REPRINT OF PAPERS ELECTROSTATICS AND MAGNETISM. Cambridge : PRINTED BY C. J. CLAY, M.A. & SOX, AT THE UNIVERSITY PRESS. KEPKINT OF PAPERS ON ELECTROSTATICS AND MAGNETISM BY SIR WILLIAM THOMSON, D.C.L., LL.D., F.R.S., F.R.S.E., FELLOW OF ST PETEK's COLLEGE, CAMBRIDGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW. SECOND EDITION. Honfcon : MACMILLAN & CO. 1884 [The rights of translation and reproduction are reserved.} KH- PREFACE TO THE FIRST EDITION. THIS volume consists chiefly of reprinted articles, on Electro- statics and mathematically allied subjects, which originally appeared at different times during the last thirty years, in the Cambridge Mathematical Journal, the Cambridge and Dublin Mathematical Journal, Liouville's Journal de Mathematiques, the Philosophical Magazine, Nichol's Cyclopaedia, the Reports of the British Association, the Transactions or Proceedings of the Royal Societies of London and Edinburgh, the Royal Institution of Great Britain, and the Philosophical Societies of Manchester and Glasgow. The remainder, constituting about a quarter of the whole, is now printed for the first time from manuscript, which, except a small part about twenty years old, entitled " Electro- magnets," has been written for the present publication, to fill up roughly gaps in the collection. The original dates of the republished articles, the dates of all new matter appearing as insertions or notes in the course of those articles, and the dates of the fresh articles have all been carefully indicated. The article on Atmospheric Electricity, extracted from Nichol's Cyclopaedia, was originally written at the request of my late friend and colleague the Editor ; and for the permission to reprint it I am indebted to his son, my colleague, Professor John Nichol, and to the Messrs. Griffin, the publishers of the Cyclopaedia. 797943 vi PREFACE. The present volume includes as nearly as may be all that I have hitherto written on electrostatics and magnetism. I have excluded from it electrical papers in which either thermo- dynamics or the kinetics of electricity is prominent. I intend that, as soon as possible, it shall be followed by a collected re- print of all my other papers hitherto published. I take this opportunity of thanking Professors Clerk Max- well and Tait for much valuable assistance which they have given me in the course of this work. WILLIAM THOMSON. YACHT "LALLA KOOKH," LAMLASH, Oct. 12, 1872. PREFACE TO THE SECOND EDITION. THIS Second Edition is substantially a reprint of the First Edition ; the only changes made being the correction of a few errata which had escaped detection in the volume as originally published. W. T. THE UNIVEKSITY, GLASGOW, March 12, 1884. CONTENTS. AETICLE I. ON THE UNIFORM MOTION OF HEAT IN HOMOGENEOUS SOLID BODIES, AND ITS CONNEXION WITH THE MATHEMATICAL THEORY OF ELECTRI- CITY. SECTIONS Temperature at any point within or without an Isothermal Surface V 110 Uniform Motion of Heat in an Ellipsoid . . . . . 11 20 Attraction of a Homogeneous Ellipsoid on a point within or without it . >, ...... >.- >...-.- . 2124 H. ON THE MATHEMATICAL THEORY OF ELECTRI- CITY IN EQUILIBRIUM. DIVISION I. ON THE ELEMENTABY LAWS OF STATICAL ELECTRICITY. Investigations of Coulomb, Poisson, and Green . ... 25 Examination of Harris's Experimental results . . . . 2635 Faraday's researches on Electrostatical Induction . . _ 3650 III. ON THE ELECTROSTATICAL CAPACITY OF A LEYDEN PHIAL AND OF A TELEGRAPH WIRE INSULATED IN THE AXIS OF A CYLINDRICAL CON- DUCTING SHEATH. Application of the Principles brought forward in the preceding Articles 5156 IV. ON THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. DIVISION. IL-r-A STATEMENT OP THE PRINCIPLES ON WHICH THE MATHEMATICAL THEORY is FOUNDED. Object of the Article 57 The two kinds of Electricity 5860 Electrical Quantity . . . . . . . . 6162 Superposition of Electric Forces 63 viii Contents. SECTIONS The Law of Force between Electrified bodies .... 64 Definition of the resultant Electric Force at a Point ... 65 Electrical Equilibrium 66 Non-conductors of Electricity 67 Conductors of Electricity 68 Electrical Density at any Point of a charged Surface ... 69 Exclusion of all Non-conductors except Air .... 70 Insulated Conductors 71 Recapitulation of the Fundamental Laws 72 Objects of the Mathematical Theory of Electricity ... 73 Actual Progress in the Mathematical Theory of Electricity . . 74 V. ON THE MATHEMATICAL THEOEY OF ELECTRICITY IN EQUILIBRIUM. DIVISION III. GEOMETRICAL INVESTIGATION WITH REFERENCE TO THE DISTRIBUTION OF ELECTRICITY ON SPHERICAL CON- DUCTORS. Object of the Article 75 Insulated Conducting Sphere subject to no External Influence . 76 Determination of the Distribution 77 Verification of Law III 78 Digression on the Division of Surfaces and Elements Object of the Digression 79 Explanation and Definition regarding Cones . 80 The Solid Angle of a Cone, or a complete Conical Surface . '. 81 Sum of all the Solid Angles round a Point 4?r . . . . 82 Sum of the Solid Angles of all the complete Conical Surfaces = 27r 83 Solid Angle subtended at a Point by a Terminated Surface . 84 Orthogonal and Oblique Sections of a Small Cone ... 85 Area of the Segment cut from a Spherical Surface by a Small Cone 86 Theorem 87 Repulsion on an Element of the Electrified Surface ... 88 INSULATED SPHERE SUBJECTED TO THE INFLUENCE OF AN ELEC- TRICAL POINT. Object 89 Attraction of a Spherical Surface of which the Density varies inversely as the Cube of the Distance from a given point . 90 92 Application of the preceding Theorems to the Problem of Electrical Influence 9395 EFFECTS OF ELECTRICAL INFLUENCE ON INTERNAL SPHERICAL AND ON PLANE CONDUCTING SURFACES . . . .- 96 112 INSULATED SPHERE SUBJECT TO THE INFLUENCE OF A BODY OF ANY FORM ELECTRIFIED IN ANY GIVEN MANNER .... 113 127 Contents. ix SECTIONS VI. ON THE MUTUAL ATTEACTION OE REPULSION BETWEEN TWO ELECTRIFIED SPHERICAL CON- DUCTORS 128142 VII. ON THE ATTRACTIONS OF CONDUCTING AND NON- CONDUCTING ELECTRIFIED BODIES .... 144148 VIII. DEMONSTRATION OF A FUNDAMENTAL PROPO- SITION IN THE MECHANICAL THEORY OF ELECTRI- CITY . /-vnqtaes 149-155 IX.- NOTE ON INDUCED MAGNETISM IN A PLATE . . 156162 X. SUR UNE PROPRIETE DE LA COUCHE ELECTRIQUE EN EQUILIBRE A LA SURFACE D'UN CORPS CON- DUCTEUR. Par M. J. LIOUVILLE 163164 Note on the preceding Paper . . . ..'>- . . 165 XL ON CERTAIN DEFINITE INTEGRALS SUGGESTED BY PROBLEMS IN THE THEORY OF ELECTRICITY . 166186 XII. PROPOSITIONS IN THE THEORY OF ATTRACTION. Parti. :*./:.*:*"-.-?". . 187198 Part II. 'V ."'.' 199205 XHL THEOREMS WITH REFERENCE TO THE SOLUTION OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS . 206 ADDITIONS TO A FEENCH TEANSLATION OF THE PKECEDING . . . 207 XIV. ELECTRICAL IMAGES. Extraite d'une lettre de M. William Thomson a M. Liouville . 208 210 Extraits de deux lettres addressees a M. Liouville. Par M. William Thomson ". . ..... . . . 211 220 Note au sujet de F Article precedent. Par M. Liouville . . , 221 230 XV. DETERMINATION OF THE DISTRIBUTION OF ELEC- TRICITY ON A CIRCULAR SEGMENT OF PLANE OR SPHERICAL CONDUCTING SURFACE, UNDER ANY GIVEN INFLUENCE 231248 XVI. ATMOSPHERIC ELECTRICITY. Preliminary Explanations 249 251 The whole Surface of the Earth electrified [generally negatively] . 252 The State of Electrification of the Air [unknown and cannot be inferred with certainty from observations of the Electric density of the Earth's Surface: observation from balloons wanted] . 253 261 Description of preliminary Experiments made to test aerial Elec- tricity, and of the Instruments employed 262 266 x Contents. EOYAL INSTITUTION LECTUEE. SECTIONS Earliest Observations of Atmospheric Electricity .... 267 268 Essential qualities of the Apparatus required for the observation of Atmospheric Electricity 269 Description of the Divided King Reflecting Electrometer . . 270273 Description of the Common House Electrometer .... 274 276 Description of the Portable Electrometer 277 Burning Match and Water-dropping Collectors .... 278 279 Remarks on the origin, nature, and changes of Terrestrial Atmo- spheric Electricity 280291 Kew Self-recording Atmospheric Electrometer, with specimen of the results 292293 ON ELECTRICAL FREQUENCY 294 ON THE NECESSITY FOR INCESSANT RECORDING AND FOR SIMUL- TANEOUS OBSERVATIONS IN DIFFERENT LOCALITIES TO INVESTI- GATE ATMOSPHERIC ELECTRICITY 295 OBSERVATIONS ON ATMOSPHERIC ELECTRICITY 296300 ON SOME REMARKABLE EFFECTS OF LlGHTNING OBSERVED IN A FARM HOUSE NEAR MONIMAIL 301 XVII. SOUND PRODUCED BY THE DISCHARGE OF A CONDENSER . 302304 XVIII. MEASUREMENT OF THE ELECTROSTATIC FORCE PRODUCED BY A DANIELL'S BATTERY. Preliminary Explanation .' .' "."'". .... 305 306 Absolute Electrometer . . . . ;* . . . 307 Reduction to absolute measure, of the readings of torsional Electro- meters, by means of the Absolute Electrometer .... 308 313 General results of the Weighings 314316 Postscript corrected results 317 319 XIX. MEASUREMENT OF THE ELECTROMOTIVE FORCE REQUIRED TO PRODUCE A SPARK IN AIR BETWEEN PARALLEL METAL PLATES AT DIFFERENT DIS- TANCES. Description of the Experiments 320 321 Table I., Measurements by Absolute Electrometer .... 322323 Table II. , Measurements by Portable Electrometer .... 324 Table III., The two series compared Additional Experiments, Tables IV. and V 325 Table VI., Summary of results reduced to Absolute Measure . . ^ 326 APPENDIX Explanation of Terms. Measurement of quantities of electricity. Electric density. Resultant electric force at any point in an insulating fluid. Relation between Electric density Contents. xi SECTIONS on the surface of a conductor and electric force at points in the air close to it. Electric pressure from the surface of a conductor balanced by Air. Collected formulae. Electric potential. In- terpretation of measurement by Electrometer. Kelation between Electrostatic force and variation of Electric potential. Stratum of Air between two parallel or nearly parallel plane or curved metallic surfaces maintained at different potentials . . . 327 338 Absolute Electrometer .^ 339 Additional Experiments ... . , . . . 340 XX. EEPOET ON ELECTEOMETEES AND ELECTEOSTATIC MEASUEEMENTS. Definition Eequisites for accurate Electrometry .... 341 342 Classification of Electrometers I. Eepulsion Electrometers II. Symmetrical Electrometers III. Attracted Disc Electrometer 343 Divided Eing Electrometer described, and adjustments explained . 344 357 Absolute Electrometer ... . . . . . . 358363 New Absolute Electrometer .... .V -. ; . 364367 Portable Electrometer . ; , '" . ,. ;. .:':. .- 368378 Standard Electrometer . . . ... . . . . 379382 Long-range Electrometer . ;. . . .. '." . . 383384 Idiostatic and Heterostatic Electrometers ". \ *. .; . 385 Concluding remarks regarding Electrometers ... . . . 386390 XXI. ATMOSPHEEIC ELECTEICITY. New Apparatus for observing Atmospheric Electricity . . . 391 Description and results of simultaneous observations made at two stations at different elevations in the University of Glasgow -, . 392 Effect of pudden changes of wind . . . 1 > . - * 393 395 Changes observed during a thunder-storm. . . -. ,- . .. ' 396 Effects observed in the neighbourhood of an escape of high-pressure steam 397399 XXIL NEW PEOOF OF CONTACT ELECTEICITY ... 400 XXIII. ELECTEOPHOEIC APPAEATUS AND ILLUSTEATIONS OF VOLTAIC THEOEY. On a Self-acting Apparatus for multiplying and maintaining Elec- tric Charges, with applications to illustrate the Voltaic Theory . 401407 On a Uniform Electric Current Accumulator .... 408 411 On Volta-Convection by Flame 412415 Xll Contents. SECTIONS On Electric Machines founded on Induction and Convection, Electric Keplenisher, Potential-Equalizer, with applications . 416 426 On the Reciprocal Electrophorus 427429 XXIV. A MATHEMATICAL THEORY OF MAGNETISM. Introduction 430433 PABT FIRST. ON MAGNETS AND THE MUTUAL FORCE BETWEEN MAGNETS. CHAPTER I. PRELIMINARY DEFINITIONS AND EXPLANATIONS. Definition of a Magnet 434435 Action of the Earth on a Magnet sensibly a couple ; Directive tendency, Magnetic Axis, Dip, Polarity . . . . ~ '-*- 436 446 Distribution of Magnetism in a Magnet 447 451 CHAPTER n. ON THE LAWS OF MAGNETIC FORCE, AND ON THE DIS- TRIBUTION OF MAGNETISM IN MAGNETIZED MATTER. Mutual Action between two thin uniformly and longitudinally Magnetized Bars . . 452 453 Strength of a Magnet, Unit Strength, Magnetic Moment, Inten- sity and Direction of Magnetization . . --,.'- . . 454 462 CHAPTER III. ON THE IMAGINARY MAGNETIC MATTER BY MEANS OF WHICH THE POLARITY OF A MAGNETIZED BODY MAY BE REPRE- SENTED 463475 CHAPTER IV. DETERMINATION OF THE MUTUAL ACTIONS BETWEEN ANY GIVEN PORTIONS OF MAGNETIZED MATTER. Explanations vX . . .... .. , .- . .. . 476 478 "Resultant Magnetic Force at any Point " . . * ". 479 480 The "Potential" 481484 Potential at a point P due to a given Magnet .... 485 501 On the Expression of Mutual Action between two Magnets by means of the Differential Coefficients of a Function of their relative Positions . . " . . . . . . . 502 503 CHAPTER V. ON SOLENOID AL AND LAMELLAR DISTRIBUTIONS OF MAGNETISM. Explanations 504 Definitions and Explanations regarding Magnetic Solenoids . 505 Definitions and Explanations regarding Magnetic Shells . . 506 Solenoidal and Lamellar Distributions of Magnetisms ; . 507 Complex Lamellar and Complex Solenoidal Distributions of Magnetism 508509 Action of a Magnetic Solenoid and of a Complex Solenoid . . - 510 511 Potential at any point due to a Magnetic Shell Action of Magnetic Shells . . . 512 Criterion of a Solenoidal Distribution of Magnetism ... 513 Contents. Xlll SECTIONS Criterion of a Lamellar Distribution of Magnetism . . . 514 Kesultant Force, due to a lamellarly-magnetized Magnet, on any external or internal point 515 523 CHAPTER VI. ON ELECTROMAGNETS. Introductory Eemarks .... ..... ,,. 524 Investigation of the Action between two Galvanic Arcs, or be- tween a Galvanic Arc and a Magnetic Pole . 525 530 Unit of strength for an Electric Current . , . . . . . 531 533 Hypothesis of Matter flowing . ... . . , . 534 Division of Electromagnets into three Classes . . . ... _ 535 Linear Electromagnets . . . . . ... . . 536 Superficial Electromagnets " % ., . ... . . 537 Solid Electromagnets -. 538 Analytical Investigation of the Conditions to which the Distri- bution of Galvanism in Solid and Superficial Electromagnets is subject . . . . . . \ ' ' '. . -./. . 539543 Applications .... . . '. . .' 544 A similar Synthetic Solution indicated *.- * -~ . . . 545 Electromagnets and their respective equivalent Polar Magnets Eules for Direction . . . . '. - . . ; -' ; . 546 550 Eemarks and Additions . . . .-. . . ' .' ' . 551 553 Original Investigation of 517 referred to in 518 , . . . .' 554 XXV. ON THE POTENTIAL OF A CLOSED GALVANIC CIECUIT OF ANY FOEM . 555-560 XXVI. CHAPTER VII. ON THE MECHANICAL VALUES OF DISTRIBU- TIONS or MATTER AND OF MAGNETS. Mechanical Values of Distributions of Matter . ." " ;'. . 561 563 Polar Magnets . ; : ; ' ; ; .' ir .' .' . 564568 Electromagnets 569 572 XXVII. CHAPTER VIII. HYDROKINETIC ANALOGY 573583 XXVIII. CHAPTER IX. INVERSE PROBLEMS. Definition Divided into two Classes . . . ., -. . ... . * 584 Class I. Force given for every point of space .. . . . 585 588 Class II. Force or component of force given through some portion of space . % . . .* . . . . . 589601 XXIX.- ON THE ELECTEIC CUEEENTS BY WHICH THE PHENOMENA OF TEEEESTEIAL MAGNETISM MAY BE PEODUCED %.*,;. 602603 CHAPTER X. MAGNETIC INDUCTION. XXX. ON THE THEOEY OF MAGNETIC INDUCTION IN CEYSTALLINE AND NON-CEYSTALLINE SUBSTANCES. xiv Contents, SECTIONS Explanations and Definitions. Force at any point due to a Magnet. Total magnetic force at a point. "A Field of magnetic force." " A line of magnetic force." "A nniforni field of magnetic force." Resultant Distribution of Magnetism . . . ... . 604 605 Axioms of Magnetic Force 606 Laws of Magnetic Induction according to Poisson's Theory . . 607 609 Conclusions from these Laws 610 619 Appendix Quotations from Poisson regarding Magne-Crystallic action Explanation. Demonstration 620 624 XXXI. MAGNETIC PEEME ABILITY AND ANALOGUES IN ELECTEOSTATIC INDUCTION, CONDUCTION OF HEAT AND FLUID MOTION 625-631 XXXII. DIAGEAMS OF LINES OF FOECE ; TO ILLUSTEATE MAGNETIC PERMEABILITY 632-633 XXXIII. ON THE FOECE S EXPERIENCED BY SMALL SPHEEES UNDEE MAGNETIC INFLUENCE; AND ON SOME OF THE PHENOMENA PEESENTED BY DIA- MAGNETIC SUBSTANCES. Attraction of Ferromagnetics 634642 Repulsion of Diamagnetics 643 646 XXXIV. REMARKS ON THE FOECES EXPERIENCED BY INDUCTIVELY MAGNETIZED FEEEOMAGNETIC OE DIAMAGNETIC NON-CEYSTALLINE SUBSTANCES. Faraday's Law of Attractions and Eepulsions .... 647 653 Experimental illustrations of Faraday's Law .... 654 664 On the Stability of Small Inductively magnetized bodies in Posi- tions of Equilibrium 665 On the relations of Ferromagnetic and Diamagnetic Magnetization to the magnetizing force 666 668 XXXV. ABSTEACT OF TWO COMMUNICATIONS On certain Magnetic Curves ; with applications to Problems in the Theories of Heat, Electricity, and Fluid Motion . . . 669 On the Equilibrium of elongated Masses of Ferromagnetic Sub- stances in uniform and varied Fields of Force .... 669 XXXVI. REMARQUE S SUR LES OSCILLATIONS d'aiguilles non cristallise*es de faible pouvoir inductif paramag- n6tique ou diamagnetique, et sur d'autres phenomenes mag- netiques produits par des corps cristallises'ou non cristallises; from the "Comptes Rendus" of the French Academy, 1854, first half-year ^ 670 XXXVII. ELEMENTARY DEMONSTRATION OF PROPOSI- TIONS IN THE THEORY OF MAGNETIC FORCE . 671-673 Contents. xv SECTIONS Examination of the Action experienced by an infinitely tliin, uniformly and longitudinally Magnetized Bar, placed in a Non- uniform Field of Force, with its length direct along a line of Force 674688 XXXVHI. COBBESPONDENCE WITH PEOFESSOB TYNDALL. Letter to Professor Tyndall on the "Magnetic Medium," and on the effects of Compression 689 693 Letter from Professor Tyndall to Professor W. Thomson on Beciprocal Molecular Induction . . ; .- . . . 694= Letter from Professor W. Thomson to Professor Tyndall, on the Beciprocal Action of Diamagnetic Particles . .- . - . 695 696 XXXIX. INDUCTIVE SUSCEPTIBILITY OF A POLAB MAG- NET . . . :' 697699 XL. GENEBAL PBOBLEM OF MAGNETIC INDUCTION . 700732 XLI.^HYDEOKINETIC ANALOGY FOB THE MAGNETIC INFLUENCE OF AN IDEAL EXTBEME DIAMAGNETIC. On Forces experienced by Solids immersed in a Moving Liquid . 733 740 Extracts from two Letters to Professor Guthrie .... 741743 Beport of an Address on the Attractions and Bepulsions due to Vibration, observed by Guthrie and Schellbach. Hydrokinetic Analogy for Extreme Diamagnetic 744 750 XLII. GENEBAL HYDBOKINETIC ANALOGY FOB INDUCED MAGNETISM. Permeability in Hydrokinetic Analogy 751 756 Kinetic Energy a Minimum 757 758 Analogy of Force 759763 I. ON THE UNIFORM MOTION OF HEAT IN HOMOGENEOUS SOLID BODIES, AND ITS CONNEXION WITH THE MATHEMATICAL THEORY OF ELECTRICITY.* (Art. in. of complete list in Mathematical and Physical Papers, Vol. i.) [From Cambridge Mathematical Journal, Feb. 1842. Reprinted Philosophical Magazine (1854, first half-year).] [Since the following article was written,-)- the writer finds that most of his ideas have been anticipated by M. Chasles in two Me'moires in the Journal de Mathematiques ; the first, in vol. in., on the Determination of the Value of a certain Definite Integral, and the second, in vol. v., on a new Method of Determining the Attraction of an Ellipsoid on a Point with- out it. , In the latter of these Memoires, M. Chasles refers to a paper, by himself, in the twenty-fifth Cahier of the Journal de VEcole Poly technique, in which it is probable there are still further anticipations, though the writer of the present article * [Note added June 1854.] This paper first appeared anonymously in the Cambridge Mathematical Journal in February 1842. The text is reprinted without alteration or addition. All the footnotes are of the present date (March 1854). The general conclusions established in it show that the laws of distribution of electric or magnetic force in any case whatever must be identical with the laws of distribution of the lines of motion of heat in certain perfectly defined circumstances. With developments and applications con- tained in a subsequent paper (11. below) on the Elementary Laws of Statical Electricity (Cambridge and Dublin Mathematical Journal, Nov. 1845), they constitute a full theory of the characteristics of lines of force, which have been so admirably investigated experimentally by Faraday, and complete the analogy with the theory of the conduction of heat, of which such terms as "conducting power for lines of force" (Exp. Res. 27972802) involve the idea. t [Note added June 1854.] This preliminary notice was written some months later than the text which follows, and was communicated to the editor of the journal to be prefixed to the paper, which had been in his hands since the month of September 1841. The ideas in which the author had ascertained he had been anticipated by M. Chasles, were those by which he was led to the determination of the attraction of an ellipsoid given in the latter part of the paper. He found soon afterwards that he was anticipated by the same author in an enunciation of the general theorems regarding attraction ; still later he found that both an enunciation and demonstration of the same general theorems had been given by Gauss, whose paper ap- T. E. 1 2 Uniform Motion of Heat and [i. has not had access to so late a volume of the latter journal. Since, however, most of his methods are very different from those of M. Chasles, which are nearly entirely geometrical, the following article may be not uninteresting to some readers : ] 1. If an infinite homogeneous solid be submitted to the action of certain constant sources of heat, the stationary temperature at any point will vary according to its position; and through every point there will be a surface, over the whole extent of which the temperature is constant, which is therefore called an isothermal surface. In this paper the case will be considered in which these surfaces are finite, and consequently closed. 2. It is obvious that the temperature of any point without a given isothermal surface, depends merely on the form and temperature of the surface, being independent of the actual sources of heat by which this temperature is produced, provided there are no sources without the surface. The temperature of an external point is consequently the same as if all the sources were distributed over this surface in such a manner as to produce the given constant temperature. Hence we may consider the temperature of any point without the isothermal surface, as the sum of the temperatures due to certain constant sources of heat, distributed over that surface. peared shortly after M. Chasles' enunciations; and after all, he found that these theorems had been discovered and published in the most complete and general manner, with rich applications to "the theories of electricity and magnetism, more than ten years previously, by Green! It was not until early in 1845 that the author, after having inquired for it in vain for several years, in consequence of an obscure allusion to it in one of Murphy's papers, was fortunate enough to meet with a copy of the remarkable paper ("An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism," by George Green, Nottingham, 1828) in which this great advance in physical mathematics was first made. It is worth remarking, that, referring to Green as the originator of the term, Murphy gives a mistaken definition of "potential." It appears highly probable that he may never have had access to Green's essay at all, and that this is the explanation of the fact (of which any other explanation is scarcely conceiv- able), that in his Treatise on Electricity (Murphy's Electricity, Cambridge, 1833) he makes no allusion whatever to Green's discoveries, and gives a theory in no respect pushed beyond what had been done by Poisson. All the general theorems on attraction which Green and the other writers referred to, demonstrated by various purely mathematical processes, are- seen as axiomatic truths in approaching the subject by the way laid down in the paper which is now republished. The analogy with the conduction of heat on which these views are founded, has not, so far as the author is aware, been noticed by any other writer. I.] Mathematical Theory of Electricity. 3 3. To find the temperature produced by a single source of heat, let r be the distance of any point from it, and let v be the temperature at that point. Then, since the temperature is the same for all points situated at the same distance from the source, it is readily shown that v is determined by the equation z dv . -r*-r- = A. dr Dividing both members by r 2 , and integrating, we have =- + a r Now let us suppose that the natural temperature of the solid, or the temperature at an infinite distance from the source, is zero : then we shall have (7=0, and consequently = 7 ................................ 4. Hence that part of the temperature of a point without an isothermal surface which is due to the sources of heat situated on any element, dco*, of the surface, is a?1 , where r t is the r i distance from the element to that point, and p l a quantity measuring the intensity of the sources of heat at different parts of the surface. Hence, the supposition being still made that there are no sources of heat without the surface, if v be the temperature at the external point, we have ......................... . the integrals being extended over the whole surface. The quantity p l must be determined by the condition v = v 1 .... .......................... (3), for any point in the surface, v being a given constant tem- perature. 5. Let us now consider what will be the temperature of a point within the surface, supposing all the sources of heat by which the surface is retained at the temperature v 1 to be distri- buted over it. Since there are no sources in the interior of the surface, it follows that as much heat must flow out from the interior across the surface as flows into the interior, from the sources of heat at the surface. Hence the total flux of heat from the original surface to an adjacent isothermal surface in 12 4 Uniform Motion of Heat and [i. the interior is nothing. Hence also the flux of heat from this latter surface to an adjacent isothermal surface in its interior must be nothing ; and so on through the whole of the body within the original surface. Hence the temperature in the interior is constant, and equal to v lt and therefore, for points at the surface, or within it, we have Now, if we suppose the surface to be covered with an attrac- tive medium, whose density at different points is proportional to pj, -j- II * will b e the attraction, in the direction of the axis of cc, on a point whose rectangular co-ordinates are x, y, z. Hence it follows that the attraction of this medium on a point within the surface is nothing, and consequently p is proportional to the intensity of electricity in a state of equi- librium on the surface, the attraction of electricity in a state of equilibrium being nothing on an interior point. Since, at the surface, the value of II L is constant, and since, on that account, its value within the surface is constant also, it follows, that if the attractive force on a point at the surface is perpendicular to the surface, the attraction on a point within the surface is nothing. Hence the sole condition of equi- librium of electricity, distributed over the surface of a body, is, that it must be so distributed that the attraction on a point at the surface, oppositely electrified, may be perpendicular to the surface. 6. Since, at any of the isothermal surfaces, v is constant, it follows that -r- , where n is the length of a curve which cuts an all the surfaces perpendicularly, measured from a fixed point to the point attracted, is the total attraction on the latter point; and that this attraction is in a tangent to the curve n, or in a normal to the isothermal surface passing through the point. For the same reason also, if p l represent a flux of heat, and not an electrical intensity, -=- will be the total flux of heat at the variable extremity of n, and the direction of this flux will be I.] Mathematical Theory of Electricity. 5 along n, or perpendicular to the isothermal surface. Hence, if a surface in an infinite solid be retained at a constant tempera- ture, and if a conducting body, bounded by a similar surface, be electrified, the flux of heat, at any point, in the first case, will be proportional to the attraction on an electrical point, similarly situated, in the second; and the direction of the flux will correspond to that of the attraction. 7. Let -j-t be the external value of =- at the original dn^ dn surface, or the attraction on a point without it, and indefinitely near it. Now this attraction is composed of two parts ; one the attraction of the adjacent element of the surface; and the other the attraction of all the rest of the surface. Hence, calling the former of these a, and the latter b, we have dv, --* Now, since the adjacent element of the surface may be taken as infinitely larger, in its linear dimensions, than the distance from it of the point attracted, its attraction will be the same as that of an infinite plane, of the density p r Hence a is inde- pendent of the distance of the point from the surface, and is equal to STT/DJ. Hence Now, for a point within the surface, the attraction of the adja- cent element will be the same, but in a contrary direction, and the attraction of the rest of the surface will be the same, and in the same direction. Hence the attraction on a point within the surface, and indefinitely near it, is ^7rp l -f- b ; and conse- quently, since this is equal to nothing, we must have b = 27rp 1 , and therefore dv. --^r ?l ......................... () - Hence p 1 is equal to the total flux of heat, at any point of the surface, divided by 4>ir. 8. It also follows that if the attraction of matter spread over the surface be nothing on an interior point, the attraction on an exterior point, indefinitely near the surface, is perpendicular to the surface, and equal to the density of the matter at the part of the surface adjacent to that point, multiplied by 4?r. 6 Uniform Motion of Heat and [i. 9. If v be the temperature at any isothermal surface, and p the intensity of the sources at any point of this surface, which would be necessary to sustain the temperature v, we have, by (5), dv which equation holds, whatever be the manner in which the actual sources of heat are arranged, whether over an isothermal surface or not ; and the temperature produced in an external point by the former sources, is the same as that produced by the latter. Also, the total flux of heat across the isothermal surface, whose temperature is v, is equal to the total flux of heat from the actual sources. From this, and from what has been proved above, it follows that if a surface be described round a conducting or non-conducting electrified body, so that the attraction on points situated on this surface may be every- where perpendicular to it, and if the electricity be removed from the original body, and distributed in equilibrium over this surface, its intensity at any point will be equal to the attraction of the original body on that point, divided by 4?r, and its attraction on any point without it will be equal to the attraction of the original body, on the same point.* If we call E the total expenditure of heat, or the whole flux across any isothermal surface, we have, obviously, 'dv dn *' 10. Now this quantity should be equal to the sum of the expenditures of heat from all the sources. To verify this, we must, in the first place, find the expenditure of a single source. Now the temperature produced by a single source is, by (1), v = , and hence the expenditure is obviously equal to * [Note added June 1854. After having established this remarkable theorem in the manner shown in the text, the author attempted to prove it by direct integration, but only succeeded in doing so upwards of a year later, when he obtained the demonstration published in a paper, "Propositions in the Theory of Attraction" (Camb. Math. Jour. Nov. 1842), which appeared almost contemporaneously with a paper by M. Sturm in Liouville's Journal, containing the same demonstration; exactly the same demonstration, as the author afterwards (in 1845) found, had been given fourteen years earlier by Green.] I.] Mathematical Theory of Electricity. dv dr -7- x 4vrr 2 , or to 4nrA. If A p^a**, this becomes Hence the total expenditure is ff4t7rp da)*, or II ~~ da>*, which agrees with the expression found above. The following is an example of the application of these principles : . Uniform Motion of Heat in an Ellipsoid. 11. The principles established above afford an easy method of determining the isothermal surfaces, and the corresponding temperatures, in the case in which the original isothermal sur- face is an ellipsoid. The first step is to find p^ which is proportional to the quantity of matter at any point in the surface of an ellipsoid, when the matter is so distributed that the attraction on a point within the ellipsoid is nothing. Now the attraction of a shell, bounded by two concentric similar ellipsoids, on a point within it, is nothing. If the shell be infinitely thin, its attraction will be the same as that of matter distributed over the surface of one of the ellipsoids in such a manner that the quantity on a given infinitely small area at any point is proportional to the thickness of the shell at the same point. Let a lt b lf c t be the semi-axes of one of the ellipsoids, a i + Sa 1 , b + S6 1} c x -f Sc t those of the other. Let also p^ be the perpendicular from the centre to the tangent plane at any point on the first ellipsoid, and Pi + $Pi ^ ne perpendicular from the centre to the tangent plane at a point similarly situated on the second. Then ^ is the thickness of the shell, since, the two ellipsoids being similar, the tangent planes at the points similarly situated on their surfaces are parallel. Also, on account of their similarity, CN ^7 ^ , - = - r 1 = * -^, and consequently the thickness of the shell a, b, c, Pl ' is proportional to p lt Hence we have, by (5), 1 dv. , , . = P *iPi ..................... ( a )> ^ ' where k^ is a constant, to be determined by the condition v = v lt at the surface of the ellipsoid. 12. To find the equation of the isothermal surface at which the temperature is v l + dv lt let - dv l = C, in (a). Then we have Uniform Motion of Heat and [i. G ^p^dn^ = r , or p^^ = # 13 where t is an infinitely small con- stant quantity ; and the required equation will be the equation of the surface traced by the extremity of the line dn^ drawn externally perpendicular to the ellipsoid. Let x', y t z' be the co-ordinates of any point in that surface, and x, y, z those of the corresponding point in the ellipsoid. Then, calling a lt /3 15 since l is infinitely small, and therefore also a i xx\ whence x' =,'(i_4)=^ V a?) 1 + ,' In a similar manner we should find *' y= - and z = 1+- 1 ? But 2 + ^i H 2 = Ij and hence we have H : 7rr 2 + < - f\f __ 1 __ 3 __ I __ 1 * + * *~ for the equation to the isothermal surface whose temperature isv 1 + dv 1) and which is therefore an ellipsoid described from the same foci as the original isothermal ellipsoid. In exactly the same manner it might be shown that the isothermal surface whose temperature is t> t + dv t + dv^, is an ellipsoid having the same foci as the ellipsoid whose temperature is ^ + dv v and consequently, as the original ellipsoid also. By continuing this I.] Mathematical Theory of Electricity. 9 process it may be proved that all the isothermal surfaces are ellipsoids, having the same foci as the original one. 13. From the form of the equation found above for the iso- thermal ellipsoid whose temperature is v -f dv^ it follows that 6 l or p^dn^ is = a^da^ where da 1 is the increment of a lt correspond- ing to the increment dn t of n r Hence, if a be one of the semi-axes of an ellipsoid, a + da the corresponding semi-axis of another ellipsoid having the same foci, dn the thickness at any point of the shell bounded by the two ellipsoids, and p the perpendicular from the centre to the plane touching either ellipsoid at the same point, we have dn a /1A -j- = - (6). da p 14. All that remains to be done is to find the temperature at the surface of any given ellipsoid, having the same foci as the original ellipsoid. For this purpose, let us first find the value of -7- at any point in the surface of the isothermal ellipsoid whose semi-axes are a, b, c. Now we have, from (a), dv where "k is constant for any point in the surface of the isothermal ellipsoid under consideration, and determined by the condition that the whole flux of heat across this surface must be equal to the whole flux across the surface of the original ellipsoid. Now 7 the first of these quantities is equal to ^Trkffpdca? (dco* being an element of the surface), or to 4?r K- ffSpdaf, since r\ ^ = . But ffSpdo) 2 is equal to the volume of a shell a p bounded by two similar ellipsoids, whose semi-axes are a, b, c, and a -f 8a, b + Sb, c + 8c, and is therefore readily shown to be 5* 7 equal to 4?r abc. Hence 4-7T -~- ffSpdco*, or 4i7rkffpda) 2 , is a oa equal to 4*Vkabc. In a similar manner we have, for the flux of heat across the original isothermal surface, 4V 2 & 1 a 1 & 1 c 1 , and therefore ^Vkabc =4V 2 ^ 1 a 1 ^ 1 c 1 , which gives k = k V 1 . 1 abc 10 Uniform Motion of Heat and [i. Hence we have dv 15. The value of v may be found by integrating this equation. To effect this, since a, b, c are the semi-axes of an ellipsoid passing through the variable extremity of n, and having the same foci as the original ellipsoid, whose axes are 15 b lt C 1? we have a 2 a* = b 2 b 2 = c 2 c 2 ; which gives b 2 = a 2 f 2 (d). where f = < - b?, \ , ,, o-j =/cosec (^ J " which we may do with propriety if f be the greater of the two quantities / and g y since a is always greater than either of them, as we see from (d). On this assumption, equation (e) becomes where c = 7 / I.] Mathematical Theory of Electricity. 11 18. Determining from this the values of C and k t by the conditions mentioned above, we find (7=0, and hence the expression for v becomes 0== v rErr (*) 19. The results which have been obtained may be stated as follows : If, in an infinite solid, the surface of an ellipsoid be retained at a constant temperature, the temperature of any point in the solid will be the same as that of any other point in the surface of an ellipsoid described from the same foci, and passing through that point ; and the flux of heat at any point in the surface of this ellipsoid will be proportional to the perpendicular from the centre to a plane touching it at the point, and inversely pro- portional to the volume of the ellipsoid. 20. This case of the uniform motion of heat was first solved by Lame, in his Me'moire on Isothermal Surfaces, in Liouville's Journal de Mathematiques, vol. ii. p. 147, by showing that a series of isothermal surfaces of the second order will satisfy the equation d*v d*v d*v_~ da? + ~ihf + di?~ ' provided they are all described from the same foci. The value which he finds for v agrees with (e), and he finds, for the flux of heat at any point, the expression KA or, according to the notation which we have employed, where v is the greater real semi-axis of the hyperboloid of one sheet, and p the real semi-axis of the hyperboloid of two sheets, described from the same foci as the original ellipsoid, and passing through the point considered. Hence a 2 , v 2 , p 2 are the three roots of the equation ^, f . * 3 =1 U II f 2 U Q 2 ' 12 Uniform Motion of Heat and [i. or Hence and V + a Therefore, 4 - t/y - (a 2 - 6 2 ) 2 2 + 2 (a 2 - 6 2 )(a 2 - c 2 ) a 4 - (a 2 - 6 2 ) (a 2 -c 2 ) - (6 2 + c 2 )^ 2 - (a 2 -c*)y z V + 6V - {(6 2 + c 2 )* 2 + (a 2 + c 2 )/ + (a 2 + & 2 )/}; which is readily shown, by substituting for a 2 6 2 + aV + 6V its equal (a 2 6 2 + aV + 6V) ( -. - 2 + j* + - 2 ) , to be equal to 5- . \d o c / p Hence the expression for ^- , given above, becomes ^- dv -j . , dn * a6c ^ which agrees with (c). Attraction of a Homogeneous Ellipsoid on a Point within or without it. 21. If, in (c), we put \ = 1 , the value of -y- at any point a^ an will be the attraction on that point of a shell bounded by two similar concentric ellipsoids, whose semi-axes are a,, a,V(l - e 2 ), a^(l - e'\ and a t + da^ (a, + rfaj V(l - O, K + cfej V(l - e /2 ) where a 2 - 6 2 = a/ 6^ = a x V ) , and a 2 -c 2 = a 1 2 -c 1 2 = a ) 2 e /2 J" the density of the shell being unity. Now this attraction is in I.] Mathematical Theory of Electricity. 13 a normal drawn through the point attracted to the surface of the ellipsoid, whose semi-axes are a, b, c. If we call a, /3, 7 the angles which this normal makes with the co-ordinates so, y, z of the point attracted, we have oc a px cos a = - = / (x y z ywt% j -i i a P!J P z and similarly, cos p =-^~ , cos 7=^-2-. 6 c Hence, calling d4, rfjB, c?(7 the components of the attraction parallel to the axes of co-ordinates, we have, from (c), i. - dA = d C = ab c b c .(2). 22. The integrals of these expressions, between the limits a t = and a l = a/, are the components of the attraction of an ellipsoid whose semi-axes are a/, &/, c/, or a/, a^(l-e z ) t a/V(l 'O on ^ e P omt (^ #> s: )- Now, by (1), we may express each of the quantities 6, c, 6 1? c x , in terms of a and a lf and the equation !? + F + ? =:1 ' or ? + a 2 -6V + 2 -^V = 1 "" (8)> enables us to express either of the quantities a, a in terms of the other. The simplest way, however, to integrate equations (2), will be to express each in terms of a third quantity, -; .............. ................. w- Eliminating a from (3), by means of this quantity, we have , f M -y w-v l , Hence a t da, = |^ 2 + ^_ & j + ^_ e 'j\ du = ( ~4 + |i + ~i ) a>*u*du = a*p~*u~ 3 du. \CZ> C / Also, from (4), we have a = ; from which we find, by (1), u 14 Uniform Motion of Heat. [i. 6 = ^ V(l - eV), c = - 1 V(l - e'V). By (1) also, \ = a t V(l - O> % w Cj = a t V(l - e ' 2 )- Making these substitutions in (2), and inte- grating, we have, calling CL the value of of the surface in the neigh- bourhood of the point. Coulomb's demonstration of this theorem may be found in a preceding paper in the Mathematical Journal, Vol. iii. p. 74 (above, i. 7). He gives it himself, in his sixth memoir on Electricity (Histoire de VAcademie, 1788, p. 677), in connexion with an investigation of the theory of the proof plane in which, by an error that is readily rectified, he arrives at the result that a small insulated conducting disc, put in contact with an elec- trified conductor at any point, and then removed, carries with it as much elec- tricity as lies on an element of the conductor at that point equal in area to the two faces of the disc ; the quantity actually removed being only half of this. This result, however, does not at all affect the experimental use which he makes of the proof plane, which is merely to find the ratios of the intensities at different points of a charged conductor. As the complete theory of this valuable instrument has not, so far as I am aware, been given in any English work, I annex the following remarkably clear account of it, which is ex- tracted from Pouillet's Traite de Physique: " Quand le plan d'epreuve est tangent a une surface, il se confond avec 1'element qu'il touche, il prend en quelque sorte sa place relativement a 1'electricite', ou plutot il devient lui- m6me 1'e" lament sur lequel la fluide se repand; ainsi, quand on retire ce plan, on fait la meme chose que si Ton avait de'coupe' sur la surface un element de meme epaisseur et de meme etendue que lui, et qu'on 1'eut enleve' pour le porter dans la balance sans qu'il perdit rien de 1'dlectricitd qui le IL] Elementary Laws of Statical Electricity. 17 even in some very singular phenomena, of the experimental results with the theoretical deductions. For a complete ac- count of the experiments we must refer to Coulomb's fifth memoir (Histoire de V Academic, 1787), and for the mathe- matical investigations to the first and second memoirs of Poisson (Mtmoires de Vlnstitut, 1811), or to the treatise on Electricity in the Encyclopaedia Metropolitana, where the sub- stance of Poisson's first memoir is given. The mathematical theory received by far the most complete development which it has hitherto obtained, in Green's Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism,* in which a series of general theorems were demonstrated, and many interesting applications made to actual problems. ( Of late years some distinguished experimentalists have begun to doubt the truth of the laws established by Coulomb, and have made extensive researches with a view to discover the laws of certain phenomena which they considered incompatible with his theory. The most remarkable works of this kind couvre; une fois separe de la surface, cet element n'aurait plus dans ses dif- f^rents points qu'une e'paisseur electrique moitid moindre, puisque la fluide devrait se rdpandre pour en couvrir les deux faces. Ce principe posd, 1'ex- pe'rience n'exige plus que de 1'habitude et de la dexte'rite': apres avoir touche* un point de la surface avec le plan d'e'preuve, on 1'apporte dans la balance, ou il partage son electricity avec le disque de 1' aiguille qui lui est gale, et Ton observe la force de torsion une distance connue. On re'p^te la me'me experience en touchant un autre point, et le rapport des forces de torsion est le rapport des repulsions electriques ; on en prend la racine carre'e pour avoir le rapport des dpaisseurs. Ainsi le ge'nie de Coulomb a donnd en me'me temps aux mathe'maticieiis la loi fondamentale suivant laquelle la matiere electrique s'attire et se repousse ; et aux physiciens une balance nouvelle, et des principes d'expdrience au moyen desquels ils peuvent en quelque sorte sonder 1'dpaisseur de 1' e'lectricite' sur tous les corps, et determiner les pressions qu'elle exerce sur les obstacles qui Farce 1 tent." To this explanation it should be added that, when the proof plane is still very near the body to which it has been applied, the effect of mutual influence is such as to make the intensity be insensible at every point of the disc on the side next the conductor, and at each point of the conductor which is under the disc. It is only when the disc is removed to a considerable distance that the electricity spreads itself symmetrically on its two faces, and that the intensity at the point of the conductor to which it was applied, recovers its original value. It was the omission of this consideration that caused Coulomb to fall into the error alluded to above. * Nottingham, 1828. t This memoir of Green's has been unfortunately very little known, either in this country or on the Continent. Some of the principal theorems in it T. B. 2 18 On the Mathematical Theory of Electricity. [n. have been undertaken independently by Mr Snow Harris and Mr Faraday, and in their memoirs, published in the Wulo sophical Transactions, we find detailed accounts of their re- searches. All the experiments, however, which they have made, having direct reference to the distribution of electricity in equilibrium, are, I think, in full accordance with the laws of Coulomb, and must therefore, instead of objections to his theory, be considered as confirming it. As, however, many have believed Coulomb's theory to be overturned by these investigations, and as others have at least been led to entertain doubts as to its certainty or accuracy, the following attempt to explain the apparent difficulties is made the subject of the first of a series of papers in which various parts of the mathe- matical theory of electricity, and corresponding problems in the theories of magnetism and heat, will be considered. 26. We may commence by examining some experimental results published in Mr Harris's first memoir On the Elemen- tary Laws of Electricity* After describing the instruments employed in his researches, Mr Harris gives the details of some experiments with -reference to the attraction exercised by an insulated electrified body on an uninsulated conductor placed in its neighbourhood. The first result which he an- have been re-discovered within the last few years, and published in the following works : Comptes Rendus for Feb. llth, 1839, where part of the series of theorems is announced without demonstration, by Chasles. Gauss's memoir on " General Theorems relating to Attractive and Ee- pulsive Forces, varying inversely as the square of the distance," in the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839, Leipsic, 1840. (Translations of this paper have been published in Taylor's Scientific Memoirs for April 1842, and in the Numbers of Liouville's Journal for July and August 1842.) Mathematical Journal, vol. iii., Feb. 1842, in a paper " On the Uniform Motion of Heat, etc." (i. above). Additions to the Connaissance des Terns for 1845 (published June 1842), where Chasles supplies demonstrations of the theorems which he had previously announced. I should add that it was not till the beginning of the present year (1845) that I succeeded in meeting with Green's Essay. The allusion made to his name with reference to the word "potential" (Mathematical Journal, vol. iii. p. 190), was taken from a memoir of Murphy's, " On definite Integrals with Physical Applications," in the Cambridge Transactions, where a mistaken definition of that term, as used by Green, is given. * Philosophical Transactions, 1834. II.] Elementary Laws of Statical Electricity. 