurement of the in- ternal resistance of, 173. Chattock (A. P.) on an electrolytic theory of dielectrics, 461. Chree (C.) on rotating elastic solid cylinders of elliptic section, 70, 154. Circuits, on the equivalent resist- ance, self-induction, and capacity of parallel, with harmonic im- pressed electromotive force, 271. Clark (G. M.) on the determination of low temperatures by platinum- thermometers, 515. Cole (Prof. G. A. J.) on the litho- physes in the obsidian of Lipari, 138. Colour-blindness, on, 100, 439. Condenser, on oscillations that occur in the charging of a, 389. INDEX. Conductivity of insulating bodies, on the, 580. Conductor, on the effects of self- induction and distributed static capacity in a, 528. Crawford (J.) on the geology of Nicaragua, 384. Crehore (Dr. A. C.) on the equiva- lent resistance, self-induction, and capacity of parallel circuits with harmonic impressed electromotive force, 271; on the effects of self- induction and distributed static capacity in a conductor, 528. Croft (W. B.) on breath figures, 180. Cylinders, on rotating elastic solid, of elliptic section, 70, 154. Density, on the determination of the critical, 507. Devonian limestone of 8S. Devon, on the, 135. Dewar (Prof.) on the spectrum of liquid oxygen, and the refractive indices of liquid oxygen, nitrous oxide, and ethylene, 205; on the electrical resistance of pure metals, alloys, and non-metals at the boiling-point of oxygen, 326. Dielectric constant of conducting liquids, on the, 388. Dielectrics, on an electrolytic theory of, 461. Dimensions of physical quantities to directions in space, on the relation of the, 234. Donders’ (Prof.) theory of colour- blindness, 439. Donnan (F. G.) on Raoult’s law of the lowering of vapour-pressure, 411. Drift-beds of the N. and Mid-Wales coast, on the, 134. Du Bois (H. E. J.G.) on the mathe- matical theory of ferromagnetism, 307. Dynamical problems, on graphic solution of, 443. Edgeworth (Prof. F. Y.) on the law of error and correlated averages, 190, 429, 518. Edser (E.) on an instrument for measuring magnetic fields, 186. Elastic solid cylinders of elliptic section, on rotating, 70, 154. Electrical force at the electrodes, and the electrification of a gas in the glow-discharge, on the, 219. Phil. Mag. 8. 5. Vol. 34. No. 211. Dec. 1892. 533 Electrical resistance of the human body, on the, 218; of metals, alloys, and non-metals at the boil- ing-point of oxygen, on the, 326. Electrolytes, on the specific induc- tive capacity of, 344. Electrolytic polarization, on, 140. theory of dielectrics, on an, 461. Electromotive force, on an apparent relation of, to gravity, 307. Electrostatic theorems, on the “ elas- tic medium ” method of treating, 18. Emerald mines of Egypt, on the, 137. Error, on the law of, and correlated averages, 190, 429, 518. Etbai, on the Geology of the Nor- thern, 137. Ethylene, on the refractive indices of liquid, 209. ~ Ewing (Prof.) on joints in magnetic circuits, 520, Ferromagnetism, on the mathemati- cal theory of, 307. Fisher (Rev. O.) on theories to ac- count for glacial submergence, 337. Fleming (Prof. J. A.) on the elec- trical resistance of pure metals, alloys, and non-metals at the boiling-point of oxygen, 326. Floyer (EK. A.) on the geology of the Northern Etbai, 137. Fluid surfaces, on the instability of cylindrical, 177. Fluids, on the stability of the flow of, 59. Frey (M. vy.) on the electrical resist- ance of the human body, 218. Gas, on the electrification of a, in the glow-discharge, 219. Gases, on the heat of dissolution of, in liquids, 35; on the dissociation of, 143. Gauge for the appreciation of ultra- visible quantities, on a, 415. Geological Society, proceedings of the, 180, 305, 383. Gibson (W.) on the gold-bearing rocks of the Southern Transvaal, 135. Giltay (J. W.) on the use of a per- manently magnetized core in the telephone, 460. Glacial submergence, on theories to account for, 337. Glow-discharge, on the electrification of a gas in the, 219. 