19 nounces is that, when other circumstances remain the same, the attraction varies as the square of the quantity of electricity with which the insulated body is charged. It is readily seen, as was first remarked by Dr Whewell in his Report on the Theories of Electricity, etc.* that this is a rigorous deduction from the mathematical theory, following from the fact that the quantity of electricity induced upon the uninsulated body is proportional to the charge on the electrified body by which it is attracted. 27. The remaining results have reference to the force of attraction at different distances, and with bodies of different forms opposed. As these are generally very irregular (such as " plane circular areas backed by small cones "), we should not, according to Coulomb's theory, expect any very simple laws, such as Mr Harris discovers, to be rigorously true. Accord- ingly, though they are announced by him without restriction, we must examine whether the experiments from which they have been deduced are of a sufficiently comprehensive character to lead to any general conclusions with respect to electrical action. Now, in the first place, we find that in all of them the attraction is " independent of the form of the unopposed parts " of the bodies, which will be the case only when the intensity of the induced electricity on the unopposed parts of the un- insulated body is insensible. According to the mathematical theory, and according to Mr Faraday's researches " on induction in curved lines/' which will be referred to below, the intensity never absolutely vanishes at any point of the uninsulated body : but it is readily seen that in the case of Mr Harris's experiments, it will be so slight on the unopposed portions that it could not be perceived without experiments of a very refined nature, such as might be made by the proof plane of Coulomb, which is in fact, with a slight modification, the instrument employed by Mr Faraday in the investigation. Now to the degree of approximation to which the intensity on the unopposed parts may be neglected, the laws observed by Mr Harris when the opposed surfaces are plane may be readily deduced from the mathematical theory. Thus let v be the British Association Report for 1837. 22 20 On the Mathematical Theory of Electricity. [n. potential in the interior of the charged body, A ; a quantity which will depend solely on the state of the interior coating of the battery with which in Mr Harris's experiments A is connected, and will therefore be sensibly constant for different positions of A relative to the uninsulated opposed body, B. Let a be the distance between the plane opposed faces of A and B, and let S be the area of the opposed parts of these faces, which will in general be the area of the smaller, if they be unequal. When the distance a is so small that we may en- tirely neglect the intensity on all the unopposed parts of the bodies, it is readily shown from the mathematical theory that (since the difference of the potentials at the surfaces of A and B is v) the intensity of the electricity produced by induction at any point of the portion of the surface of B which is opposed to A, is -. . Hence the attraction on any small element w, 4-Tra of the portion S of the surface of B, will be in a direction (v \ a T ) .* Hence the whole attraction on B is This formula expresses all the laws stated by Mr Harris as results of his experiments in the case when the opposed surfaces are plane. 28. When the opposed surfaces are curved, for instance when A and B are equal spheres, we can make no approximation analogous to that which has led us to so simple an expression in the case of opposed planes; and we find accordingly that no such simple law for the attraction in this case has been announced by Mr Harris. He has, however, found that it is expressed with tolerable accuracy by the formula F= * c(c-2a)' where c is the distance between the centres of the spheres, a the radius of each, k a constant, which will dependjm a and on the charge of the battery with which A is in communica- * See VH. below. II. Elementary Laws of Statical Electricity. 21 tion. Though, however, this formula may give results which do not differ very much from observation within a limited range of distances, it cannot, according to any theory, be con- sidered as expressing the physical law of the phenomenon. For, according to it, when the balls are very distant, F ulti- mately varies as -^ . Now it is clear that the law of force must ultimately become the inverse cube of the distance, since the quantity of electricity induced upon B will be ultimately in the inverse ratio of the distance, and the attraction between the balls as the product of the quantities of electricity directly, and as the square of the distance inversely, and hence the formula given by Mr Harris cannot express the law of force when the balls are very distant. In the experiments by which his formula is tested, the force of attraction is measured by means of an ordinary balance and weights : the only com- parison of results which he publishes is transcribed in the following table : Dist. of Centres. Measured Force in Grains. Valuesof 15c ^- 2 >. c(c-2) Cl = 2-3 15 15 c 2 = 2-5 8-25 + 8-28 C 3 = 2-8 4-6 + 4-62 c 4 = 3-0 3-5- 3-45 29. From this table we see that the formula is verified in three cases to the extent of accuracy of the experiments. Comparisons extended to a much wider range of distances would be required to establish it, and it would be necessary to take precautions to prevent the experimental results from being influenced by disturbing causes. In the experiments made by Mr Harris, we find that no precautions have been taken to avoid the dis- turbing influence of extraneous conductors, which, according to the descriptions and drawings he gives of his instruments, seem to exist very abundantly in the neighbourhood of the bodies operated upon, being partly metal in connexion with the insulated system with which the body A communicates, and partly uninsulated metal, in the fixed parts of the electro- 22 On the Mathematical Theory of Electricity. [n. meter, and in the movable parts by which B is supported. The general effect produced by the presence of such bodies in disturbing the observed law of force, must be to make it diminish less rapidly with the distance when A and B are separated by a considerable interval : and it is probably owing, at least in part, to such disturbing causes that Mr Harris's results nearly agree, as far as they go, with a formula which would ultimately give for the law of force the inverse square of the distance between A and B, instead of the inverse cube. 30. The determination by the mathematical theory of the attraction or repulsion between two electrified conducting spheres has not hitherto, so far as I am aware, been attempted, and would present considerable difficulty by means of the formulae ordinarily given for such problems. It may, however, very readily be effected by means of a general theorem on the attraction between electrified conductors, which will be given in a subsequent paper.* Thus, if F(c) be the force of attraction, corresponding to the distance c between the centres, in the particular case when the two spheres are equal (the radius of each being unity), and the potential in the interior of one of them is nothing (as will be the case when the body is un- insulated), the potential in the interior of the other being v, I have found the following formulae, which express F(c) by a converging series : where * [Note added March 1854. The enunciation of the ''general theorem" alluded to, the investigation founded on it, by which the author first arrived at the conclusion made use of here, and another demonstration of the same conclusion, founded on the method of electrical images, and strictly synthe- tical in its character, are published, with comprehensive numerical results, in the Philosophical Magazine for April 1853.] II.] Elementary Laws of Statical Electricity. 31. These formulae enable us to calculate Q t , Q 2 , Q 3 , Q^ etc., and then P lf P 2 , P 3 , P 4 , etc., successively, by a simple and uniform arithmetical process, for any particular value of c, I have thus calculated the values of M in five cases, the first four of which are those examined by Mr Harris, and have obtained the following results, each of which is true to five places of decimals : c. v-*F(c). 2-3 0-32926 2-5 0-17423 2-8 0-09168 3-0 0-06592 4-0 0-02075 32. To compare these with Mr Harris's measurements, we may calculate the value of the potential in his battery, during the observations, by means of his first result, and thence find the attraction for the other three cases by means of the calcu- lated values of v~ 2 F(c). Thus we have v~* x 15 = '3293, which gives fl 2 = 45'56, and hence P(2'5) = 7'94, F(2'8) = 418, These numbers differ considerably from Mr Harris's results, but in the direction indicated by the considerations mentioned above. 33. The most important part of the researches of Mr Harris is that in which he investigates the insulating power of air of different densities. The result at which he arrives is, that the intensity necessary to produce a spark depends solely on the density of the air, and not otherwise on the pressure or tem- perature. He thus shows that the conducting power of flame, of heated bodies, and of a vacuum, are due solely to the rare- faction of the air in each case. He also shows that the in- tensities necessary to produce a spark are in the simple ratios of the densities of the air. 24 On the Mathematical Theory of Electricity. [n. 34. In a subsequent memoir, by the same author,* we find additional experiments on the elementary principles of the theory of electricity. The first series which is described, was made for the purpose of testing the truth of Coulomb's law, that the repulsion of two similarly charged points is inversely as the square of the distance, and directly as the product of the masses. In experiments of this kind in which accurate quantitative results are aimed at, many precautions are neces- sary. Thus all conducting bodies, except those operated upon, must be placed beyond the reach of influence, and the distance between the repelling bodies must be considerable with refer- ence to their linear dimensions, so that the distribution of electricity on each may be uninfluenced by the presence of the other. Also the bodies should be spheres, so that the attrac- tion may be the same as if the whole electricity of each were collected at its centre; and the distance to be measured will then be the distance between the centres. These conditions have been expressly mentioned by Coulomb, and they have been fulfilled, as far as possible, in his researches, as we see by the descriptions of the experiments made, which we find in his memoirs. He has thus arrived by direct measurement at the law, which we know by a mathematical demonstration, f founded upon independent experiments, to be the rigorous law of nature, for electrical action. None of these precautions, however, have been taken in the experiments described in Mr Harris's * Philosophical Transactions, 1836. t See Murphy's Electricity, p. 41, or Pratt's Mechanics, Art. 154. [Note added March 1854. Cavendish demonstrates mathematically that if the law of force be any other than the inverse square of the distance, electricity could not rest in equilibrium on the surface of a conductor. But experiment has shown that electricity does rest at the surface of a conductor. Hence the law of force must be the inverse square of the distance. Caven- dish considered the second proposition as highly probable, but had not ex- perimental evidence to support this opinion, in his published work (An attempt to explain the phenomena of Electricity by means of an Elastic Fluid). Since his time, the most perfect experimental evidence has been obtained that electricity resides at the surface of a conductor; in such facts, for instance, as the perfect equivalence in all electro -statical relations of a hollow metallic conductor of ever so thin substance, or of a gilt non-con- ductor (possessing a conducting film of not more than ^ 00 ^ 000 of an inch thick) and a solid conductor, of the same external form and dimensions ; the minor premise of his syllogism is thus demonstrated, and the conclusion is therefore established.] ii.] Elementary Laws of Statical Electricity. 25 memoir, and the results are accordingly unavailable for the accurate quantitative verification of any law, on account of the numerous unknown disturbing circumstances by which they are affected. The phenomena which he observes, however, afford qualitative illustrations of the mathematical theory of a very interesting nature, as may be seen from the following examples of his results : (a) When the distance between the bodies is great with reference to their linear dimensions, the repulsion is inversely as the square of the distance, and directly as the product of the masses. (6) When the distance is small, the action becomes ap- parently irregular. Thus if the quantities of electricity on the two bodies be equal, the force, which is always of repulsion, does not increase so rapidly when the bodies approach, as if it followed the law of the inverse square of the distance. (c) If the charges be unequal, the repulsion ceases at a certain distance, and at all smaller distances there is attraction between the bodies. 35. These results are, with all their peculiarities, in full ac- cordance with the theory of Coulomb, which indicates that, if the quantities of electricity be equal, and the bodies equal and similar, there will be repulsion in every position : but if there be any difference, however small, between the charges, the repulsion will necessarily cease, and attraction commence, before contact takes place, when one body is made to approach the other. Unless, however, the difference of the charges be sufficiently considerable, a spark may pass between the bodies, and render the charges equal, before attraction commences. In Mr Harris's experiments, in which the bodies seem to have been nearly oblate spheroids, the attraction is generally sensible before the distance is small enough to allow a spark to pass, if the charge on one be double of that on the other. Mr Harris next proceeds to investigate the theory of the proof plane, and to examine whether it can be considered as indicating with certainty the intensity of electricity at any part of a charged body, and, principally from an experiment made on a charged non-conductor (a hollow sphere of glass), comes to a negative conclusion. It should be remembered, 26 On the Mathematical Theory of Electricity. [n. however, that, the proof plane having never been applied to determine the intensity at points of the surface of a charged non-conductor, such conclusions in no way interfere with adopted ideas. Since there can be no manner of doubt as to the theory of this valuable instrument, as we find it explained by M. Pouillet,* nor as to the experimental use of it made by Coulomb, it is unnecessary to enter more at length on the subject here. 36. Mr Faraday's researches on electrostatical induction, which are published in a memoir forming the eleventh series of his Experimental Researches in Electricity, were under- taken with a view to test an idea which he had long possessed, that the forces of attraction and repulsion exercised by free electricity, are not the resultant of actions exercised at a dis- tance, but are propagated by means of molecular action among the contiguous particles of the insulating medium surrounding the electrified bodies, which he therefore calls the dielectric. By this idea he has been led to some very remarkable views upon induction, or, in fact, upon electrical action in general As it is impossible that the phenomena observed by Faraday can be incompatible with the results of experiment which constitute Coulomb's theory, it is to be expected that the difference of his ideas from those of Coulomb must arise solely from a different method of stating, and interpreting physically, the same laws : and farther, it may, I think, be ' shown that either method of viewing the subject, when carried sufficiently far, may be made the foundation of a mathematical theory which would lead to the elementary principles of the other as consequences. This theory would accordingly be the expres- sion of the ultimate law of the phenomena, independently of any physical hypothesis we might, from other circumstances, be led to adopt. That there are necessarily two distinct elementary ways of viewing the theory of electricity, may be seen from the following considerations, founded on the principles developed in a previous paper in this Journal.^ * See foot-note on 25. t On the Uniform Motion of Heat, and its Connexion with the Mathe- matical Theory of Electricity (i. above). ii.] Elementary Laws of Statical Electricity. 27 37. Corresponding to every problem relative to the distribu- tion of electricity on conductors, or to forces of attraction and repulsion exercised by electrified bodies, there is a problem in the uniform motion of heat which presents the same analytical conditions, and which, therefore, considered mathematically, is the same problem. Thus, let a conductor A, charged with a given quantity of electricity, be insulated in a hollow con- ducting shell, B, which we may suppose to be uninsulated. According to the mathematical theory, an equal quantity of electricity of the contrary kind will be attracted to the interior surface of B (or the surface of B, as we may call it to avoid circumlocution), and the distribution of this charge, and of the charge on A, will take place so that the resultant attraction at any point of each surface may be in the direction of the normal. This condition being satisfied, it will follow that there is no attraction on any point within A, or without the surface of B, that is, on any point within either of the conducting bodies. The most convenient mathematical expression for the condition of equilibrium, is that the potential at any poinfr P* must have a constant value when P is on the surface of A, and the value nothing when P is on the surface of B\ and it will follow from this that the potential will have the same constant value for any point within A, and will be equal to nothing for any point without the surface of B. If A be subject to the influence of any uninsulated con- ductors, we must consider such bodies as belonging to the shell in which A is contained, and their surfaces as forming part of the surface of B : in such cases this surface will gene- rally be the interior surface of the walls of the room in which A is contained, and of all uninsulated conductors in the room. If, however, we have to consider the case in which A is subject to no external influence, we must suppose every part of the surface of B to be very far from A. The most general problem we can contemplate in electricity (exclusively of the case in which the insulating medium is heterogeneous, and exercises a special action, which will be alluded to below), is to determine * The term used by Green for the sum of the quotients obtained by divid- ing the product of each element of the surfaces of A and B, and its electrical intensity, by its distance from P. 28 On the Mathematical Theory of Electricity. [n. the potential at any point when A, instead of being a single conductor, is a group of separate insulated conductors charged to different degrees, and when there are non-conductors elec- trified in a given manner, placed in the insulating medium, in the neighbourhood. The conditions of equilibrium will still be that the potential at each surface due to all the free electricity must be constant, and the theorems stated above will still be true: thus the attraction will be nothing in the interior of each portion of A, and without the surface of B\ and the whole quantity of induced electricity on the latter surface will be the algebraic sum of the charges of all the interior bodies with its sign changed. When the potential due to such a system is determined for every point, the component of the resultant force at any point P, in any direction PL, may be found by differentiation, being the limit of the difference between the values of the potential at P, and at a point Q, in PL, divided by PQ, when Q moves up towards and ultimately coincides with P, and the direction of the force, on a negative particle, being that in which the potential increases. By Coulomb's theorem, the intensity at any point in one of the conducting surfaces is equal to the attraction (on a negative unit) at that point, divided by 4?r. 38. Now if we wish to consider the corresponding problem in the theory of heat, we must suppose the space between A and B, instead of being filled with a dielectric medium (that is a non-conductor for electricity), to be occupied by any homo- geneous solid body, and sources of heat or cold to be so dis- tributed over the terminating surfaces, or the interior surface of B and the surface of A, that the permanent temperature at the first surface may be zero, and at the second shall have a certain constant value, the same as that of the potential in the case of electricity. If A consist of different isolated portions, the temperature at the surface of each will have a constant value, which is not necessarily the same for the different por- tions. The problem of distributing sources of heat, according to these conditions, is mathematically identical with the problem of distributing electricity in equilibrium on the surfaces of A and B. In the case of heat, the permanent temperature at any point replaces the potential at the corresponding point in the II.] Elementary Laws of Statical Electricity. 29 electrical system, and consequently the resultant flux of heat replaces the resultant attraction of the electrified bodies, in direction and magnitude. The problem in each case is deter- minate, and we may therefore employ the elementary principles of one theory, as theorems, relative to the other. Thus, in the paper in which these considerations are developed, Coulomb's fundamental theorem relative to electricity is applied to the theory of heat ; and self-evident propositions in the latter theory are made the foundation of Green's theorems in elec- tricity.* Now the laws of motion for heat which Fourier lays down in his Theorie Analytique de la Chaleur, are of that simple elementary kind which constitute a mathematical theory properly so called ; and therefore, when we find corresponding laws to be true for the phenomena presented by electrified bodies, we may make them the foundation of the mathematical theory of electricity : and this may be done if we consider them merely as actual truths, without adopting any physical hypothesis, although the idea they naturally suggest is that of the propagation of some effect by means of the mutual action of contiguous particles; just as Coulomb, although his laws naturally suggest the idea of material particles attracting or repelling one another at a distance, most carefully avoids making this a physical hypothesis, and confines himself to the consideration of the mechanical effects which he observes and their necessary consequences.-)- 39. All the views which Faraday has brought forward, and illustrated or demonstrated by experiment, lead to this method of establishing the mathematical theory, and, as far as the analysis is concerned, it would, in most general propositions, be even more simple, if possible, than that of Coulomb. (Of course the analysis of particular problems would be identical in the two methods.) It is thus that Faraday arrives at a knowledge of some of the most important of the general * It was not until some time after that paper was published, that I was able to add the direct analytical demonstrations of the theorems, which are given in the papers on " General Propositions in the Theory of Attraction," Cairib. Math. Jour. , vol. iii. pp. 189, 201 (xii. below), and which I have since found are the same as those originally given by Green. f See first foot note on 25. 30 On the Mathematical Theory of Electricity. [n. theorems, which, from their nature, seemed destined never to be perceived except as mathematical truths. Thus, in his theory, the following proposition is an elementary principle : Let any portion a of the surface of A be projected on B, by means of lines (which will be in general curved) possessing the property that the resultant electrical force at any point of each of them is in the direction of the tangent : the quantity of electricity produced by induction on this projection is equal to the quantity of the opposite kind of electricity on .* The lines thus defined are what Faraday calls the "curved lines of inductive action." For a detailed account of the experiments by which these phenomena are investigated, reference must be made to Mr Faraday's own memoirs, published in the Philo- sophical Transactions, and in a separate form in his Experi- mental Researches. 40. The hypothesis adopted by Faraday, of the propagation of inductive action, naturally led him to the idea that its effects may be in some degree dependent upon the nature of the insulating medium or dielectric, by which, according to this view, it is transmitted. In the second part of his memoir he describes a series of researches instituted to put this to the test of experiment, and arrives at the following conclusions : * This theorem may be proved as follows : Let S be any closed surface, containing no part of the electrified bodies within it, which we may conceive to be described between A and B ; let P be the component in the direction of the normal, of the resultant force at any point of the surface S, and let ds be an element of the surface at the same point. Then it may be easily proved (see Garrib. Math. Jour., vol. iii. p. 204) that _ ffPds = (a), the integrations being extended over the entire surface. Now let S be supposed to consist of three parts; the portion a, of the surface of A ; its projection j3, on the interior surface of B ; and the surface generated by the curved lines of projection. The value of P at each point of the latter portion of S will be nothing, since the tangent at any point of a line of pro- jection is the direction of the force. Hence, if [ffPds] and (ffPds) denote the values offfPds, for the portions a and /8 of S, the equation (a) becomes But if p be the intensity of the distribution on the surface A or .B, at any point, we have, by Coulomb's theorem, P which is the theorem quoted in the text. II.] Elementary Laws of Statical Electricity. 31 41. If the dielectric be air, the inductive action is quite inde- pendent of its density or temperature (which, as Mr Faraday remarks, agrees perfectly with previous results obtained by Mr Harris) ; and in general, if the dielectric be any gas or vapour capable of insulating a charge, the inductive action is invariable. Hence he concludes that " all gases have the same power of, or capacity for, sustaining induction through them (which might have been expected when it was found that no variation of density or pressure produced any effect)." When the dielectric is solid, the induction is greater than through air, and varies according to the nature of the sub- stance. Numbers which measure the "specific inductive capacities" of the dielectrics employed (sulphur, shell lac, glass, etc.) are deduced from the experiments. 42. To express these results in the language of the mathe- matical theory, let us recur to the supposition of a body, A, charged with a given quantity of electricity, and insulated in the interior of a closed conducting shell, B. The potential of the system at the interior surface of B, and at every point without this surface, will be nothing ; at the surface and in the interior of A it will have a constant value, which will depend on the form, magnitude, and relative position of the surfaces A and B, on the quantity of electricity on A, and, according to Faraday's discovery, on the dielectric power of the insulating medium which fills the space between A and B. If this be gaseous, neither its nature nor its state as to temperature, pressure, or density will affect the value of the potential in A but if it be a solid substance, such as sulphur or shell lac, the value of the potential will be less than when the space is occupied by air, and will vary with the nature of the insulating solid. 43. The result in the case of a gaseous dielectric is what would follow from Coulomb's theory, if we consider gases to be quite impermeable to electricity, and to be entirely unaffected by electrical influence. The phenomena observed with solid dielectrics, which agree with the circumstance observed by Nicholson, that the dissimulating power of a Leyden phial depends on the nature of the glass of which it is made, as well as on its thickness, have been by some attributed to a slight degree of conducting power, or of penetrability, pos- 32 On the Mathematical Theory of Electricity. [IT. sessed by solid insulators. This explanation, however, seems to be very insufficient ; and besides, Faraday has estimated the nature of the effects of imperfect insulation by independent experiments, and has established, in what seems to be a very satisfactory manner, the existence of a peculiar action in the interior of solid insulators when subjected to electrical influ- ence. As far as can be gathered from the experiments which have yet been made, it seems probable that a dielectric, sub- jected to electrical influence, becomes excited in such a manner that every portion of it, however small, possesses polarity exactly analogous to the magnetic polarity induced in the sub- stance of a piece of soft iron under the influence of a magnet. By means of a certain hypothesis regarding the nature of mag- netic action,* Poisson has investigated the mathematical laws of the distribution of magnetism, and of magnetic attractions and repulsions. These laws seem to represent in the most general manner the state of a body polarized by influence, and therefore, without adopting any particular mechanical hypo- thesis, we may make use of them to form a mathematical theory of electrical influence in dielectrics, the truth of which can only be established by a rigorous comparison of its results with experiment. 44. Let us therefore consider what would be the effect, accord- ing to this theory, which would be produced by the presence of a solid dielectric, (7, placed in the space between A and B, the rest of which is occupied by air. The action of C, when excited by the influence of the electricities on A and B, may (as Poisson has shown for magnetism) be represented, whether * Faraday adopts the corresponding hypothesis to explain the action of a solid dielectric, which he states thus: "If the space round a charged glohe were filled with a mixture of an insulating dielectric, as oil of turpentine or air, and small globular conductors, as shot, the latter being at a little dis- tance from each other, so as to be insulated, then these in their condition and action exactly resemble what I consider to be the condition and action of the particles of the insulating dielectric itself. If the globe were charged, these little conductors would all be polar; if the globe were discharged, they would all return to their normal state, to be polarized again upon the re- charging of the globe." (Experimental Researches, 1679.) The results of the mathematical analysis of such an action are given in the text. It may be added that the value of the coefficient k will differ sensibly from unity if the volume occupied by the small conducting balls bear a finite ratio to that occupied by the insulating medium. II.] Elementary Laws of Statical Electricity. 33 on points within or without C, by a certain distribution of positive electricity on one portion of the surface of C, and of an equal quantity of negative electricity on the remainder. The condition necessary and sufficient for determining this distribution may (as can be shown from Poisson's analysis) be expressed as follows. Let R be the resultant force on a point P without (7, and R on a point P f without C, due to the electrified surfaces A and B, and to the imagined distribu- tion on G. If P and P ' be taken infinitely near one another, and consequently each infinitely near the surface of (7, the component of R' in the direction of the normal must bear to the component of R in the same direction a constant ratio depending on the capacity for dielectric induction of the ; ; /. . , J _ matter of (7.* The components of R and R' in the tangent plane will of course be equal and in the same direction, and, if p be the intensity of the imagined distribution on the surface of C, in the neighbourhood of P and P' , the difference of the normal components will be 4>7rp, as is evident from Coulomb's theorem, referred to above. 45. Let us now suppose C to be a shell surrounding A, and let 8 and $', its interior and exterior surfaces, be surfaces of equilibrium in the system of forces due to the action of A and B, and of the polarity of C. It may be shown that the same surfaces S, S f , would necessarily be surfaces of equilibrium, if C were removed and the whole space were filled with air; and consequently, that the whole series of surfaces of equi- * From this it follows that, in the case of heat, G must be replaced by a body whose conducting power is k times as great as that of the matter oc- cupying the remainder of the space between A and B. [Note added March 1854. The same demonstration, of course, is applic- able to the influence of a piece of soft iron, or other "paramagnetic" (i.e., substance of ferro- magnetic inductive capacity), or to the reverse influence of a diamagnetic on the magnetic force in any locality near a magnet in which it can be placed, and shows that the lines of magnetic force will be altered by it precisely as the lines of motion of heat in corresponding thermal circum- stances would be altered by introducing a body of greater or of less conduct- ing power for heat. Hence we see how strict is the foundation for an analogy on which the conducting power of a magnetic medium for lines of force may be spoken of, and we have a perfect explanation of the condensing action of a paramagnetic, and the repulsive effect of a diamagnetic, upon the lines of force of a magnetic field, which have been described by Faraday. (Exp. ResearcJies, 2807, 2808.)] T. E, 3 34 On the Mathematical Theory of Electricity. [n, librium, commencing with A and ending with B, will be the same in the two cases. Hence the resultant force due to the excitation of the dielectric C (or to the imagined distributions of electricity on 8 and S' which produce it), on points within 8 or without 8', must be such as not to alter the distributions on A and B when the quantity on A is given ; and is therefore nothing. Accordingly, let Q be the total force on a point indefinitely near 8, and within it ; Q' the total force on a point without />', but indefinitely near it. Since the forces on points without 8 and within 8' indefinitely near the former points are, according to the law stated above, ~ and -y- , it follows* fc /c that the intensities of the imagined distributions on 8 and 8'. in the neighbourhood of the points considered, are Hence, if U, U' be the potentials at 8, S', due to A and B alone, and v the potential at any point P, it follows that the potential at P, due to the polarity of the dielectric, is or - l_ w + 1-f U', \ kj \ kj or - (l - j] v + (l - T) v, that is, 0, \ K/ \ KJ according as P is within S, within 8' and without S, or without 8'. Hence the total potential will be, according to the position of P, v - or v. Hence the sole effect of the dielectric C, on the state of A and B, is to diminish the potential in the interior of the former by the quantity * See Green's Essay, Art. 12; or above, i. 8.. II.] Elementary Laws of Statical Electricity. 35 If the whole space between A and B be occupied by the solid dielectric, the surfaces S and A will coincide, as also, S' and B, and therefore U=V, U' = 0. Hence the potential in the V interior of A will be -=- , K or the fraction j of the potential, with the same charge on A, /c and with a gaseous dielectric. From this it follows that, when the dielectric is solid, it would require, to produce a given potential in the interior of A, k times the charge which would be necessary to produce the same potential when the dielectric is gaseous, and therefore the body A in a given state, denned by the potential in its interior, produces on the interior surface of B, by induction, through the solid dielectric, a quantity of electricity k times as great as through a gaseous dielectric. On this account Faraday calls the property of a dielectric measured by Jc, its " specific inductive capacity." 46. In Faraday's experiments an apparatus (which is in fact a Leyden phial, in which any solid or fluid may be substituted for the glass dielectric of an ordinary Leyden phial) is used, corresponding to the case we have been considering, in which A is a conducting sphere (2*33 inches in diameter), and B a concentric spherical shell surrounding it (the distance between the surfaces of A and B being '62 of an inch). In the shell B there is an aperture into which a shell-lac stem is fixed ; a wire, attached to A, passes through the centre of this stem to the outside of the shell, and supports a ball of metal, M , which is thus insulated and connected with A. It may be shown that in such an apparatus the state of the ball A and of the shell B will approximately be not affected by the aperture in the latter, or by the wire supporting M, and that the distribu- tion of electricity on M will be approximately the same as if. the wire supporting it and the conductors A and B were re- moved. Hence the sole relation between A and M will be that the potentials in their interiors are the same ; and there- fore the latter, which is accessible, may be taken as an index of the state of the former. 47. To determine the specific inductive capacity of any di- electric, Faraday uses two apparatus of the kind just described, 32 36 On the Mathematical Theory of Electricity. [il. precisely equal and similar, in one of which the space between A and B is filled with air, and in the other with the dielectric to be examined. One of these apparatus is charged, and the intensity measured : the balls M, M' in the two are then made to touch and separate again, and the remaining intensity on the first (which is equal to the intensity imparted to the second) is measured. If this be found to differ from half the original intensity, it will follow that the specific inductive capacity of the substance examined differs from that of air, which is unity, and its value may be determined by means of a simple expression from the experimental data. To investi- gate this, let us first suppose each apparatus to be charged, and let it be required to find the intensity on the balls after they are made to touch, and then removed from mutual influence ; and let the dielectrics be any two substances, whose inductive capacities are k, k' . Let p, p be the intensities before, and a the common intensity after contact. Then, denoting by Q, Q' the quantities of electricity constituting the charges before, and q, q after contact, we shall have, by the principles already , , kp er q cr q developed, *=_. -= ~> = 7v' Q kp p Q p Q Also Hence we deduce - 1. For sulphur Faraday finds the value to be rather more than 2*2 ; for shell-lac, about 2 ; and for flint-glass, greater than 176. 50. The commonly received ideas of attraction and repul- sion exercised at a distance, independently of any intervening medium, are quite consistent with all the phenomena of elec- trical action which have been here adduced. Thus we may consider the particles of air in the neighbourhood of electrified bodies to be entirely uninfluenced, and therefore to produce no effect in the resultant action on any point: but the particles of a solid non-conductor must be considered as assuming a polarized state when under the influence of free electricity, so as to exercise attractions or repulsions on points at a distance, which, with the action due to the charged surfaces, produce the resultant force at any point. It is, no doubt, possible that such forces at a distance may be discovered to be produced entirely by the action of contiguous particles of some inter- vening medium, and we have an analogy for this in the case of heat, where certain effects which follow the same laws are undoubtedly propagated from particle to particle. It might also be found that magnetic forces are propagated by means of a second medium, and the force of gravitation by means of a third. We know nothing, however, of the molecular action by which such effects could be produced, and in the present state of physical science it is necessary to admit the known facts in each theory as the foundation of the ultimate laws of action at a distance. ST PETER'S COLLEGE, Nov. 22, 1845. III. ON THE ELECTEO-STATICAL CAPACITY OF A LEYDEN PHIAL AND OF A TELEGEAPH WIRE INSULATED IN THE AXIS OF A CYLINDRICAL CONDUCTING SHEATH.* [From the Philosophical Magazine, 1855, first half-year.] 51. The principles brought forward in the preceding articles On the Uniform Motion of Heat, etc., enable us with great ease to investigate the " capacity "f of a Leyden phial with either air, or any liquid or solid dielectric, and of other analogous arrange- ments, such as the copper wires in gutta-percha tubes under water, with which Faraday has recently performed such re- markable experiments. J 52. Thus, for a Leyden phial, let us suppose a portion S of the surface of a conductor A to be everywhere so near the surface of a conductor A' t that the distance between them at any point is a small fraction of the radii of curvature of each surface in the neighbourhood; and let z be the distance between them at a particular position, P. Then, by the analogy with heat, it is clear that if the two surfaces be kept at different electrical potentials, V and V, the potentials at equidistant points in any line across from one to the other will be in arithmetical V V progression. Hence - will be the rate of variation of the z potential perpendicularly across in the position P. If, in the first place, the dielectric be air, the electric force in the air * Communicated as an Additional Note to two papers (i. and n. above) " On the Uniform Motion of Heat in Homogeneous Solid Bodies, and its connexion with the Mathematical Theory of Electricity," and "On the Mathematical Theory of Electricity in Equilibrium;" only not in time to be appended to the reprints of those papers which appeared in the Philosophical Magazine, June and July 1854 (1854, i. and n.). t Defined (Philosophical Magazine, June 1853) for any conductor (subject or not to the influence of other conductors), as the quantity of electricity which it takes to charge it to unit potential. J Described in a lecture at the Royal Institution, Jan. 20, 1854, and subsequently published in the Philosophical Magazine (1854, i. p. 197). in.] Electro- Statical Capacity of a Leyden Phial, etc. 39 between the two about the position P will consequently be F- V , and therefore the electrical density (according to the theorem proved in the first article) on one surface must be 1 V V 1 V V + j- - , and on the other - -: . The quantity of 4?r z 4-7T z J electricity in the position P, on an area ds of the surface S, is 1 F- V therefore -r ds> and therefore the whole quantity on S is F- V [ds 4-7T J Z ' which is Green's general expression for the electrification of either coating of a Leyden phial. If the thickness of the dielectric be constant and equal to r, it becomes F- V'S 4?r r' 53. Now if A' be uninsulated, we have F' = ; and then, S to charge S to the potential F, it takes the quantity F x . Hence the " capacity " of S is |T 4-7TT ' If instead of air there be a solid or liquid dielectric of inductive capacity, k, occupying the space between the two surfaces, the quantity of heat conducted across, in the analogous thermal circumstances, would be k times as great as in the case cor- responding to the air dielectric, with the same difference of temperatures ; and in the actual electrical arrangement, the quantity of electricity on each of the conducting surfaces would be k times as great as with air for dielectric and the same dif- ference of potentials. The expression for the capacity of an actual Leyden phial is therefore kS_ 4-7TT' k being the inductive capacity of the solid non-conductor of which it is formed, r its thickness, and S the area of it which is coated on each side. 54. To investigate the capacity of a copper wire in the cir- cumstances experimented on by Faraday, let us first consider the analogous circumstances regarding the conduction of heat ; that is, let us consider the conduction of heat that would take place 40 On the Electro- Statical Capacity of a [in. across the gutta-percha, if the copper wire in its interior were kept continually at a temperature a little above that of the water which surrounds it. Here the quantity of heat flowing outwards from any length of the copper wire, the quantities flowing across different surfaces surrounding it in the gutta- percha, and the quantity flowing into the water from the same length of gutta-percha tube, in the same time, must be equal. But the areas of the same length of different cylindrical surfaces are proportional to their radii, and therefore the flow of heat across equal areas of different cylindrical surfaces in the gutta- percha, coaxial with the wire, must be inversely as their radii. Hence, in the corresponding electrical problem, with air as the dielectric instead of gutta-percha, if R denote the resultant electrical force at any point P in the air between an insulated, electrified, infinitely long cylindrical conductor, and an un- insulated, coaxial, hollow cylindrical conductor surrounding it, and if x be the distance of P from the axis, we have 7? A R= x' where A denotes a constant. But if v be the potential at P ; by the definition of " potential " we have dv -7- = jK. dx Hence dv _ __ A ^ dx x ' and, by integration, v = A log x + C. Assigning the constants A and C so that the potential may have the value V at the surface of the wire, and may vanish at the hollow conducting surface round it, if r and r denote the radii of these cylinders respectively, we have and 55. Taking x = r, we find by this the electric force in the air ill.] Leyden Phial and of a Telegraph Wire. 41 infinitely near the inner electrified conductor ; and dividing the value found, by 4?r (according to the general theorem), we have for the electrical density on the surface of the conductor. Multiplying this by 2?rH, the area of a length I of the surface, we find VI for the whole quantity of electricity on that length. Hence, if k be the specific inductive capacity of gutta-percha, the electri- city resting on a length I of the wire in the actual circumstances will amount to . kl , 7 2 ~T' v ' l s- Or if 8 denote the surface of the wire, we have, for the quantity of electricity which it holds, and therefore its capacity is the same as that of a Leyden phial with an equal area of coated glass of thickness equal to / r T r log - , if I denote the specific inductive capacity of the ft/ T glass. 56. In the case experimented on by Mr Faraday, the diameter of the wire was yg-th of an inch, and the exterior diameter of the gutta-percha covering was about four times as great. Hence the thickness of the equivalent Leyden phial must have been / 1 , 71 los 1 * 4 = T, ' 09 o 6 Tn ' 9Q-A& ft/ )*w A/ *O \/O As the surface of the wire amounted to 8300 square feet, we may infer that if the gutta-percha had only the same induc- tive capacity as glass (and it probably has a little greater), the insulated wire, when the outer surface of the gutta-percha was uninsulated, would have had an electrical capacity equal to that of an ordinary Leyden battery of 8300 square feet of coated glass -^d of an inch thick. INVEECLOY, ABBAN, June, 1854. IV. ON THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. (Art. xxxviu. of complete list in Mathematical and Physical Papers, Vol. i.) II. A STATEMENT OF THE PRINCIPLES ON WHICH THE MATHE- MATICAL THEORY OF ELECTRICITY IS FOUNDED. [Cambridge and Dublin Mathematical Journal, March, 1848.] 57. This paper may be regarded as introductory to some others which will follow, containing various investigations in the Theory of Electricity. The fundamental mathematical prin- ciples of the phenomena of Electricity in Equilibrium are stated and explained in as concise a manner as seems consistent with clearness. To avoid lengthening the paper and unnecessarily distracting the attention of the reader, no details are given with reference to the experiments which have been, or which might be, made for establishing the various propositions asserted; and, for the same reasons, scarcely any allusion is made to the his- tory of the subject. With regard to the nature of .the evidence on which the mathematical theory of electricity rests, the reader is referred to the preceding paper " On the Elementary Laws of Statical Electricity," where, besides some general ex- planations on the subject, the works containing accounts of the actual experimental researches of principal importance are indicated. That paper is marked as the first of a series which it was my intention to publish in this Journal, and of which the second now appears. In this series it will not be attempted to adhere to a systematic course of investigations such as might constitute a complete treatise on the subject; and my only reason for publishing this introductory article is for the sake of reference in other papers, there being no published work in which the principles are stated in a sufficiently concise and correct form, independently of any hypothesis, to be altogether satisfactory in the present state of science. The Two Kinds of Electricity. 58. If a piece of glass and a piece of resin are rubbed together and then separated, it is found that they attract one another IV.] Fundamental Laws and Principles. 43 mutually. The term electricity* has been applied to the agency developed in this operation; the excitation of the bodies, to which the attractive force is due, is called electrical, and the bodies so excited are said to be electrified, or to be charged with electricity. If second pieces of glass and resin be rubbed together and then separated, and placed in the neighbourhood of the first pair of electrified bodies, it may be observed (1) That the two pieces of glass repel one another. (2) That each piece of glass attracts each piece of resin. (3) That the two pieces of resin repel one another. Hence it is inferred that the two pieces of glass possess elec- trical properties which differ in their characteristics from those of the resin ; and the two kinds of electricity thus indicated are called vitreous and resinous, after the substances on which they are developed. Bodies may in various ways be made electric ; but the characteristics presented are always those of either vitreous electricity or resinous electricity. 59. An electrified body exerts no force, whether of attraction or of repulsion, upon any non-electric matter. When in any case bodies not previously electrified are observed to be attracted, or urged in any direction, by an electrical mass, it is because the bodies have become electrically excited by influence. 60. If a small piece of glass and a small piece of resin, which have been electrified by mutual friction, be placed successively in the same position in the neighbourhood of an electrified body, they will be acted upon by equal forces, in the same line, but in contrary directions. Hence the two bodies are said to be equally charged with the two kinds of electricity respectively. Electrical Quantity. 61. The force between two electrified bodies depends, ceteris paribus, on the amounts of their charges, or on the quantities of electricity which they possess. If a small piece of glass and a small piece of resin be electrified by mutual friction to such an extent that, when separated and placed at a unit of distance, they attract one another with a unit of force, the quantity of electricity possessed by the former * From ijXfKTpov, amber, on account of such phenomena having been first observed with amber as one of the substances rubbed together. 44 On ike Mathematical Theory of Electricity. [iv. is said to be unity ; tKe latter, possesses what may be called a unit of resinous electricity. If m bodies, each possessing a unit of vitreous electricity, be incorporated together, the single body thus composed is charged with m units of the same kind of electricity: It is said to possess a quantity of electricity equal to m, or its electrical mass is m. A similar definition is applicable with reference to the measure- ment of resinous electricity. 62. If two bodies possessing equal quantities of vitreous and resinous electricity be incorporated, the single body thus com- posed will be found either to be non-electric, or to be in such a state that, without the removal of any electricity of either kind from it, it may, merely by an alteration in the distribution of what it already possesses, be deprived of all electrical symptoms. Thus it appears that a body either vitreously or resinously electrified, may be deprived of its charge merely by supplying it with an equal quantity of the other kind of electricity. In consequence of this fact, we may establish a complete system of algebraic notation with reference to electrical quantity, whether of vitreous or resinous electricity, by adopting as universal the law that the total quantity of electricity possessed by two bodies, or the quantity possessed by one body made up of two, is equal to the sum of the quantities with which they are separately charged. Thus let m be the quantity of elec- tricity with which a vitreously electrified body is charged, and let m' be the quantity contained by a body equally charged with resinous electricity. We must have m + m = 0, and therefore m is equal to m. Now it is usual to regard vitreous electricity as positive ; and we must therefore regard the other kind as negative ; so that a body possessing m units of resinous electricity is to be considered as charged with a quantity - m of electricity. The Superposition of Electrical Forces. 63. If a body, electrified in a given invariable manner, be placed in the neighbourhood of any number of electrified bodies, it will experience a force which is the resultant of the forces that would be separately exerted upon it by the different bodies iv.] Fundamental Laws and Principles. 45 if they were placed in succession in the positions which they actually occupy, without any alteration in their electrical con- ditions. This law is true even if any number of the bodies considered be merely different parts of one continuous mass. COR. 1. The total mechanical action between two electrified bodies, whether parts of one continuous mass or isolated bodies, is the resultant of the forces due to the mutual actions between all parts of either body and all parts of the other, if we conceive the two bodies to be arbitrarily divided each into parts in any manner whatever. Cor. 2. We may, in any electrical problem, imagine the charge possessed by a body to be divided into two or more parts, each distributed arbitrarily with the sole condition that the sum of the quantities of electricity in any very small space of the body due to the different distributions shall be equal to the given quantity of electricity in that space, according to the actual distribution of electricity in the body ; and we may consider the force actually exerted upon any other electrified body as equivalent to the resultant of the forces due to these partial distributions. The Law of Force between Electrified Bodies. 64. The force between two small electrified bodies varies inversely as the square of the distance between them. COR. If two small bodies be charged respectively with quantities m and m of electricity, they will mutually repel with, a force equal to -^- ; (an action which will be really attractive when m and m have unlike signs, as would be the case were the bodies dissimilarly electrified). For two units, placed at a distance unity, repel with a force equal to unity, and therefore if placed at a distance A, they will repel with a force 2 ; and the expression for the repulsion between m units and m units is deduced from this, according to the principle of the superposition of forces, by multiplying by mm'. 46 On the Mathematical Theory of Electricity. [iv. Definition of the Resultant Electrical Force at a Point. 65. Let a unit of negative electricity be conceived to be con- centrated at a point P in the neighbourhood of an electrified body or group of bodies, without producing any alteration in the previously existing electrical distribution. The force exerted upon this electrical point is what we shall throughout under- stand as the resultant force at P due to the electricity of the body or bodies considered. COR. If R be the resultant force at P in any case, then the force actually exerted upon an electrical mass m, concen- trated at P, will be equal to mR. Electrical Equilibrium. 66. When a body held at rest is electrified, and when, being either subject to electrical action from other bodies, or entirely isolated, the distribution of its charge remains permanently unaltered, the electricity upon it is said to be in equilibrium. Electrical equilibrium may be disturbed in various ways. Thus if a body charged with electricity in equilibrium be touched, or even approached by another electrified body, the equilibrium may be broken, and can only be restored after a different distribution has been effected, by a motion of electricity through the body or along its surface : or if a body be initially electrified in any arbitrary manner, whether by friction or other- wise, it may be that, as soon as the exciting cause is removed, the electricity will either gradually become altered from its initial distribution, by moving slowly through the body, or will suddenly assume a certain definite distribution. The laws which regulate the distribution of electricity in equilibrium on bodies in various circumstances have been the subject of most important experimental researches ; and having been established with perfect precision by Coulomb, and placed beyond all doubt by verifications afforded in subsequent ex- periments, they constitute the foundation of an extremely in- teresting branch of the Mathematical Theory of Electricity. In connexion with these laws, and before stating them, it will be convenient to explain the nature of the distinction which is drawn between the two great classes of bodies in nature, called Conductors of Electricity, and Non-Conductors of Electricity. IV.] Fundamental Laws and Principles. 47 Non-Conductors of Electricity. 67. A body which affords such a resistance to the transmis- sion of electricity through it, or along its surface, that, if it be once electrified in any way, it retains permanently, without any change of distribution, the charge which it has received, is called a Non-Conductor of Electricity. No body exists in nature which fulfils strictly the terms of this definition; but glass and resin, besides many other substances, are such that they may, within certain limits and subject to certain restrictions, be considered as non-conductors. Conductors of Electricity. 68. A very extensive class of bodies in nature, including all the metals, many liquids, etc., are found to possess the property that, in all conceivable circumstances of electrical excitation, the resultant force at any point within their substance vanishes. Such bodies are called Conductors of Electricity, since they are destitute of the property, possessed by non-conductors, of retaining permanently, by a resistance to every change, any distribution of electricity arbitrarily imposed ; the only kind of distribution which can exist unchanged for an instant on a conductor being such as satisfies the condition that the resultant force must vanish in the interior. It is found by experiment that the electricity of a charged conductor rests entirely on its surface, and that the electrical circumstances are not at all affected by the nature of the interior, but depend solely upon the form of the external conducting surface. Thus the electrical properties of a solid conductor, of a hollow conducting shell, or of a non-conductor enclosed in an envelop, however thin (the finest gold leaf, for instance), are identical, provided the external forms be the same. A hollow conductor never shows symptoms of electricity on its interior surface, unless an electrified body be insulated within it ; in which case the interior surface will become elec- trified by influence or by induction, in such a way as to make the total resultant force at any point in the conducting matter vanish, by balancing, for any such point, the force due to the electricity of the insulated body. 48 On the Mathematical Theory of Electricity. [iv. It has been frequently assumed that electricity penetrates to a finite depth below the surface of conductors ; and, in accord- ance with certain hypothetical ideas regarding the nature of electricity, the "thickness of the stratum" at different points of the surface of a conductor has been considered as a suitable term with reference to the varying or uniform distribution of electricity over the body. All the conclusion with reference to this delicate subject which can as yet be drawn from experiment, is that the "thickness," if it exist at all, must be less than that of the finest gold leaf ; and in the present state of science we must regard it as immeasurably small. It may be conceived that the actual thickness of the excited stratum at the surface of an electrified conductor is of the same order as the space through which the physical properties of the pervading matter change continuously from those of the solids to those which characterize the surrounding air. Electrical Density at any Point of a Charged Surface. 69. In this, and in all the papers which will follow, instead of the expression "the thickness of the stratum," Coulomb's far more philosophical term, Electrical Density, will be employed with reference to the distribution of electricity on the surface of a body ; a term which is to be understood strictly in accordance to the following definitions, without involving even the idea of a hypothesis regarding the nature of electricity. The electrical density of a uniformly charged surface is the quantity of electricity distributed over a unit of surface. The electrical density at any point of a surface, whether the distribution be uniform or not, is the quotient obtained by dividing the quantity of electricity distributed over an infinitely small element at this point, by the area of the element. Exclusion of all Non-Conductors except Air. 70. In the present paper, and in some others which will follow, no bodies will be considered except conductors; and the air surrounding them, which will be considered as offering a resistance to the transference of electricity between two detached conductors, but as otherwise destitute of electrical properties. A full development of the mathematical theory, of the internal electrical polarization of solid or liquid non-con- IV.] Fundamental Laws and Principles. 49 ductors, subject to the influence of electrified bodies, discovered by Faraday (in his Experimental Researches on the specific in- ductive capacities of non-conducting media), must be reserved for a later communication.* Insulated Conductors. 71. A conductor separated from the ground, and touched only by air, is said to be insulated. Insulation may be practically effected by means of solid props of matter, such as glass, shell- lac, or gutta percha;-(- and if the props be sufficiently thin, it is found that their presence does not in any way alter or affect the electrical circumstances, and that their resisting power, as non-conductors of electricity, prevents any alteration in the quantity of electricity possessed by the insulated body ; so that however the distribution may be affected by the influence of surrounding bodies, it is only by a temporary breaking of the insulation that the absolute charge can be in- creased or diminished. If an insulated uncharged conductor be placed in the neigh- bourhood of bodies charged with electricity, it will become "electrified by influence," in such a manner that its resultant electrical force at every internal point shall counterbalance the force due to the exterior charged bodies: but, in accordance with what has been stated in the preceding paragraph, the total quantity of electricity will remain equal to nothing; that is to say, the two kinds of electricity produced upon it by influence will be equal to one another in amount. Recapitulation of the Fundamental Laws. 72. The laws of electricity in equilibrium in relation with conductors may if we tacitly take into account such principles * The results of this Theory were explained briefly in a paper entitled " Note sur les Lois Ele"mentaires de 1'Electricite" Statique" (published, in 1845, in Liouville's Journal), and more fully in the first paper of the present series, on the "Mathematical Theory of Electricity" (n. above). A similar view of this subject has been taken by Mossotti, whose investigations are published in a paper entitled "Discussione Analitica sulP Influenza che 1'Azione di un Mezzo Dielettrico ha sulla Distributione dell' Elettricita alia Superficie di piu Corpi Elettrici Disseminati in Esso" (Vol. xxiv. of the Memorie delta Societa Italiana delle Scienze Eesidente in Modena, dated 1846). t It has been recently discovered by Faraday that gutta percha is one of the best insulators among known substances (Phil. Mag., March, 1848). T. E. 4" 50 On the Mathematical Theory of Electricity. [iv. as the superposition of electrical forces, and the invariableness of the quantity of electricity on a body, except by addition or subtraction (in the extended algebraic sense of these terms) be considered as fully expressed in the three following pro- positions : I. The repulsion between two electrical points is inversely proportional to the square of their distance. II. Electricity resides at the boundary of a charged conductor. III. The resultant force at any point in the substance of a conductor, due to all existing electrified bodies, vanishes. It has been proved by Green that the second of these laws is a mathematical consequence of the first and third ; and it has been demonstrated by La Place* that the first law may be in- ferred from the truth, in a certain particular case, of the second and third. The three laws were, however, first announced by Coulomb, as the result of his experimental researches on the subject. Objects of the Mathematical Theory of Electricity. 73. The varied problems which occur in the mathematical theory of electricity in equilibrium may be divided into the two great classes of Synthetical and Analytical investigations. In problems of the former class, the object is in each case the determination either of a resultant force or of an aggregate electrical mass, according to special data regarding distributions of electricity : in the latter class, inverse problems, such as the determination of the electrical density at each point of the surface of a conductor in any circumstances, according to the laws stated above, are the objects proposed. It has been proved (by Green and Gauss) that there is a determinate unique solution of every actual analytical problem of the Theory of Electricity in relation with conductors. The demonstration of this with reference to the complete Theory of Electricity (including the action of solid non-conducting media discovered by Faraday), as well as with reference to the Theories of Heat, Magnetism, and Hydrodynamics, may be deduced from two theorems proved in the Cambridge and Dublin Mathemati- cal Journal for 1847, "Regarding the Solution of certain Partial * [Originally by Cavendish, as I learned after the first publication of this paper. See footnote of March 1854 on 34 above.] IV.] Fundamental Laws and Principles. 51 Differential Equations" (xm. below, or Thomson and Tait's Natural Philosophy, App. A.). The full investigation of any actual case of electrical equi- librium will generally involve both analytical and synthetical problems ; as it may be desirable, besides determining the distribution, to 'find the resulting electrical force at points not in the interior of any conductor, or to find the total mechanical action due to the attractions or repulsions of the elements of two conductors, or of two portions of one conductor ; and besides, it is frequently interesting to verify synthetically the solutions obtained for analytical problems. Actual Progress in the Mathematical Theory of Electricity. 74. In Poisson's valuable memoirs on this subject, the dis- tribution of electricity on two electrified spheres, uninfluenced by other electric matter, is considered ; a complete solution of the analytical problem is arrived at ; and various special cases of interest are examined in detail with great rigor. In a very elaborate memoir by Plana*, the solution given by Poisson is worked out much more fully, the excessive mathematical difficul- ties in the way of many actual numerical applications of interest being such as to render a work of this kind extremely important. The distribution of electricity on an ellipsoid (including the extreme cases of elliptic and circular discs, and of a straight rod), and the results of consequent synthetical investigations are well known. The analytical problem regarding an ellipsoid subject, to the influence of given electrical masses, has been solved by M. Liouville, by the aid of a very refined mathematical method suggested by some investigations of M. Lame with reference to corresponding problems in the Theory of Heat. Green's Essay on Electricity and his other papers on allied subjects contain, besides the solution of several special problems of interest, most valuable discoveries with reference to the general Theory of Attraction, and open the way to much more extended investigations in the Theory of Electricity than any that have yet been published. GLASGOW COLLEGE, March 4, 1848. * Turin Academy of Sciences, tome vii. Serie u. published separately in a quarto volume of 333 pages : Turin, 1845. 4 2 V._ ON THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. (Art. xxxvin. of complete list in Mathematical and Physical Papers, Vol. i.) III. GEOMETRICAL INVESTIGATIONS WITH REFERENCE TO THE DISTRIBUTION OF ELECTRICITY ON SPHERICAL CONDUCTORS.* [Cambridge and Dublin Mathematical Journal, March, May, and Nov. 1848, Nov. 1849, Feb. 1850.] 75. There is no branch of physical science which affords a surer foundation, or more definite objects for the application of mathematical reasoning, than the theory of electricity. The small amount of attention which this most attractive subject has obtained is no doubt owing to the extreme difficulty of the analysis by which even a very limited progress has as yet been made ; and no other circumstance could have totally excluded from an elementary course of reading, a subject which, besides its great physical importance, abounds so much in beautiful illustrations of ordinary mechanical principles. This character of difficulty and impracticability is not however inseparable from the mathematical theory of electricity: by very elemen- tary geometrical investigations we may arrive at the solution * The investigations given in this paper ( 75 127) form the subject of the first part of a series of lectures on the Mathematical Theory of Electricity given in the University of Glasgow during the present session [1847 8]. They are adaptations of certain methods of proof which first occurred to me as appli- cations of the principle of electrical images, made with a view to investigating the solutions of various problems regarding spherical conductors, without the explicit use of the differential or integral calculus. The spirit, if not the notation, of the differential calculus must enter into any investigations with reference to Green's theory of the potential, and therefore a more extended view of the subject is reserved for a second part of the course of lectures. A complete exposition of the principle of electrical images (of which a short account was read at the late meeting of the British Association at Oxford) has not yet been published; but an outline of it was communicated by me to M. Liouville in three letters, of which extracts are published in the Journal de Mathematiques (1845 and 1847, vols. x., xii.). [See xiv. below.] A full and elegant exposition of the method indicated, together with some highly interesting applications to problems in geometry not contemplated by me, are given by M. Liouville himself, in an article written with reference to those letters, and published along with the last of them. I cannot neglect the present opportunity of expressing my thanks for the honour which has thus been conferred upon me by so distinguished a mathematician, as well as for the kind manner in which he received those communications, imperfect as they were, and for the favour- able mention made of them in his own valuable memoir. v.] Geometrical Investigations regarding Spherical Conductors. 53 of a great variety of interesting problems with reference to the distribution of electricity on spherical conductors, including Poisson's celebrated problem of the two spheres, and others which might at first sight be regarded as presenting difficulties of a far higher order. The object of the following paper is to present, in as simple a form as possible, some investigations of this kind. The methods followed, being for the most part synthetical, were suggested by a knowledge of results founded on a less restricted view of the theory of electricity; and it must not be considered either that they constitute the best or the easiest way of ad- vancing towards a complete knowledge of the subject, or that they would be suitable as instruments of research in endeavour- ing to arrive at the solutions of new problems. Insulated Conducting Sphere subject to no External Influence. 76. We may commence with the simplest possible case, that of a spherical conductor, charged with electricity and insulated in a position removed from all other bodies which could influence the distribution of its charge. In this, as in the other cases which will be considered, the various problems, of the analytical and synthetical classes, alluded to in a previous paper (iv. 73), will be successively subjects of investigation. Thus let us first determine the density at any point of the surface, and then, after verifying the result by showing that the laws .( 72) are satisfied, let us investigate the resultant force at an external point. Determination of the Distribution. 77. Let a be the radius of the sphere, and E the amount of the charge. According to Law II., the whole charge will reside on the surface, and, on account of the symmetry, it must be uniformly distributed. Hence, if p be the required density at any point, we have E Verification of Law III. 78. The well-known theorem, that the resultant force due to a uniform spherical shell vanishes for any interior point, consti- tutes the verification required in this case. This theorem was 54 On the Mathematical Theory of Electricity [v. first given by Newton, and is to be found in the Principia; but as his demonstration is the foundation of every synthetical investigation which will be given in this paper, it may not be superfluous to insert it here; and accordingly the passage of the Principia in which it occurs, translated literally, is given here. Newton, First Book, Twelfth Section, Prop.LXX. Theorem XXX. If the different points of a spherical surface attract equally with forces varying inversely as the squares of the distances, a particle placed within the surface is not attracted in any direction. Let HIKL be the spherical surface, and P the particle within it. Let two lines HK, IL, intercepting very small arcs HI, KL, be drawn through P; then on account jy^^*:^-^ of the similar triangles HPI, KPL (Cor. * 3, Lemma VII. Newton), those arcs will be proportional to the distances HP, LP\ and any small elements of the spherical surface at HI and KL, each bounded all round by straight lines passing through P [and very nearly coinciding with HK\, will be in the duplicate ratio of those lines. Hence the forces exercised by the matter of these elements on the particle P are equal ; for they are as the quantities of matter directly, and the squares of the distances, inversely ; and these two ratios com- pounded give that of equality. The attractions therefore, being equal and opposite, destroy one another : and a similar proof shows that all the attractions due to the whole spherical sur- face are destroyed by contrary attractions. Hence the particle P is not urged in any direction by these attractions. Q, E. D. Digression on the Division of Surfaces into Elements. 79. The division of a spherical surface into infinitely small elements will frequently occur in the investigations which follow : and Newton's method, described in the preceding de- monstration, in which the division is effected in such a manner that all the parts may be taken together in pairs of opposite elements with reference to an internal point; besides other V.] Geometrical Investigations regarding Spherical Conductors. 55 methods deduced from it, suitable to the special problems to be examined; will be repeatedly employed. The present digression, in which some definitions and elementary geometrical pro- positions regarding this subject are laid down, will simplify the subsequent demonstrations, both by enabling us, through the use of convenient terms, to avoid circumlocution, and by affording us convenient means of reference for elementary principles, regarding which repeated explanations might other- wise be necessary. Explanations and Definitions regarding Cones. 80. If a straight line which constantly passes through a fixed point be moved in any manner, it is said to describe, or generate, a conical surface of which the fixed point is the vertex. If the generating line be carried from a given position con- tinuously through any series of positions, no two of which coincide, till it is brought back, to the first, the entire line on the two sides of the fixed point will generate a complete conical surface, consisting of two sheets, which are called vertical or opposite cones. Thus the elements HI and KL, described in Newton's demonstration given above, may be considered as being cut from the spherical surface by two opposite cones having P for their common vertex. The Solid Angle of a Cone, or of a complete Conical Surface. 81. If any number of spheres be described from the vertex of a cone as centre, the segments cut from the concentric spherical surfaces will be similar, and their areas will be as the squares of the radii. The quotient obtained by dividing the area of one of these segments by the square of the radius of ~ the spherical surface from which, it is cut, is taken as the measure of the solid angle of the cone. The segments of the same spherical surfaces made by the opposite cone, are re- spectively equal and similar to the former. Hence the solid angles of two vertical or opposite cones are equal : either may be taken as the solid angle of the complete conical surface, of which the opposite cones are the two sheets. 56 On the Mathematical Theory of Electricity. [v. Sum of all the Solid Angles round a Point = 4?r. 82. Since the area of a spherical surface is equal to the square of its radius multiplied by 4-Tr, it follows that the sum of the solid angles of all the distinct cones which can be de- scribed with a given point as vertex, is equal to 4?r. Sum of the Solid Angles of all the complete Conical Surfaces = 2?r. 83. The solid angles of vertical or opposite cones being equal, we may infer from what precedes that the sum of the solid angles of all the complete conical surfaces which can be described without mutual intersection, with a given point as vertex, is equal to 2?r. Solid Angle subtended at a Point by a Terminated Surface. 84. The solid angle subtended at a point by a superficial area of any kind, is the solid angle of the cone generated by a straight line passing through the point, and carried entirely round the boundary of the area. Orthogonal and Oblique Sections of a Small Cone. 85. A very small cone, that is, a cone such that any two positions of the generating line contain but a very small angle, is said to be cut at right angles, or orthogonally, by a spherical surface described from its vertex as centre, or by any surface, whether plane or curved, which touches the spherical surface at the part where the cone is cut by it. A very small cone is said to be cut obliquely, when the section is inclined at any finite angle to an orthogonal section ; and this angle of inclination is called the obliquity of the section. The area of an orthogonal section of a very small cone is equal to the area of an oblique section in the same position, multiplied by the cosine of the obliquity. Hence the area of an oblique section of a small cone is equal the quotient obtained by dividing the product of the square of its distance from the vertex, into the solid angle, by the cosine of the obliquity. Area of the Segment cut from a Spherical Surface by a Small Cone. 86. Let E denote the area of a very small element of a v.] Geometrical Investigations regarding Spfierical Conductors. 57 spherical surface at the point E (that is to say, an element every part of which is very near the point E), let o> denote the solid angle subtended by E at any point P, and let PE, produced if necessary, meet the surface again in E' : then, a denoting the radius of the spherical surface, we have 2a.co.PE 2 For, the obliquity of the element E } considered as a section of the cone of which P is the vertex and the element E, a section ; being the angle between , the given spherical surface and another de- scribed from P as centre, with PE as radius ; is equal to the angle between the radii, EP and EC, of the two spheres. Hence, by con- sidering the isosceles triangle ECE' t we find that the cosine of the obliquity is equal to ^ -^ , or to -= , and we arrive at j&O LQj the preceding expression for E. 87. Theorem.* The attraction of a uniform spherical surface on an external point is the same as if the whole mass were collected at the centre. Let P be the external point, C the centre of the sphere, and CAP a straight line cutting the spherical surface in A. Take / in <7P, so that CP, CA, CI may be continual proportionals, and let the whole spherical surface be divided into pairs of opposite elements with reference to the point I. Let H and H' denote the magnitudes of a pair of such * This theorem, which is more comprehensive than that of Newton in his first proposition regarding attraction on an external point (Prop. LXXI.), is fully established as a corollary to a subsequent proposition (Prop. LXXIII. Cor. 2). If we had considered the proportion of the forces exerted upon two external points at different distances, instead of, as in the text, investigating the absolute force on one point, and if besides we had taken together all the pairs of elements which would constitute two narrow annular portions of the surface, in planes perpendicular to PC, the theorem and its demonstration would have coincided precisely with Prop. LXXI. of the Principia. 58 On the Mathematical Theory of Electricity. [v. elements, situated respectively at the extremities of a chord HH' ; and let &> denote the magnitude of the solid angle sub- tended by either of these elements at the point /. We have ( 85) a>.IH' 2 ~cos CHI' cos CHI' Hence, if p denote the density of the surface ( 69), the attrac- tions of the two elements H and H' on P are respectively to IH 2 a, 1H' p cos CHI ' PH 2 ' and p cos CH'I ' PH' 2 ' Now the two triangles P CH, HCI have a common angle at C, - and, since PC: CH :: CH : CI, the sides about this angle are proportional. Hence the triangles are similar ; so that the angles CPH and CHI are equal, and IH_CH_ a*_ HP~ CP ~ CP' In the same way it may be proved, by considering the triangles PCH',H'CI,tlia,t the angles CPH' and CH'I are equal, and that IH' = CH' _ a H'P CP~CP' Hence the expressions for the attractions of the elements H and H' on P become G) a 2 , ft> a* , and p cos CHI-CP' cos CH'I - which are equal, since the triangle H CH' is isosceles ; and, for the same reason, .the angles CPH, CPH', which have been proved to be respectively equal to the angles CHI, CH'I, are equal. We infer that the resultant of the forces due to the two elements is in the direction PC, and is equal to To find the total force on P, we must take the sum of all * -From this we infer that the ratio of IH to HP is constant, whatever be the position of H on the spherical surface, a well-known proposition. (Thomson's Euclid, vi. Prop. G.) v.] Geometrical Investigations regarding Spherical Conductors. 59 the forces along PC due to the pairs of opposite elements; and, since the multiplier of co is the same for each pair, we must add all the values of co, and we therefore obtain ( 83), for the required resultant, > gpr The numerator of this expression; being the product of the density into the area of the spherical surface ; is equal to the mass of the entire charge ; and therefore the force on P is the same as if the whole mass were collected at C. Q. E. D. COK. The force on an external point, infinitely near the surface, is equal to 4?rp, and is in the direction of a normal at the point. The force on an internal point, however near the surface, is, by a preceding proposition, equal to nothing. Repulsion on an element of the Electrified Surface. 88. Let , at P, be taken. The area of this element will be equal to cosCHP' and therefore the repulsion along HP, which it exerts on the element a at P, will be equal to po-p* or ^__ D v n cosCHP' *cosCHP p Now the total repulsion on the element at P is in the direction CP- } the component in this direction of the repulsion due to ft the element H, is c co . p V ; and, since all the cones corresponding to the different elements of the spherical surface lie on the same side of the tangent plane at P, we deduce, for the resultant repulsion on the & element cr, 27T/>V. From the corollary to the preceding proposition, it follows that 60 On, the Mathematical Theory of Electricity. [v. this repulsion is half the force which would be exerted on an external point, possessing the same quantity of electricity as the element at the point T\ and let elements E and E' be taken subtending at S the same solid angles respectively as the elements K and K'. By this means we may divide the whole spherical surface into pairs of conjugate elements, E y E' , since it is easily seen that when we have taken every pair of elements, K, K r , the whole surface will * If, in geometrical investigations in which diagrams are referred to, the distinction of positive and negative quantities be observed, the order of the letters expressing a straight line will determine the algebraic sign of the quantity denoted: thus we should have, universally, if A, B be the extremities of a straight line, AB^-BA, each member of this equation being positive or negative according to the conventional direction in which positive quantities are estimated. In the present instance, lengths measured along the line SP in the direction from S towards P, or in corresponding directions in the continua- tion of this line on either side, are, in both figures, considered as positive. Hence, in the first figure ST will be positive; but when / is less than a, ST must be negative on account of the equation SP . STf 2 -a z . Hence the second figure represents this case ; and, if we wish to express the circumstances without the use of negative quantities, we must change the signs of both members of the equation, and substitute for the positive quantity - ST its equivalent TS, so that we have SP . TS = a 2 -/ 2 , as the most convenient form of the expression, when reference is made to the second figure. See above (Symbolical Geometry, 4), in volume of the Cambridge and Dublin Mathe- matical Journal for 1848, where the principles of interpretation of the sign - in geometry are laid down by Sir William K. Hamilton [or Tait's Quaternions, % 20, 1868]. 62 On the Mathematical Theory of Electricity. [v. have been exhausted, without repetition, by the deduced ele- FlG. 1. FIG. 2. ments, E, E'. Hence the attraction on P will be the final re- sultant of the attractions of all the pairs of elements, E E' . Now if p be the electrical density at E, and if F denote the attraction of the element E on P, we have According to the given law of density we shall have \ TU< '<% Ht^ **<* o^ P = s>^Jl ,c^u< ^^ 1 7* C-yyv* S * \* ' *^ where \ is a constant. Again, since SEK is equally inclined to the spherical surface at the two points of intersection, we have ( 85, 86) Wv^ *x. K \*j.T AS -p 8K*' KK' and hence \ SE* 2aa>.TK* TK* EP* ' KK' ' SE . SK*. EP* Now, by considering the great circle in which the sphere is cut by a plane through the line SK, we find that /fir* "1 \ Of TF CUT 1 -fl 2\ (tig. 1) bK . bE =J -a \ //2 o\ T^Cf Q ~C* 2 ^*2 ( * V /* (tig. 2) A& . bh = a -/ J and hence SK. SE = P. ST, from which we infer that the triangles KST, PSE are similar; so that TK: SK:: PE: SP. Hence TK 2 and the expression for F becomes _ KK" SE.SP*' v.] Geometrical Investigations regarding Spherical Conductors. 63 Modifying this by (2) we have (fig. 2) ^ Similarly, if F' denote the attraction of E' on P, we have | (fig.D (fig. 2) F' Now in the triangles which have been shown to be similar, the angles TKS, EPS are equal ; and the same may be proved of the angles TK'S, E'PS. Hence the two sides SK, SK' of the triangle KSK' are inclined to the third at the same angles as those between the line PS and directions PE, PE f of the two forces on the point P; and the sides SK, SK' are to one another as the forces, F, F', in the directions PE, PE'. It follows, by " the triangle of forces," that the resultant of F and F' is along PS, and that it bears to the component forces the same ratios as the side KK' of the triangle bears to the other two sides.^ Hence the resultant force due to the two elements E and E' , on the point P, is towards S, and is equal to &> (/' ~ a 2 ) . SP* (f - a a The total resultant force will consequently be towards and we find, by summation ( 83) for its magnitude, (f*~a*)SP*' Hence we infer that the resultant force at any point P, separated from S by the spherical surface, is the same as if a quantity of matter equal to -^ - ^ were concentrated at the / ~ a point 8. 91. To find the attraction when 8 and P are either both without or both within the spherical surface. Take in CS (fig. 3), or in C8 produced through 8 (fig. 4), a point S , such that 64 On the Mathematical Theory of Electricity. Then, by a well-known geometrical theorem (see note on 87), if E be any point on the spherical surface, we have 8E f SJB a' Hence we have X SE* \a 3 Hence, p being the electrical density at E, we have if Hence, by the investigation in the preceding paragraph, the attraction on P is towards S lt and is the same as if a quantity E . P FIG. 3. FIG. 4. X, . 4-Tra of matter equal to -^ $ were concentrated at that point ; f l Ji ~ a being taken to denote CS r If for/j and \ we substitute their values, -j. and -~ , we have the modified expression A a A X . 4-Tra for the quantity of matter which we must conceive to be collected at S^ 92. PROP. If a spherical surface be electrified in such a way that the electrical density varies inversely as the cube of the distance from an internal point 8 (fig. 4), or from the corre- sponding external point S lt it will attract any external point, as if its whole mass were concentrated at S, and any internal v.] Geometrical Investigations regarding Spherical Conductors. 65 point as if a quantity of matter greater than the whole mass in the ratio of a to /were concentrated at 8 r Let the density at E be denoted, as before, by -^-^ . Then, * if we consider two opposite elements at E and E' which sub- tend a solid angle co at the point S, the areas of these elements , . /ff n .. to . 2a . SE 2 , w . 2a . E' 2 , being ( 96) - ^7^ - and - -^r^ - , the quantity of elec- tricity which they possess will be ^ *~*lf. I*) fy z -r^j **^ - X . 2q . a? / 1 1 \ X. 2a.ft> SE.E'S' Now SE.E'S is constant (Euc. m. 35), and its value is a 2 -/ 2 . Hence, by summation, we find for the total quantity of elec- tricity on the spherical surface X. 4-Tra ^T 2 ' Hence, if this be denoted by m, the expressions in the preced- ing paragraphs, for the quantities of electricity which we must suppose to be concentrated at the point S or S lt according as P is without or within the spherical surface, become respectively , a m, and -* m. Q. E. D. Application of the preceding Theorems to the Problem of Electrical Influence. 93. PROB. To find the electrical density at any point of an insulated conducting sphere (radius a) charged with a quantity Q (either positive, or negative, or zero) of electricity, and placed with its centre at a given distance / from an electrical point M possessing m units of electricity. If the expression for the electrical density at any point E of the surface be X and & being constants; the force exerted by the electrified surface on any internal point will be the same as if the con- stant distribution k, which ( 78) exerts no force on an internal point, were removed; and therefore ( 90) will be T. E. 5 66 On the Mathematical Theory of Electricity. [v. the same as if a quantity of matter equal to -' -- .j were collected at the point M. Hence, if the condition .(6) be satisfied, the total attraction on an internal point, due to the electrified surface and to the influencing point, will vanish. Hence this distribution satisfies the condition of equilibrium ( 72) ; and to complete the solution of the pro- M posed problem it only remains to de- termine the quantity k, so that the total quantity of electricity on the surface may have the given value Q. Now ( 92) the total mass of the distribution, depending on the term , 8 in the expression for the density, since M is an external point, is equal to a \. 4tira Hence, adding 4?ra 2 ^, the quantity depending on the constant term k, we obtain the entire quantity, which must be equal to Q ; and we therefore have the equation From equations (b) and (c) we deduce a)m , j and & = 4t7ra 4>7ra Hence, by substituting in (a), we have d (./ a) 1 . J f A\ r~ ,* 13rc~ r ^2 \^-)> as the expression of the required distribution of electricity. This agrees with the result obtained by Poisson, by means of an investigation in which the analysis known as that of " Laplace's coefficients," is employed. 94. To find the attraction exerted by the electrified conductor on any external point. v.] Geometrical Investigations regarding Spherical Conductors. G7 We may consider separately the distributions corresponding to the constant and the variable term in the expression for the electrical density at any point of the surface. The attraction of the first of these on an external point is ( 87) the same as if its whole mass were collected at the centre of the sphere : the attraction of the second on an external point is ( 92) the same as if its whole mass were collected at an interior point /, taken in M C so that MI. MG = a 2 . Hence, according to the investigation in the preceding paragraph, we infer that the conductor attracts any external point with the same force as would be produced by quantities Q + -^ m, and ^ m of J J electricity, concentrated at the points C and / respectively. COR. The resultant force at an external point infinitely near the surface is in the direction of the normal, and is equal to 47r/>, if p be the electrical density of the surface, in the neigh- bourhood. 95. To find the mutual attraction or repulsion between the influencing point, M } and the conducting sphere. According to what precedes, the required attraction or repul- sion will be the entire force exerted upon m units of electricity at the point M, by Q + ^m at G and -^m at a point /, taken in CM, at a distance -^ from C. Hence, if the required attraction be denoted by F (a quantity which will be negative if the actual force be of repulsion], we have F=- COR. 1. If Q be zero or negative, the value of F is neces- sarily positive, since / must be greater than a ; and therefore there is a force of attraction between the influencing point 52 68 On the Mathematical Theory of Electricity. [v. and the conducting sphere, whatever be the distance between them. COR. 2. If Q be positive, then for sufficiently large values of/, F is negative, while for values nearly equal to a, F is positive. Hence if an electrical point be brought into the neighbourhood of a similarly charged insulated sphere, and if it be held at a great distance, the mutual action will be repulsive; if it then be gradually moved towards the sphere, the repulsion, which will at first increase, will, after attaining a maximum value, begin to diminish till the electrical point is moved up to a certain distance where there will be no force either of attraction or repulsion ; if it be brought still nearer to the conductor, the action will become attractive and will continually augment as the distance is diminished. If the value of Q be positive, and sufficiently great, a spark will be produced between the nearest part of the conductor and the influencing point, before the force becomes changed from repulsion to attraction. ST PETER'S COLLEGE, July 7, 1848. EFFECTS OF ELECTRICAL INFLUENCE ON INTERNAL SPHERICAL, AND ON PLANE CONDUCTING SURFACES. 