7A 534 er at Guttannen, on the so-called, 38. Gold-bearing rocks of the Southern Transvaal, on the, 135. Gore (Dr. G.) on an apparent rela- tion of electromotive force to gravity, 307. Graphic solution of dynamical prob- lems, on, 443. Gravity, on an apparent relation of electromotive force to, 307. Griffiths (EK. H.) on the determina- tion of low temperatures by plati- num-thermometers, 515. Guppy (R. J. L.) on the Tertiary microzoic formations of Trinidad, 305. Heat, on the absorption of radiant, by alum, 141; on the mechanical equivalent of, 142. Heilborn (E.) on the physical signi- ficance of 6 in Van der Waals’ equation, 459. Hobson (B.) on the basalts and andesites of Devonshire, 385. Holmes (J. V.) on railway-sections 7 ee Upminster and Romford, Hughes (R. E.) on the action of dried hydrochloric-acid gas on Iceland spar, 117. Human body, on the electrical re- sistance of the, 218. Hutchins (C. C.) on the absorption of radiant heat by alum, 141. Hydrocarbons, on the illuminating- power of, 120, 355. Hydrochloric-acid gas, on the action of dried, on Iceland spar, 117. Hydrogen, on the supposed detection of the line-spectrum of, in the oxyhydrogen flame, 371. Iceland spar, on the action of dried hydrochloric-acid gas on, 117. Interference bands of approximately homogeneous light, on _ the, 407. —— methods, on the application of, to spectroscopic measurements, 280. Inwards (R.) on an instrument for drawing parabolic curves, 57. Tons, on the colour of, 457. Irving (Rey. A.) on the Bagshot _beds of Bagshot Heath, 306. Jager (G.) on the dissociation of INDEX. gases, 143; on the magnitude of molecules, 220. Johnson Pasha (EK. A.) on the geo- logy of the Nile valley, 306. Kelvin (Lord) on graphic solution of dynamical problems, 443. Lampa (A.) on the absorption of light in turbid media, 144. Lea (M. C.) on the disruption of the silver haloid molecule by mecha- nical force, 46. Light, on the absorption of, in turbid media, 144; on the intensity of, reflected from water and mercury at nearly perpendicular incidence, 309; on the interference bands of approximately homogeneous, 407. Line spectrum of hydrogen in the oxyhydrogen flame, on Pliicker’s supposed detection of, 371. Liquid, on the instability of a eylin- der of viscous, under capillary force, 145. Liquids, on the heat of dissolution of gases in, 35; on the dielectric constant of conducting, 388; on the boiling-points of different, at equal pressures, 570, Lithophyses in the obsidian of Lipari, on the, 138. Liveing (Prof.) on the spectrum of liquid oxygen, and the refractive indices of liquid oxygen, nitrous oxide, and ethylene, 205; on Pliicker’s supposed detection of the line-spectrum of hydrogen in the oxyhydrogen flame, 371. Magnetic circuits, on joints in, 320. fields, on an instrument for measuring, 1&6. Magnetization, on the theory of, 531. . Medium, on the influence of obstacles arranged in rectangular order on the properties of a, 481. Mercury, on the intensity of light reflected from, at nearly perpen- dicular incidence, 318. Metals, on the electrical resistance of, at the boiling-point of oxygen, 326. Michelson (A. A.) on the application of interference methods to spec- troscopic measurements, 280. Miculescu (M. C.) on the mechanical equivalent of heat, 142. INDEX. 