96. In the preceding articles of this series certain problems with reference to conductors bounded externally by spherical surfaces have been considered. It is now proposed to exhibit the solutions of similar problems with reference to the dis- tribution of electricity on concave spherical surfaces, and on planes. The object of the following short digression is to define and explain the precise signification of certain technical terms and expressions which will be used in this and in subsequent papers on the Mathematical Theory of Electricity. External and Internal Conducting Surfaces. 97. DEF. 1. A closed surface separating conducting matter V.] Geometrical Investigations regarding Spherical Conductors. 69 within it from air* without it, is called an external conducting surface. DEF. 2. A closed surface separating air within it from conduct- ing matter without it is called an internal conducting surface. Thus, according to these definitions, a solid conductor has only one " conducting surface," and that " an external conduct- ing surface." A conductor containing within it one or more hollow spaces filled with air, possesses two or more "conducting surfaces;" namely, one " external conducting surface," and one or more " internal conducting surfaces." A complex arrangement, consisting of a hollow conductor and other conductors insulated within it, presents several external and internal conducting surfaces; namely, an "external conducting surface" for each individual conductor, and as many "internal conducting surfaces" as there are hollow spaces in the different conductors. 98. In any arrangement such as this, there are different masses of air which are completely separated from one another by conducting matter. Now among the General Theorems alluded to in 73, it will be proved that the bounding sur- face or surfaces of any such mass of air cannot experience any electrical influence from the surfaces of the other masses of air, or from any electrified bodies within them. Hence any statical phenomena of electricity which may be produced in a hollow space surrounded continuously by conducting matter, whether this conducting envelope be a sheet even as thin as gold leaf, or a massive conductor of any external form and dimensions, will depend solely on the form of the internal conducting surface. 99. PROP. An internal conducting surface cannot receive a charge of electricity independently of the influence of electrified bodies within it. 100. The demonstration of this proposition depends on what precedes, and on one of the General Theorems, already alluded to ( 73), by which it appears that it is impossible to distribute a charge of electricity on a closed surface in such a manner that * See 70, excluding all non-conductors except air, or gases. 70 On the Mathematical Theory of Electricity. [v. there may be no resultant force exerted on external points, and consequently impossible, with merely a distribution of electricity on an internal conducting surface, to satisfy the condition of electrical equilibrium with reference to the conducting matter which surrounds it. The preceding proposition ( 99) is fully confirmed by ex- periment (Faraday's Experimental Researches, 1173, 1174). In fact, the certainty with which its truth has been practically demonstrated in a vast variety of cases, by all electrical experimenters, may be regarded as a very strong part of the evidence on which the Elementary Laws as stated above ( 72) rest. 101. It might be further stated that the total quantity of electricity produced by influence on an internal conducting surface is necessarily equal in every case to the total quantity of electricity on the influencing electrified bodies insulated within it. This will also be demonstrated among the General Theorems ; but its truth in the special case which we are now to consider, will, as we shall see, be established by a special demonstration. Electrical Influence on an Internal Spherical Conducting Surface. 102. In investigating the effects of electrical influence upon an external, or convex, spherical conducting surface ( 93, D4, 95), we have considered the conductor to be insulated and initially charged with a given amount of electricity. In the present investigation no such considerations are necessary, since, according to the statements in the preceding paragraphs, it is of no consequence, in the case now contemplated, whether the conductor containing the internal conducting surface be insulated or not; and it is impossible to charge this internal surface initially, or to charge it at all, independently of the influence of electrified bodies within it. With the modifications and omissions necessary on this account, the preceding investi- gations are applicable to the case now to be considered. 103. PROB. To find the electrical density at any point of an internal spherical conducting surface with an electrical point insulated within it. V.] Geometrical Investigations regarding Spherical Conductors. 71 Let m denote the quantity of electricity in the electrical point M] f its distance from C the centre of the sphere, and a the radius of the sphere. If the expression for the electrical density at any point E of the internal M & M / surface be ME 3 ' (\ a constant) ; the force exerted by the electrified spherical surface on any point without it will ( 90) be the same as if a quantity of matter equal to -^ ^ were collected at the point a J M. Hence if we take X such that \.4-7ra ^r/*=- m ' the total resultant force, due to the given electrical point and to the electrified surface, will vanish at every point external to the spherical surface, and consequently at every point within the substance of the conductor; so that the condition of electrical equilibrium ( 72), in the prescribed circumstances, is satisfied. We conclude, therefore, that the required density at any point E, of the internal spherical surface is given by the equation __ p ~ lira 'ME*" This solution of the problem is complete, since it satisfies all the conditions that can possibly be prescribed, and it is unique, as follows from the general Theorem referred to in * We cannot here, as in (a) of 93, annex a constant term, since in this case there would result a force due to a corresponding quantity of electricity, concentrated at the centre of the sphere on all points of the conducting mass. t For if there were two distinct solutions there would be two different dis- tributions on the spherical surface, each balancing on external points the action of the internal influencing body, and therefore each producing the same force at external points. Hence a distribution, in which the electrical density at each point is equal to the difference of the electrical densities in those two, would produce no force at external points. But, by the theorem alluded to, no dis- tribution on a closed surface of any form can have the property of producing no force on external points; and therefore the hypothesis that there are two distinct solutions is impossible. The theorem made use of in this reasoning is susceptible of special analytical 72 On the Mathematical Theory of Electricity. [v. COB. The total quantity of electricity produced by the in- fluence of an electrical point within an internal spherical con- ducting surface is equal, but of the opposite kind to that of the influencing point. This follows at once from the investigation of 92; from which we also deduce the conclusion stated below in the next section. 104. The entire electrical force, which vanishes for all points external to the conducting surface, may, for points within it, be found by compounding the force due to the given influencing point M (charged, by hypothesis, with a quantity m of elec- tricity) with that due to an imaginary point I, taken in CM produced, at such a distance from G that CM.CI=a z , and charged with a quantity of electricity equal to j m. COR. The resultant force at an internal point infinitely near the surface, is in the direction of the normal, and is equal to 4<7r/9, if p be the electrical density of the surface in the neigh- bourhood. 105. The mutual attraction between the influencing point M, and the surface inductively electrified will be found as in 95, provided the uniform supplementary distribution which was there introduced be omitted. Hence, omitting the term of (B) which depends on this supplementary distribution; or simply, without reference to (E), considering the mutual force between m at M and ^ m at /, a force which is necessarily attractive as the two electrical points M and / possess opposite kinds of electricity ; we obtain as the expression for the required attraction. demonstration (with the aid of the method in which "Laplace's coefficients" are employed) for the case of a spherical surface; but such an investigation would be inconsistent with the synthetical character of the present series of papers, and I therefore do no more at present than allude to the general theorem. v.] Geometrical Investigations regarding Spherical Conductors. 73 Electrical Influence on a Plane Conducting Surface of infinite extent. 106. If, in either the case of an external or the case of an internal spherical conducting surface, the radius of the sphere be taken infinitely great, the results will be applicable to the present case of an infinite plane ; and it is clear that from either we may deduce the complete solution of the problem of deter- mining the distribution of electricity, produced upon a con- ducting plane, by the influence of an electrical point. The "supplementary distribution," which, in the case of a convex spherical conducting surface, must in general be taken into account, will, in the case of a sphere of infinite radius, be a finite quantity of electricity uniformly distributed over a surface of infinite extent, and will therefore produce no effect ; and the same results will be obtained whether we deduce them from the case of an external or of an internal spherical surface. 107. Let M be an electrical point possessing a quantity m of electricity placed in the neighbourhood of a conductor bounded on the side next If by a plane LL' which we must conceive to be indefi- nitely extended in every direction ; it is required to determine the electrical density at any point E of the conduct- ing surface. Draw MA perpendicular to the plane, and let its length be denoted by p. We may, in the first place, conceive that instead of the plane sur- face we have a spherical conducting surface entirely enclosing the air in which M is insulated; and, suppos- ing the shortest line from M to the spherical surface to be equal to p, we should have, according to the notation of 103, f a ~P- Hence the expression (A) becomes %ap - p z m _ / p p z \ m 4-rra ' ME* 4W ME* 74? On the Mathematical Theory of Electricity. [v. In this, let a be supposed to be infinitely great ; the second term within the vinculum will vanish, and we shall have simply mp for the required electrical density at the point E of the in- finite plane electrified inductively through the influence of the point M. COR. The total amount of the electricity produced by in- duction is equal in quantity, but opposite in kind, to that of the influencing point M. We have seen already that the same proposition is true in general for internal spherical surfaces inductively electrified; but it does not hold for an external spherical surface, even if we neglect the "supplementary distribution," as it appears from the demonstration of 92, that the amount of the distribution expressed by the first term (that which varies inversely as the cube of the distance from the influencing point) of the value of p in equation (A) of 93, is equal to ^m. The infinite plane may, as we have seen, be regarded as an extreme case of either an external or an internal spherical surface ; and the proposition which is in general true for internal, but not true for external spherical surfaces, holds in this limiting intermediate case. 108. To determine the resultant force at any point in the air, before the conducting plane, it will be only necessary, as in 104, to compound the action of the given electrical point with that of an imaginary point /. To find this point, we must produce MA beyond A to a distance AI, determined by the equation CM. CI= a 2 ; which, if we denote AI by p', becomes (a -p) (a +p f ) = a 2 . From this we deduce a and thence, in the case of a = oo , we deduce p =p. Again, for the quantity of electricity to be concentrated at /, we have the expression m, or, when a =00 , m m. ap v.] Geometrical Investigations regarding Spherical Conductors. 75 Hence the force at any point before the plane will be ob- tained by compounding that due to the given electrical point M y with a force due to an imaginary point 7, possessing an equal quantity of the other hand of electricity, and placed at an equal distance behind the plane in the perpendicular MA produced. 109. If reference be made to the general demonstration ( 90) on which all the special conclusions with reference to the effects of electrical influence on convex, concave, or plane conducting surfaces depend, we see that the geometrical construction em- ployed fails in the case of a sphere of infinite radius, becoming nugatory in almost every step: we have however deduced conclusions which are not nugatory, but, on the contrary, assume a remarkably simple form for this case ; and we may regard as rigorously established the solution of the problem of electrical influence on an infinite plane which has been thus obtained. 110. It is interesting to examine the nugatory forms which occur in attempting to apply the demonstrations of 90 and 92, to the case of an infinite plane ; and it is not difficult to derive a special demonstration, free from all nugatory steps, of the following proposition. Let LL' be an infinite "material plane," of which the "density" in different positions varies in- versely as the cube of the distance from a point $, or from an equidistant point S t , on the other side of the plane. The resultant ' p i E. force at any point P is the same as if the whole matter of the plane were concentrated sf~ at 8' 9 and the resultant force at any point Pj, on the other side of the plane, is the same as if the whole matter were collected at 8 t . 111. In the course of the demonstration (in that part which corresponds to the in- vestigation in 93) it would appear that, if the density at any point E of the plane is given by the expression X 76 On the Mathematical Theory of Electricity. [v. the entire quantity of matter distributed over the infinite extent of the plane is given by the expression P This proposition and that which precedes it * contain the simplest expression of the mathematical truths on which the solution of the problem of electrical influence on an infinite plane depends, and we might at once obtain from them the results given above. For an isolated investigation of this case of electrical equilibrium, this would be a better form of solu- tion : but I have preferred the method given above, since the solution of the more general problem, of which it is a particular case, had been previously given. 112. The case of electrical influence which has been considered * The two propositions may be analytically expressed as follows : Let 0, the point in which S/S^ cuts the plane, be origin of co-ordinates, and let this line be axis of z. Then, taking OX, OY in the plane, let the co- ordinates of P be (a;, y, z). Let also those of E be (, 17, 0) ; so that we have X } ~ Hence the proposition stated in the text ( 111), that the entire quantity of matter distributed over the infinite extent of the plane is equal to , is thus expressed : 27T\ r r J -oo J - P This equation may be very easily verified, and so an extremely simple analytical demonstration of one of the theorems enunciated above is obtained. Again, the proposition with reference to the attraction of the plane may, according to the well-known method, be expressed most simply by means of the potential. This must, in virtue of the enunciation in 110, be equal to the potential due to the same quantity of matter, collected at the point S, or the point S lt according as the attracted point is separated from the former or from the latter by the plane. Hence we must have 27T\ f" f 00 J -oo J - the positive or negative sign being attached to z in the denominator of the second member, according as z is given with a positive or negative value. This equation (of which a geometrical demonstration is included in 107 and 108, in connexion with 90) is included in a result (the evaluation of a certain multiple integral), of which three different analytical demonstrations were given in a paper On certain Definite Integrals suggested by Problems in the Theory of Electricity, published in March 1847 in this Journal, vol. ii. p. 109 (ix. below). v.] Geometrical Investigations regarding Spherical Conductors. 77 might at first sight appear to be of a singularly unpractical nature, since a conductor presenting on one side a plane surface of infinite extent in every direction would be required for fully realizing the prescribed circumstances. If, however, we have a plane table of conducting matter, or covered with a sheet of tinfoil, or if we have a wall presenting an uninterrupted plane surface of some extent, the imagined circumstances are, as we readily see, approximately realized with reference to the in- fluence of any electrical point in the neighbourhood of such a conducting plane, provided the distance of the influencing point from the plane be small compared with its distance from the nearest part where the continuity of the plane surface is in any way broken. FOBTBREDA, .BELFAST, Oct. 17, 1849, INSULATED SPHERE SUBJECT TO THE INFLUENCE OF A BODY OF ANY FORM ELECTRIFIED IN ANY GIVEN MANNER. 113. The problem of determining the distribution of elec- tricity upon a sphere, or upon internal or plane spherical conducting surfaces, under the influence of an electrical point, was fully solved in 89... 112 of this series of papers. On the principle of the superposition of electrical forces ( 63) we may apply the same method to the solution of corresponding problems with reference to the influence of any number of given electrical points. 114. Thus let M, M', M" be any number of electrical points possessing respectively m, m', m" units of electricity, at dis- tances f t /', /" from C the centre of a sphere insulated and charged with a quantity Q of electricity. The actual distribution of electricity on the spheri- cal surface must be such that the force due to it at any internal point shall be equal and opposite to the force due to the electricity at M, M' t M". Now if there were a distribution of electricity on the spherical surface such that the density at any point E would be TT^O, the force due to this at any internal point 78 On the Mathematical Theory of Electricity. [v. would ( 90) be the same as that due to a quantity -'_ concentrated at the point H\ and therefore if we take the force at internal points due to this distribution would be equal and opposite to the force due to the actual electricity of M. We might similarly express distributions which would respectively balance the actions of M' , M", etc., upon points within the sphere; and thence, by supposing all those distri- butions to coexist on the surface, we infer that a single dis- tribution such that the density at E is equal to 4ara M'E 3 would balance the joint action of all the electrical points M, M' y M", on points within the sphere. Again, from 92, we infer that the total quantity of electricity in such a dis- tribution is a a , a Hence, unless the data chance to be such that Q is equal to this quantity, a supplementary distribution will be necessary to constitute the actual distribution which it is required to find. The amount of this supplementary distribution will be .a , . a which must be so distributed as to produce no force on internal points. 115. Taking then the distribution found above, which balances the action of the electricity at M, M', etc., on points within the sphere, and a uniform supplementary distribution ; and superimposing one on the other, we obtain a resultant electrical distribution in which the density at any point E of the surface of the sphere is given by the equation 1 (/"-a*) + and we draw the following conclusions : v.] Geometrical Investigations regarding Spherical Conductors. 79 (1) The total force at any internal point, due to this distri- bution and to the electricity of M t M', etc., vanishes. (2) The entire quantity of electricity on the spherical sur- face is equal to Q. Hence this distribution of the given charge on the sphere satisfies the condition of electrical equilibrium under the in- fluence of the given electrical points M, M', etc. ; and ( 73) it is therefore the distribution which actually exists upon the spherical conductor in the prescribed circumstances. 116. The resultant force at any external point may be found as in the particular case treated in 94. Thus, if we join MC, M r C } M" (7, and take in the lines so drawn, points /, /', /" respectively, at distances from C such that the resultant action due to the actual electricity of the spherical surface will, at any external point, be the same as if the sphere were removed, and electrical points /, /', etc., substituted in its stead, besides (except in the case when the supplementary dis- tribution vanishes) an electrical point at C: and the quantities of electricity which must be conceived for this representation, to be concentrated at these points, are respectively a , -7,ra, at / a , T , .(2). and <)++ + etc.,at C 117. By means of these imaginary electrical points we may give another form to the expression for the distribution on the spherical surface, which in many important cases, especially that of two mutually influencing spherical surfaces, is extremely convenient. For (as in 94, Cor.) it is readily seen that the first term, in the expression for p multiplied by 4?r, or a ME*' is the resultant force at E, due to If and /, and that this force is in the direction of a normal to the spherical surface through 80 On the Mathematical Theory of Electricity. [v. E' and that similar conclusions hold with reference to the other similar terms of (2). Again, the last term, Pz-> e ^ Ct > Denote the masses of the series of which the first is at the centre of A ; and let f v f\, / 2 , / 2 , etc., denote the distances of these points from the centres of A and B alter- * The potential at any point in the neighbourhood of, or within, an electrified body, is the quantity of work that would be required to bring a unit of positive electricity from an infinite distance to that point, if the given distribution of electricity were maintained unaltered. Since the electrical force vanishes at every point within a conductor, the potential is constant throughout its interior. t Cambridge and Dublin Mathematical Journal, Feb. 1850 (v. above, 127). 88 On the Mutual Attraction or Repulsion [vi. nately ; and, again, let q iy q\, q z , q z , ..., denote the masses, and ffi> 9\> 9v 9 v > *ke distances of the successive points of the other series from the centres of B and A alternately. These quantities are determined by using the following equations, and giving n successively the values 1, 2, 3, ... : /, = <), ft-.tia f _J!_ '-_., J n- r.-f ' P~ Pn h Pn+l P n a The two series of imaginary electrical points thus specified, would, if they existed, produce the same action in all space external to the spherical surfaces as the actual distributions of electricity do, those (p lt qf lt pi, Js J t Js+1 J t+1 J t+8-l f and _i2j _ _ _ ^ +j . Now = : and and are each independent of u and u v u v v ; hence the following notation may be adopted conveniently : u v (13). Then, taking n to denote t + s in the preceding equations, we have - = -- f C ~fn-t~~9t C ~fn-t~l~9t Hence we have (rf n V \ c fn-t+i 9t) from which we conclude that *.., ,_, ( C -/,-^,) 2 =1 flP.-*,,, and, by using this and transformations similarly obtained for. the other parts of the expression for F } we obtain t=n t=n-l 133. The quantities P n , Q n , 5 B which occur in this expression, may be determined successively for successive values of n in the following manner: By substituting, in (8), for p n , p' n > q n > % n their values by (13), and eliminating f nJ f nt g n , 9 ' n > we find 90 On the Mutual Attraction or Eepulsion cP n = aS n _, + bS n , cQ n = bS n . [VI. from which we derive S Sn-^-X (6). By giving n the values 1 and 2 in (13) and (8), we find 1 By these equations we have directly the values of the first two terms of each of the sets of quantities P^P,, P 8 , etc., Q lf Q 2 , Q 8 , etc., and $,, >Sf 2 , >S> 3 , etc.; and the others may be calculated suc- cessively by the preceding equations. 134. The polynomials which constitute the numerators of the successive terms of the second member of (15) may also be calculated successively, by means of equations obtained in the following manner. We have by (c), (6), and (a), t _ 2 + etc. and similarly we find + etc -) - vi.] between two Electrified Spherical Conductors. -+ etc. 91 and Hence, if we put 2/s ' t=l in terms of which notation the expression (15) for F becomes we have #'n +1 = c- a -y ab ab c z -a 2 -b* ab to). G;-(QU-^& Also we have directly from (e) and (c), 1 , -^ 2 - "i =: k~Zi> ^a = 4 cfb* 135. These equations enable us to calculate successively the values of S\, 8' 8' 3 , etc., P\, P\, P' s , etc., and Q\, Q' z , Q' 5 , etc., after the values of S 19 S t , etc., P I} P 2 , etc., and Q if Q z , etc., have been found. 92 On the Mutual Attraction or Repulsion [vi. 136. The solution of (b) as equations of finite differences with reference to n, and the determination of the arbitrary constants of integration by (c), leads to general expressions for S n , P n , and Q n ; and by using these in (g), integrating the equations so obtained, and determining the arbitrary constants by means of (A), general expressions for S' n , F n , and Q' n are obtained. The expression for F may therefore be put in the form of an infinite series, with a finite expression for the general term. Further, the value of this series may be expressed, by means of analysis similar to that which Poisson has used for similar purposes, in terms of a definite integral. I do not, however, in the present communication give any of this analysis, except for the case of two spheres in contact which is discussed below, because, except for cases in which the spheres are very near one another, the series for F is rapidly convergent, and the terms of it may be successively calculated with great ease, by regular arithmetical processes, for any set of values of c, a, and b, by using first the equations (c), to calculate 8 lt $ 2 , P v P 2 , Q v Q 2 ; then (b) with the values 2, 3, etc., successively substi- tuted for n, to calculate $ 3 , $ 4 , etc., and P 3 , P 4 , etc., and Q 3 , Q^ etc. ; then (h) and (g) to calculate by a similar succession of processes, the values of S\, /S" 2 , $' 3 , etc., F v P' 2 , P 3 , etc., and Q\, Q\, Q' y etc. 137. The following is the method, alluded to above, by which I first arrived at the solution of this problem in the year 1845. 138. The " mechanical value " of a distribution of electricity on a group of insulated conductors, may be easily shown to be equal to half the sum of the products obtained by multiplying the quantity of electricity on each conductor into the potential within it.* Hence, if D and E denote the quantities of elec- tricity on the two spheres in the present case, and if W denote the mechanical value of the distribution of electricity on them, we have W = % (Du + Ev). * This proposition occurred to me in thinking over the demonstration which Gauss gave of the theorem that a given quantity of matter may be distributed in one and only one way over a given surface so as to produce a given potential at every point of the surface, and considering the mechanical signification of the function on the rendering of which a minimum that demonstration is founded. It was published, I believe, by Helmholtz in 1847, in his treatise Ueber die Erhaltung der Kraft, by the translation of which, in the last number of the New Scientific Memoirs, a great benefit has been conferred on the British scientific public. vi.] between two Electrified Spherical Conductors. 93 Now if the two spheres, kept insulated, be pushed towards one another, so as to diminish the distance between their centres from c to cdc, the quantity of work that will have to be spent will be F '. dc, since F denotes the repulsive force against which this relative motion is affected. But the mechanical value of the distribution in the altered circumstances must be increased by an amount equal to the work spent in producing no other effect but this alteration. Hence F. dc = dW, and therefore where u and v are to be considered as varying with c, and D and E as constants. Now, according to the notation expressed in (13), we have +etc. = ... (17). Determining 1 - and -y- by the differentiation of these equa- dc dc tions, and using the results in (16), we find This expression agrees perfectly with (/), given above ; since, by differentiating the equations (6) and (c) with reference to c, we find that the quantities denoted above by S\, S' 2 , S' s , etc., P 19 P' 2 , P' 3 , etc, Q\, Q',, Q' 3> etc., and expressed by the equations (g) and (h), are equal respectively to 139. The series (/) or (18) for F becomes divergent for the case of two spheres in contact, but the doubly infinite series from which this was derived in the first of the two investiga- tions given above, is convergent when the terms are properly grouped together; and its sum may be expressed by means of a definite integral in the following manner : On the Mutual Attraction or Repulsion [VI. 140. Since the two spheres are in contact, the potentials within them must be equal, that is, we must have u=v. For the sake of simplicity, let us suppose the radii of the two spheres to be equal, and let each be taken as unity. Then we shall have a = b = 1, and c = 2; and the terms of doubly infinite series (9) in this case are easily expressed,* in very simple forms, by equations (8). Thus we find 1 1.2 1.3 1.4 5 2 2.3 5 2 3.2 1.5 6 2 2.4 6 2 30 .3 i etc. etc. etc. etc. 4 2 2.2 4 2 3.1 5 2 4.1 5 2 6 2 4.2 6* 5.1 6 2 ~I 7a If we add the terms in the vertical columns, we find 2.3.4 3.4.5 -etc.), 3 2 4 2 which is a diverging series, and is the same as we should have found by using the form (/) or (18). But if we add the terms in the horizontal lines, we find the following convergent series for F: etc. (1 + 6)'' * From equations (8) we find, in this case, 2n-l 2n-2 2n ' i-l v Hence P& p',q' t P.P't and then, by (9), we obtain the expression for F in this particular case, given in the text. vi.] between two Electrified Spherical Conductors. 95 Hence, since (1 + 0)~* =1-20 + 36* - etc, we have or, by actual integration, x (log 2 - J) = v 2 . J x (-69315 --25) = v 2 x -073858. The quantity of electricity on each sphere being equal to the sum of the masses of the imaginary series of points within it, is, according to the formulae for ft, q\,p 2 , q' 2 , etc., Hence we have the following expression for the repulsion be- tween the two spheres, in terms of Q the quantity of electricity on each, (Iog2) 2 141. If a; denote the distance at which two electrical points, containing quantities equal to the quantities on the two spheres, must be placed so as to repel one another with a force equal to the actual force of repulsion between the spheres, we have (v . log 2) 2 _ v x* Using the value for F found above, we obtain x = -- iSA = 2-550. If the electrical distribution on each surface were uniform, this distance would be equal to 2, the distance between the centres of the spheres ; but it exceeds this amount, to the extent shown by the preceding result, because in reality the electrical density on each conductor increases gradually from the point of contact to the remotest points of the two surfaces. P.S. The calculation by the method shown in the preceding paper, of the various quantities required for determining the force between two spheres of equal radii (each unity), insulated 96 On the Mutual Attraction or Repulsion [VI. with their centres at distances 2'1, 2'2, 2*3, etc., up to 4, has been undertaken, and is now nearly complete. GLASGOW COLLEGE, March 21, 1853. 142. The following numerical results have been calculated (by means of the formulae established above) for application to the theory of a new electrometer which I have recently had con- structed to determine electrical potentials in absolute measure, from the repulsions of uninsulated balls in the interior of a hollow insulated and electrified conductor, by means of a bifilar or torsion balance bearing a vertical shaft which passes through a small aperture to the outside of the conductor : TABLE I. Showing the Quantities of Electricity on two equal Spheri- cal Conductors, of radius r, and the mutual force between them, when charged to potentials u and v respectively. Col. 1. Cols. 2 and 3. Cols. 4 and 5. Col. 6. Ratio of the poten- Dis- tance from centre For determining the quantities of electricity, For determining the mutual force, tials when there is neither attraction nor repulsion, to centre =cr. D=(Iu-Jv)r E=(Iv-Ju)r. being repulsion when positive, and attraction when negative. p= Z~(l2~ 1 ) l_B , (B 2 U P~A' T \A* / c. I. J. A. B. P. t -073 858 2-0 J+ -693147 CO CO A + ^x -073858 1 V A 2-1 1-58396 88175 1-13844 1-17439 77828 2-2 1-43131 72378 52852 56350 69637 2-3 1-34827 63395 32917 36357 63553 2-4 1-29316 57202 23159 26464 58975 2-5 1-25324 52537 17432 20630 55888 2-6 1-22218 48819 13696 16787 51699 2-7 1-19755 45746 11082 14090 47805 2-8 1-17738 43140 09174 12073 46049 2-9 1-16056 40886 07720 10526 43667 3-0 1-14629 38908 06592 09299 41567 3-1 1-13404 37151 05693 08304 39672 3-2 1-12340 35571 04963 07481 37947 3-3 1-11410 34150 04363 06791 36376 3-4 1-10588 32852 03863 06203 34939 3-5 1-09859 31663 03441 05697 33615 3-6 1-09208 30569 03084 05257 32418 3-7 1-08623 29557 02775 04872 31263 3-8 1-08095 28617 02509 04531 30211 3-9 1-07617 27742 02278 04229 29233 4-0 1-07182 26924 02075 03958 28318 VL] between two Electrified Spherical Conductors. 97 TABLE II. Showing the Potentials in two equal Spherical Conductors, and the mutual force between them, when charged with quantities D and E of electricity respectively. Col. 1. Cols. 2 and 3. Cols. 4 and 5. Col. 6. For determining the mutual Dis- tance from centre to For determining the potentials, U= Yf2- J~2 ' D + f2-j2 ' ) r force, I Jr 2 ' where Ratio of the quan- tities when thereis neither attraction nor repulsion, centre t n i *f T\\ -^ JQ T =cr. '-V/2-J2- fa-ja- A' B(I*+J*)-2AIJ e= I~Jp ' ^ (fa-jap 1 J f\ c. /2 -' /a ' ja-ja a. 0. u. 2-0 i v I V 00 a+ i x -153726 - /153726 2 -693147 2 -693147 V a 2-1 91482 50926 15375 22668 39102 2-2 93869 47467 08263 15251 29435 2-3 95220 44782 05444 12186 23580 2-4 96142 42528 03955 10309 19944 2-5 96829 40599 02997 09038 16908 2-6 97354 38888 02342 08078 14476 2-7 97771 37348 01849 07341 12786 2-8 98105 35946 01500 06710 11318 2-9 98376 34658 01222 06186 09971 3-0 98598 33467 01010 05731 08877 3-1 98782 32361 00842 05333 07944 3-2 98934 31327 00708 04981 07139 3-3 99067 30366 00599 04666 06442 3-4 99178 29462 00510 04382 05839 3-5 99272 28612 00437 04126 05298 3-6 99351 27810 00378 03891 04868 3-7 99423 27054 00326 03679 04349 3-8 99484 26338 00283 03484 04061 3-9 99537 25659 00247 03305 03736 4-0 99583 25015 00216 03139 03444 T. E. VII. ON THE ATTEACTIONS OF CONDUCTING AND NON-CONDUCTING ELECTKIFIED BODIES. (Art. vii. of complete list in Mathematical and Physical Papers, Vol. i.) [From the Cambridge Mathematical Journal, May 1843.] 144. In measuring the action exerted upon an electrified body, by a quantity of free electricity distributed in any manner over another body, the methods followed in the cases in which the attracted body is conducting and non-conducting are different. Now, the only difference between the state of a conducting body and that of a non-conducting body is, that the electricity is held upon a conducting body by the pressure of the atmosphere (to a certain extent at least), while on a non- conducting body it is held by the friction of the particles of the body. 145. To find the attraction of an electrical mass E, on a non- conducting electrified body A, the obvious way is to proceed as in ordinary cases of attraction, considering the electricity on A as the attracted mass. In finding the action on a conducting body A t the method followed is to consider its electricity as exerting no pressure upon the particles of the body, but disturbing its equilibrium, by making the pressure of the air unequal at different parts of its surface. These two methods of measuring the action of E on A should obviously lead to the same result, since the action must be the same, whether A be conducting or non- conducting, the distribution remaining the same. It is the object of the following paper to show that they do lead to the same result. 146. We must first find the pressure of an element of the electricity of A, on the atmosphere. Let ds be the area of the element, and pds its electrical mass. Let ds form part of another element Qi> Qa> e *c., to be substituted, the system being constructed thus. Each of the imaginary magnets is equal and similar to Q, and similarly magnetized ; Q occupies the place of Q, and the others are similarly placed behind it, along a line perpen- dicular to the plate, the distance between corresponding points of each consecutive pair being equal to twice the thickness of ix.] Note on Induced Magnetism in a Plate. 105 the plate. The intensities of the successive magnets decrease in a geometrical progression, of which the common ratio is m 2 (a quantity measuring (45) the inductive capacity for magnet- ism of the plate), commencing with that of Q', which is equal to 1 m 2 , if the intensity of Q be unity. It is hardly necessary to point out the analogy between this and the corresponding result in optics, in which the illumination produced through a plate of glass, by a candle, is found to be due to the candle itself, with diminished brightness, and to a row of images behind it, with intensities decreasing in a geometrical progres- sion, which arise from successive internal reflections. 158. If the iron plate be infinitely thin, all the images, Q i} Q 2 , etc., will coincide with Q' ; and, since the sum of their intensities is unity, the total effect will be the same as that of Q, which will therefore be unaffected by the interposition of the screen. The same will be the case if the distance of Q be infinitely great, and the thickness of the screen finite ; but in this case, at least as far as the present result can show us, the dimensions of the planes which bound the plate must be infinitely great compared with the distance of Q. 159. The result which I have stated is applicable also to the imaginary case in which, instead of being a magnet, Q is a mass of positive or negative magnetism.* Thus, let Q be a unit of positive magnetism collected in a point, which case is investigated by Green. To express the action analytically, let Q be taken as origin of co-ordinates, a line perpendicular to the plate as axis of x, and the plane through this line, and P, as plane of (x, y). Then denoting by a the thickness of the plate, and considering Q as a positive unit of matter, we shall have, for the total potential at P, due to Q and the plate, * This expression does not imply any hypothesis of a magnetic matter or of a fluid or fluids, but it is merely used for brevity in consequence of the principle established by Coulomb, Poisson, and Ampere, that the action of a magnetized body of any kind, or of a collection of electric " closed currents," may always be represented by an imaginary positive and negative distribution of matter, of which the whole mass is algebraically nothing. By an element of positive or negative magnitude, we merely mean a portion of this imagined matter. 106 Note on Induced Magnetism in a Plate. [ix. 160. For all magnetic bodies m is between and 1, the former limit being its value when the inductive capacity for magnetism is nothing, and the latter being never attained, though it is approached in such bodies as iron, of which the inductive capacity is great. In the extreme case of m = 1, the laws of induction in a magnetic body degenerate into those of electrical equilibrium on the surface of a conductor of electricity. If in the expression for F we put m = 1, one of the factors vanishes and the other becomes infinite, but the ultimate value of the product is nothing, which shows that the effect of the plate is to destroy all action behind it. This we know to be the case when an infinite conducting screen of any form is placed before an electrified body. 161. In the case when the plate is of iron, the value of m is nearly unity. Hence, as the series is multiplied by 1 m 2 , it might be imagined that, if we " neglect small quantities of the order (1 g) compared with those which are retained," (1 g being, in Green's notation, a quantity of the same order as 1 m), an approximate result would be obtained by putting m = 1 in the successive terms of the series within the vin- culum. And it is thus that Green, having, in the investigation, neglected quantities multiplied by (1 gf, arrives at the result, 3 As, however, this series has an infinite sum, it is clear that no value of m can be sufficiently near to unity to render the approximation admissible. If instead of Q we were to sub- stitute a magnet, or any collection of positive and negative particles, such that the sum of the masses is zero, the series for the potential, deduced from Green's expression, would con- verge : and the same remark is applicable to the series which would be found for the attraction of the system on a point beyond the screen, even when Q is a positive point, by differ- entiating the expression for F. Notwithstanding this, the approximation is still inadmissible; since, if we expand the rigorous expression in either case in ascending powers (1 m), we find that, though the first term is finite, the coefficients of all the terms which follow it are infinite. ix.] Note on Induced Magnetism in a Plate. 107 162. Although the method by which I obtained the rigorous solution is quite distinct from that followed by Green, being independent of any mathematical process, it may be satis- factory to show that the result can be deduced from his own analysis, and even with greater ease than his solution is ob- tained after making unnecessary approximation. By a very remarkable investigation, in which he extends Laplace's well-known analysis for spherical co-ordinates to the case when the radius of the sphere becomes infinite, Green arrives (Essay on Electricity, p. 64) at the following expression for the total potential at P, due to the positive unit of matter Q, and to the interposed plate, before making any approxima- tion : Let ra = o Then we have, by expansion, and by changing ^ + 9 the order of the integration, ' V 4 Y a + etc.)cos V + 0V + (* + 2a) 2 + /3y + + 4a) 2 + fftf + 6tC ' J = - (1 m*) S o 1 a /i > where xi x + 2/a, \ / M & I .* /-*i -10. ^ LM * * ' which agrees with the expression given above. ST PETEK'S COLLEGE, Oct. lth, 1845. X. SUE UNE PROPRIETE DE LA COUCHE ELECTRIQUE EN EQUILIBRE A LA SURFACE D'UN CORPS CONDUCTEUR. Par M. J. LIOUVILLE. (Art. xxrv. of complete list in Mathematical and Physical Papers, Vol. i.) [From the Cambridge and Dublin Mathematical Journal, Nov. 1846.] 163. La me'thode la plus ge'ne'rale que Ton connaisse pour former des couches electriques, en equilibre a la surface de corps conducteurs, consiste a eonsiderer une masse M\ et le potentiel, v _ f/Y/0/ V, O dx' dy dz' ~JJJ~ A de cette masse, par rapport a un point quelconque (x, y, z), dont la distance au point (#', y , z), ou a 1'ele'ment /(x'^'^'ldtc'dy'dz', est de'signe'e par A. Prenons ensuite une surface de niveau ou d'equilibre relativement a 1'attraction de la masse M, et qui entoure cette masse, c'est a dire prenons une surface fermee (A), contenant la masse M dans son inte*rieur, et pour tous les points de laquelle V conserve une valeur constante. En fin dV soit -7 ds la variation infiniment petite que V e'prouve lorsqu'on passe d'un point de cette surface a un point exte'rieur infini- ment voisin situd sur la normale a une distance ds. C'est la de'rive'e -p , multipli^e si Ton veut par une constante, qui reglera la loi des densites de 1'e'lectricite' en Equilibre sur un corps conducteur termini par la surface (A). Plusieurs geo- metres sont parvenus, chacun de leur cote', a ce beau the'oreme; mais c'est George Green qui 1'a, je crois, donne le premier dans un excellent me'moire public* en 1828, sous ce titre : An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Je me propose de montrer que la couche electrique en e'quilibre ainsi obtenue a pre'cisement le meme centre de gravite que la masse M. 164. Pla9ons 1'origine des coordonnees x, y, z, au centre de x.] Propri^te de la Gouche Electrique en Equilibre. 109 gravit^ de la masse M ; et de'signons par a? x une quelconque des coordonne'es du centre de gravitE de la couche Electrique, laquelle sera fournie par la formule dV, ou les integrations s'appliquent a la surface (A) dont Moment est reprEsente par dco. II s'agit de prouver que x^ = 0. D'apres 1'expression de F, on a suivant que le point (a?, y, z) appartient ou non a la masse M. Pour plus de simplicite, ecrivons toujours en regardant la fonction f(x, y, z) comme nulle hors de la masse M ; et combinons cette Equation avec cette autre de forme analogue #17, ffU da? + dy 2 ou nous supposons que U est une fonction de x, y, z, qui reste finie et continue ainsi que ses de'rive'es dans tout 1'espace intErieur a (A). Nous aurons d*V d 2 U tfV d*U d*V Multiplions par dx dy dz, et intEgrons dans tout Tespace interieur a (A). En conservant a ds et a dw la meme signifi- cation que ci-dessus, on trouve, apres des transformations bien connues : -dco - U ~da> = **MUf(x, y, z) dxdydz. Mais 1'Equation en U est satisfaite par U=x\ nous avons done: dco - \\x ^-dco - 47r///^/(^ y, z) dxdydz. as jj cts L'intEgrale triple du second membre, divise'e par M, donne 1'abscisse du centre de gravitE de la masse M. Ce centre Etant a 1'origine des coordonnees, 1'integrale dont nous parlons est 110 Propri&tf de la Couche Electrique en Equilibre [x. ffdx F-T- dco Test aussi. D'abord on peut faire sortir V du signe /, puisque, sur la sur- face (A), V est constant. Observons ensuite que -j- a pour valeur le cosinus de Tangle a. que la normale ds fait avec 1'axe des x. Notre inte'grale deviendra done : F/Jcos adv. Or 1' integrate // cos adco est nulle, d'apres un the'oreme connu, comme compose'e d'elements deux a deux e'gaux et de signes r r fig contraires. Ainsi 1 1 T -r- dco = 0. II reste done finalement JJ ds et Ton en conclut x l = 0, ce qu'il fallait demontrer. TODL, 4 Juillet 1846. NOTE ON THE PRECEDING PAPER. By WILLIAM THOMSON. [Extracted from a Letter to M. LIOUVILLK.] 165. ". . .The demonstration which you have given has led me to this other theorem, that the mass M } and the shell surround- ing it, have the same principal axes, through any point. To demonstrate this, let U=yz in the formula which you have given. Then, since, if we denote by K the constant value of F at the shell, we have we find which proves the proposition enunciated. If we take U= a?, we find d 2 U d*V d*U d*V yz j dco = 4>7rfffyz ./(#, y, z) dxdydz (1), -j n j--T^- -j-o--^-0-- ^-^ - Bar rfic 2 c??/ 2 c?2/ ^ ds? * See xii. below, 200, (8). x.] d la Surface dun Corps Conducteur. Ill from which, observing that we deduce Let A, B, C be the moments of inertia of the mass If round the axes of co-ordinates, and A v B v C v those of the shell, round the same axes, it being supposed that the quantity of matter of the shell is the same as that of If;* the preceding equation, and the two others which correspond relatively to the axes of y and z, are with this notation, , C^Q + C ......... (2),f where Q, = ^-fjj(V- K) dxdydz, is a quantity which is independent of the position of the origin. From equations (2), we have A demonstration of your theorem and of the theorems ex- pressed by the equations (1) and (3) may be arrived at by com- paring the expressions for the equal potentials J produced by the mass M, and the shell at very distant points." [| ST PETEB'S COLLEGE, July 15, 1846. * In this case the "density" of the distribution at any point of the shell will be equal to j-.-~ . See i. above, 7. 4:7T (tS t If the origin be taken at the centre of gravity, and the axes of co-ordinates principal axes of M (and therefore of the shell, according to the proposition enunciated above), these equations show that the "central ellipsoid" (see note to p. 202 of Cambridge and Dublin Mathematical Journal, 1846) for the shell is confocal with that for the body M. J A shell constructed round the mass M, in the manner described by M. Liouville, with a quantity of matter equal to M , exerts the same force upon points without the shell, as was proved first by Green (see also i. above, 9) ; and since the potential of each vanishes at an infinite distance, it follows that the two bodies produce equal potentials at every point without the shell. || [See Thomson and Tait's Natural Philosophy, 539.] XI. ON CEETAIN DEFINITE INTEGRALS SUGGESTED BY PROBLEMS IN THE THEORY OF ELECTRICITY. (Art. xxvin. of complete list in Mathematical and Physical Papers, Vol. i.) [From the Cambridge and Dublin Mathematical Journal, March 1847.] 166. It follows from the solution of the problem of the dis- tribution of electricity on an infinite plane,* subject to the influence of an electrical point, that the value of the double integral, ''I* <(f - f + (i - y'T + *}* STT is A direct analytical verification of this result is therefore interest- ing in connexion with the physical problem. In the following paper the multiple integral ^^..^ is considered, and its value is shown to be a result of which the one mentioned above is a particular case. Several distinct demonstrations of this theorem are given, and some other formulae, which have occurred to me in connexion with it, are added. 167. The first part of the following paper, which is a transla- tion, with slight alterations, of a memoir in Liouville's Journal, ~f* contains a demonstration suggested to me by a method followed by Green in proving the remarkable theorem in Art (5) of his Essay on Electricity. In the second part some formulae are given which, in the case of two variables, are such as would * See above, 111, footnote. t 1845, p. 137, "Demonstration d'un Theoreme d'Analyse" (April 1845). XI.] Problems in the Theory of Electricity. 113 occur in the analysis of problems in heat and electricity, with reference to a body bounded in one direction by an infinite plane, if the methods indicated by Fourier were followed ; and from them the value of the multiple integral mentioned above is deduced. In ill. the evaluation is effected by a direct process of reduction, suggested by geometrical considerations*. PAET I. 168. Let the value of the multiple integral, which, if we use a very convenient notation analogous to that of factorials, may be written thus, [U _ x 4- i be denoted by U. Let u + u' = a, it being understood that u and u' are taken as positive. Then, if we assume i 1 we have -2(,-l)^=[/_"J*'f It is easily seen that the second member of this equation vanishes when v = oo , and that it does not become infinite, even when one of the values 0, 2u, or a is assigned to u. Hence the preceding equation may be written But we have When we take the integral with respect to v between the * See "Extrait d'une lettre & M. Liouville, etc." Liouville's Journal, 1845, p. 364 (xiv. 210, below). T. E. 8 114 On certain Definite Integrals suggested by [xi. limits oo and u, the first term vanishes, since at each limit R = 0. Thus the preceding equation is reduced to 169. Now we have -y-y- + 2 ~jr > for all values of f x , f 2 ..., provided v be not equal to a. Hence this equation is satisfied for all the values of the variables between the limits of the integration in the preceding ex- 72 TV pression, and we may therefore employ it to eliminate TT* we thus obtain ~2(s-i)wf7=r rr 17^'^+^^ Taking one of the terms of the second member, and integrating by parts, we have r /T J _oo \J 00 J r rr 1- J CO \_J CO J since the integrated parts vanish at each limit. By applying a similar process to each term under the sign S, we find But, if we denote by Q and Q' the two parts of R, in equation (1), so that R = Q-Q',w& have for all values of the variables v, t , etc., within the limits of integration ; hence there remains To determine the value of this expression it may be remarked XL] Problems in the Theory of Electricity. 115 that the quantity under the integral signs vanishes for all values of the variables which differ sensibly from those ex- pressed by v = > ?i = ^i> ? 2 = ^2> etc -> and moreover, that if we consider separately the terms of the second member, each is found to be a converging integral: it follows that, if we denote by P the value which R r receives when the variables have these values assigned, we have ..^, (3), where the limits of integration must be such as to include the values 0, x 1} x 2 , etc., but are otherwise entirely arbitrary. By considering separately the different terms of this expres- sion,* and integrating each with respect to the variable to which it is related, without yet assigning the limits of the integration, we find - 2 (.- 1) U U-P... 2 ...+ \ _ __ V ~r*~ 15 dv~ ~ The integrations in equation (3) may be extended to all the values of the variables which satisfy the condition and the limits in (4) will then be such as to include all the values which satisfy the equation v 2 + flj 2 + 1> 2 2 + . . . + v a * = a 2 , or r 2 = a 2 , etc. If in the integrations we only take the positive values of the variables v, v v v 2 , etc., which satisfy the limiting condition, we must multiply each integral by 2* +1 ; and we may then simply take, in the successive terms the second member of (4), dQ_ s-l dQ__s-l ~dv~ ~~ V ~ " Thus we have 82 116 On certain Definite Integrals suggested by [XI. u U= (//. . . wfo, [J_J i _ rr I* dz^...dz 8 - X'Y + W'p- 1 ) \J _> J (1 + < + *,*+...+ * i rr ir^-di^... -x'Y + w' 2 p-D 'Jo Jo '" (1 + t + , + ...+ i)*( 1 __ >n* r W-idh - xj + w /2 ^-D r(Ja) Jo (1 + ^) i(8+1) Also, when w = oo , the value of F is nothing. 174. Thus we see that V has the same value as the expression when M = 0, and when ^ = 00; which enables us to infer that for all positive values of u, provided u be taken as positive ; for the second member of this equation satisfies equation (7) for all positive values of u, and for any values of the other variables, and at the limits u = and u = has the same value as V, and therefore, by a theorem of Green's*, in the memoir referred to, must be equal to V for all positive values of u. [Included in Theorem 2 of xm. below.] 118 On certain Definite Integrals suggested by [xi. 175. From what has been proved above we may deduce the solution of the following problem : Having given for all values of f t , f a ..., the value of the multiple integral * - 2 where w' and // are any unknown functions of x^ t x^ . . . #/, let it be required to find the value of where #, , # 8 . . . #, are any given quantities, and w a given positive quantity. Denoting the expression (a) by , and the expression (b) by 0, we have, from the theorem established above, But, by hypothesis, is given for all values of f 1? f 2 ... f 5 ; and therefore this equation expresses the solution of the problem. We may also deduce from the theorem (5) the expression 6 - ( s _ 1)^^+1) [j_ oo j {^(f-^^^p- 1 ) " by means of which may be determined when the value, M^, of cZ r* 2 - corresponding to u = is given. 176. For the particular case of u = 0, the theorem (d) is in- cluded in a theorem given by Green, in which the number n in the exponent of the denominator may differ from the number s of variables, the sole condition being that n s + 1 must be positive ; but it is only in the case of n = s that a general theorem such as (d), by means of which the general value of is obtained from the value -f when u = 0, can be established. du XL] Problems in the Theory of Electricity. 119 177. Let us now apply these formulas to the case of 5 = 2: we may in this case conveniently replace x v # 2 , u by x, y, z, and ?i> ?2> by ft *! Equations (c) and (d) become = f" f d % dr > 27r J _ J _ ~{(f _ , + 7? _ 72 + sii ' * where "^ denotes the value of when a; = f, 2/ = 77, # = 0. 178. The first of these theorems maybe deduced from a very general theorem given by Green in his essay on Electricity and Magnetism [ (5) eq. (6)]. The second may be demonstrated in the following manner: Let x' } y', z be considered as the co-ordinates of a point P', where there is situated a quantity of matter p' dx dy dz, in the volume dx dy f dz. Then will be the potential on a point P (x, y, z\ above the plane of sc, y which we may regard as hori- zontal, due to a quantity of matter, M t (=Mp'dx'dy'dz f ) situated below this plane. Now it follows from a theorem, first, so far as I am aware, given by Gauss, for a surface of any form, that there is a determinate distribution of matter upon the plane (xy) which will produce this same potential on points above the plane. Let k be the density of this distribution at a point II (f , 77) of the plane, so that r r J - J -{(- which gives t^ dz '*} -J -oo {( - xf + (n - y? + ^ ' Let z = ; then denoting by k and ( ~ J the values of k and - at the point (x, y, 0), we find = _k r r zd ^ dz Q ] -J -oo {( - a)* + (77 - y)* + 120 On certain Definite Integrals suggested by [xi. since the value of the integral in the second member is 2?r, whatever be the value of z. Hence we conclude that and equation (/) is established. 179. It should be remarked that the total quantity of matter distributed over the plane xy must be equal to the mass M, which it represents : this is readily verified from the preceding formulae. 180. The same formulae admit of an interesting application in the theory of heat. Thus let < be the permanent tempera- ture of a point P in an infinite homogeneous solid, heated by constant sources distributed below the plane (xy), (the case in which some of the sources are in this plane being of course included). If the temperature at any point II in the plane (xy) be given, the formula (e) enables us to find the temperature at any point above the plane. 181. As an example, let us suppose that the sources of heat are such that the temperature of a portion A of the plane (xy), between two lines parallel to OF and at equal distances, a, on its two sides, has a constant value c, and the temperature of the remainder of the plane zero. In this case the formula (e) will give, for the temperature at a point (x, y, z) above the plane, C / . _-t w ~"r" Cv _i vG (M \ = - tan tan 7T\ Z Z ) = - tan' 7T From this we conclude that the isothermal surfaces which corre- spond to this case are circular cylinders, which intersect the plane (xy) in the two parallel lines bounding A. The application to this example, and all others in which the isothermal surfaces are cylindrical, may be made directly by putting s = I in the general formulae. XL] Problems in the Theory of Electricity. 121 PART II. 182. I now proceed to find the values, which will be denoted by Fand W, of the integrals rr is anrl IV7/wj" L'-ooJ L J L " ^ ~(2m 2 )' ' where the symbols [cos m%\ s , [cos mx\ s denote the products cos m x f t . cos m 2 2 . . cos m s f s , cos m^ . cos m 2 # 2 . . cos m^; and the notation is in other respects the same as before. By means of the formula [cos mf 4- sin m . \J( l)] s = cos (Sm) 4- sin (2m|f) . *J( 1), it is easily shown that F rr I'j^ooss^ (a) |_y oo _| \^^ ~T" U ) Hence, by a suitable linear transformation, in which one of the assumptions is Smf = 77 (Sm 2 )^, we have [if //, denote (Sm 2 )^] Now, by means of Liouville's theorem*, we find r r i 5 - 1 [dt;]*- 1 ^-^ r zfdf.? LJ-.J (?+ v+sn- "rj(- i)Jo (F+^ + Differentiating with respect to u, by which the further reduction of the integral will be facilitated, we have dV_, ~du- (8 " Now * See Cambridge Mathematical Journal, Feb. 1841, p. 221 [or Gregory's Examples, Ed. 1841, p. 469]. 122 On certain Definite Integrals suggested by [xi. dV 47r* (s - 1 Hence -j- = frr? : du 1 i (5 - - From this, by integration with respect to u, we deduce the value of V: thus we have the result -00 J 183. To evaluate the integral W we may in the first place reduce it to a double integral by a process similar to that in- dicated above, for obtaining the expression (c) ; and we thus find W= where r denotes (2# 2 )i If we take ra = p cos ^, ?i = p sin ^-, this becomes 4wri(*-l) T* r* 77 dtidpp 8 - 2 cos s ~ 2 ^ cos (rp sin S-)e- w . . .(6). */0 -/O Now we have C S r Sn 6 ~ PM = acos cos Considering first the case where 5 is even, let/= ^s 1 ; we thus find (JZ du* + C S and, by substitution in (6), we have 47T^- 1 ) f 00 /"^, /(^ 2 rf 2 w= iw=i)L L ^ dp + cos sm -P Ti(* - 1) j(s~ i) ^ 2 ^v (M* + -l) 12 02 / -I \2 ' XI.] Problems in the Theory of Electricity. 123 In the second case, when s is odd, let/ =4(5 - 1) in (c); then, making use of the result in (6), we have $ d 2 x^*- 1 ) r /* = C + fc) Jo Jo -D / d 2 ^ d* \*('-D r sin (rp) TV T~2 + 7T 2 - ~ - 1) \cfor r/ Bence, whether 5 be odd or even, we conclude that 184. The investigation which we have just gone through, of the integrals (F), (W) constitutes the verification of "Fourier's theorem " in a particular case. For, by this theorem, we have, if F (oc^ % 2 ...) be a function which remains the same when the signs of any of the variables are changed, CO and if we take the result of the integrations with respect to f 1? f 2 ..., is given by (F), and the second member thus becomes a multiple integral with respect to m 1} m 2 ..., which is shown by (W) to be equal to the first member. Conversely, if we assume Fourier's theorem, we may deduce the value W, by means of it, from that of V. The integrals V and W are also con- nected by means of another case of Fourier's theorem, found by taking, in (e), In this way, after the value of W has been found, that of V may be deduced. 185. The formulae (F) and (TF) may be applied to evaluate the multiple integral u, and we shall thus obtain the result of the investigation in I. in a different manner. 124 On certain Definite Integrals suggested by [xi. By means of the equation obtained by differentiating (TF) with respect to u, we find u 1 ) w*(-i> r J (5 - 1) 00 Ijldmf [cos m (f - Making this substitution, for one of the factors of the expression under the integral signs U, we have rr _ L_ [f T ~ 2-*(- 1) irK- 1 ) rio- 1) L/ -oo J } " 1 ] s [cos m (f -a?) K 20 -| _ QO J [cfo]' [ ^ cos m(^ - aOfc-lWi- - , by ( F), by(F) ' which agrees with the value obtained above. PART III. 186. The value of the integral Z7may also be obtained by a direct process of reduction, as follows : By a suitable linear transformation, in which assumptions such as fc-a are made, we find where / 2 = 2 (a? - ')* Let us now assume ! = p sin sin t sin ^ 2 ...... cos ^ s _ 2 , f, = /o sin sin t sin a ...... sin ^ 2 , XI.] Problems in the Theory of Electricity. 125 from which we deduce* sn a transformation given first by Green. Equation (a) is thus reduced to TTfr f 8 - z Jo .' where 1T S _ 2 denotes the product T sm*-*0d6 . r^-^ede ...... (*"de. ' o 7o ./o Let p = u tan JS- ; we thus get ir fir I - ' _ _ {2 ( / 2 + w' 2 -f %*) + 2 (/ 2 -f w /2 - M*) cos ^ - 4^/sin ^ cos ^j^*- 1 ) and we may now conveniently assume 4- w - u) cos - - wsn ^ cos $ = 2{(f + u*- u'*f + 4^ 2 / 2 l* cos 6> = 2hk cos (9, and sin sin ^- = sin 9 sin 0, from which we deduce sn c)r = sn the expression for 7 becomes 77 _ i TT l v r sn*- sn " 5 *~ 2 Jo Jo (F s ~ 2 r_ Vo (^ 2 - 2^ cos (9 + Let A sin (^r - ^) = k sin >/r ; by means of this transformation, observing that h > k, we readily find or which is the same as the result previously obtained. ST PETEB'S COLLEGE, Oct. 3, 1846. * See Cambridge Mathematical Journal, Nov. 1843, p. 24, First Series; [or Green, "Attraction of Ellipsoids, " 6, Comb. Phil. Trans., May, 1833.] XII. PROPOSITIONS IN THE THEORY OF ATTRACTION. (Art. VI. of complete list in Mathematical and Physical Papers, Vol. i.) [From the Camb. Math. Jour., Nov. 1842 and Feb. 1843.] 187. Let x, y, z be the co-ordinates of any point P in an attracting or repelling body M\ let dm be an element of the mass, at the point P, which will be positive or negative accord- ing as it is attractive or repulsive ; let #', y ', z be the co-ordinates of an attracted point P' ; let , [dm and let v I -r- , the integral including the whole of M. This expression has been called by Green the potential* of the body M, on the point P f } and the same name has been employed by Gauss (in a Me'moire on "General Theorems relating to Attractive and Repulsive Forces, in the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839, Leipsic 1840, edited by M. Gauss and Weber)f. By a known theorem, the com- ponents of the attraction of M on P', in the directions of x, y, z, are dv dv dv and if dy be the element of any line, straight or curved, which passes through P', the attraction in the direction of this element is -j-f . Hence it follows that if a surface be drawn dy through any point P' for every point of which the potential has the same value, the attraction on every point in the surface is wholly in the direction of the normal. Surfaces for which the potential is constant are therefore called, by Gauss, surfaces of equilibrium. It has been shown in a former paper (i. above), * [" This I found in a reference to his memoirs, in Murphy's first memoir on " definite integrals. Ever since I have been trying to see Green's memoir, but " could not hear of it from anybody till to-day, when I have got a copy from " Mr Hopkins. Jan. 25, 1845." (Private note which I find written on p. 190 of vol. iii. of my copy of the Camb. Math. Jour.)] t Translations of this paper have been published in Taylor's Scientific Memoirs for April, 1842, and in the Numbers of Liouville's Journal for July and August, 1842. xii.] Propositions in the Theory of Attraction. 127 that if M, instead of an attractive mass, were a group of sources of heat or cold in the interior of an infinite homogeneous solid, v would be the permanent temperature produced by them at P. In that case, the surfaces of equilibrium would be iso- thermal surfaces. 188. When the attraction of (positive or negative) matter, as for instance electricity, spread over a surface is considered, the density of the matter at any point is measured by the quantity of matter on an element of the surface, divided by that element. The principal object of this paper is to prove the following theorems : If upon E, one of the surfaces of equilibrium enclosing an attracting mass, its matter be distributed in such a manner that its density at any point P is equal to the attraction of M on P; then (1) The attraction of the matter spread over E, on an external point, is equal to the attraction of M on the same point multi- plied by 4-7T. (2) The attraction of the matter on E } on an internal point, is nothing. 189. These theorems were proved in a previous paper (i. 5, 9), from considerations relative to the uniform motion of heat; but in the following they are proved by direct integration : Let u be the potential of M, on the point P, (soyz) in E. The components of the attraction of M on P, in the directions of x, y, z, are du du du ~Tx' ~Ty* ~dz'> and hence, if a, /3, 7 be the angles which a normal to E at P makes with these directions, the total attraction on P is du du ' du \ du if dn be an element of the normal through P. This is therefore the expression for the density at P of the matter we have supposed to be spread over E. Let ds be an element of E at P ; let v' be the potential of E, on a point P, (x'y'z'}, either within or without E ; and let A be the distance from P to P. Then 128 Propositions in the Theory of Attraction. [xii. du du du \ du _ cosa+ cosff + . the brackets enclosing the integrals denoting that the integra- tions are to be extended over the whole surface E. Now for ds, we may choose any one of the expressions, , dvdz i dxdz , dxdii ds = , ds = - 7, , ds = * . cos a cosp cos y Hence any integral of the form {f(A cosa + B cos /3 -f G cos 7) ds} may be transformed into the sum of the three integrals, (!$Adydz\ (tfBdxdz), (NCdxdy), by using the first, second, and third of the expressions for ds in the first, second, and third terms of the integral respectively. Hence, if A = Q+, B = ^+, C = ^, dx r dy r dz T ({ d( l> * j \ ( f"3T W* ) \Jdn r J or o i J 7 cos a + 7 cos p + -7^- cos 7 yds}-, dx dy dz / ' j the limits of the integrations relative to y and z, x and 2, x and y, being so chosen as to include the whole of the surface considered. 190. Making use of this transformation in (a) we have f du dydz du dxdz du dxdy^C\ f ,. AT dudydz .. . -~ fd*u 1 du d Now A 191. Hence, if the integrals in the second member include every point in the space contained between E, and another surface of equilibrium, E t , without E, and which we shall sup- pose to be also without P', we have dudydz] ( CCdu dydz} [fffcPu 1 du d IN , , , Sr*-\ t - Wdx-^-\ = JJj te A + dx dx A) **** the accent denoting that, in the term accented, the integrals are XIL] Propositions in the Theory of Attraction. 129 to be extended over the surface E t . Modifying in a similar manner the second and third terms of v, we have du , du , du , as . f , -y- as (ft/ffu d*u cfu du d 1 du d 1 dud 1 \ = JJJ teajp+5 S ' A + d A + ^S ' AJ ^ dy Now, for all points without M, d 2 u d*u } by a known theorem ; and such points only are included in the integrals in the second member of (c). Also, by integration by parts, = ll u i^y^ -IS! W ^l Modifying similarly the two remaining terms of the second member of (c), we have du -. dljj dlj, dlj,\] d 2 l Now, since E and ^ are surfaces of equilibrium, u is con stant for each. Again, except when P coincides with P 7 , at which point w has the value u. Hence, the value of the integrals, 2 1 J 2 1 d 2 1 is only affected by these elements, for which u = u, and hence u may be taken without the integral sign, as being constant and equal to u. If, therefore, for brevity, we put T. E. 9 130 Propositions in the Theory of Attraction. [xn. = or according as the integrals refer to E, or to E y) and 1 tf 1 d 2 the integrations including every point between E and E' ; equa- tion (c) becomes du j T-as, ,'T, ("} UK (c ). Now it is obvious that, at a great distance from M, the surfaces of equilibrium are very nearly spherical. Let E' be taken so far off that it may be considered as spherical, without sensible error, and let 7 be the distance of any point in E' from the centre, a fixed point in M, or, which is the same, the radius of the sphere. Then ^- , or -=- , is the attraction of M, dn dy on a point in E ', and is therefore equal to z , and therefore, by the known expression for the potential of a uniform spherical shell, on an interior point, du j j :1 -Bfffefo) M A 4f7ry = 4s7r(u) / It now only remains to determine the integrals (h], (h) lt and k. By putting, in (6), ^ = 1, ^ )= 'T> we ^^ the following transformation, for (h), jl , f A , fdA ds h = I -5 as = I -= -T-. . J c?w J rfw A 2 Now let the point (xyz) be referred to the polar co-ordinates, 7, 0, $. Then, if P be pole, 7 = A. Also, if ^ be the angle between A and dn } the expression for ds is , A 2 sin 0d0cfa> . c^A 5 = - r r , or, since cos ty = -7- , 7A dA c?/i Hence ^ = -si xii.] Propositions in the Theory of Attraction. 131 If P' be within the surface to which the integrals refer, the limits for 6 are and TT, and for <, and STT, and in that case, h == 4<7r ; therefore, since P' is always within E t , ' If P' be without the surface considered, then, for each value of 6 } we must take the sum of the expressions - sin 6dOd, and - sin 6 (- dO) dfa and, therefore, each element of the integral is destroyed by another equal to it, but with a contrary sign, and the value of the complete integral is therefore zero. Hence, according as P' is without or within E, (h) = 0, or (h) = -47r .................... (A). Again, to find the value of k, we have, by dividing it into three terms, and integrating each once, = (h\ - (h) = - 4?r - 0, or = - 4?r + 4-rr ; and, therefore, according as P' is without or within E, & = _47T, or & = ...................... (k). Hence, making use of (/), (g), (h), (k), in (c")- t we have v' = 4<7ru, when P' is without E. ........ .... (1), i/ = 4w(tt), when P' is within ^. ............. (2). From the first of these equations it follows that the attrac- tion of E, on a point without it, is the same as that of M t multiplied by 4?r; and since the second shows that the potential of E on internal points is constant, we infer that the attraction of E on internal points is nothing. These theorems, along with some others which were also proved in the previous paper in this Journal, already referred to, had, I have since found, been given previously by Gauss. One of the most important of these is the following : If a mass Jfbe wholly within or wholly without a surface, an equal mass may be distributed over this surface [in the former case, or a certain less mass may be distributed over it in the latter case] in such a manner that its attraction, in the former case on 92 132 Propositions in the Theory of Attraction. [xn. external points, and in the latter on internal, will be equal to the attraction of M on the same points. This theorem, which was proved from physical considerations in the paper On the Uniform Motion of Heat, etc., is proved analytically in Gauss's Memoir e, but the same method is used in both to infer from it the truth of propositions (1) and (2). From Prop. (2) it follows that, if E be the surface of an electrified conducting body, the intensity of the electricity at any point will be proportional to the attraction of M on the point. Hence we have the means of finding an infinite number of forms for conducting bodies, on which the distribution of electricity can be determined. Thus, if M consists of a group of material points, m v m 2 , etc., whose co-ordinates are x l9 y^ z l5 ; a? 2 , y z , z z , etc.: the general equation to the surfaces of equilibrium is 2U T7Z N2 , /. ... \! , /_ _ \2)i."~ tC. A, and the intensity of electricity at any point of a solid body, bounded by one of them, will be the value of dy at the point. To take a simple case : Let there be only two material points, of equal intensity. The surface will then be a surface of revolution, and will be symmetrical with regard to a plane perpendicular, through its point of bisection, to the line joining the two points, and would probably very easily be constructed in practice. We should thus have a simple method of verifying numerically the mathematical theory of electricity. PAET H. [From the Cambridge Mathematical Journal, February 1843.] 199. I shall now prove a general theorem, which comprehends the propositions demonstrated in Part I., along with several others of importance in the theories of electricity and heat. Let M and M t be two bodies, or groups or attracting or re- pelling points ; and let v and v l be their potentials on xyz\ let R and R^ be their total attractions on the same point ; and xii.] Propositions in the Theory of Attraction. 133 let 6 be the angle between the directions of R and R v and a/97, a i^i7i> the angles which they make with xyz. Let S be a closed surface, ds an element, corresponding to the co- ordinates xyz ; and P and P x the components of R^R^ in a direction perpendicular to the surface at ds. Then we have dv cos = cos a cos a t + cos /3 cos /3 t 4- cos 7 cos y l ; , dv dv. dv dv. dv dv. a hence, T~^ +^~ T-^^"^-^ = ^^i cos ^ dx dx dy dy dz dz Hence dv dv. dvdv. where we shall suppose the integrals to include every point in the interior of S. Now, by integration by parts, the second member may be put under the form, f where the double integrals are extended over the surface 8, and the triple integrals, as before, over every point in its interior. If we transform the first term of this by (b), Part I., and observe that -j- = P, it becomes dn -Jfvfds. d?v d z v 6? v , x except when xyz is a point of the attracting mass. If this be the case, and if k be the density of the matter at the point, we have d*v d*v d?v A 7 A "1 T-i + J~2 + -j- z + 47TA; = da? dy' dz* I ......... ^ therefore ff? + ^ + f?) ^%^ + 47r^m = 134 Propositions in the Theory of Attraction. [xn. Hence (a) is transformed into ///RZ^ cos Odxdydz = ^fffv^m - jfaPds (3) ; similarly, by performing the integration in (a), on the terms dv dv dv . , r dv. dv. dv. -j- > -j- > -j- > instead of l - , -=-* , -^ , dx dy dz dx dy dz we should have found fffBE l cos edxdydz = faffjvdm^ - ffvPfo (4). 200. If the triple integrals in (a) were extended over all the space without S } or over every point between $, and another surface, $,, enclosing it, at an infinite distance, it may be shown, as in Part I., that the superior values of the double in- tegrals in (6), corresponding to $ /} vanish. Hence, the inferior values being those which correspond to S, we have, instead of (3) and (4), ! cos edxdydz = ^Trfjfv^m + /Jt^Pcfe (5), t cos edxdydz = 4>7rfffvdm 1 + jjvP^ds (6). It is obvious that v and v t in these equations may be any functions, each of which satisfy equations (c) and (d), whether we consider them as potentials or temperatures, or as mere analytical functions with the restriction that, in (5) and (6), v and v l must be such as to make jjvfds and ffvP^s vanish at S, [and (a condition the necessity for which has been dis- covered by Helmholtz),* that, in (3) and (4), if 8 be multiply continuous, v and v t must be single-valued functions through- out it]. If each of them satisfy (c) for all the points within the limits of the triple integrals considered, dm and dm^ will each vanish ; but if there be any points within the limits, for which either v or v 1 does not satisfy (c), the value of dm or dm^ at those points will be found from (d). 201. Thus let v l = 1, for every point. Then we must have dm^ = 0. Also R! = 0, P t = 0. Hence (3) becomes (7), * [See Helmholtz; Crelle's Journal, 1858 (Wirbelbewegung), translated by Tait, Phil. Mag. 1867, I. (Vortex-Motion) ; or Thomson (Vortex-Motion, 54... 58), Trans. Royal Society of Edinburgh, 1868.] XIL] Propositions in the Theory of Attraction. 135 if m be the part of M within S. This expression is independent of the quantity of matter without S, and if m = it becomes JfPds = ........................ (8). If M be a group of sources of heat in a solid body, P will be the flux across a unit of surface, at the point xyz. Hence the total flux of heat across S is equal to the sum of the ex- penditures from all the sources in the interior ; and if there be no sources in the interior, the whole flux is nothing. Both these results, though our physical ideas of heat would readily lead us to anticipate them, are by no means axiomatic when considered analytically. In exactly a similar manner, Poisson* proves that the total flux of, heat out of a body during an instant of time is equal to the sum of the diminutions of heat of each particle of the body, during the same time. This follows at once from (7). For if we suppose there to be no sources of heat within $, but the temperature of interior points to vary with the time, on account of a non-uniform initial distribution of heat, we have d?v d*v d?v dv Hence, by (d), we must use -j-dxdydz y instead of and therefore (7) becomes It was the- analysis used by Poisson, in the demonstration of this theorem, that suggested the demonstrations given in Part I. of propositions (1) and (2). 202. As another example of the application of the theorem expressed by (3) and (4), let v t be the potential of a unit of mass, concentrated at a fixed point, xyz. Hence, M v = 1 and dm t 0, except when xyz, at which dm^ is supposed to be situated, coincides with xyz \ and, if A be the distance of xyz from x'y'z, v*= -r . Hence, according as x'y'z is without or within S, O, or S$vdm, = v' Mdm^v .. ........ (e), See Theorie de la Chaleur, p. 177. 136 Propositions in the Theory of Attraction. [xn. the triple integrals being extended over the space within $. Now let us suppose M to be such, that v has a constant value (v) at 8. Then jjvP.ds = (v) JJP.cfe, which, by (7), is = 0, or to 4-7T (v), according as x'yz is without or within S. Hence, by comparing (3) and (4), we have, in the two cases, dm and 4*- = -^ M therefore = 47r(i;) ............................... (10). These are the two propositions (1) and (2) proved in Part I., which are therefore, as we see, particular cases of the general theorem expressed by (3) and (4).* 203. If v = v lt and if both arise from sources situated with- out S, (3) becomes fffR*dxdydz = ffvPds ................ (11), a proposition given by Gauss. If v have a constant value (v) over S, we have ffvPds = (v) ffPds = 0, by (8), hence JJfffdxdydg = 0. Therefore _R = and v = (v) for interior points. Hence, if the potential produced by any number of sources have the same value over every point of a surface which contains none of them, it will have the same value for every interior point also. If we consider the sources to be spread over S, it follows that v = (t>) at the surface is a condition which implies that the attraction on an interior point will be nothing. Hence the sole condition for the distribution of electricity over a conducting surface, is that its attraction shall be everywhere perpendicular to the surface, a proposition which was proved from indirect considerations, relative to heat, in a former paper.-f * It may be here proper to state that these theorems, which were first demonstrated by Gauss, are the subject of a Memoire by M. Chasles, in the Additions to the Connaissance des Temps for 1845, published in June, 1842. In this Me"moire he refers to an announcement of them, without a demonstra- tion, in the Comptes Rendus des Seances de VAcademie des Sciences, Feb. 11, 1839, a date earlier than that of M. Gauss's Memoire, which was read at the Royal Society of Gottingen in March, 1840. t Sae i. above, 5. XIL] Propositions in the Theory of Attraction. 137 204. In exactly a similar manner, if none of the sources be without S, by means of (5) and (7), it may be shown that HSE*dxdydz=irM(v) (12); the triple integrals being extended over all the space without $. Hence a quantity of matter p can only be distributed in one way on S, so as to make (v) be constant. For if there were two distributions of /-t, each making (v) constant, there would be a third, corresponding to their difference, which would also make (t>) constant. The whole mass in the third case would be nothing. Hence, by (12), we must have jjjR^dxdydz Q, and therefore R for external points ; and, since (v) is constant at the surface, R must be = for interior points also. Now this cannot be the case unless the density at each point of the surface be nothing, on account of the theorem of Laplace, that, if p be the density at any point of a stratum which exerts no attraction on interior points, its attraction on an interior point close to the surface will be 4?r/D. This important theorem, which shows that there is only one distribution of electricity on a body that satisfies the condition of equilibrium, was first given by Gauss. It may be readily extended, as has been done by Liouville,* to the case of any number of electrified bodies, influencing one another, by supposing S to consist of a number of isolated portions, which will obviously not affect the truth of (5) and (6). Then, if we suppose v to have the constant values, (v), (v)', etc., at the different surfaces, and the quantities of matter on these surfaces to be M, M' t etc., we should have, instead of (11), JSJKdxdydz = 4?r {M (v) + M' (v}' + etc.} (13), and from this it may be shown, as above, that there is only one distribution of the same quantities of matter, M, M', etc., which satisfies the conditions of equilibrium. 205. If both M and M t be wholly within S, by comparing (5) and (6), or if both be without $, by comparing (3) and (4), we have ffPv.ds^ffPjids (14). * See Note to M, Chasles' Memoire in the Connaissance des Temps for 1845. 138 Propositions in the Theory of Attraction. [xn. Now let 8 be a sphere, and let r6(f> be the polar co-ordi- nates, from the centre as pole, of any point in the surface to which the potentials v and v l correspond. Then we shall have P -7- , P l = -7- 1 , and we may assume ds = r* sin 6ddd(f). Hence (14) becomes fir r2ir fjv fir r2v ,j, fLfrMMf~j v^smedOdti ..... (15). JoJo Vr JoJo dr This equation leads at once to the fundamental property of Laplace's coefficients. For if v and v l be of the forms Y m r m , Y n r n , m and n being any positive or negative integers, zero included, and Y m and Y n being independent of r, we have, by substitution in (15), r r Y * T n sin eded 4> = n r r r JOJO J J J If m be not = n, this cannot be satisfied unless /T JoJo ,(16). OJO This is the* fundamental property of Laplace's coefficients. There are some other applications of the general theorem which has been established, especially to the Theory of Elec- tricity, which must, however, be left for a future opportunity. * [For a justification of this use of the definite article, see Murphy's Elec- tricity, Chap. i. Props, i. and n., Cambridge 1833.] XIII. THEOREMS WITH REFERENCE TO THE SOLUTION OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS. (Art. xxxvi. of complete list in Mathematical and Physical Papers, Vol. i.) [From the Cambridge and Dublin Mathematical Journal, Jan. 1848.] 206. Theorem 1. It is possible to find a function V, of x, y, z* which shall satisfy, for all real values of these variables, the differential equation a being any real continuous or discontinuous function of x, y, z, and p a function which vanishes for all values of x, y, z, exceed- ing certain finite limits (such as may be represented geo- metrically by a finite closed surface), within which its value is finite, but entirely arbitrary. Theorem 2. There cannot be two different solutions of equa- tion (A) for all real values of the variables. 1. (Demonstration). Let Z7be a function of x,y, z, given by the equation 77 _ fff pdx'dy'dz - - the integrations in the second member including all the space for which p' is finite ; so that, if we please, we may conceive the limits of each integration to be oo and + oo , as thus all the values of the variables for which p' is finite will be included, and the amount of the integral will not be affected by those values of the variables for which p vanishes, being included. Again, V being any real function of x, y, z, let * The case of three variables, which includes the applications to physical problems, is alone considered here; although the analysis is equally applicable whatever be the number of variables. 140 Theorems with reference to the Solution of [xm. .00 p .00 , f dv Irf 0y . dv ldu ^ HJ JrS-S-ffi) + (*^--dy) It is obvious that, although V may be assigned so as to make Q as great as we please, it is impossible to make the value of Q less than a certain limit, since we see at once that it cannot be negative. Hence Q, considered as depending on the arbitrary function V, is susceptible of a minimum value ; and the calculus of variations will lead us to the assigning of V according to this condition. Thus we have fff(f dV ldU\ dSV ( dV 1 dU\ dSV i a ^ --- -j- }- a ^r- + a ^ --- -j- - a -^- JJJ(\ dx a dyj dx \ dy a dy J dy r dV ldU\ dSV} , , , + a-, ---- T- a j Y dxdydz. \ dz a dz ) dz j Hence, by the ordinary process of integration by parts, the integrated terms vanishing at each limit,* we deduce d f <,dV dU But by a well-known theorem (proved in Pratt's Mechanics, and in the treatise on Attraction in Earnshaw's Dynamics), we have &U d*U d*U Hence the preceding expression becomes We have, therefore, for the condition that Q may be a maximum or minimum, the equation, d t ,dV\ d f ,dV\ d f ,dV a +* + a to be satisfied for all values of the variables. * All the functions of as, y, z contemplated in this paper are supposed to vanish for infinite values of the variables. XIIL] certain Partial Differential Equations. 141 Now it is possible to assign F so that Q may be a minimum, and therefore there exists a function, V, which satisfies equa- tion (A). 2. (Demonstration). Let F be a solution of (A), and let V l be any different function of x, y, z, that is to say, any function such that F t F, which we may denote by , does not vanish for all values of x, y, z. Let us consider the integral Q x , obtained by substituting F t for Fin the expression for Q. Since / dV t ldU\ z f dV IdU\* / dV ldU\ d6 z dtf [a -y- 1 --- 7- = [a-j --- -T- +2[a-j ---- r~ a-^r + a --> \ dx a dx J \ dx a dx J \ dx a.dx J dx dx z we have n n , offff/' dV ^ dU \ d( t> ( dV ldU \ d( t> Q =Q+2 H a -5 --- -j- a-^ + a-j --- -j- a-^ J J J (V dx a. dx J dx \ dy a dy ) dy Now, by integration by parts, we find )3r-. dxdydz J -J -ooJ co dx a dx J dx d the integrated term vanishing at each limit. Applying this and similar processes with reference to y and z, we find an expression for the second term of Q lf which, on account of equation (A), vanishes. Hence which shows that Q l is greater than Q. Now the only pecu- liarity of Q is, that F, from which it is obtained, satisfies the equation (A), and therefore F, cannot be a solution of (A). Hence no function different from F can be a solution of (A). The analysis given above, especially when interpreted in various cases of abrupt variations in the value of a, and of infinite or evanescent values, through finite spaces, possesses very important applications in the theories of heat, electricity, magnetism, and hydrodynamics, which may form the subject of future communications. EmNBARNET, DUMBARTONSHIRE, Oct. 9, 1847. 142 Theorems with reference to the Solution of [XIIT. ADDITION TO A FRENCH TRANSLATION OF THE PRECEDING. [From Liouville's Journal de MatMmatiques, 1847.] 207. Dans les applications qui presentent le plus d'interet, il faut considerer des transitions subites dans la valeur de a. Par example, si a a une valeur constante dans tout 1'espace ex- te'rieur a une surface fermee S, dans I'inte'rieur de laquelle a est infinie, notre analyse convient au cas d'un corps conducteur 8 soumis a 1'influence d'une masse electrique donn^e (Jffpdxdydz), et cette application ne presente aucune difficult^. On en tire, en effet, les demonstrations donne'es par Green, que la solution analytique du probleme de la distribution d'electricite dans ces circonstances est possible et qu'elle est unique. Dans une application a 1'hydrodynamique, ou a un certain probleme de magne'tisme, il faut considerer un espace dans lequel la valeur de a soit z^ro. L'interpretation du r^sultat ne pr^sente aucune difficulte, mais il est plus difficile de bien comprendre comment la demonstration telle que je 1'ai donnee plus haut se prte ^, ce cas. En essayant de 1'expliquer nettement, j'ai trouve' une demonstration directe du th^oreme suivant, qui renferme le r^sultat dont il s'agit : "II est possible de trouver une fonction Fqui s'eVanouisse pour les valeurs infmiment grandes des variables x, y, z, et satisfasse a 1'dquation tfV _ dx? dy* dt? "" ) pour tous les points extdrieurs a une surface fermee S } avec cette condition dV -F fa-*> dans laquelle F est une fonction arbitraire des coordonn^es d'un point sur la surface 8, et dn est 1' element d'une normale ext^rieure a la surface en ce point." Pour le demontrer, consid^rons 1'inte'grale XIII.] certain Partial Differential Equations. 143 relative a 1'espace exterieur a S. Parmi toutes les fonctions V qui verifient la condition ou A est une quantite* quelconque, il y en a une pour laquelle rint^grale Q est un minimum. Une fonction F, ainsi de'ter- mine'e, satisfait aux Equations fY_ df + dz>- dr ,F -7 = CJP an (ou c est une constante), comme on s'en assure par le calcul des variations. Suivant les valeurs de A, c aura des valeurs proportionnelles ; on peut prendre A telle que c 1. De la on conclut le the'oreme e'nonce. II serait facile d'aj outer une demonstration, que la solution du probleme de la determination de V sous ces conditions est unique.* * [Provided S is a simply continuous surface. If S be a multiply continuous surface, as, for instance, the inner boundary of an endless tube (a finite tube with its ends united, so as to constitute a circuit), we may add to V the velocity- potential of a liquid moving through it irrotationally (Thomson and Tait's Natural Philosophy, 184 190 ; Thomson, Vortex Motion, 54... 58) without violating the conditions prescribed in the text. Compare above, 200, footnote.] XIV. ELECTRIC IMAGES. EXTKAIT D'UNE LETTEE DE M. WILLIAM THOMSON A M. LIOUVILLE. (Art. xix. of complete list in Mathematical and Physical Papers, Vol. I.) [From Liouville's Journal de Mathematiques, 1845.] " CAMBRIDGE, 8 Octobre 1845. 208. "...Pendant mon sejour a Paris, je vous ai parle du principe des images pour la solution de quelques problemes relatifs a la distribution de 1'electricite. II y a une foule de problemes auxquels je ne pensais pas alors, et ou j'ai trouve' plus tard qu'on peut 1'appliquer. Par exemple, on parvient ainsi a exprimer algebriquement la distribution d'electricite' sur deux plans conducteurs qui se coupent sous un angle -. , quand un point electrique est pose* dans 1'espace entre les deux plans. (L'ide'e est analogue a celle du kaleidoscope de Brewster.) Quand il y a trois plans qui se coupent perpen- diculairement, ou quand il y a un plan qui coupe perpendicu- lairement deux plans qui se coupent sous un angle - , on peut i egalement trouver la distribution sous I'influence d'un point electrique donnd On peut aussi exprimer tres-facilement la distribution sur les parois inteVieures d'un parallelipipede rect- angulaire creux, soumis a 1'influence d'un point electrique pose* en dedans, en se servant des inte'grales de'finies. "Soient C le centre d'une sphere S; Q, Q' deux points pris sur un m6me rayon CA et sur son prolongement, de telle maniere que CQ.CQ' = (Li 2 ; et P un point quelconque sur la surface S. On a, comme on sait, PQ' AQ" On peut, a cause de ce the'oreme, appeler Q et Q' points recipro- ques relatifs a la sphere S, dont chacun est Vintage de 1'autre xiv.] Electric Images. 145 dans la sphere. Suivant cette definition, 1'image d'une ligne ou surface sera le lieu des images de points pris sur cette ligne ou surface. Ainsi, on trouve que 1'image d'un plan ou d'une sphere est toujours une sphere (le plan etant compris sous cette designation). Les images de deux spheres se coupent sous le meme angle, reel ou imaginaire, que les surfaces donn^es. "Soient Q, Q' deux points re'ciproques, relativement a une sphere 8, et q, q', s leurs images et 1'image de la sphere 8 dans une autre sphere donnde. Les points q, q' seront re'ciproques relativement a la sphere s. 209. " A 1'aide de ces thdoremes, je parviens facilement a determiner les images successives d'un point quelconque (qui n'est pas ne'cessairement dans la ligne qui passe par leurs centres), dans deux spheres qui se coupent sous un angle donne. Quand cet angle est imaginaire, je parviens ainsi a exprimer la distribution de 1'electricite' sur les deux spheres, sous I'influence d'un point quelconque, charge* d'eiectricite, au moyen des series de M. Poisson (qui convergent comme des series ge'ome'triques). Quand Tangle d'intersection est reel et rrp compris dans 1'expression - , on parvient ainsi a exprimer algebriquement la distribution d'une quantite donnde d'elec- tricite sur la surface exterieure des spheres, qui n'est soumise a aucune influence ou qui Test a celle d'un point donne. S'il y a trois surfaces sphe'riques qui se coupent perpendiculairement, on exprime algebriquement, par les memes principes, la distri- bution sur la surface exterieure. Je parviens aussi a deter- miner les temperatures stationnaires dans Tinte'rieur d'une 7T lentille dont les deux surfaces se coupent sous un angle - , la temperature de chaque point de ces surfaces etant donnde. 210. "Si Ton veut determiner la distribution d'eiectricite sur une surface donn^e S, sous Tinfluence d'un point quelconque Q, on reduit, par les me"mes principes, le probleme a la determina- tion de la distribution, sans aucune influence, sur 1'image de S dans une sphere decrite du centre Q, avec un rayon quelconque, Une application generale de ce theoreme conduit a une demon- stration rigoureuse du theoreme de M. Gauss, qu'on peut pro- duire, au moyen d'une distribution determinee de matiere sur T. E. 10 146 Electric Images. [xiv. une surface fermee quelconque, une valeur donnee du potentiel a chaque point de la surface. II y a aussi beaucoup d'applica- tions spe'ciales [see below, 218... 220] qu'on peut faire de ce theoreme aux cas dans lesquels S est une sphere, un disque circulaire, ou un segment d'une surface spherique fait par un plan. J'en ai aussi de'duit une demonstration geometrique du the'oreme que vous avez publid dans le nume'ro d'avril 1845 de votre Journal (voir page 137), dont voici 1'expression analy- tique * * *" [see above, XL 167, 186]. EXTRAITS DE DEUX LETTRES ADRESSEES A M. LIOUVILLE, PAR M. WILLIAM THOMSON. [From Liouville's Journal de Mathtmatiques, 1847.] "CAMBRIDGE, 2Gjuin 1846. 211. "... Les recherches sur lesquelles je vous ai e'crit, le 8 octobre 1845, m'ont conduit a Temploi d'un systeme nouveau de coordonne'es orthogonales tres-commode dans quelques pro- blemes des theories de la cbaleur et de 1'electricite'. Les sur- faces eoordonne'es dans ce systeme sont les surfaces engendrees par la rotation, autour d'un axe convenable, d'un systeme de coordonne'es curvilignes dans un plan, et les plans ineVidiens. En effet, soit M un plan meridien quelconque ; les coordonne'es d'un point P dans ce plan sont deux cercles qui se coupent a angle droit en ce point, et dont le premier passe par deux points fixes A, A', dans 1'axe de revolution X'X, tandis que le second est la courbe orthogonale de la se'rie entiere des cercles qui passent par les points A, A'. On de'montre facilement que cette courbe est un cercle qui passe par deux points imaginaires B, E' y dans la droite TOY perpendiculaire a X 'OX, a des dis- tances aux deux cote's de dont chacune est ^gale a a */ 1, a etant la valeur des distances ^gales A'O, OA. En effet, la premiere s^rie est exprime'e par liquation (!) a? + i/ 2 - 2uy = a 2 , u etant un parametre variable, et Ton en de'duit (2) # 2 + 2/ 2 -2^ = -a 2 , pour liquation de la courbe orthogonale. xiv.] Electric Images. 