535 Molecular configurations, on the probabilities of, 51. Molecules, on a method of deter- mining the maguitude of, 220. Mountains, on the formation of, 299, Natanson (Dr. L.) on the probabili- ties of molecular configurations, 51. Nile Valley, on the geology of the, 306. Nitrous oxide, on the refractive indices of liquid, 209. Ogilvie (Miss M. M.) on the geology of the S. Tyrol, 383. Olenellus-zone in the N.W. High- lands, on the, 181. Ostwald (Prof. W.) on the colour of ions, 457. Oxygen, on the spectrum and refrac- tive index of liquid, 205. Parabolic curves, on an instrument for drawing, 57. Paulson (A. W.) on auroras observed at Godthaab, 143. - Peach (B. N.) on the Olenellus-zone in the N.W. Highlands, 131. Physical quantities, on the relation of the dimensions of, to directions in space, 234. Pickering (S. U.) on the heat of dissolution of gases in liquids, 35. Platinum-iridium and _ platinum- rhodium, on the thermoelectrics of, 376. Platinum-thermometers, on the de- termination of low temperatures by, 515. Pleistocene deposits of the Sussex coast, on the, 133. Polarization, on electrolytic, 140. Pole (Dr. W.) on colour-blindness, 100, 439, Postlethwaite (J.) on the dioritic picrite of White Hawse and Great Cockup, 383. Prestwich (Dr. J.) on the raised beaches of the South of England, 130, 152. Pump, on an air-mercury, for raising mercury in mercurial pumps, 115. Raised beaches of the South of Eng- land, on the, 130, 1382. Raoult’s law of the lowering of vapour-pressure, on, 411. Rayleigh (Lord) on the stability of the flow of fluids, 59; on the in- stability of a cylinder of viscous liquid under capillary force, 145 ; on the instability of cylindrical fluid surfaces, 177 ; on the inten- sity of light reflected from water and mercury at nearly perpen- dicular incidence, 309; on the interference bands of approxi- mately homogeneous light, 407 ; on the influence of obstacles arraneed in rectangular order upon the properties of a medium, 481. Reade (T. M.) on the drift-beds of the N. and Mid-Wales coast, 134. Refractive indices of liquid oxygen, nitrous oxide, and ethylene, on the, 205. Reid (C.) on the Pleistocene deposits of the Sussex coast, 185. Resistance of cells, on the measure- ment of the internal, 173. , on the equivalent, of parallel circuits with harmonic impressed electromotive force, 271. Richmond (H. D.) on the geology of the Nile valley, 806. Robb (Dr. W. L.) on oscillations that occur in the charging of a condenser, 389. Rock, on the fusion constants of igneous, l. Rosa (Prof. E. B.) on the specific inductive capacity of electrolytes, 344. Rowland (H. A.) on the theory of the transformer, 54. Rudski (M. M. P.) on the level of no strain in acooling homogeneous sphere, 299. Self-induction and distributed static capacity in a conductor, on the effects of, 528. Silver haloid molecule, on the dis- ruption of the, by mechanical force, 46. Smith (EH. W.) on the measurement of the internal resistance of cells, 173. Smith (F. J.) on an air-mercury pump, forraising mercury In mer- curial pumps, 115. Spectroscopic measurements, on the application of interference methods to, 280. Spectrum of liquid oxygen, on the, 205. 536 Sphere, on the level of no strain in a cooling homogeneous, 299, Stansfield (H.) on an instrument for measuring magnetic fields, 186. Stoney (Dr. G. J.) on the apprecia- tion. of ultra-visible quantities, 415. Stschegtiaeff (W.) on the dielectric constant of conducting liquids, 388. Surfaces, on the instability of cylin- drical fluid, 177. Tate (J.) on recent borings in the Tees salt-district, 385. Tees salt-district, on recent borings in the, 385. Telephone, on the use of a per- manently magnetized core in the, 276, 460. Temperature, on the measurement of high, 1; on the determination of low, by platinum-thermome- ters, 515. Thermoelectries of platinum-iridium and platinum-rhodium, 376. Thomas (G. L.) on the determina- tion of the critical density, 507. Transformer, on the