147 212. " Posons u = a cot 0, v = a J 1 . cot ^ ; 6 sera Tangle que la tangente du cercle (1), au point A ou A' t fait avec 1'axe X'X, et ^ sera Tangle imaginaire que la tangente du cercle (2), au point B ou B', fait avec Y' Y. Pour avoir la se'rie entiere des cercles (1), il faudrait donner a u toutes les valeurs re'elles de oo a oo , ou a 6 toutes les valeurs de a TT ; et, pour la serie (2), il faudrait donner a v toutes les valeurs de a a oo , et de GO a a. On peut conside'rer un point P comme determine' sans ambigui'te' par les coordonne'es 9, ty (en prenant 6 + TT au lieu de 6 pour Tautre point d'intersection des memes cercles). Les Equations de transformation, entre les coordon- nee's (x, y) et (6, ty) d'un meme point P, sont (3) fl^. + y* - 2ay cot = a 2 , (4) x* + y* - 2axcot^J^l = -a 2 . On en de'duit sin^/r J 1 cos i|r cos 6 ' sin cos i|r cos ^ ' cos ^Ir + cos 6 =a Dans les applications physiques, il s'agit d'exprimer la distance A, entre deux points P, P' en fonction des nouvelles coordon- ne'es. On trouve facilement, a Taide des formules donnees ci-dessus, dans le cas de P et P' dans un meme plan me'ridien M, (COS ^r COS 6) (COS ty' COS 6') ' Pour le trois coordonne'es d'un point dans Tespace, je prends 0, i/r qui fixent sa position dans un plan me'ridien, et Tangle que ce plan fait avec un plan me'ridien fixe. Je trouve main- tenant, pour la distance entre deux points quelconques P, P', . 2 _ 2 2 cos (i|r- 1|/) - [cos 6 cos & + sin sin 0' cos ( j> . " Les surfaces repr^sentees par liquation r = constante sont des spheres engendre'es par la revolution d^une s^rie de cercles autour de la droite qui contient leurs centres. Sup- posons que 1'espace entre deux de ces spheres (quand chaque sphere est en dehors de 1'autre, cet espace sera 1'espace infini en dehors des deux spheres), dont les Equations sont r = a, r = a l} soit rempli d'un milieu solide homogene, que les temperatures de tous les points de chaque surface soient donn^es, et qu'il s'agisse de determiner la temperature stationnaire d'un point quelconque dans le solide; on r^soudra ce probleme avec beaucoup de facilite* au moyen de 1'analyse de Laplace, en employant les coordonne'es que j'ai indiqu^es. Dans le cas particulier d'une temperature constante pour chaque sphere, on parvient, apres quelques reductions, a trouver la solution que Poisson a donnee pour le probleme correspondant de deux spheres eiectrisees. 213. " II y a un systeme nouveau et tres-remarquable de coordonnees, qu'on trouve en posant r cos 6 = % , r sin 6 cos (j> = 77, r sin sin < = r, 0, (j) appartenant au systeme explique ci-dessus. Dans ce systeme (f, 77, f), les surfaces coordonnees sont des spheres orthogonales qui passent par un point fixe, et qui touchent, par xiv.] Electric Images. 149 consequent, trois plans orthogonaux menes par ce point. Je suis parvenu a consid^rer ces systemes de coordonne'es en cherchant les images des series de surfaces des systemes (polaire et rectangulaire) ordinaires, dans des spheres convenablement dispose'es. " L'application du systeme (f, 77, f) aux problemes de physique, pour le cas de deux systemes qui se touchent 1'un 1'autre, en donne les solutions avec beaucoup de facilite*; mais il est plus simple de faire directement la recherche de ces coordonne'es, que de les de'duire du systeme (r, 0, <). En effet, soient les Equations de trois spheres qui se coupent a un point P (elles se coupent aussi a 1'origine 0). Je prends f, ij, f pour fc_ 1 -M ^ les coordonne'es de ce point (il faudrait substituer - - - - a a a dans ces equations, au lieu de 97, f, pour retrouver les coordon- nees f, rj } findiqudes ci-dessus). De ces Equations on tire -' " ~ et liquation devient, pour les nouvelles coordonnees, f ^ *(W , *W . ~~" ~ oh Pour exemple de 1'emploi qu'on peut faire de ce systeme de coordonnees, supposons que la temperature d'un point (a, 77, f) est une fonction donned F (77, ) des coordonnees 77, f de sa 150 Electric Images. [xiv. position sur la sphere a, et que la temperature d'un point (!> *7> ?) est -^1 fo f)> et a 1 ). Ces reductions faites, 1'expression (b) se trouve re'duite a XIV.] Electric Images. 151 et jTg A /00 v = ~ I I <#wd^ cos WT; cos wf tei-Ji* h [*<,) - 6 -M i)] suivant les deux cas. Liquation (I) se rdduit a if-Fj a cause de la valeur qu'on trouve pour 1'intdgrale de'fmie qui y est contenue*. 215. " L'expression pour v, dans le second cas, se trouve reduite en serie convergente, si Ton substitue pour _ h(a la se'rie et puis, pour chaque terme, sa valeur, suivant la formule citde dans le cas (I). On trouve ainsi + ou 7 = 2 (a - o^). * Les int^grales d^finies (c) et (I) sont des cas particuliers de deux integrates multiples dont j'ai trouv6 les valeur s en cherchant une demonstration de la formule (5), tome X de votre Journal, page 141. J'ai trouve", [above, 182, formula (F)J, en effet, n-l [ f 00 dp 1 dp2... J -oo J -co '" : et /oo' /"GO -ff. J -oo " .cos m l x 1 cos m^x 2 ... d'ou Ton d6duit immediatement les integrales cit6es. 152 Electric Images. [xiv. De cette expression on deduit facilement la distribution d'elec- tricite' sur deux spheres qui se toucheat. 216. "Le cas (I) correspond a deux spheres dont 1'une, (a), est en dedans de 1'autre, (aj. Dans le cas (II), le solide consid^re remplit 1'espace entier en dehors des deux spheres, et la temperature est ze'ro a une distance in rime. 217. "II y a une interpretation pour le nouveau systeme de coordonne'es (r, 6) dans un plan, qui est tres-simple. En effet, soient A, A' deux points fixes, et P un point quelconque dont il s'agit d'exprimer la position. Cela peut se faire au moyen de Tangle APA', que j'appelle 0, et de la raison r de AP a AP. Quand 6 a une valeur constante, le lieu de P est un cercle qui passe par les points A, A' ; et quand r a une valeur constante, le lieu de P est un cercle, dont le centre est dans le prolonge- ment de AA' t d'un cote' ou de 1'autre, suivant que cette valeur est plus grande ou plus petite que 1'unite, et qui a la propri^te de couper a angle droit tout cercle de'crit par les points A, A. " Posons maintenant, pour expliquer le second systeme, r cos = % , r sin 9 = rj. Le lieu de P, quand f a une valeur constante, sera tel que, si Ton mene, de A, AD perpendiculaire a A'P, la raison DP-=r AP sera constante, et Ton trouve ainsi que ce lieu est un cercle qui touche en A' une droite perpendiculaire a A' A ; et Ton trouve semblablement que le lieu de P, quand 77 a une valeur constante, est un cercle qui touche A' A au point A'." "KNOCK, le 16 septembre 1846. 218. "...Depuis que je vous ai ecrit la derniere fois, j'ai consid^re* le probleme de la distribution d'electricite sur le segment d'une couche sph^rique infiniment mince, fait par un plan, ce corps e'tant compose' de matiere conductrice, et j'ai trouve*, en expression finie, la solution complete, en sup- posant que le corps possede une quantit^ donn^e d'electricite' et que la distribution se fait sous I'innuence de masses e'lec- triques donne'es. J'avais Tintention de rddiger de suite pour vous un petit Memoire sur ces recherches, mais j'ai rencontre quelque difficult^ dans 1' exposition de la mdthode suivie, et comme je suis a present tres-occupe' (les cours a Glasgow commencent le l er novembre, et il me faudra beaucoup de xiv.] Electric Images. 153 preparation), il me faut diffdrer cette tache*. Je me bornerai pour le moment aux e'nonce's de quelques-uns des re'sultats. 219. " Soit 8 le corps conducteur sur lequel il s'agit de de*- terminer la distribution. Pour premier cas, soit Q un point en dehors de $, sur la meme surface spherique dont S fait partie, et supposons que S soit mis en communication avec le sol par un fil conducteur infiniment mince (ainsi le potentiel dans S sera toujours zdro, quels que soient les corps electrises qui en soient voisins). II s'agit de determiner la distribution d'elec- tricite sur 8 sous I'influence d'une quantite donn^e d'electricite negative Q, concentred au point Q. Je d^montre que 1'intensite d'electricite a la meme valeur aux points voisins des deux cote's de la couche S, et, en denotant par , z et f, ?;,.'.., f deux groupes con- tenant un nombre egal ou inegal de variables, les premieres sc, y,..., z y inddpendantes, les autres f, ^..., ffonctions des pre- mieres, en sorte que soit encore p = ty (x, y, . . ., z). Designons d'ailleurs par f, ?/',.> %> P ce ^ ue deviennent les fonctions f, 17,..., f, p, quand on j remplace x, y,..., z par #', y',... } z. Cela posd, on demande de determiner les fonctions /, F,..., 1 7"- quelconques. II n'y aurait de changement que dans quelques details, et seulement si le nombre des variables e'tait different dans les deux groupes. Au surplus, nous n'aurons besoin plus tard que du cas ou ce nombre est le meme de part et d'autre, et ne surpasse pas trois, ce qui nous permettra d'in- terpre'ter geome'triquement les resultats de notre analyse. Donnons a x ', y' > z des valeurs particulieres a? , y , z a volonte, et repr^sentons par p , f , rj 0) f les valeurs correspon- dantes de p' t f ', if, %'. Liquation (1) nous donnera (a-0 1 +(y-yJ' 1 p ~ 2 - Mais, pour plus de simplicity, nous mettrons partout *?+*?<>> ?+ ? 0> + *v y + 2/o> ^+^o> au lieu de f> / fc x > y> z > et de meme f + ?<, ' + ^ e *v au li eu x/ > Q ^ c -> ce q u i ne change rien aux differences f ' f , a/ a;, etc. La valeur de 2 deviendra et 1'equation (1) subsistera telle qu'elle est. En faisant on aura et en portant ces valeurs dans liquation (1), on trouvera aise*- ment i 2+ 4_ 2 (|i; + ^v . t n P P VP P pp p p/ y y z Maintenant donnons a x, y f , z quatre systemes de valeurs connues a volonte, a chacun desquels re'pondront des valeurs ddtermindes de /, f, r), f, p', et nous aurons ainsi quatre equations du premier degr^ qui fourniront les valeurs de I i I I P" p 21 p 2 ' p 2 ' considdrees comme quatre inconnues, en fonction lineaire de 156 Electric Images. [xiv. x y z 1 7' ?' 7' ?' En designant done par A, B, C, D des constantes, et par P, Q, jR, $ des polynomes du premier degre' en x, y, z, ces valeurs seront de la forme ? P v D Q ? IT 1 S> J 2 = ^+-5> -l = -S-!- : 4, -^=0+-!, - 2 =.D + - 2 . p 2 r 2 p 2 r* 2 p 2 r 2 p 2 r 2 En faisant la somme des Carre's des trois premieres, on trouve une valeur de -5 qui doit etre e'gale a celle que donne la quatrieme Equation. Ainsi les deux fonctions doivent etre egales. Mais la premiere devient une fonction entiere quand on la multiplie par r 2 . II faut done que la seconde le devienne aussi, et que, par consequent, P 2 4- Q 2 + -R 2 soit e'galement divisible par r 2 . Le quotient ne peut eVidem- ment etre qu'une constante, puisque le nume'rateur et le de*- nominateur sont du meme degrd Soit m 2 cette constante, et P 2 + ^ + B* = mV = m 2 (x 2 + 2/ 2 + /). P } Q, R eHant des polynomes du premier degrd, je fais P = m (ax + ly + cz + g), Q = m (ax + Vy + cz + g'\ R=m(a"x+V'y+c"z+g"), et j'en conclus par la comparaison des deux membres, d'une part, a 2 + a /2 + a" 2 = 1, ab + M + a"V = 0, c 2 + c /2 + c //2 = 1, 6c 4- 5V + "c" = 0, Equations d'ou r^sultent, comme on sait, les Equations inverses a 2 + 6 2 +c 2 =1, aa' +66 r +cc x = 0, a /2 + 6 /2 +c /2 =l, cui" + W + cc"= 0, a" 8 +&""+(!" > = 1, a'a"+VV'+c'c"= 0; et, d'autre part, Y x = 0, c^r + c^ r + c"g" = 0, y + 6v =0, ^ + ? ?o au f, 17, ?, et a?-a? , 2/-y , ^-^ au lieu de ^ ^ ^ Ce change- ment fait, on aura les formules les plus ge'ne'rales qui puissent satisfaire a liquation (1). Nous avons done le thdoreme suivant : 158 Electric Images. [xiv. Les formules g^neVales qui peuvent satisfaire a 1'equation (1) s'obtiendront en posant d'abord z = a" (as - x ) + V'(y - y Q ] + c"(z - z.}, les coefficients a, 6, etc., ve'rifiant les Equations de condition v + a V = 0, c + c ' 2 + et enfin w R^ciproquement, on peut d^montrer que 1'^quation (1) est satisfaite de cette maniere, et trouver la valeur de p qui con- vient. D'abord, des trois dernieres formules on conclut facilement te tvj.r' ^j.r^ rv (^ - u)*+ (v- ^) 2 + (w- wf . (?-?) +to -^) +(?-?) = (tt i + ^ + l ^ ) ( tt ' + les trois pr^c^dentes donnent de mme enfin, a cause des Equations de condition entre a, b, etc., on trouve (X' - X)' + (T' - Y)' + (Z' - Z) z = (/ - ) 2 + (/ - y) 2 + (/ - z) 2 . II vient done, en effet, la valeur de ^9 2 etant 2= ( m valeur qu'on pourra aisdment exprimer en x, y, z, en observant que le produit (x 2 + Y 2 + z 2 ) (w 2 + v 2 + w 2 ) est ^gal a (4 2 + J5 2 + s ) (x 2 + Y 2 + z 2 ) + 2.4mx + 25^ Y + 2 <7mz + m 2 , xiv.] Electric Images. 159 et que x, Y, z sont connus en fonction de x, y, z. La valeur qu'on trouvera ainsi peut se mettre sous la forme fljj, y v z^ e'tant des constantes dont voici les valeurs: m (Aa + Ba f + Ca") Si done nous regardons plus tard x, y, z comme etant les co- ordonnees rectangulaires d'un point quelconque, on voit que la quantite p sera proportionnelle a la distance de ce point (x, y } z) a un point fixe (x v y^ z^). II est aisd aussi de s'assurer que 222. Pour avoir explicitement f, rj, f en a?, y, z, il suffira de remplacer u, v, w, x, Y, z par leurs valeurs. La premiere sub- stitution fournit ,. ^(x 2 +r* + z 2 ) + mx * ^ (A*+B*+ O 2 ) (X 2 + Y 2 + z 2 ) + 2^mx+ 25mY+ 2(7mz+ m 2 ' Le denominateur est precisement la valeur de mp 2 dont on vient de donner 1' expression en x, y } z, savoir, mp 2 = (A* + B* + C z ) [(x - x$ + (y- 2/1 ) 2 + (z- z^]. II ne reste done plus qu'a chercher le numerate ur. Le calcul deviendra d'ailleurs fort simple si Ton retranche des deux mem- bres la quantite* car alors le second membre pourra se r^duire a une fraction ayant pour num^rateur un polynome du premier degr^ en X, Y, z, et, par consequent aussi, en x, y, z. En de'signant done par X un tel polynome, et posant, pour abre'ger, ^ + J. 2 + 5 2 +0 2== ^' -\r on pourra ecrire f f = -g , 160 Electric Images. [xiv. y 7 et de meme ^ - if = -* , f- = - 2 , 97, f e*tant des constantes, et Y", ^ des fonctions lineaires de x, y, z. Les polynomes X, T, Z s'obtiendraient sans peine par ce qu'on vient de dire ; mais on les trouve sous une forme plus commode en ope'rant comme il suit. II est ais^ de voir qu'en attribuant une valeur infinie a une ou plusieurs des quantites x, y, z, ou, si Ton veut, en faisant X* + f + Z* = CO , t v o * M, P* ona f = e, v = y, ?=?, ^ Si done on introduit cette hypothese de a; 2 + 2/ 2 + -Z 2 = o> dans liquation g^ndrale il viendra d'ou, en efFa^ant les accents, ^xo m Mais, d'un autre cote, done c'est-a-dire = ( - *f + (y - yi )* + (z - ,y. De la, par un calcul tout semblable a celui qu'on a effectue dans le num^ro pre'ce'dent pour liquation P 2 + G 2 + ^ 2 =m 2 (^ 2 + 2/ 2 +/), on conclut qu'en reprdsentant par a, /3, 7, a', etc., des constantes assujetties aux Equations de condition a 2 + a' 2 + a" 2 =1, a/3 + a^ + a"/3" = 0, /3 //2 = 1, a 7 + aV + a V= 0, + 7 //2 =1, 07+ Y+ /SV= 0, du meme genre que celles entre a, b, etc., on devra prendre xiv.] Electric Images. 161 (y-yj + y (z-z^ Y= a' (x - x,) + & (y - Vl ) + y (z - z,), Et, re'ciproquement, il est facile de verifier qu'en adoptant ces valeurs de X, Y, Z, les formules nZ qui rdsultent de notre analyse en faisant, pour abr^ger, m entraineront liquation demandde (1) dont la solution ge'ne'rale est exprime'e ainsi d'une maniere nouvelle et plus simple. En effet, on trouve d'abord pus (X'-Z) 2 + (F- F) s + (^-^) 2 =(^- ir ) 2 + (2/'-y) 2 + (/-^, a cause des Equations de condition entre a, y8, etc. Et de la on tire c'est-a-dire 1'equation (1), en prenant _(- .)+ (y - y.) 8 _ n 223. On pourrait former inversement les valeurs de x, y, z en f, 77, f ; mais il est clair sans calcul, et a priori, que ces valeurs doivent s'exprimer par des formules du meme genre que celles qui donnent f , 77, f en #, y, z. En effet, p e'tant une fonction de x, y, z, on peut concevoir cette quantite' comme fonction de f, 77, f. Soit done = 1 ' = 1 GT CT CT dtant une certaine fonction de f, 77, et r' la m^rne fonction de f ', 77', '. L'e'quation (1) se changera dans liquation nouvelle d'une forme toute semblable a Tequation (1) elle-meme, et qui, T. E. 11 162 Electric linages. [xiv. par consequent, donnera x, y, z en f, rj, f de la meme maniere que liquation (1) a donnd f , 97, f en x, y, z. 224. On voit que, par 1'echange des lettres x, y, z et f, 77, f les unes dans les autres, une solution particuliere de 1'equation (1), je veux dire une solution dans laquelle les constantes auraient des valeurs particulieres, en donnera une autre, la plupart du temps differente, quoique rentrant toujours, bien entendu, dans le type general indique' tout a 1'heure. II est aise aussi de voir que deux solutions donndes en fournissent une troisieme. Supposons, en effet, qu'en prenant pour f , 97, q des fonctions de U, F, TT, on ait et que, de meme, en prenant pour U, V, W, p, des fonctions de x, y, z, on ait (U'~ uy+ (v- V) 2 + ( w- F) 2 = ( a?/ ~ a? )' + (y / _zy)'+( / -^ > t il est clair qu'on pourra exprimer aussi q, f, rj, en x, y, z, et qu'il viendra + > ? ?> q ue nous regarderons comme des coordonnees rectangulaires prises par rapport aux memes axes. Cette trans- formation est a rayons vecteurs r^ciproques, comme nous 1'avons vu n 225. Elle s'opere en portant sur les rayons vecteurs menes de 1'origine actuelle des longueurs inversement propor- tionnelles a ces rayons vecteurs; 1'ancienne figure se trouve ainsi changed en celle qui resulte des extr^mit^s de toutes ces longueurs. Passer ensuite de f f , y if, ? ? a f , 77, n'est qu'un simple deplacement de 1'origine, les axes restant paralleles a eux-memes; cela ne produit dans la figure trans- formed aucune alteration. Nos formules du n 222 resultent done d'une transformation par rayons vecteurs rdciproques, combin^e avec des change- raents ordinaires de coordonnees. De telles transformations en nombre quelconque donnent toujours naissance a une equa- tion de la forme (1), et Interpretation geometrique des formules par lesquelles nous avions d'abord lie (n 221) x, y, z et f, rj, f semblait en demander deux, relatives a deux origines diflerentes, 1'une pour le passage de x, Y, z a u, v, w, 1'autre pour le passage de u, v, w a f, rj, ; mais on voit, par ce qui precede, et gr^ce aux formules plus simples du n 222, qu'une seule transforma- tion suffit pour conduire au resultat le plus general ; il etait important de le demontrer. 227. Les considerations geometriques dont nous venons de faire usage, pour interpreter les formules qui conduisent a 1'equation (1), donnent lieu a des consequences remarquables dont nous allons dire quelques mots. Dans les deux figures que determinent respectivement les coordonnees x, y, z.et les coordonnees f, 77, considerons, d'une part, deux points quel- 106 Electric Images. [xiv. conques w, m, et, d'autre part, les points correspondants JJL, p. Solent D la distance des deux premiers, A celle des deux autres, en sorte que V = (x-xf+(y'-yf +(*-*)*, A=( _)+(,,'-,)+ or-?) 1 . L'equation (1), qui pourra s'ecrire _ ~p*p"' ~pp" A D ' fournit une relation entre la distance A de deux points /A, // dans 1'une des figures et les quantite's D, p, p. Nous venons de dire que D est la distance des deux points m, m correspon- dants dans 1'autre figure ; quant a p et p ', ce sont, a un facteur constant pres, les distances des points m, m a un certain point fixe. Toute relation metrique entre deux ou plusieurs dis- tances A dans 1'une des figures fournira done immediatement une relation analogue dans 1'autre figure. Mais il ne faut pas croire que les divers points correspondants a ceux de la droite A soient sur la droite D ; cela arrive pour les points extremes par la definition meme de ces droites, mais n'a pas lieu, en general, pour les points interme'diaires. En general, la suite des points correspondants a ceux d'une droite de la premiere figure forme dans la seconde figure une circonference de cercle, laquelle ne se re'duit a une ligne droite que dans un cas par- ticulier, celui oil son rayon est infini. Ayant en , rj, 1'equation d'une surface ou les equations d'une ligne appartenant a la premiere figure, il suffit de substi- tuer a , 77, f leurs valeurs pour former en x, y, z I'e'quation de la surface ou les Equations de la ligne correspondante. On trouve bien facilement, de cette maniere, que les plans et des spheres se transforment en des spheres qui peuvent se rdduire a des plans quand le rayon devient infini ; que, de meme, des droites et des circonferences de cercle se transforment en des circonfe'rences de cercle, etc. Mais, pour suivre le me'canisme de ces transformations, il suffit de considerer la transformation par rayons vecteurs reciproques, qui combinee avec des change- ments de coordonnees donne, comme on Fa vu, la transforma- tion la plus generale. Soit done xiv.] Electric Images. 167 _ nx _ nx n % n a? + 2/ 2 + z* r 2 ' ~~ a + tf -f- f* ~~ n 2 ' ny _ n y nrj _ nij , _ ^ __ w> nf wf D nD nD PP *J(x* + y*+z z )(x'* + y'*+z'*) rr 1'ense mble des formules relatives a la transformation par rayons vecteurs re'ciproques. On en conclut immediatement ce que nous venons d'avancer, concernant les plans et les spheres, les droites et les circonferences de cercle, Mais on voit, de plus, et meme sans calcul, que les plans qui passent par le point 0, origine des rayons vecteurs. sont les seuls qui restent des plans dans la transformation; avant et apres, leur position est la meme, quoique leurs divers points, bien entendu, se soient de'places pour se substituer les uns aux autres, ceux qui e'taient loin de 1'origme en e'tant a present devenus voisins, et vice versa. Tout autre plan se transforme en une sphere passant par le point (ou la transformation amene tous les points situe's a 1'infini) et ayant son centre sur la perpendiculaire au plan menee du point ; la perpendiculaire et le diametre de la sphere ont tin produit e'gal a la constante n, et se ddduisent ainsi facilement 1'tme de 1'autre. II est inutile d'ajouter que deux spheres qui correspondent a deux plans paralleles se touchent au point 0. De meme, deux spheres ainsi posees se transformeraient en deux plans paralleles. Mais une sphere qui ne passe pas par le point doit rester une sphere, puis- qu'elle ne peut acquerir aucun point a 1'infini. Les droites passant par le point restent des droites, et conservent leur position invariable. Toute autre droite donne lieu a une cir- confe'rence de cercle dont le plan est determine' par la droite et par le point 0, et dont le centre est situe' sur la perpendiculaire abaissee du point sur la droite ; le diaraetre est le quotient de la constante n par cette perpendiculaire. Les circonferences provenant de droites paralleles sont toutes tangentes a une parallele menee par le point a ces droites. On peut voir, enfin, que la transformed d'une circonference est une droite 168 Electric Images. [xiv. quand la circonfe'rence passe par le point 0, et, dans tout autre cas, reste une circonference. Une proprie'te remarquable de ce genre de transformation censiste en ce que les deux triangles forme's par trois points infiniment voisins quelconques de la figure primitive et les trois points correspondants de sa transformee sont semblables Fun a 1'autre, en sorte que si deux lignes se coupent dans 1'une des deux figures sous un certain angle, les lignes correspon- dantes de 1'autre figure se couperont sous le meme angle*. La demonstration de cette proprie'te' repose sur 1'equation (1), a laquelle nous avons donne la forme ~~PP" Supposons, en effet, que les deux points m, m', ou (#, y, z\ (a/, y f , z), soient infiniment voisins, et que leur distance D soit represented par ds. Representons par dcr celle des deux points correspondants ^ p. Comme p et p n'auront -pas de difference sensible, il nous viendra 7 ds do- = . P* Les elements dcr, ds ont done en chaque lieu un rapport con- stant qui depend de p et change, en general, d'un lieu a 1'autre. Considerons un troisieme point m' infiniment voisin des deux premiers, et ddsignons par ds et ds" ses distances a m et a m ; dcr , dcr" etant les distances correspondantes dans la seconde figure, on aura encore j dcr = o- . P Done dcr : da : dcr" ::ds:ds' : ds". Ainsi, le triangle infinitesimal mm'm" est semblable au triangle * De la similitude des triangles infiniment petits correspondants, il resulte encore que la figure transformee est semblable a la figure primitive, ou a sa symetrique, dans ses elements infiniment petits. En s'en tenant au premier cas, qui est proprement celui de nos formules, ou nous prenons naturellement la constante n positive, on aura, a trois dimensions, une sorte de representations des corps, analogue au trac des cartes geographiques [those according to the " stereographic projection"], pour lesquelles le rapport de similitude des elements correspondants est variable aussi d'un lieu & 1'autre. XIV.] Electric Images. 169 correspondant p,p '//' L'angle de ds avec ds est, par conse*- quent, le mme que celui de da avec da. Cette demonstration, on le voit, n'exige pas meme que 1'dquation (1) ait lieu pour deux points situ^s a une distance finie ; elle demande seule- ment que cette equation ait toujours lieu pour deux points infiniment voisins. On doit en dire autant d'un theoreme que je vais etablir, et qui n'est qu'un corollaire de la proposition pre'ce'dente. Une surface appartenant a Tune des deux figures tant donnde, representez-vous les lignes de courbure de cette sur- face, et les deux series de surfaces deVeloppables, orthogonales entre elles et a la surface donn^e, qui sont formers par les normales successives. Dans la seconde figure, les series de surfaces correspondantes resteront orthogonales entre elles et a la transformed de la surface donn^e ; par suite, en vertu du beau theoreme de M. Ch. Dupin, elles traceront encore sur cette transformed des lignes de courbure. Ces lignes de cour- bure rdsulteront ainsi des lignes de courbure de la premiere surface donnee, et seront imme'diatenient connues si les autres le sont. II sera ais^ d'appliquer ce th^oreme aux surfaces du second degre, comme aussi aux systemes triples de surfaces orthogonales que M. Serret a indiqu^s dans une Note recente*, et qui, par notre transformation, en donneront d 'autres non moins curieux, etc. Proposons-nous, par exemple, de trouver les lignes de cour- bure de la surface enveloppe des spheres qui touchent trois spheres donn^es, probleme que M. Ch. Dupin a re'solu jadis dans la Correspondance sur VEcole Poly technique, tome I, page 22. Soient et P les points d'intersection de ces trois spheres ; prenons le point pour origine, et operons une transformation par rayons vecteurs reciproques, ce qui nous fournira une seconde figure d'ou nous reviendrons aisement a la premiere- Dans la seconde figure, les trois spheres donn^es seront rem- placees par trois plans qui se couperont en un point II correspondant au second point P d'intersection de nos trois spheres. La surface enveloppe des spheres tangentes a ces trois plans sera (en se bornant a un des angles solides et a son * Page 241 du present volume [Liouville's Journal, 1847]. 170 Electric Images. [xiv. oppose') celle d'un cone droit a base circulaire ayant son sommet au point II, et circonscrit a une quelconque des spheres tan- gentes aux trois plans. Les lignes de courbure de cette surface conique sont : 1 les generatrices rectilignes qui passent toutes par le point II : dans le retour a la premiere figure, ces droites deviendront des cercles passant tous par le point P, dont les tangentes en P feront toutes le meme angle avec la tangente au cercle dans lequel se transforme 1'axe du cone, d'ou resultera un nouveau cone droit, et passant toutes aussi avec des cir- constances semblables par le point ; 2 des cercles, dont les plans sont tous paralleles entre eux et perpendiculaires a 1'axe du cone, et qui, lors du retour a la premiere figure, deviendront des cercles coupant a angle droit ceux qui resultent de generatrices rectilignes. Les lignes de courbure de la surface enveloppe des spheres tangentes a trois spheres donne'es sont done des circonferences de cercle. On de'montre avec la meme facilite le theorem e de M. Dupin concernant la courbe que trace sur chacune des trois spheres donne'es la sphere variable qui les touche. En effet, quand les trois spheres donne'es sont remplace'es par trois plans, il est clair que la suite des points suivant lesquels la sphere variable touche un quelconque des plans est une ligne droite passant par le point d'intersection II. Done, en revenant aux trois spheres donne'es, la courbe demandee est une circonfe'rence de cercle qui passe par les points et P. II peut arriver, bien entendu, que les points et P soient imaginaires ; mais il n'y a alors aucun changement essentiel a faire dans ce que nous venons de dire, et nos conclusions subsistent. La circonstance d'une origine imaginaire aurait plus d'inconvenient s'il s'agissait de resoudre le probleme d'une sphere tangente a quatre autres, en le ramenant au probleme tres-simple de trouver une sphere tangente a une sphere donnee et a trois plans donne's ; mais on y reme'dierait en augmentant d'une meme quantite* les rayons des quatre spheres donnees, ce qui ne change pas la position du centre de la sphere tangente. De meme, en se bornant a considerer des points tous situe's dans un plan passant par 1'origine 0, on ramenera la determi- nation du cercle tangent a trois autres a celle d'un cercle qui touche un cercle donne et deux droites donnees. xiv.] Electric Images. 171 En g^ndral, les systemes de spheres ou de cercles, et sp&nale- ment de spheres ou de cercles passant par un point donne, jouissent de proprietes curieuses dont beaucoup deviennent intuitives par la transformation dont nous venons de nous occuper. On pent appliquer en particulier cette remarque aux theoremes que M. Miquel a donnes dans son Me*moire sur les angles curvilignes*. Pour nous borner au cas le plus simple, il est Evident que, dans un triangle ABC forme par trois arcs de cercles passant tous par un meme point 0, la somme des angles vaut 2 droits, puisque notre transformation rend ce triangle rectiligne sans alte'rer ses angles. 228. Le passage des relations metriques d'une figure a 1'autre, dans la transformation par rayons vecteurs re'ciproques, en allant des coordonnees , 97, aux coordonnees x, y, z, s'opere a 1'aide de la formule ou simplement A = , , en posant n = 1, ce qui n'a aucun inconvenient. Mais en designant par 1'origine, dans la seconde figure seulement, et en employant les autres lettres A, B, etc., pour repr^senter a la fois les points de la premiere figure et les points correspondants de la seconde figure, cette formule revient a dire que, dans toute relation entre des distances AB, ED, etc., il faut rem placer chaque distance telle que AB par ^ ^. Voila done une regie pratique tres-commode ; cette regie convient aussi bien au cas du plan qu'a celui de 1'espace. Deux exemples suflfiront. Que des droites partant d'un point fixe A coupent chacune un cercle en deux points B et C, B f et C f , etc., on aura AB x AC = AB' xAC'= constante. Done, dans la figure transformed, AB AC AB' AC' OA.OB X OA.OC X OA . OB X OA . OG f ' et par consequent, AB AC Tome ix. de ce Journal page 20 [Liouville's Journal, 1844]. 172 Electric Images. [xiv. D'ailleurs les points A, B, C, qui e'taient en ligne droite, se trouvent a present sur une circonference de cercle passant par le point 0. Nous voyons par la que les cercles passant par deux points fixes A, coupent un cercle donne en deux points B, C tels, que le rapport des produits des distances AB x A C et OB x OC a une valeur constante pour tous ces cercles. Que les cotes BC, AC, AB d'un triangle rectiligne ABC soient coupes en trois points A', B f , C' par une transversale, on aura AC'* BA' x CB' = BC' x CA' x AB'. Done, dans la figure transformed, AC' BA CB' BC' CA 1 AB OA.OC'* OB. OA*OC.OB'~ OB.OC'* OC.OA' X OA.OB" ce-qui redonne AC' x BA 1 x CB' = BC' x CA x AB. Mais cette relation s'applique a present a un triangle curviligne ABC forme' par trois cercles qui passent tous au point et dont les cote's sont coupes en A', B', C' par un quatrieme cercle passant aussi au point 0. II est, du reste, inutile d'aj outer que AC', BA', etc., sont les plus courtes distances des points A et C', B et A', etc., et non des segments mesures sur les cote's du triangle curviligne. On ge'ne'raliserait aisement de la meme maniere le theorem e relatif a un polygone gauche coup^ par un plan. Mais en voila assez sur ce sujet. 229. Etant donne'es deux spheres qui ne se coupent pas, on pent toujours placer 1'origine sur la droite qui joint leurs centres, en un point re'el tel, qu'apres la transformation par rayons vecteurs re'ciproques, ces deux spheres seront con- centriques. Prenons la droite des centres pour axe des #; designons par h la distance inconnue du point au centre de la premiere sphere, et par h + l sa distance au centre de la seconde sphere ; soient k, k' les rayons. Les Equations des deux spheres seront, avant la transformation, et apres la transformation, qui consistera a remplacer x, y, z par xiv.] Electric Images. 173 g y * elles deviendront h Pour que le centre soit le meme a present, il faut et il suffit h h+l F^fc 2 ~( + J) 2 --&' a ' d'ou lh* + (I 2 + A; 2 - &' 2 ) A + ta 2 = 0, Equation du second degre qui donnera pour h deux valeurs, en posant G=(l-k et il est aise* de voir que G sera positive si les deux spheres qu'on a donnees d'abord ne se coupent pas. 230. Ce the'oreme pourra etre utile en ge'ome'trie ; mais il aura surtout une application importante dans les questions de physique mathe'matique. Essayons ici d'indiquer rapidement 1'usage, en ce genre de questions, de la transformation ge'ne'rale qui donne 1'dquation (1). La Lettre de M. Thomson nous servira de guide ; nous y ajouterons quelques deVeloppements. La ge'ne'ralite plus ou moins grande de la solution par laquelle on satisfait a 1'^quation (1) ne change en rien la marche & suivre, qui reste la meme dans tous les cas. Et d'abord de Tdquation 1 _pp' &-~D on peut conclure, avec M. Thomson, que, si une fonction U de f, 97, f satisfait a liquation cette meme fonction, divise'e par p et exprimde en x, y, z, ve*rifiera liquation de mme forme *.p~ l = dz* 174- Electric Images. [xiv. De la une liaison entre deux problem es distincts concern ant tous deux I'e'quilibre de temperature dans les corps homogenes, mais relatifs a deux systemes dont Fun resulte de 1'autre par la transformation qui lie f , ??, % a x, y, z. Que le premier systeme soit forme de deux spheres qui ne se coupent pas, que la tempe'rature soit donn^e en chaque point de leurs surfaces, et demandons quelle est la loi des tempe'ra- tures permanentes dans 1'espace compris entre elles, si 1'une est inte'rieure a 1'autre, ou dans 1'espace infini exterieur a toutes deux, si 1'une est en dehors de 1'autre, en ajoutant dans ce dernier cas la condition que la temperature soit nulle a Tinfini. On ramenera cette question au cas tres-facile de deux spheres concentriques. Cela requite du thdoreme e'tabli ci-dessus et en montre toute 1'importance. En indiquant cette application a la th^orie de la chaleur, M. Thomson ajoute, du reste, avec raison qu'elle s'etend d'elle-meme a la theorie de 1'electricitd Dans la theorie de I'electricite' ou du magnetism e, et, en general, dans la thdorie de 1'attraction, la quantite* que G. Green et M. Gauss nomment potentiel, c'est-a-dire la quantitd qu'on obtient en faisant la somme des elements attractifs ou repulsifs d'une masse divise's par leurs distances a un point, joue un role capital. On connait le probleme de M. Gauss : " Distribuer sur une surface donn^e une masse attractive ou repulsive, de telle sorte que le potentiel ait en chaque point de la surface une valeur donnde." On a re'solu ce probleme pour differentes surfaces, en particulier pour Tellipsoide. Or la solution relative & une surface quelconque donne la solution pour toutes les surfaces qui se deduisent de celle-la par une transformation pour laquelle I'e'quation (1) ait lieu. Ay ant, en effet, liquation ' **- if- A pour la premiere surface, on aura pour la second e surface une equation du meme genre, rempla^ant par leurs nouvelles valeurs A et da. On a A^ PP" Quant a dco', j'observe que les e'le'ments lindaires correspondants da- et ds sont lids par la formule , ds xiv.] Electric Images. 175 Done entre deux elements superficiels correspond ants dco, da, on aura dco 4 - , dco' = -~ ; par suite, ce qui re'sout le probleme de M. Gauss pour la surface trans- formee. On peut voir aussi que les Equations design e'es par (^4), (B), (C) dans mes Lettres a M. Blanchet*, et qui sont d'un si grand usage dans la plupart des questions physico-mathe'matiques concernant 1'ellipsoide, ont leurs analogues, qu'on en deMuit immediatement pour les surfaces transformers de rellipsoide-f-. On peut conside'rer encore liquation df dp drf ' d? '' et lui faire subir la transformation de f , 77, f en x, y, z. A cause de 1'equation da = ds P Z> qui peut s'e'crire on trouve, par des formules connues, que la quantite tig. UTJ a est e'gale a f , 1 dU , 1 dU & a J (i . o ~7 &* . p dy dy * Voyez le tome XI de ce Journal. t Parmi ces surfaces, il faut distinguer celle que donne la transformation par rayons vecteurs reciproques, en mettant 1'origine au centre mme de l'ellipsoide. On salt qu'elle est aussi le lieu des pieds des perpendiculaires abaissees du centre sur les plans tangents un autre ellipsoi'de dont les axes ont pour valeurs les inverses des valeurs des axes de Pellipsoide donne". Une propriete analogue a lieu dans le plan, pour la lemniscate par exemple, qui peut ainsi 6tre engendr6e de deux mani^res diffe'rentes au moyen d'une hyperbole equilatere, circonstance dont M. Chasles a tire un heureux parti dans ses recherches sur les arcs egaux de la lemniscate (Comptes Rendus de V Academic des Sciences, tome XXI, seanee du 21 juillet 1845). 176 Electric Images. [xiv. c'est-a-dire a i(d*U d*U d 2 U\ _ 9 fdU dp dUdp dU dp\ P \ dx* dy* dz* ) ^ \dx dx dy dy dz dz) 5 (d* .p~ l U d? . p~ l U d? . p' 1 ou enfin a p ( -7-3 1 -~^ 1 -- en se rappelant que cfi pi cfi Par la on voit d'abord que 1'equation CPU cPU d*U_ d? + di? d? ~ revient a celle-ci : } . l . =0 dx* dy* dz* ce que nous savions deja. On voit ensuite que liquation d?U_tfU ffU tfU dt* * cZp + drf + d? se transforme en ou, mieux encore, en ^P^U^ <(d*.p- l U d*.p- l U' d?.p*U\ dt P ( dx* d dz* )' R^ciproquement, cette derniere Equation, ou le coefficient p varie proportionnellement a la distance du point (#, y> z) a un point fixe, se ramene a Tequation dt* d^ drf qui est a coefficients constants, resultat qui trouve une applica- tion utile dans la thdorie du son. On peut enfin aj outer que les Equations aux differences partielles /rf^\ a . /^iTN 1 , (dU\* Q xiv.] Electric Triages. 177 sont des transforme'es Tune de 1'autre, ce qui pourra servir dans les questions de dynamique, ou MM. Hamilton et Jacobi ont introduit de telles Equations aux differences partielles. On me pardonnera, je 1'espere, ces developpements que j'ai cru pouvoir donner, a la suite des deux Lettres si interessantes de M. Thomson, sans le gener dans ses recherches. Mon but sera rempli, je le repete, s'ils peuvent aider a bien faire com- prendre la haute importance du travail de ce jeune geometre, et si M. Thomson lui-meme veut bien y voir une prenve nouvelle de I'amitie' que je lui porte et de I'estime que j'ai pour son talent. T. E. 12 XV. DETERMINATION OF THE DISTRIBUTION OF ELECTRI- CITY ON A CIRCULAR SEGMENT OF PLANE OR SPHERICAL CONDUCTING SURFACE, UNDER ANY GIVEN INFLUENCE. [Jan. 1869. Not hitherto published.] 231. The electric density at any point of the surface of an insulated conducting ellipsoid, electrified and left undisturbed by external influence, is ( 11) simply proportional to the dis- tance of the tangent plane from the centre. If we take p=^kp as the expression of this law, and call q the whole quantity of electricity communicated, we have ( 14) ^irkabc = q ; so that the formula for the electric density, p, at any point P of the surface in terms of p, the distance of the tangent plane from the centre, and a, b, c the three semi-axes, is or, in terms of rectangular co-ordinates of the point P, 232. To find the "electrostatic capacity" ( 51, footnote) of the charged ellipsoid, let V denote the potential at its surface. We have, by 15 (e), v _ _____ ^ ___ , q v - q 22 2 2 ' ' and therefore the capacity is the reciprocal of the definite integral which appears in this formula. 233. By taking c = we fall on the case of an infinitely thin plane elliptic disc : for which we have xv.] Distribution of Electricity, etc. and therefore p = ^ n . . 179 W- Putting 6 = a in this, we have, far an infinitely thin circular disc, where a denotes the radius of the disc, and p the electric density on either side of it, at a distance r from the centre. This result was first given by Green, near the conclusion of his paper " On the Laws of the Equilibrium of Fluids analogous to the Electric Fluid" (Transactions of the Cambridge Philosophical Society for Nov. 12, 1832); from which I make the following extract: 234. " Biot (Traite de Physique, tome ii. p. 277) has related " the results of some experiments made by Coulomb on the " distribution of the electric fluid when in equilibrium upon a " plate of copper 10 inches in diameter, but of which the thick- "ness is not specified. If we conceive this thickness to be " very small compared with the diameter of the plate, which " was undoubtedly the case, the formula just found ought to be "applicable to it, provided we except those parts of the plate " which are in the immediate vicinity of its exterior edge. As " the comparison of any results mathematically deduced from " the received theory of electricity with those of the experi- " ments of so accurate an observer as Coulomb must always be " interesting, we will liere give a table of the values of the " density at different points on the surface of the plate, calcu- lated by means of the formula (29), together with the cor- " responding values found from experiment : Distances from the Plate's edge. Observed Densities. -Calculated Densities. 5 in. . . . 4 1, 1,001 1 1,020 3 1,005 1,090 2 1,17 1,250 i : : : : : : 1,52 1,667 ,5. ... 2,07 2,90 2,294 infinite "We thus see that the differences between the calculated "and observed densities are trifling; and, moreover, that the 122 180 Distribution of Electricity on Circular [xv. " observed are all something smaller than the calculated ones, " which, it is evident, ought to be the case, since the latter " have been determined by considering the thickness of the " plate as infinitely small, and consequently they will be some- " what greater than when this thickness is a finite quantity, as " it necessarily was in Coulomb's experiments." 235. In this case (3) of 232 becomes du 7T ,_. 2a Hence the capacity is . But [ 232 (3)] the capacity of a globe is numerically equal to its radius ; and therefore the capacity of an infinitely thin disc is less than that of a globe of equal radius, in the ratio of 1 to ~-, or 1 to 1'571. Caven- dish found the ratio 1 to 1*57, by experiment * ! 236. The expression (5), 233, for the electric density at any point P on either side of an infinitely thin circular disc of con- ducting material electrified and left free from disturbing influence, may '^ be put into a form more convenient for geometrical investigation, thus : Let C be the centre of the disc, so that CA = a, CP = r, according to previous notation. Hence BP = a + r; PA = a r ; * My authority for this statement is the following entry which I find written in pencil on an old memorandum-book : " PLYMOUTH, Mond., July 2, 1849. " Sir William Snow Harris has been showing me Cavendish's unpublished "MSS., put in his hands by Lord Burlington, and his work upon them; a " most valuable mine of results. I find already the capacity of a disc " (circular) was determined experimentally by Cavendish as - that of a I'O i " sphere of same radius. Now we have rdr _ !\J f, "capacity of disc = t^" ^ ^ = .. It is much to be desired that those manuscripts of Cavendish should be published complete; or, at all events, that their safe keeping and accessi- bility should be secured to the world. XV.] Segment of Spherical Conducting Surface. 181 a 2 - r * = BP.PA = KP. PL ; if KL be any chord through P; and (5), with - substituted 7T for q according to (6), becomes P = - - ...(7) * 237. Consider a plane disc, S', thus electrified, to a potential which we shall, for a moment, denote by V ; and, following the suggestion of 210, take its image relatively to a spherical surface of radius E described from any point Q as centre. This image will ( 207) be a spherical segment, 8, electrified ( 210 and 238) as an infinitely thin conducting surface under the influence of a quantity V R of electricity concentrated at Q; and (compare 213) the spherical surface of which S is a part will pass through Q. The reader wall have no difficulty in verifying these statements for himself ; but if he desires it, he will find some further information and examples in Thom- son and Tail's Natural Philosophy, 512... 518. Thus ( 515 of that work) if p be the electric density on either side of the disc at P\, and p that on either side of its image at P, we have ~~ ^ ' /Q\ . P =s ~Qp P ......................... W' and if v and v be the potentials at any point II', and II the image of II', due respectively to the disc S' and its image, we have (Thomson and Tait, 516) This shows that, as the potential due to 8 f has a constant value, F', at all points of S', the potential due to will be, at different points of S, inversely as their distances from Q ; and if we take q = -RV, and denote by V the potential due to electricity distributed over the two sides of S, we have and so see that S is electrified as an infinitely thin conducting sheet of the same figure would be if connected with the earth by an infinitely fine wire, and inductively electrified only by 182 Distribution of Electricity on Circular [XV. the influence of a quantity q of electricity insulated at Q. Now, with our present notation (7) gives 27T 2 V// P . P A" 27T 2 v'^'P' PA' if K'L' be any chord through P of tlie circle bounding the plane disc $'. 238. Let K, P, ami L be the images of K' t P', L', so that -/TL, the image of K'L ', is the arc in which $ is cut by the plane through Q and A"'Z'. We have ( 207) * Hence K'Q : P Q :: PQ : KQ; and therefore the triangles K'P 'Q, PKQ are similar ; and therefore 'Q.K'Q PK 't4 i g constant, it follows that T p constant*; a theorem of geometry given above ( 228) by M. Liouville. Each mem- * As a particular case let Q be either pole of the fixed circle. In this case LQ = KQ, a constant. Hence LP . KP is constant; that is, the product cf the two chords from any fixed point P on a spherical surface to the two points in which any fixed circle on the surface is cut hy a plane through P and one of the poles of that circle, is constant, however the plane he varied. This is the simplest extension to spherical surfaces of the elementary geo- metrical theorem (Euc. in. '65) for the constancy of the rectangle under the two parts of a varying chord of a fixed circle through a fixed point, already used in the text ( 236). xv.] Segment of Spherical Conducting Surface. 183 ber of (12) may be altered in form thus : bisect L'K' in M, and arc LK in N. We have L'P'. FK r = MK fZ - MP' Z ] LP.KP = NK Z -NP Z \ .................. (14), LQ.KQ = NQ Z -NK Z } equations of which the last two are very easily proved from the formula sin (a /3) sin (a + <3) = sin 2 a sin 2 /?, by taking for a and /3 the angles subtended by \NK and \NP at the centre of the circle QKPL ; and again, by taking for a and ft the angles subtended by %NQ and %NK at the same point. 239. Using (13) in (11), and the result in (8), we find .J__JL /LQ.KQ p ~2>ir*QP*V LP.KP ' and modifying by (14), .-1..JL /N7rfPQ 2 \/ a?-CP 2 ' Now calling the centre of the spherical surface, let COP be denoted by rj ; COQ by 0', the value of either of these when P or Q is at the lip of the bowl, by a ; and the angle between the planes of COP and COQ, by (f) : so that we have a 2 =J/ 2 (l-cosa), CP' = /*(! -cos 1;), C(> 2 = i/ 2 (l - cos 0) and PQ 2 = %f* (1 COST; cos 6 sin 77 sin 9 cos ) ; xv.] Segment of Spherical Conducting Surface. 185 and we may take do- = %f* sin 0d0d = t, A = l- cos rj cos 0, and B = sin ij sin 0, we find * _ x r __ ^ 1 - cos 77 cos 0- sin 77 sin cos /o <4-.-B + ( and therefore = F _ T^^sin ^ V(cos a - cos 0) s97-cosa)J a cos 7j cos(T~ Lastly, putting V(cos a cos 6) = z t we find / ff ^ sin V(cos - cos 6) _ /V(eosa+i) ^^ J a cos TJ cos 6 J cos 77 cos a 4- ^ 2 - . cos7;-cosaj Hence we have, in conclusion, V I / cosa + 1 / cosa + 1 ) P 5~^7iA/~ -- ton A / 1-...(18). 27r 7 IV cos ^ - cos a V cos V ~ cos aj or, with /and a as above, and r to denote the chord CP t and the same, with the addition of 2^ ........................... ( 20 >> gives ( 240) the electric density on the convex side ; which are exactly the results stated above in 220. Twenty-two years ago these and the very simple formula (17) were communicated by me to M. Liouville without proof, and were published in his Journal. From that time till now they have not been proved, or even noticed, so far as I am aware, by any other writer. 242. Numerical results, calculated from the preceding for- mulae (19) and (20), are shown in the following tables : 186 Distribution of Electricity on Circular [XV. Plane Disc. Curved Disc. Arc 10. Curved Disc. Arc 20. Bason. Arc 900. Concave. Convex. ^- Concave. Convex. Concave. Convex. 9136 1-0685 8636 1-1364 4459 1-5541 9457 1-0826 8776 1-1504 4469 1-5551 9920 1-1289 9236 1-1964 4828 1-5910 1-0858 1-2227 1-0165 1-2893 5566 1-6648 1-2722 1-4091 1-2884 1-5611 7065 1-8147 1-7386 1-8755 1-6652 1-9379 1-0933 22015 Mean. Mean. Mean. 0000 i-oooo -oooo i-oooo 0142 1-0141 -0140 1-0010 0607 1-0605 -0600 1-0369 1547 1-1542 -1529 1-1106 3416 1-3407 -4247 1-2606 8091 1-8071 -8016 1-6474 Bowl. Arc 180. Bowl. Arc 270. Bowl. Arc 3400. Concave. Convex. Concave. Convex. Concave. Convex. 1202 1-8798 0135 1-9865 0001 1-9999 1266 1-8862 0144 1-9874 0002 1-9999 1418 1-9014 0176 1-9906 0002 2-0000 1779 1-9375 0253 1-9983 0004 2-0001 2570 2-0166 0451 2-0181 0009 2-0006 4959 2-2555 1195 2-0925 0042 2-0040 Mean. 1-0000 1-0064 1-0216 1-0577 1-1366 1-3757 Mean. 1-0000 1-0009 1-0041 1-0118 1-0316 1-1060 Mean. 1-0000 1-0000 1-0001 1-0002 1-0007 1-0041 It is remarkable how slight an amount of curvature produces a very sensible excess of electric density on the convex side in the first two cases (10 and 20) of curved discs ; yet how nearly the mean of the densities on the convex and concave sides at any point agrees with that at the corresponding point on a plane disc shown in the first column. The results for bowls of 270 and 340 illustrate the tendency of the whole charge to the convex surface, as the case of a thin spherical conducting surface with an infinitely small aperture is approached. xv.] Segment of Spherical Conducting Surface. 187 The constant coefficient for each case has been taken so as to make the mean of the electric densities on the convex and concave sides unity at the middle point (as in Green's numbers, 234 above, for the plane disc). The six points for which the electric densities are shown in the tables below are (not the six points to which Coulomb's observations and Green's numbers quoted in 234 refer, but) the middle point, and the five points dividing the arc from the middle to the edge or lip into six equal parts. 243. A second application of the principle stated in 210, and used in 237... 239, allows us to proceed from the solu- tion now found for the electrification of an uninfluenced bowl to determine the electrification of a bowl or disc under the influence of electricity insulated at a point Q (not, as in the solution of 239, necessarily in the spherical surface or plane of the bowl or disc, but) anywhere in the neighbourhood. Con- sider the image, $, of an uninfluenced electrified bowl, $', relatively to a spherical surface described from any point Q in its neighbourhood, as centre, with radius R. Let D' be the point on the spherical surface of $' continued, which is equi- distant from the lip (so that D' and the middle point of the conducting surface S' are the two poles of the circle constitut- ing the lip) ; D'K' P L' the circle in which S', and the con- tinuation of its spherical surface, are cut by the plane through Z)', Q, and any point P of 8 at which it is desired to find the electric density; and DKPL the image of D'K' PL'. In the annexed diagrams two cases are illustrated ; in one of which 8 is spherical and concave towards the influencing point, Q ; in the other, S is plane. Using now for 8' all the notation of 240, 241, but with accents added, and taking advantage of 238, footnote, we see that 188 Distribution of Electricity on Circular [xv. and /" - a' 2 = D'K 2 = D ' L* = UK' . D'L. Hence (19) becomes ,_ V ( /D'K'.DL _, /D'K' .D'L'\ p ~ 2', the unoccupied pole of the lip of the original bowl S'} may be found, without reference to S ', by construction from 8 and Q supposed given ; thus : From (22) of 243 we have KD : DL :: KQ : QL ................ (27), xv.] Segment of Spherical Conducting Surface. 189 where K and L may be the points in which the lip of the bowl 8 is cut by any plane through QD'D. Let, for instance, this plane pass through the centre of one of the spherical surfaces. It must also pass through the centre of the other, and bisect each bowl ; and if E, F be the points in which it cuts the lip of S, (26) applied to the present case gives HD:DF::EQ:QF. Hence (Euclid, VI. 3) the lines bisecting the angles ELF, EQF cut X. % ^./ the base EF in the same point; and D must be in the circle which is the locus of all points in the plane EFQ fulfilling this condition, being found by the well-known construction, thus : Bisect the angle EQC by QA, meeting EF in A. Draw QB perpendicular to QA, and let it meet EF produced, in R On BA as diameter describe a circle, which is the required locus ; and D is the point in which this circle cuts the unoccupied part of the spherical or plane surface of S. 245. D being found by this simple construction, the solution of the problem is complete, without reference to $', thus : To find the electric density at any point P, draw a plane through QDP y and let it meet the lip in K and L. Measure DK, DL, PK, PL, PQ, and DQ, and calculate by (25) and (26). But we have an important simplification from the geometrical theorem of 238, which shows that DK.DL_Dk.Dl PK.PL~ Pk.Pl" if k, I be points in which the lip is cut by any plane whatever through PD. Choose, for instance, the plane through PD, and C the middle point of S. Then, as D, k } P, C, I lie all on one circle, and C is the middle point of the arc kPl, we have (as above, in 238) Dk . Dl= CD 2 - Ck* = CD 2 - a* , Pk.Pl=Ck z - CP* = a* - CP* 190 Distribution of Electricity on Circular [xv, where, as before, a denotes the chord from the middle point to the lip. Using this in (28) and (25) we have, finally, gh(f-K)(PQ ICV-* JP /CD>-d>l) - *r?P L-DQV a? - CP*\ j ( for the density on the side remote from Q; h and/ h being the shorter and longer distance. 246. For the case in which S is a plane disc, or/= oo , this becomes [PQ /CD* -a? ,-JPQ /CD* -a* 3 \I>Q\/ tf=cp*~ \DQ\f a 2 - CP* . . ( and the addition (26) to it to give the electric density on the side next to Q, 27TPQ' ' Also, as EFD is a straight line in this case, (27) gives (31). QE, QF are to be calculated immediately from data of whatever form, specifying the position of Q', and from them and CD found by this formula, DQ is to be calculated. Thus explicitly we have every element required for calculating electric densities by (30). 247. For the case of Q in the axis of the disc, D is infinitely CD distant, so that CD = oo , DQ = oo , and -= = 1. And if for OP we put r, (30) and (31) give, for the density on remote side, and for the density on near side, If P be at the centre of the disc, and if we take q = 2-Tr 2 , these ibecome for remote side, p = ~ (--tan' 1 -) ] *** a) ............ <35); for near side, p 7T w xv.] Segment of Spherical Conducting Surface. 191 from which the following numerical results have been calcu- lated, with a, the radius of the disc taken as unity : Distance of Induced Electric Density at middle Influencing of Disc Point On remote side On near side h- p = A- 1 -cot~ 1 (^~ 1 ). p + ffft-2. 2 1651 78-7049 4 1218 19-7567 6 1655 8-8921 8 1957 5-1044 1-0 2146 3-3562 1-2 2250 2-4067 1-4 2293 1-8322 1-6 2296 1-4568 1-8 2273 1-1969 2-0 2232 1-0086 3-0 1946 5437 248. These numbers show that the distance at which the in- fluencing point, if restricted to the axis of the disc, must be held to render the induced electric density at the middle on the far side a maximum is about 1*5 times the radius. But the characteristics (1) of the zero electric density on the far side, and infinite on the near side, when the influencing point is infinitely near the disc ; (2) the proportionality of the latter to h~* for very small distances ; and (3) the ultimate vanishing of the difference between the two sides as the influencing point is removed to an infinite distance, and the approximation of each to Green's result for a plane uninfluenced disc electrified to a potential equal to qh ( 234, above), is better illustrated by the formulae themselves, (35) ; (33), (34) ; and (30), (31) ; than by any numerical results calculated from them, however elaborately. It would be interesting to continue the analytical investigation far enough to determine the electric potential at any point in the neighbourhood of a disc electrified under in- fluence, and so to illustrate further than is done by the numbers and formulas already obtained, the theory of electric screens, and of Faraday's celebrated "induction in curved lines" (Experi- mental Researches in Electricity, 1161, 1232; Dec. 21, 1837) ; but I am obliged to leave the subject for the present, in the hope that others may be induced to take it up. XVI. ATMOSPHERIC ELECTRICITY. [From Nichol's Cyclopedia, 2d Ed. (I860).] 249. It may be premised, to avoid circumlocution in this article, that every body in communication with the earth by means of matter possessing electric conductivity enough to prevent its electric potential* from differing sensibly from that of the earth, will be called part of the earth. Moist stone, and rock of all kinds, and all vegetable and animal bodies, in their natural conditions, except in circumstances of extraordinary dryness, possess, either superficially or throughout their sub- stance, the requisite conductivity to fulfil that condition. On the other hand, various natural minerals and artificial com- pounds, such as glass, various vegetable gums, such as India- rubber, gutta percha, rosin, and various animal products, such as silk and gossamer fibre, when either in a very dry natural or in an artificially dried atmosphere, resist electrical conduc- tion so strongly that they may support a body, or otherwise form a material communication between it and the earth, and yet allow it to remain charged with electricity to a potential sensibly differing from the earth's, for fractions of a second, for minutes, for hours, for days, or even for years, without any fresh excitation or continued source of electricity. Again, air, whether dry or saturated with vapour of water, and probably all gases and vapours, unless ruptured by too strong an electro- motive force, are very thoroughly destitute of conductivity that is to say, are very perfectly endowed with the property of resisting the tendency of electricity to pass and establish * Two conducting bodies are said to be of the same electric potential when, if put in conducting communication with the two electrodes of an electrometer, no electric effect is produced. When, on the other hand, the electrometer shows an effect, the amount of this effect measures the difference of potentials between the two bodies thus tested. Difference of potentials is also called electromotive force. XVI.] Atmospheric Electricity. 193 equality of potential between two bodies not otherwise materi- ally connected. 250. Hence, when "the surface of the earth" is spoken of, the surface separating the solids and liquids of the earth from the air will be meant; and when the more qualified ex- pression " outer surface of the earth " is used, inner surfaces of vesicles, or the surfaces bounding completely enclosed spaces of air, must be understood to be excluded. Thus, the surface of a mountain peak; the surface of a cave, up to the inmost recesses of the most intricate passages ; the surface of a tunnel ; the surface of the sea, or of a lake or river ; all the surface of a sheet of unbroken water in such a fall as that of Niagara ; the surface of blades of grass and flowers, and of soil below ; in a wood, the surface of soil, and of trunks and leaves of trees; the surface of any animal resting on the earth ; the outside of the roof of a house; the whole inside surface of a room with an open window ; all belong to the outer surface of the earth. 251. On the other hand, the moon, meteoric stones, birds or insects flying, leaves or fruit falling, seed wafted through the air, spray breaking away from a cascade or from waves of the sea, the liquid particles of a cloud or a fog, present surfaces not belonging to the earth, and between which and the earth's surface differences of potential, and lines of electric force, may and generally do exist. 252. The whole surface of the earth, as defined above ( 250), is at every moment electrified in every part, with the exception of neutral lines dividing portions which are negatively (resin- ously) from portions which are positively (vitreously) electrified. The negatively electrified portions are of very much greater extent, at all times, than those positively electrified ; and there may be times when the whole surface is negatively electrified, because in all localities in which electrical observations have been hitherto made, with possibly one remarkable exception*, the earth's surface is always found negative, day and night, * At Guajara station, on the Peak of Teneriffe, "During the whole period of "observation, by day and night, the electricity was moderate in quantity, and " always resinous. This was during the period of N.E. trade wind, and within "its influence, though above its clouds." [Professor Piazzi Smyth's Account of the Teneriffe Astronomical Experiment, Philosophical Transactions, 1858, and separate publication ordered by the Lords of the Admiralty.] The "electricity" here referred to was that acquired by an insulated conductor carrying a burning T. E. 13 XVI. ATMOSPHERIC ELECTRICITY. [From Nichol's Cyclopaedia, 2d Ed. (I860).] 249. It may be premised, to avoid circumlocution in this article, that every body in communication with the earth by means of matter possessing electric conductivity enough to prevent its electric potential* from differing sensibly from that of the earth, will be called part of the earth. Moist stone, and rock of all kinds, and all vegetable and animal bodies, in their natural conditions, except in circumstances of extraordinary dryness, possess, either superficially or throughout their sub- stance, the requisite conductivity to fulfil that condition. On the other hand, various natural minerals and artificial com- pounds, such as glass, various vegetable gums, such as India- rubber, gutta percha, rosin, and various animal products, such as silk and gossamer fibre, when either in a very dry natural or in an artificially dried atmosphere, resist electrical conduc- tion so strongly that they may support a body, or otherwise form a material communication between it and the earth, and yet allow it to remain charged with electricity to a potential sensibly differing from the earth's, for fractions of a second, for minutes, for hours, for days, or even for years, without any fresh excitation or continued source of electricity. Again, air, whether dry or saturated with vapour of water, and probably all gases and vapours, unless ruptured by too strong an electro- motive force, are very thoroughly destitute of conductivity that is to say, are very perfectly endowed with the property of resisting the tendency of electricity to pass and establish * Two conducting bodies are said to be of the same electric potential when, if put in conducting communication with the two electrodes of an electrometer, no electric effect is produced. When, on the other hand, the electrometer shows an effect, the amount of this effect measures the difference of potentials between the two bodies thus tested. Difference of potentials is also called electromotive force. XVI.] Atmospheric Electricity. 195 of the distribution of electricity within a non-conducting mass, it may be remarked, that a determination of the normal com- ponent of the force all round a closed surface is just sufficient to show the aggregate quantity of electricity possessed by all the matter situated within it*. Hence observation in positions all round a mass of air is necessary for determining the quantity of electricity which it contains; and, therefore, the balloon must be put in requisition if knowledge of the distribution of electricity through the atmosphere is to be sought for. 254. Without leaving the earth, however, although we cannot thoroughly investigate the electrification of the air, we can make important inferences about it from observations of the electric density over the earth's surface, by a principle of judg- ing which may be thus explained : If the earth were simply an electrified body, placed in a perfectly insulating medium of indefinite extent, and not sensibly influenced by any other electrified matter, or by reflex influence from any conductor or dielectric in its vicinity, its electricity would be distributed over its surface according to a perfectly definite law, depend- ing solely on the form of the surface, and deducible by a sufficiently powerful mathematical analysis from sufficiently perfect data of "geometry" (in the primitive sense of the term), or of what, in more modern language, is called geodesy. If the surface of the earth were truly spherical, this law would simply be uniform distribution. A truly elliptic oblateness of the earth would give, instead of uniformity, a distribution of electric density in simple proportion to the perpendicular distance between a tangent (that is horizontal) plane through any point and the earth's centre ; according to which the electric density at the equator would be greatest, and would exceed that at either pole, where it would be least, by ^ : a differ- ence which, for the present, we may disregard. 255. The whole amount of electricity over the surface of any great region of mountainous country, or of forest land, * Let N be the normal component of the force at any point of a closed surface, ds an element of the surface, / the sign of integration for the whole surface, and Q the whole quantity of electricity within it. Then, by a well- known theorem of Green's, rediscovered as alluded to in a preceding note, we have Q=LfNds. 132 196 Atmospheric Electricity. [xvi. or of soil and vegetation of any kind, or of streets and houses in a town, or of rough sea, would be very approximately the same as that on an area of unruffled ocean, equal to the " reduced" area of the irregular surface ; but the distribution of the elec- tricity over hill and valley, over the leaves and trunks of trees, and the surfaces of plants generally, and on the soil beneath them, over the roofs, perpendicular walls, and overhanging or overshaded surfaces of buildings, and the surfaces of streets and enclosed courts between them, and over the hollows and crests of waves in a stormy sea, would be extremely irregular, with, in general, greater electric density on the more prominent and convex portions of surfaces, and less on the more covered and concave quite insensible, indeed, in any such position 'as the interior of a cave, or the soil below trees in a forest even where considerable angular openings of sky are presented, or the roof or floor of a tunnel, or covered chamber, even although open to a considerable angle of sky. 256. If thus a perfect electro-geodesy gave a " reduced " electric density equal over the whole earth, we might infer that the electrification of the earth is not influenced by any elec- tricity in the air. According to what has been stated above, there might in that case be either no electricity in the air, from the earth's atmosphere to the remotest star, and the lines of electric force rising from the earth might either be infinite or terminate in the surfaces of the moon, meteoric stones, sun, planets, and stars; or there might be, at any distance con- siderably exceeding the height of the highest mountain, a uni- formly electrified stratum of equal quantity and opposite kind to the earth's, balancing through all the exterior space the force due to the terrestrial electricity, and limiting the manifestations of electric force to the atmosphere within it ; or there might be any of the infinite variety of distributions of electricity in space round the earth, by which the electric density at the earth's surface would be uninfluenced. 257. But, in reality, the electric density varies greatly, even in serene weather, over the earth's surface at any one time, as we may infer from (1.) the facts (established for Europe, .and probably true in all the temperate zones of both hemi- spheres), that in any one place the electric density of the xvi.] Atmospheric Electricity. 197 surface observed during serene weather is much greater in winter than in summer, and that it varies according to some- thing of a regular periodicity with the hours of the day and night; and (2.) the consideration that there is often serene weather of day and night, and of summer and winter, at one and the same time, in different temperate portions of the earth. We may, therefore, consider it as quite established that, even in serene weather, the electrification of the earth's surface is largely influenced by external electrified matter. Although we cannot ( 253) discover the exact locality and distribution of this influencing electricity from its, effects at the earth's surface alone, yet it is possible, from the character of the distribution of the terrestrial electric density as influenced by it, to assign a superior limit to its height*. If at any one instant the electric density reduced to the sea level were distributed according to a simple "harmonic" law, or, more generally, according to a certain definite character of non-abruptness of variation easily specified in mathematical language *)-, the external influencing electricity might be at any distance, however great, for all we could discover by observations near the earth's surface. But. little as we know yet regarding the diurnal law of electric variation in serene weather, it is, we may say with almost perfect certainty, not such as could give at any instant a dis- tribution over the whole earth possessing any such gradual character as that referred to; and, therefore, we may, in all probability, from the character of the diurnal variation itself, say that its electric origin is not at a distance of many radii from the surface. On the other hand, when we consider that in temperate regions the velocity with which the earth's surface * If at any instant the co-efficients of the series of "Laplace's functions," expressing the terrestrial electric density reduced to the sea level, converged ultimately with less rapidity than the geometrical series 1, , ^v- we might be sure that there is electricity in the air at some distance from the centre of the earth, not exceeding m times the radius of the earth's surface. For the principles on which this assertion is founded, see a short article, entitled ."Note on Certain Points in the Theory of Heat," Cambridge Mathe- matical Journal, November 1843. t For instance, if in simple proportion to the cosine of the angular distance by a series of "Laplace's more rapidly than any geometrical series 198 Atmospheric Electricity. [xvi. is carried round in its diurnal course is from 500 to 900 miles per hour, we see clearly that any law of diurnal electric varia- tion, established on observations even so frequent as once every hour, could not possibly fix the locality of the origin to within 100 miles of the surface ; and as we have as yet nothing to go upon in the way of published observations more frequent than three or four times a day, towards establishing either the ex- istence or the character of the diurnal law, we cannot consider it as proved by observation that the influencing electricity which produces it is even as near as the 50 or 100 miles limit which is commonly (but in the opinion of the writer of this article, most unreasonably) assigned as an end to the earth's atmosphere. 258. The great suddenness of the electric variations during broken weather, and their close correspondence with beginnings, changes, and cessations of rain, hail, or snow, compel us (by a common sense estimate founded on an unconscious application of the mathematical law stated in the footnotes to the preced- ing 257) to believe that their origin agrees in position with that of the showers, and to give it a " local habitation" and a name Thundercloud. 259. The writer of this article has observed extremely rapid variations of terrestrial electrification during perfectly serene weather. Thus, in a calm summer night, with an unvarying cloudless sky overhead, and not the faintest appearance of auroral light to be seen, he has, in a temporary electric observa- tory in the Island of Arran, found large variations (as much as from a certain degree to double and back) in the course of a minute of time. The influencing electricity by which these variations were produced, cannot possibly (unless on the ex- tremely improbable hypothesis of their being due to highly electrified extra-terrestrial matter moving very rapidly with reference to the earth) have been very far removed from the earth's surface. It is not impossible, and we have as yet nothing to make it decidedly improbable, that they were due to fluctuations up and down of aerial strata, perhaps those of the great atmospheric currents, in high regions of the atmo- sphere. Judging, however, from still more recent observations referred to below ( 262), we may think it more probable that these remarkable variations in the observed electric force were XVI.] Atmospheric Electricity. 199 due chiefly to positively or negatively electrified masses moving along within a few miles of the locality of observation. 260. Keturning to the subject of the distribution of elec- tricity over the earth's surface at any instant, we may remark, that if over an area of several miles in diameter, of perfectly level bare country, or of sea, the electrical density is sensibly uniform, we could not, without going up in a balloon, and observing the electric force at points in the air above, form any judgment whatever as to the distance from the earth at which the influencing electricity is situated. If, on the other hand, we find a very sensible variation in the electric density between two points of a piece of level open country, or at sea ; not many miles apart, we may infer as quite certain that there is influencing electricity not many miles up in the air, and not uniformly distributed in level strata. Nothing can be easier than to make this trial only to observe simul- taneously with similar instruments, similarly placed, at two neighbouring stations, in a suitable locality and most interest- ing and important results are to be derived from it, as soon as arrangements can be made for continuing the requisite observa- tions day and night, during various vicissitudes of weather, especially during a time of perfect serenity. 261. Corresponding statements apply to a mountainous country, with this modification, that a very varied, instead of a uniform distribution of electric density, is, in such a locality, as explained above in 255, the natural consequence of freedom from the disturbing influence of near electrified masses of air or cloud. The problem of accurately determining, from purely geometric data ( 256), this undisturbed distribution over even the smoothest hillside, would infinitely transcend human mathe- matical power, although an approximate solution may be readily given for any piece of country over the whole of which both the inclination and the ratio of the height above the general level to the radius of curvature of the surface are small. For a rugged mountainous country, the most perfect geometric data, and the most strenuous mathematical efforts, could scarcely lead us towards an approximate estimate of the inequalities of electric density which different localities must present without any disturbance from near electrified atmosphere. Hence, in a 200 Atmospheric Electricity. [xvi. mountainous country unless we find electricity strong in some locality where from the configuration of the surface, we correctly judge it ought to be weak if undisturbed, or weak where it ought to be strong, or unless, at least, we find some very decided devia- tion from any such amount of difference between two stations as, without being able to make a precise calculation, we can estimate for the difference due to figure we cannot judge as to the influence of aerial electrification from simultaneous absolute determinations at any one instant alone. But of one thing we may be sure, that although the absolute amounts of the electrification at any two stations not far apart may differ largely, they must remain in an absolutely constant propor- tion to one another, if there is no electrified air or cloud near. 262. Hence, if we find observations made simultaneously by two electrometers in neighbouring positions, in a mountainous country, to bear always the same mutual proportion, we may not be able to draw any inference as to electrified air ; but if, on the contrary, we find their proportion varying, we may be perfectly certain that there are varying electrified masses of air or cloud not far off. A first application of this test is described in the following extract from the Proceedings of the Literary and Philosophical Society of Manchester for October 18, 1859 : " The following extract of a letter received from Professor W. " Thomson, F.R.S., Glasgow, Honorary Member of the Society, " etc., was read by Dr Joule : ' I have a very simple " domestic " apparatus by which I can 'observe atmospheric electricity in an easy way. It consists 'merely of an insulated can of water set on a table or window ' sill inside, and discharging by a small pipe through a fine nozzle ' two or three feet from the wall. With only about ten inches ' head of water and a discharge so slow as to give no trouble in ' replenishing the can with water, the atmospheric effect is ' collected so quickly that any difference of potentials between ' the insulated conductor and the air at the place where the ' stream from the nozzle breaks into drops is done away with at ' the rate of five per cent, per half second, or even faster. Hence ' a very moderate degree of insulation is sensibly as good as ' perfect, so far as observing the atmospheric effect is concerned. ' It is easy, by my plan of drawing the atmosphere round the ' insulating stems by means of pumice-stone moistened with XVI.] Atmospheric Electricity. 201 ' sulphuric acid, to insure a degree of insulation in all weathers, ' by which there need not be more than five per cent, per hour ' lost by it from the atmospheric apparatus at any time. A little ' attention to keep the outer part of the conductor clear of ' spider lines is necessary. The ' apparatus I employed at In- ' vercloy stood on a table beside ' a window on the second floor, ' which was kept open about ' an inch to let the discharg- * ing tube project out without ' coming in contact with the Fm - 1 - ' frame. The nozzle was only about two feet and a half from ' the wall, and nearly on a level with the window sill. The ' divided ring electrometer stood on the table beside it, and ' acted in a very satisfactory way (as I had supplied it with a ' Leyden phial, consisting of a common thin white glass shade ' which insulated remark- ' ably well, instead of the 1 German glass jar the ' second of the kind which ' I had tried, and which ' would not hold its charge * for half a day). I found * from 13J to 14 of torsion ' required to bring the index ' to zero, when urged aside ' by the electromotive force ' of ten zinc-copper water ' cells. The Leyden phial ' held so well, that the sensi- ' bility of the electrometer, ' measured in that way, did ' not fall more than from ' 13i to 13i in three days. ' The atmospheric effect ' ranged from 30 to above ' 420 during the four days 1 which I had to test it ; that F IG . 2. 202 Atmospheric Electricity. [xvi. * is to say, the electromotive force per foot of air, measured hori- ' zontally from the side of the house, was from 9 to above 126 zinc- ' copper water cells. The weather was almost perfectly settled, ' either calm, or with slight east wind, and in general an easterly ' haze in the air. The electrometer twice within half an hour went ' above 420, there being at the time a fresh temporary breeze ' from the east. What I had previously observed regarding the ' effect of east wind was amply confirmed. Invariably the ' electrometer showed very high positive in fine weather, before 'and during east wind. It generally rose very much shortly ' before a slight puff of wind from that quarter, and continued 'high till the breeze would begin to abate. I never once ' observed the electrometer going up unusually high during fair ' weather without east wind following immediately. One even- ' ing in August I did not perceive the east wind at all, when ' warned by the electrometer to expect it ; but I took the ' precaution of bringing my boat up to a safe part of the beach, 'and immediately found by waves coming in that the wind ' must be blowing a short distance out at sea, although it did ' not get so far as the shore. I made a slight commencement ' of the electrogeodesy which I pointed out as desirable at the ' British Association, and in the course of two days, namely, ' October 10th and llth, got some very decided results. Mac- * farlane, and one of my former laboratory and Agamemnon ' assistants, Russell, came down to Arran for that purpose. Mr ' Russell and I went up Goatfell on the 10th instant, with the ' portable electrometer (see Fig. 3), and made observations, while ' Mr Macfarlane remained at Invercloy, constantly observing ' and recording the indications of the house electrometer. On 'the llth instant the same process was continued, to observe ' simultaneously at the house and at one or other of several ' stations on the way up Goatfell. I have not yet reduced all ' the observations ; but I see enough to leave no doubt whatever ' but that cloudless masses of air at no great distance from the ' earth, certainly not more than a mile or two, influence the ' electrometer largely by electricity which they carry. This I ' conclude because I find no constancy in the relation between 'the simultaneous electrometric indications at the different ' stations. Between the house and the nearest station the rela- xvi.] Atmospheric Electricity. 203 ' tive variation was least. Between the house and a station about 'halfway up Goatfell, at a distance estimated at two miles and ' a half in a right line, the number expressing the ratio varied 'from about 113 to 360 in the course of about three hours. On 'two different mornings the ratio of a house to a station about ' sixty yards distant on the road beside the sea was 97 and 96 'respectively. On the afternoon of the llth instant, during a ' fresh temporary breeze of east wind, blowing up a little spray as ' far as the road station, most of which would fall short of the 'house, the ratio was 108 in favour of the house electrometer ' both standing at the time very high the house about 350. 'I have little doubt but that this was owing to the negative 'electricity carried by the spray from the sea, which would 'diminish relatively the indications of the road electrometer'." 263. The electrometers referred to in the preceding extract were on two different plans. The first, or "divided ring electrometer," consists of (1.) A ring of metal divided into sectors, of which some one or more are insulated and con- nected with the conductor to be electrically tested, and the remainder connected with the earth. (2.) An index of metal supported by a glass fibre, or a wire, stretched in the line of the axis of the ring, and capable of having its fixed end turned through angles measured by a circle and pointer. (3.) A Leyden phial, with its insulated coating electrically connected with the index. (4.) A case to protect the index from currents of air, and to keep an artificially dried atmosphere round the insulating supports glazed to allow the index to be seen from without, but with the inner surface of the glass screened (electrically) by wire cloth, perforated metal, or tinfoil, to do away with irregular reflections on the index. In the instru- ment represented in the drawing (No. 2) above, the ring is divided only into two parts, which are equal, and separated by a space of air about ^ of an inch. Each of these half rings is supported on two glass pillars ; and by means of screws acting on a foot which bears these pillars, it is adjusted and fixed in its proper position. The index is of thin sheet aluminium, and projects in only one direction from the glass fibre bearing it. A stiff vertical wire, rigidly connected with it, nearly in the prolongation of the fibre, bears a counterpoise considerably 204 Atmospheric Electricity. [xvi. below the level of the index, and heavy enough to keep the index horizontal. A thin platinum wire hooked to the lower end of this vertical wire, dips in sulphuric acid in the bottom of the Leyden phial. The Leyden phial is charged either posi- tively or negatively ; and is found to retain its charge for months, losing, however, gradually, at some low rate, less generally than one per cent, per day of its amount. The index is thus, when the instrument is in use, kept in a state of charge corresponding to the potential of the inside coating of the phial. When one of the half rings is connected with the earth, and a charge of electricity communicated to the other, the index moves from or towards the latter, according as the charge communicated to it is of the same or the opposite kind to that of the index. This instrument, as an electroscope, possesses extreme sensibility much greater than that of any other hitherto constructed ; and by the aid of the torsion arrangement, it may be made to give accurate metrical results. There are some difficulties in the use of it, especially as regards the comparison of the indi- cations obtained with different degrees of elec- trification of the index, and the reduction of the results to absolute measure, hither- to obviated only by a daily application of Delmann's method of reference to a zinc-copper water battery, which Delmann himself ap- plies once for all, to one of his electrometers (unless his glass fibre breaks, when he must make a fresh deter- mination of the sensibility of the instrument with its new fibre). The high sensi- bility of the divided ring FIG. 3. Portable Atmospheric Electrometer. XVI.] Atmospheric Electricity. 205 electrometer renders this test really very easy, as not more than from ten to twenty cells are required ; and a comparison with a few good cells of Daniell's may be made by its aid, to ascertain the absolute value and the constancy of the water cells. The difficulty thus met is altogether done away with in another kind of electrometer, also "heterostatic," of which only one has yet been constructed the electrometer of the portable apparatus shown in the third drawing. In it the index is attached at right angles to the middle of a fine platinum wire, firmly stretched between the inside coatings of two Leyden phials, and consists simply of a very light bar of aluminium, extend- ing equally on the two sides of the supporting wire. It is repelled by two short bars of metal, fixed on the two sides of the top of a metal tube, which is supported by the inside coat- ing of the lower phial, and has the fine wire in its axis. A conductor of suitable shape, bearing an electrode, to connect with the body to be tested, insulated inside the case of the instrument, in the neighbourhood of the index, and when elec- trified in the same way, or the contrary way, to the inside coatings of the Leyden phials, causes, by its influence, the repulsion between the index and the fixed bars to be diminished or increased. The upper Leyden phial is moveable about a fixed axis, through angles measured by a pointer and circle, and thus the amount of torsion, in one-half of the bearing wire, required to bring the index to a constant position, in any case, is measured. The square root of the number of degrees of torsion measures the difference of potentials between the conductor tested and the inner coating of the Leyden phial. In using the instrument, the conductor tested is first put in connexion with the earth, and the torsion required to bring the index to its fixed position is read off. This is called the zero, or earth reading. The tested conductor is then electrified, and the torsion reading taken. In the atmospheric application, this is called the air reading. The excess positive or negative of its square root, above that of the zero reading, measures the electromotive force between the earth and the point of air tested. This result, when positive shows vitreous, when nega- tive resinous potential in the air; if the index is resinous. By the aid of Barlow's table of square roots, the indications of the 206 Atmospheric Electricity. [xvi. instrument may thus be reduced to definite measure of potential, almost as quickly as they can be written down. Once for all, the sensibility of the instrument can be determined by com- parison with an absolute electrometer, or a galvanic battery. In the portable apparatus a burning match is used instead of the water-dropping system, which the writer finds more con- venient than any other for a fixed apparatus to reduce the insulated conductor to the same potential as the air at its end. 264. As has been remarked above ( 252), it is the electrifica- tion of the earth's surface which has either directly or virtually been the subject of measurement in all observations on atmo- spheric electricity hitherto made. The methods which have been followed may be divided into two classes (1.) Those in which means are taken to reduce the potential of an insulated conductor to the same as that of the air, at some point, a few feet or yards distant from the earth. (2.) Those in which a portion of the earth (see above, 253) is insulated, removed from its position, and tested by an electrometer, in a different position, or under cover. The first method was very imperfectly carried out by Beccaria with his long " exploring wire," stretched between insulating supports, or elevated portions of buildings, tree tops, or other prominent positions of the earth (see above, 249) ; also, very imperfectly by means of " Volta's lantern" an enclosed flame, supported on the top of an insulated conduc- tor. On the other hand, it is put in practice very perfectly, by means of a match, or flame burning in the open air, on the top of a well insulated conductor a plan adopted, after Volta's suggestion, by many observers ; also, even more decidedly, by means of the water-dropping system described in the preced- ing extract which has recently occurred to the writer, and has been found by him both to be very satisfactory in its action, and extremely easy and convenient in practice. The principle of each of these methods of the first class may be explained best by first considering the methods of the second class, as follows : 265. If a large sheet of metal were laid on the earth in a perfectly level district, and if a circular area of the same metal were laid upon it, and, after the manner of Coulomb's proof plane, were lifted by an insulated handle, and removed XVL] Atmospheric Electricity. 207 to an electrometer within doors, a measure of the earth's elec- trification, at the time, would be obtained ; or, if a ball, placed on the top of a conducting rod in the open air, were lifted from that position by an insulating support, and carried to an electrometer within doors, we should also have, on precisely the same principle, a measure of the earth's electrification at the time. If the height of the ball in this second plan were equal to one- sixteenth of the circumference of the disc (compare 235) used in the first plan, the electrometric indications would be the same, provided the diameter of the ball is small, in comparison with the height to which it is raised in the air, and the electrostatic capacity of the electrometer is small enough not to take any considerable proportion of the electricity from the ball in its application. The idea of experimenting by means of a disc laid flat on the earth, is merely suggested for the sake of illustra- tion, and would obviously be most inconvenient in practice. On the other hand, the method, by a carrier ball, instead of a proof plane, is precisely the method by which, on a small scale, Faraday investigated the distribution of electricity induced on the earth's surface (see above, 249), by a piece of rubbed shell- lac ; and the same method, applied on a suitable scale, for test- ing the natural electrification of the earth in the open air, has given, in the hands of Delmann of Creuznach, the most accurate results hitherto published in the way of electro-meteorological observation*. 266. If, now, we conceive an elevated conductor, first belong- ing to the earth ( 249), to become insulated, and to be made to throw off, and to continue throwing off, portions from an exposed position of its own surface, this part of its surface will quickly be reduced to a state of no electrification, and the whole conductor will be brought to such a potential as will allow it to remain in electrical equilibrium in the air, with that portion of its surface neutral. In other words, the potential throughout the insulated conductor is brought to be the same as that of the * Through some misapprehension, Mr Delmann himself has not perceived that his own method of observation really consists in removing a portion of the earth, and bringing it insulated with the electricity which it possessed in situ, to be tested within doors, otherwise, he could not have objected, as he has, to Peltier's view. 208 Atmospheric Electricity. [xvi. particular equi-potential surface in the air, which passes through the point of it from which matter breaks away. A flame, or the heated gas passing from a burning match, does precisely this: the flame itself, or the highly-heated gas close to the match being a conductor which is constantly extending out, and gradually becoming a non-conductor. The drops into which the jet issuing from the insulated conductor, on the plan introduced by the writer, produce the same effects, with more pointed decision, and with more of dynamical energy to remove the rejected matter with the electricity which it carries from the neighbourhood of the fixed conductor. ROYAL INSTITUTION FRIDAY EVENING LECTURE, MAY 18, 1860. 267. Stephen Gray, a pensioner of the Charter-house, after many years of enthusiastic and persevering devotion to electric science, closed his philosophical labours, about one hundred and thirty years ago, with the following remarkable conjec- ture : " That there may be found a way to collect a greater "quantity of the electrical fire, and consequently to increase "the force of that power, which, by several of these experi- " ments, si licet magna componere parvis, seems to be of the " same nature with that of thunder and lightning." The inventions of the electrical machine and the Leyden phial immediately fulfilled these expectations as to collecting greater quantities of electric fire ; and the surprise and delight which they elicited by their mimic lightnings and thunders, and above all by the terrible electric shock, had scarcely sub- sided when Franklin sent his kite messenger to the clouds, and demonstrated that the imagination had been a true guide to this great scientific discovery the identity of the natural agent in the thunderstorm with the mysterious influence produced by the simple operation of rubbing a piece of amber, which, two thousand years before, had attracted the attention of those xvi.] Atmospheric Electricity. 209 philosophers among the ancients who did not despise the small things of nature. 268. The investigation of atmospheric electricity immediately became a very popular branch of natural science ; and the dis- covery of remarkable and most interesting phenomena quickly rewarded its cultivators. The foundation of all we now know was completed by Beccaria, in his observations on "the mild electricity of serene weather," nearly a hundred years ago. It was not until comparatively recent years that definite quan- titative comparisons from time to time of the electric quality manifested by the atmosphere in one locality were first obtained by the application of Peltier's mode of observation with his metrical electroscope. The much more accurate electrometer, and the greatly improved mode of observation, invented by Delmann, have given for the electric intensity, at an}' instant, still more precise results ; but have left something to desire in point of simplicity and convenience for general use, and have not afforded any means for continuous observation, or for the introduction of self-recording apparatus. The speaker had attempted to supply some of these wants, and he explained the construction and use of instruments, now exhibited to the meeting, which he had planned for this purpose. 269. Apparatus for the observation of atmospheric electricity has essentially two functions to perform; to electrify a body with some of the natural electricity, or with electricity produced by its influence; and to measure the electrification thus obtained. 270. The measuring apparatus exhibited, consisted of three electrometers, which were referred to under the designations of (I.) The divided ring reflecting electrometer ; (II.) The common house electrometer ; and (III.) The portable electrometer. (I.) The divided ring reflecting electrometer [compare 263, above, and 444... 