theory of the, 54. Trinidad, on the Tertiary microzoic formations of, 305. Trouton (Dr. F. T.) on the use of a permanently magnetized core in the telephone, 276. INDEX. Ultra-visible quantities, on the ap- preciation of, 415. Van der Waals’ equation, on the physical significance of 6 in, 459. Vapour-pressure, on Raoult’s law of the lowering of, 411. Volume, on the determination of the critical, 503. Warburg (L.) on the electrical force at the electrodes, and the elec- trification of a gas in the glow- discharge, 219. Wassmuth (Dr.) on the theory of magnetization, 531. Water, on the intensity of light reflected from, at nearly perpen- dicular incidence, 309. Wethered (E.) on the Devonian limestone of 8. Devon, 135. Williams (W.) on the relation of the dimensions of physical quanti- ties to directions in space, 234. Williamson (Prof. W. C.) on the passage of a foraminiferal ooze into crystalline calcite, 156. Wilson (F. R. L.) on the action of dried hydrochloric-acid gas on Iceland spar, 117. Young (Dr. S.) on the determination of the critical volume, 503; on the determination of the critical density, 507; on the boiling- points of different liquids at equal pressures, 510. END OF THE THIRTY-FOURTH VOLUME. Printed by Tayzor and Fraxcts, Red Lion Court, Fleet Street. s YaAty soa ULL FULT QIN. AUA, DIA ANAOUL- OLQIOIA-OULLOU. JO UOYWYAL AY bruamoys POU) GF ‘buy -OOTL -OOOL 006 008 -OOL 009 00¢ 00 -00¢ 4, PLT. A ) Vv Phil. Mag. S.5. Vol. =X = 13000 12000 11000 MIICROV|IOLTS Nie pre- iBN De EN oe A and pe NI spent Donte . Phil.Mag.S.5.Vol. 34. PILI. UL so ug Ute JUL Hs 2tAws (Obs) ALISNALNI ST1TAMXVIWN IA /? a ‘e's Ey “AUOYYDS11A > omg | S8O]INO]OD POLYM “NOISIA DIWOMHDIG § 4O Wnuloads YAH erm | -ayby ary “povay.Lop OMT QM. UpPa ‘MOYO ‘Pa Uuay. NOP Ap pon (PAYosngn > (peynyyp | perp (PEPOMyYD> Aypopob AY (payin TUO YY NAT Aypongpn. tb i AOU DAD FUDYYAULT MOT Peyannwe amg | MOPYPOK —— | esd Je | at S ce “AYU d ull ee “Sir * UOISIA OTUIO. Yo anyg pue Moyyax oF pottdde “AIdIONIYd TISAMXVYW-NOLMIN JZHL NO SWWYOVIG aNOTOS fe Fa COLOUR DIAGRAMS ON THE NEWTON-MAXWELL PRINCIPLE, applied to Yellow and Blue Dichromic Vision . ee (White ) Ww Fig. 1. Fig. 2. MB POLE. MR PARRY. BLUE <-} SENSATION Q Blue Verditer OBlixe Verditer B (Black) SPECTRUM OF NORMAL VISION Fig. 3. Newtons Colours. Saturated Yellow — Brillvant ; Sectaar 3 Be ‘ a ‘ UI. 5 | ob Phil.Mag.S.5.Vol.34.P fa SWS ESSSSSS TT TE OE ET TE TT EE OPO TE TT me iG POR AARG o) ASSSSSSSGSSSSSSQR A Sita See yeoalel aie ese ‘OD EERE ol A b> 2 <2 N et RORY N N ok SLIAN N G A ARSON O OOOO IR Ca PSO OR ren cx xs ras 2 % y Phil.Mas.S.5 Vol.34. Pl ii UW. VOL. OS. £2 REL INS ESSSSSSSSSSSSS SSS SNS SH Al B Mintern Bros. lth. o 2 : ’ > ate SRLS Oe eT a . We mena f Plate > te SK ee VAs NZ he & he eV * Fig. 4. the felling off of the fields at points axes of the armatures. shew the directions.) 200 100 se Gulchar Are Dynamo. Elwell Porkar Phil.Mag. S.5.Vol.34. PL. WV. ric 1120 1010 4 Tramstirmer. Crompton Dyn? re jk, © 690 or 500 :s i faa Laing Wharton a Kapp. & Doww Ship Dynamo. 0535 a | | 400 | © 517 B Thomsor-Howstow Are light Dynamo. { 1690 6EQ, 300 |& pp a es 38 oe) Fig.2. The Garves in all cases represent the falling off of the fields at pouts 200 wm directions at right angles to the axes of the armatures. (The dotted lines shew the directions.) 100 — ile | = = 0 5" 10° 15" 20" 25" 30” 35" 4.0" 45° 50" Fig elt Mintern Bros lith. a ane ae one ar » é Phil. Mag. 8. 5. Vol. 34, PL. V Phil. Mag. §. 5. Vol, 84, Pl. V Phil. Mag. §. 5. Vol. 34. Pl. VI. 160 mm EY : 1 4 Z + = % 2 , 5 f “ x ae et Say Phil. Mag. 8. 5. Vol. 84. Pl. VI. Fig 16 Mg, Fig 1G Mas a Na, £20 are No, 120 140 Phil. Mag. 8. 5. Vol. 34, Pl. 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