456; below] consists of: (1) A ring of metal divided into two equal parts, of which one is insulated, and the other connected with the metal case (5) of the instrument. (2) A very light needle of sheet aluminium hung by a fine glass fibre, and counterpoised so as to make it project only to one side of this axis of suspension. T. E. 14 210 Atmospheric Electricity. [xvi. (3) A Leyden phial, consisting of an open glass jar, coated outside and inside in the usual manner, with the exception that the tinfoil of the inner coating does not extend to the bottom of the jar, which is occupied instead by a small quantity of sulphuric acid [connected with the tinfoil by means of a platinum wire]. (4) A stiff straight wire rigidly attached to the aluminium needle, as nearly as may be in the line of the suspending fibre, bearing a light platinum wire linked to its lower end, and hanging down so as to dip into the sulphuric acid. (5) A case protecting the needle from currents of air, and from irregular electric actions, and maintaining an artificially dried atmosphere round the glass pillar or pillars supporting the insulated half- ring and the uncqated portion of the glass of the phial. (6) A light stiff metallic electrode pro- jecting from the insulated half-ring through the middle of a small aperture in the metal case, to the outside. (7) A wide metal tube of somewhat less diameter than the Leyden jar, attached to a metal ring borne by its inside coat- ing, and standing up vertically to a few inches above the level of the mouth of the jar. (8) A stiff wire projecting horizontally from this metal tube above the edge of the Leyden jar, and out through a wide hole in the case of the instrument to a convenient position for applying electricity to charge the jar with. (9) A very light glass mirror, about three-quarters of an inch diameter, attached by its back to the wire (4), and there- fore rigidly connected with the aluminium needle. (10) A circular aperture in th.e case shut by a convex lens, and a long horizontal slit shut by plate glass, with its centre im- mediately above or below that of the lens, one of them above, and the other equally below the level of the centre of the mirror. (11) A large aperture in the wide metal tube (7), on a level with the mirror (9), to allow light from a lamp outside the case, entering through the lens, to fall upon the mirror, and be xvi.] Atmospheric Electricity. 211 reflected out through the plate-glass window; and three or four fine metal wires stretched across this aperture to screen the mirror from irregular electric influences, without sensibly diminishing the amount of light falling on and reflected off it. 271. The divided ring (1) is cut out of thick strong sheet metal (generally brass). Its outer diameter is about 4 inches, its inner diameter 2J ; and it is divided into two equal parts by cutting it along a diameter with a saw. The two halves are fixed horizontally; one of them on a firm metal support, and the other on glass, so as to retain as nearly as may be their original relative position, with just the saw cut, from -^ to -fa of an inch broad, vacant between them. They are placed with their common centre as nearly as may be in the axis of the case (5), which is cylindrical, and placed vertically. The Leyden jar (3), and the tube (7), carried by its inside coating, have their common axis fixed to coincide as nearly as may be with that of the case and divided ring. The glass fibre hangs down from above in the direction of this axis, and supports the needle about an inch above the level of the divided ring. The stiff wire (4), attached to the needle, hangs down as nearly as may be along the axis of the tube (7). - [The following diagrams, placed here to facilitate comparison, represent the arrangement of " needle " and quadrants described below in 345, as substituted in the modern instrument for the bisected ring and narrow needle of the old electrometer here described] : 272. Before using the instrument, the Leyden ^phial (3) is charged by means of its projecting electrode (8). When an electrical machine is not available, this is very easily done by the aid of a stick of vulcanite, rubbed by a piece of chamois leather. The potential of the charge thus communicated to the phial, is 142 212 Atmospheric Electricity. [xvi. to be kept as nearly constant as is required for the accuracy of the investigation for which the instrument is used. Two or three rubs of the stick of vulcanite once a day, or twice a day, are sufficient when the phial is of good glass, well kept dry. The most convenient test for the charge of the phial is a proper electrometer or electroscope, of any convenient kind, kept constantly in communication with the charging elec- trode (8). [Compare 353, below.] The electrometer (II.) is to be ordinarily used for that pur- pose in the Kew apparatus. Failing any such gauge electro- meter or electroscope, a zinc-copper-water battery often, twenty, or more small cells may be very conveniently used (after the manner of Delmann) to test directly the sensibility of the re- flecting electrometer, which is to be brought to its proper degree by charging its Ley den phial as much as is required. 273. In the use of this electrometer, the two bodies of which the difference of potentials is to be tested are connected, one of them, which is generally the earth, with the metal case of the instrument, and the other with the insulated half ring. The needle being, let us suppose, negatively electrified, will move towards or from the insulated half ring, according as the poten- tial of the conductor connected with this half ring differs posi- tively or negatively from that of the other conductor (earth) connected with the case. The mirror turns accordingly in one direction or the other through a small angle from its zero posi- tion, and produces a corresponding motion in the image of the lamp on the screen on which it is thrown. 274. (II.) The common house electrometer [compare 263, above, and 374... 377, below]. This instrument consists of: (1) A thin flint-glass bell, coated outside and inside like a Leyden phial, with the exception of the bottom inside, which contains a little sulphuric acid. (2) A cylindrical metal case, enclosing the glass jar, cemented to it round its mouth outside, extending upwards about an inch and a half above the mouth, and downwards to a metal base supporting the whole instrument, and protecting the glass against the danger of breakage. (3) A cover of plate glass, with a metal rim, closing the top of the cylindrical case of the instrument. XVI.] Atmospheric Electricity. 213 (4) A torsion head, after the manner of Coulomb's balance, supported in the centre of the glass cover, and bearing a glass fibre which hangs down through an aperture in its centre. (5) A light aluminium needle attached across the lower end of the fibre (which is somewhat above the centre of the glass bell), and a stiff platinum wire attached to it at right angles, and hanging down to near the bottom of the jar. (6) A very light platinum wire, long enough to hang within one-eighth of an inch or so of the bottom of the jar, and to dip in the sulphuric acid. (7) A metal ring, attached to the inner coating of the jar, bearing two plates in proper positions for repelling the two ends of the aluminium needle when similarly electrified, and proper stops to limit the angular motion of the needle to with- in about 45 from these plates. (8) A cage of fine brass wire, stretched on brass framework, supported from the main case above by two glass pillars, and partially enclosing the two ends of the needle, and the repel- ling plates, from all of which it is separated by clear spaces, of nowhere less than one-fourth of an inch of air. (9) A charging electrode, attached to the ring (7), and pro- jecting over the mouth of the jar to the outside of the metal case (2), through a wide aperture, which is commonly kept closed by a metal cap, leaving at least one quarter of an inch of air round the projecting end of the electrode. (10) An electrode attached to the cage (8), and projecting over 214 Atmospheric Electricity. [xvi. the mouth of the jar to the outside of the metal case (2), through the centre of an aperture, about a quarter of an inch diameter. 275. This instrument is adapted to measure differences of potential between two conducting systems, namely ; as one, the aluminium needle (5), the repelling plates (7), and the inner coating of the jar ; and, as the other, the insulated cage (8). This latter is commonly connected by means of its projecting electrode (10), with the conductor to be tested. The two conducting systems, if through their projecting electrodes connected by a metallic wire, may be electrified to any degree, without causing the slightest sensible motion in the needle. If, on the other hand, the two electrodes of these two systems are connected with two conductors, electrified to different potentials, the needle moves away from the repelling plates ; and if, by turning the torsion head, it is brought back to one accurately marked posi- tion, the number of degrees of torsion required is proportional to the square of the difference of potentials thus tested. 276. In the ordinary use of the instrument, the inner coating of the Leyden jar is charged negatively, by an external applica- tion of electricity through its projecting electrode (9). The degree of the charge thus communicated, is determined by putting the cage in connexion with the earth through its elec- trode (10), and bringing the needle by torsion to its marked position. The square root of the number of degrees of torsion required to effect this, measures the potential of the Leyden charge. This result is called the reduced earth reading. When the atmosphere inside the jar is kept sufficiently dry, this charge is retained from day to day with little loss ; not more, often, than one per cent, in the twenty-four hours. In using the instrument the charging electrode (9) of the jar is left untouched, with the aperture through which it projects closed over it by the metal cap referred to above. The electrode (10) of the cage, when an observation is to be made, is connected with the conductor to be tested, and the needle is brought by torsion to its marked position. The square root of the number of degrees of torsion now required measures the difference of potentials between the conductor tested and the interior coating of the Leyden jar. The excess, positive or nega- tive, of this result above the reduced earth reading, measures xvi.] Atmospheric Electricity. 215 the excess of the potential, positive or negative, of the conduc- tor tested above that of the earth ; or simply the potential of the conductor tested, if we regard that of the earth as zero. 277. (III.) The portable electrometer [compare 263, above, and 363... 373, below] is constructed on the same elec- trical principles as the house electrometer just described. The mode of suspension of the needle is, however, essentially different ; and a varied plan of connexion between the different electrical parts has been consequently adopted as more con- venient. In the portable electrometer, the needle is firmly attached at right angles to the middle of a fine platinum wire, tightly stretched in the axis of a brass tube with apertures in its middle to allow the needle to project on the two sides. One end of the platinum wire is rigidly connected with this tube ; the other is attached to a graduated torsion head. The brass tube carries two metal plates in suitable positions to repel the two ends of the needle in contrary directions, and metal stops to limit its angular motion within a convenient range. The conducting system composed of these different parts is supported from the metal cover, or roof of the jar, by three glass stems. The torsion head is carried round by means of a stout glass bar, projecting down from a pinion centered on the lower side of this cover, and turned by the action of a tan- gent screw presenting a milled head, to the hand of the opera- tor outside. The conducting system thus borne by insulating supports is connected with the outside conductor to be tested by means of an electrode passing out through the centre of the top of the case by a wide aperture in the centre of the pinion. A wire cage, surrounding the central part of the tube and the needle and repelling plates, is rigidly attached to the interior coating of the Leyden jar. It carries two metal sectors, or "bulkheads," in suitable positions to attract the two ends of the needle, which, however, is prevented from touching them by the limiting stops referred to above. The effect of these attracting plates, as they will be called, is to increase very much the sensibility of the instrument. The square root of the number of degrees of torsion required to bring the needle to a sighted position near the repelling plates, measures the 216 Atmospheric Electricity. [xvi. difference of potentials between the cage and the conducting system, consisting of tube, torsion-head, repelling plates, and needle. The metal roof of the jar is attached to a strong metal case, cemented round the outside of the top of the jar, and enclosing it all round and below, to protect it from breakage when being carried about. There are sufficient apertures in this case, opened by means of a sliding piece, to allow the observer to see the needle and graduated circle (torsion-head), when using the instrument. On the outside of the roof of the jar a stout glass stem is attached, which supports a light stiff metallic conductor, by means of which a burning match is supported, at the height of two or three feet above the observer. This conductor is connected by means of a fine wire with the electrometer, in the manner described above, through the centre of the aperture in the roof. An artificially dried atmosphere is maintained around this glass stem, by means of a metal case surrounding it, and containing receptacles of gutta percha, or lead, holding suitably shaped pieces of pumice-stone moistened with sulphuric acid. The conductor which bears the match projects upwards through the centre of a sufficiently wide aper- ture, and bears a small umbrella, which both stops rain from falling into this aperture, and diminishes the circulation of air, owing to wind blowing round the instrument, from taking place, to so great a degree as to do away with the dryness of the in- terior atmosphere required to allow the glass stem to insulate sufficiently. The instrument may be held by the observer in his hand in the open air without the assistance of any fixed stand. A sling attached to the instrument and passing over his left shoulder, much facilitates operations, and renders it easy to carry the apparatus to the place of observation, even if up a rugged hill side, with little risk of accident. 278. The burning match in the apparatus which has just been described, performs the collecting function referred to above. The collector employed for the station apparatus, whether the reflecting electrometer or the common house electrometer is used, is an insulated vessel of water, allowed to flow out in a fine stream through a small aperture at the end of a pipe pro- jecting to a distance of several feet from the wall of the build- ing in which the observations are made. xvi.] Atmospheric Electricity. 217 279. The principle of collecting, whether by fire or by water, in the observation of atmospheric electricity, was explained by the speaker thus : The earth's surface is, except at instants, always found electrified, in general negatively, but sometimes positively. [Quotation from Nichol's Cyclopaedia, viz., 265, above, comes here in the original.] After having given so much of these explanations as seemed necessary to convey a general idea of the principles on which the construction of the instruments of investigation depended, the speaker proceeded to call attention to the special subject proposed for consideration this evening. 280. What is terrestrial atmospheric electricity ? Is it elec- tricity of earth, or electricity of air, or electricity of watery or other particles in the air ? An endeavour to answer these ques- tions was all that was offered ; abstinence from speculation as to the origin of this electric condition of our atmosphere, and its physical relations with earth, air, and water, having been pain- fully learned by repeated and varied failure in every attempt to see beyond facts of observation. In serene weather, the earth's surface is generally, in most localities hitherto examined, found negatively or resinously electrified; and when this fact alone is known, it might be supposed that the globe is merely electrified as a whole with a resinous charge, and left insulated in space. 281. But it is to be remarked that the earth, although insulated in its atmospheric envelope, being in fact a conductor, touched only by air one of the best although not the strongest of in- sulators, cannot with its atmosphere be supposed to be insulated so as to hold an electric charge in interplanetary space. It has been supposed, indeed, that outside the earth's recognised atmo- sphere there exists something or nothing in space which con- stitutes a perfect insulator ; but this supposition seems to have no other foundation than a strange idea that electric conduc- tivity is a strength or a power of matter rather than a mere non-resistance. In reality we know that air highly rarefied by the air-pump, or by other processes, as in the construction of the " vacuum tubes," by which such admirable phenomena of electric light have recently been seen in this place, becomes extremely weak in its resistance to the transference of elec- 218 Atmospheric Electricity. [xvi. tricity through it, and begins to appear rather as a conductor than an insulator. One hundred miles or upwards from the earth's surface, the air in space cannot in all probability have resisting power enough to bear any such electric forces as those which we generally find even in serene weather in the lower strata. Hence we cannot, with Peltier, regard the earth as a resinously charged conductor, insulated in space, and subject only to accidental influences from temporary electric deposits in clouds, or air round it ; but we must suppose that there is always essentially in the higher aerial regions a distribution arising from the self-relief of the outer highly rarefied air by disruptive discharge. This electric stratum must constitute very nearly the electro-polar complement to all the electricity that exists on the earth's surface, and in the lower strata of the atmosphere ; in other words, the total quantity of electricity, reckoned as excess of positive above negative, or of negative above positive, in any large portion of the atmosphere, and on the portion of the earth's surface below it, must be very nearly zero. The quality of non-resistance to electric force of the thin interplanetary air being duly considered, we might regard the earth, its atmosphere, and the surrounding medium as constitut- ing respectively the inner coating, the di-electric (as it were glass), and the outer coating of a great Leyden phial, charged negatively; and even if we were to neglect the consideration of possible deposits of electricity through the body of the di- electric itself, we should arrive at a correct view of the electric indications discoverable at any one time and place of the earth's surface. In fact, any kind of "collector," or plan for collect- ing electricity from or in virtue of the natural " terrestrial atmospheric electricity," gives an effect simply proportional to the electrification of the earth's surface then and there. The methods of collecting by fire and water which the speaker exhibited, gave definitively, in the language of the mathemati- cal theory, the "electric potential" of the air at the point occupied by the burning end of the match, or by the portion of the stream of water where it breaks into drops. If the apparatus is used in an open plane, and care be taken to eliminate all disturbance due to the presence of the electro- meter itself and of the observer above the ground, the indicated xvi.] Atmospheric Electricity. 219 effect, if expressed in absolute electrostatic measure, and divided by the height of the point tested above the ground, has only to be [according to an old theorem of Coulomb's (see footnote on 25, above), corrected by Laplace] divided by four times the ratio of the circumference of a circle to its diameter, to reduce it to an expression of the number of units, in absolute electrostatic measure, of the electricity per unit of area of the earth's surface at the time and place. The mathematical theory does away with every difficulty in explaining the various and seemingly irreconcilable views which different writers have expressed, and explanations which different observers have given of the functions of their testing apparatus. In the present state of electric science, the most convenient and generally intelligible way to state the result of an observation of terrestrial atmo- spheric electricity, in absolute measure, is in terms of the number of elements of a constant galvanic battery, required to produce the same difference of potentials as exists between the earth and a point in the air at a stated height above an open level plane of ground. Observations with the portable electro- meter had given, in ordinary fair weather, in the island of Arran, on a flat open sea beach, readings varying from 200 to 400, Daniel's elements, as the difference of potentials between the earth and the match, at a height of 9 feet above it. Hence, the intensity of electric force perpendicular to the earth's sur- face must have amounted to from 22 to 44 Daniel's elements per foot of air. In fair weather, with breezes from the east or north-east, he had often found from 6 to 10 times the higher of these intensities. 282. Even in fair weather, the intensity of the electric force in the air near the earth's surface is perpetually fluctuating. The speaker had often observed it, especially during calms or very light breezes from the east, varying from 40 Daniel's elements per foot to three or four times that amount during a few minutes ; and returning again as rapidly to the lower amount. More frequently he had observed variations from about 30 to about 40, and back again, recurring in uncertain periods of perhaps about two minutes. These gradual variations cannot but be produced by electrified masses of air or cloud, floating by the locality of observation. Again, it is well known that 220 Atmospheric Electricity. [xvi. during storms of rain, hail, or snow, there are great and some- times sudden variations of electric force in the air close to the earth. These are undoubtedly produced, partly as those of fair weather, by motions of electrified masses of air and cloud ; partly by the fall of vitreously or resinously electrified rain, leaving a corresponding deficiency in the air or cloud from which it falls ; and partly by disruptive discharges (flashes of lightning) between masses of air or cloud, or between either and the earth. The consideration of these various phenomena suggested the following questions, and modes of observation for answering them : 283. Question 1. How is electricity distributed through the different strata of the atmosphere to a height of five or six miles above the earth's surface in ordinary fair weather ? To be answered by electrical observations in balloons at all heights up to the highest limit, and simultaneous observations at the earth's surface. Q. 2. Does electrification of air close to the earth's surface, or within a few hundred feet of it, sensibly influence the observed electric force ? and if so, how does it vary with the weather, and with the time of day or year ? The first part of this question has been answered very decidedly in the affirma- tive, first, for large masses of air within a few hundred yards of the earth's surface, by means of observations made simul- taneously at a station near the seashore in the island of Arran, and at one or other of several stations at different distances, within six miles of it, on the sides and summit of Goat fell. After that it was found, by simultaneous observations made at a window in the Natural Philosophy Lecture-Room, and on the College Tower of the University of Glasgow, that the influence of the air within 100 feet of the earth's surface was always sensible at both stations, and often paramount at the lower. Thus, for example, when, in broken weather, the superficial electrification of the outside of the lecture-room, about 20 feet above the ground, in a quadrangle of buildings, was found positive, the superficial electrification of the sides of the tower, about 70 feet higher, was often found negative, or nearly zero ; and this sometimes even when the positive electrification of the sides of the building at the lower station equalled in amount xvi.] Atmospheric Electricity. 221 an ordinary fair weather negative. This state of things could only exist in virtue of a negative electrification of the circum- ambient air, inducing a positive electrification on the ground and sides of the quadrangle, but not sufficient to counter- balance the influence, on the higher parts of the tower, of more distant positively electrified aerial masses. A long continuation of such systems of simultaneous obser- vation not in a town only, but in various situations of flat and of mountainous country, on the sea coast as well as far inland, in various regions of the world will be required to obtain the information asked for in the second part of this question. Q. 3. Do the particles of rain, hail, and snow in falling through the air possess absolute charges of electricity ? and if so, whether positive or negative, and of what amounts in differ- ent conditions as to place and weather ? Attempts to answer this question have been made by various observers, but as yet without success ; as, for instance, by an " electro-pluviometer," tried at Kew many years ago. By using a sufficiently well- insulated vessel to collect the falling particles, it is quite certain that a decided answer may be obtained with ease for the cases of hail and snow. Inductive effects produced by drops splash- ing away from the collecting vessel, if exposed to the electric force of the air in an open position, or inductive effects of the opposite kind produced by drops splashing away from surround- ing walls or screens and falling into the collecting vessel, if not in an exposed position, make it less easy to ascertain the elec- trical quality of rain ; but, by taking means to obviate the disturbing effects of these influences, the speaker hoped to arrive at definite results. 284. It would have been more satisfactory to have been able to conclude a discourse on atmospheric electricity otherwise than in questions, but no other form of conclusion would have been at all consistent with the present state of knowledge. 285. The discourse was illustrated by the use of the mirror electrometer reflecting a beam of light from the electric lamp, arid throwing it on a white screen, where its motions were measured by a divided scale. The principle of the water- dropping collector was illustrated by allowing a jet of water to flow by a fine nozzle into the middle of the lecture-room, from 222 Atmospheric Electricity. [xvi. an uninsulated metal vessel of water and compressed air, and collecting the drops in an insulated vessel on the floor. This vessel was connected with the testing electrode of the reflecting electrometer ; and it was then found to experience a continually increasing negative electrification, when fixed positively elec- trified bodies were in the neighbourhood of the nozzle. If the same experiment were made in ordinary fair weather in the open air, instead of under the roof and within the walls of the lecture-room, the same result would be observed, without the presence of any artificially electrified body. The vessel from which the water was discharged was next insulated ; and other circumstances remaining unvaried, it was shown that this vessel became rapidly electrified to a certain degree of positive potential, and the falling drops ceased to communicate any more electricity to the vessel in which they were gathered. 286. The influence of electrified masses of air was illustrated by carrying about the portable electrometer, with its match burn- ing, to different parts of the lecture-room, while insulated spirit-lamps connected with the positive and negative con- ductor of an electrical machine, burned on the two sides. The speaker observed the indications on the portable electrometer ; but the potentials thus measured were seen by the audience marked on the scale by the spot of light ; the reflecting electro- meter being kept connected with the portable electrometer in all its positions, by means of a long fine wire. It was found that, when the burning match was on one side of a certain surface dividing the air of the lecture-room, the potential indi- cated was positive, and on the other side negative. 287. The water-dropping collector constructed for the self- registering apparatus to be used at Kew, had been previously set upon the roof of the Royal Institution, and an insulated wire (Beccaria's " Deferent Wire") led down to the reflecting electrometer on the lecture-room table. The electric force in the air above the roof was thus tested several times during the meeting ; and it was at first found to be, as it had been during several days preceding, somewhat feeble positive (corresponding to % a feeble negative electrification of the earth's surface, or rather housetops, in the neighbourhood). This was a not unfrequent electrical condition of .days, such as these had been Atmospheric Electricity. 223 of dull rain, with occasional intervals of heavier rain and of cessation. The natural electricity was again observed by means of the reflecting electrometer during several minutes near the end of the discourse ; and was found, instead of the weak positive which had been previously observed, to be strong positive of three or four times the amount. Upon this the speaker quoted * an answer which Prior Ceca had given to a question Beccaria had put to him "concerning the state of " electricity when the weather clears up." " ' If, when the rain "'has ceased (the Prior said to me) a strong excessivef elec- " ' tricity obtains, it is a sign that the weather will continue fair " ' for several days ; if the electricity is but small, it is a sign " * that such weather will not last so much as that whole day, "'and that it will soon be cloudy again, or even will again "'rain.'" The climate of this country is very different from that of Piedmont, where Beccaria and his friend made their observations, but their rule as to the "electricity of clearing weather" has been found frequently confirmed by the speaker. He therefore considered that, although it was still raining at the commencement of the meeting, the electrical indications they had seen gave fair promise for the remainder of this evening, if not for a longer period. There can be no doubt but that electric indications, when sufficiently studied, will be found important additions to our means for prognosticating the weather; and the speaker hoped soon to see the atmospheric electrometer generally adopted as ^a useful and convenient weather-glass. 288. The speaker could not conclude without guarding him- self against any imputation of having assumed the existence of two electric fluids or substances, because he had frequently spoken of the vitreous and resinous electricities. Dufay's very important discovery of two modes or qualities of electrification, led his followers too readily to admit his supposition of two distinct electric fluids. Franklin, ^Epirius, and Cavendish, * From Beccaria's first letter "On Terrestrial Atmospheric Electricity during Serene Weather." Garzegna di Mondavi, May 16, 1775. t i.e., vitreous, or positive. J At the conclusion of the meeting it was found that the rain had actually ceased. The weather continued fair during the remainder of. the night, and three or four of the finest days of the season followed. 224 Atmospheric Electricity. [xvi. with a hypothesis of one electric fluid, opened the way for a j lister appeciation of the unity of nature in electric phenomena. Beccaria, with his " electric atmospheres," somewhat vaguely struggled to see deeper into the working of electric force, but his views found little acceptance, and scarcely suggested in- quiry or even meditation. The eighteenth century made a school of science for itself, in which, for the not unnatural dogma of the earlier schoolmen, " matter cannot act where it is not," was substituted the most fantastic of paradoxes, contact does not exist. Boscovich's theory was the consummation of the eighteenth century school of physical science. This strange idea took deep root, and from it grew up a barren tree, exhaust- ing the soil and overshadowing the whole field of molecular investigation, on which so much unavailing labour was spent by the great mathematicians of the early part of our nineteenth century. If Boscovich's theory no longer cumbers the ground, it is because one true philosopher required more light for trac- ing lines of electric force. 289. Mr Faraday's investigation of electrostatic induction influences now every department of physical speculation, and constitutes an era in science. If we can no longer regard electric and magnetic fluids attracting or repelling at a distance as realities, we may now also contemplate as a thing of the past that belief in atoms and in vacuum, against which Leib- nitz so earnestly contended in his memorable correspondence with Dr Samuel Clarke. ^ $ 290. We now look on space as full. We know that light is propagated like sound through pressure and motion. We know that there is no substance of caloric that inscrutably minute motions cause the expansion which the thermometer marks, and stimulate our sensation of heat that fire is not laid up in coal more than in this Leyden phial, or this weight : there is potential fire in each. If electric force depends on a residual surface action, a resultant of an inner tension experienced by the insulating medium, we can conceive that electricity itself is to be understood as not an accident, but an essence of matter. Whatever electricity is, it seems quite certain that electricity in motion is heat; and that a certain alignment of axes of revolution in this motion is magnetism. Faraday's magneto- XVI.] Atmospheric Electricity. 225 optic experiment makes this not a hypothesis, but a demon- strated conclusion*. Thus a rifle-bullet keeps its point fore- most; Foucault's gyroscope finds the earth's axis of palpable rotation ; and the magnetic needle shows that more subtle rotatory movement in matter of the earth, which we call ter- restrial magnetism : all by one and the same dynamical action. 291. It is often asked, are we to fall back on facts and pheno- mena, and give up all idea of penetrating that mystery which hangs round the ultimate nature of matter ? This is a question that must be answered by the metaphysician, and it does not be- long to the domain of Natural Philosophy. But it does seem that the marvellous train of discovery, unparalleled in the history of experimental science, which the last years of the world has seen to emanate from experiments within these walls, must lead to a stage of knowledge, in which laws of inorganic nature will be understood in this sense that one will be known as essentially connected with all, and in which unity of plan through an inexhaustibly varied execution, will be recognised as a universally manifested result of creative wisdom. 292. [Postscript, with diagram, communicated to the Philoso- phical Magazine in 1861 ; but now first published.] Mr Balfour Stewart, Director of the Kew Meteorological Observatory, has, since the commencement of the present year (1861), brought into regular and satisfactory operation the self- recording atmospheric electrometer with water-dropping collec- tor, described in the preceding abstract : a specimen of the results is exhibited in the accompanying photographic curves. i ' i 1 ' i 1 V Vr j ' ' r ' ' i ' ' ' i ' * See "Dynamical Illustrations of the Magnetic and the Helicoidal Botatory Effects of Transparent Bodies on Polarized Light." By Prof. W. Thomson. Proceedings of the Royal Society, June 12, 1856. T. E. 15 220 Atmospheric Electricity. [xvi. 293. The diagram exhibits the variations of the electric force of the atmosphere, as photographically recorded by the divided ring electrometer at the Kew Observatory for two succes- sive days, commencing on the 28th of April 1861. The prepared sensitive paper was made to move vertically at a uniform rate by means of clock-work, while a spot of light (the image of a portion of a gas-flame reflected from the mirror of the divided ring electrometer) moved horizontally across it according to the continually varying electric force of the atmo- sphere, and marked the curve photographically. The datum line, showing the position the spot of light would have if the electric force were zero, is produced by an image from the same source of light reflected from a fixed mirror attached to the case of the electrometer. The numbers indicate hours reckoned from noon as zero, up to 23. The same paper is, for the sake of economy, generally used to bear the record for two days. Thus the distance of the spot of light from the datum line, on one side or other, indicates, and the photo-chemical action records, for each instant of time the electric potential, positive or negative, of the atmosphere at the point where the stream of water discharged from the insulated vessel breaks into drops. ON ELECTRICAL "FREQUENCY." [From Report of British Association, Aberdeen Meeting, 1859.] 294. Beccaria found that a conductor insulated in the open air becomes charged sometimes with greater and sometimes with less rapidity, and he gave the name of " frequency " to ex- press the atmospheric quality on which the rapidity of charg- ing depends. It might seem natural to attribute this quality to electrification of the air itself round the conductor, or to electrified particles in the air impinging upon it; but the author gave reasons for believing that the observed effects are entirely due to particles flying away from the surface of the conductor, in consequence of the impact of non-electrified particles against it. He had shown in a previous communication that when no electricity of separation (or, as it is more generally called, " frictional electricity," or " contact electricity ") is called into xvi.] Atmospheric Electricity. 227 play, the tendency of particles continually flying off from a conductor is to destroy all electrification at the part of its sur- face from which they break away. Hence a conductor insulated in the open air, and exposed to mist or rain, with wind, will tend rapidly to the same electric potential as that of the air, beside that part of its surface from which there is the most frequent dropping, or flying away, of aqueous particles. The rapid charging indicated by the electrometer under cover, after putting it for an instant in connexion with the earth, is there- fore, in reality, due to a rapid discharging of the exposed parts of the conductor. The author had been led to these views by remarking the extreme rapidity with which an electrometer j connected by a fine wire with a conductor insulated above the roof of his temporary electric observatory in the island of Arran, became charged, reaching its full indication in a few seconds, and sometimes in a fraction of a second, after being- touched by the hand, during a gale of wind and rain. The conductor, a vertical cylinder about 10 inches long and 4 inches diameter, with its upper end flat and corner slightly rounded off, stood only 8 feet above the roof, or, in all, 20 feet above the ground, and was nearly surrounded by buildings rising to a higher level. Even with so moderate an exposure as this, sparks were frequently produced between an insulated and an uninsulated piece of metal, which may have been about ^th of an inch apart, within the electrometer, and more than once a continuous line of fire was observed in the instrument during nearly a minute at a time, while rain was falling in torrents outside. ON THE NECESSITY FOR INCESSANT RECORDING, AND FOR SIMULTANEOUS OBSERVATIONS IN DIFFERENT LOCALI- TIES, TO INVESTIGATE ATMOSPHERIC ELECTRICITY. [From Report of British Association, Aberdeen Meeting, 1859.] 295. The necessity for incessantly recording the electric con- dition of the atmosphere was illustrated by reference to obser- vations recently made by the author in the island of Arran, by which it appeared that even under a cloudless sky, without any 152 228 Atmospheric Electricity. [xvi. sensible wind, the negative electrification of the surface of the earth, always found during serene weather, is constantly vary- ing in degree. He had found it impossible, at any time, to leave the electrometer without losing remarkable features of the phenomenon. Beccaria, Professor of Natural Philosophy in the University of Turin a century ago, used to retire to Garzegna when his vacation commenced, and to make inces- sant observations on atmospheric electricity, night and day, sleeping in the room with his electrometer in a lofty position, from which he could watch the sky all round, limited by the Alpine range on one side, and the great plain of Piedmont on the other. Unless relays of observers can be got to follow his example, and to take advantage of the more accurate instru- ments supplied by advanced electric science, a self-recording apparatus must be applied to provide the data required for obtaining knowledge in this most interesting field of nature. The author pointed out certain simple and easily-executed modifications of working electrometers (exhibited to the meet- ing), to render them self-recording. He also explained a new collecting apparatus for atmospheric electricity, consisting of an insulated vessel of water, discharging its contents in a fine stream from a pointed tube. This stream carries away electricity as long as any exists on its surface, where it breaks into drops. The immediate object of this arrangement is to maintain the whole insulated conductor, including the portion of the electrometer connected with it and the connecting wire, in the condition of no absolute charge ; that is to say, with as much positive electricity on one side of a neutral line as of negative on the other. Hence the position of the discharging nozzle must be such, that the point where the stream breaks into drops is in what would be the neutral line of the con- ductor, if first perfectly discharged under temporary cover, and then exposed in its permanent open position, in which it will become inductively electrified by the aerial electromotive force. If the insulation is maintained in perfection, the dropping will not be called on for any electrical effect, and sudden or slow atmospheric changes will all instantaneously and perfectly in- duce their corresponding variations in the conductor, and give their appropriate indications to the electrometer. The neces- xvi.] Atmospheric Electricity. 229 sary imperfection of the actual insulation, which tends to bring the neutral line downwards or inwards, or the contrary effects of aerial convection, which, when the insulation is good, gene- rally preponderate, and which in some conditions of the atmo- sphere, especially during heavy wind and rain, are often very large, are corrected by the tendency of the dropping to main- tain the neutral line in the one definite position. The objects to be attained by simultaneous observations in different localities alluded to were (1) to fix the constant for any observatory, by which its observations are reduced to absolute measure of electromotive force per foot of air; (2) to investigate the dis- tribution of electricity in the air itself (whether on visible clouds or in clear air) by a species of electrical trigonometry, of which the general principles were slightly indicated. A por- table electrometer, adapted for balloon and mountain observa- tions, with a burning match, regulated by a spring so as to give a cone of fire in the open air, in a definite position with refer- ence to the instrument, was exhibited. It is easily carried, with or without the aid of a shoulder-strap, and can be used by the observer standing up, and simply holding the entire apparatus in his hands, without a stand or rest of any kind. Its indications distinguish positive from negative, and are re- ducible to absolute measure on the spot. The author gave the result of a determination which he had made, with the assist- ance of Mr Joule, on the Links, a piece of level ground near the sea, beside the city of Aberdeen, about 8 A.M. on the pre- ceding day (September 14), under a cloudless sky, and with a light north-west wind blowing, with the insulating stand of the collecting part of the apparatus buried in the ground, and the electrometer removed to a distance of 5 or 6 yards, and con- nected by a fine wire with the collecting conductor. The height of the match was 3 feet above the ground, and the observer at the electrometer lay on the ground to render the electrical influence of his own body on the match insensible. The result showed a difference of potentials between the earth (negative) and the air (positive) at the match equal to that of 115 elements of Daniell's battery, and, therefore, at that time and place, the aerial electromotive force per foot amounted to that of thirty-eight Daniell's cells, or T2 cells per centimetre. 230 Atmospheric Electricity, [xvi. OBSERVATIONS ON ATMOSPHERIC ELECTRICITY. [From the Proceedings of the Literary and Philosophical Society of Manchester, March, 1862.] 296. I find that atmospheric electricity is generally negative within doors, and almost always sensible to my divided ring reflecting electrometer. I use a spirit-lamp, on an insulated stand a few feet from walls, floor, or ceiling of my lecture room, and connect it by a fine wire with the insulated half ring of the electrometer. A decided negative effect is generally found, which shows a potential to be produced in the con- ductors connected with the flame, negative relatively to the earth by a difference amounting to several times the difference of potentials (or electromotive force) between two wires of one metal connected with the two plates of a single element of Daniell's. I have tested that the spirit-lamp gives no idio- electric effect amounting to so much as the effect of a single cell. The electric effect observed is therefore not due to thermal or chemical action in the flame. It cannot be due to contact electrifications of metallic or other bodies in conductive communication with the walls, floor, or ceiling, because the potentials of such must always fall short of the difference of potentials produced by a single cell. I have taken care to distinguish the observed natural effect from anything that can be produced by electrical operations for lecture or laboratory purposes. Thus I observe generally in the morning before any electrical operations have been performed, and find ordinarily results quite similar to those observed on the Monday mornings when the electrical machine has not been turned since the previous Friday. The effect, when there has been no artificial disturbance, has always been found negative, except two or three timeSy since the middle of November; but trustworthy obser- vations have not been made on more than a quarter of the number of days. 297. A few turns of the electrical machine, with a spirit-lamp on its prime conductor, or a slightly charged Ley den phial, with its inside coating positive put in connexion with an insulated spirit-lamp, is enough to reverse the common negative indica- tion. Another very striking way in which this may be done is to put a negatively charged Leyden phial below an insulated XVI.] Atmospheric Electricity. 231 flame (a common gas-burner, for instance). The flame, becom- ing positively electrified by induction, keeps throwing off, by the dynamic power of its burning, portions of its own gaseous matter, and does not allow them to be electrically attracted down to the Leyden phial, but forces them to rise. These, on cooling, become, like common air, excellent non-conductors*, and, mixing with the air of the room, give a preponderance of positive influence to the testing insulated flame (that is to say, render the air potential positive at the place occupied by this flame). 298. Half an hour, or often much more, elapses after such an operation, before the natural negatively electrified air becomes again paramount in its influence on the testing flame. 299. That either positive or negative electricity may be carried, even through narrow passages, by air, I have tested by turning an electric machine, with a spirit-lamp on its prime conductor, for a short time in a room separated from the lecture room by an oblique passage about two yards long and then stopping the machine and extinguishing the lamp ; so as to send a limited quantity of positive electricity into the air of that room. When the lecture-room window was kept open, and the door leading to the adjoining room shut, the testing spirit- lamp showed the natural negative. When the window was closed, and a small chink (an inch or less wide) opened of the door, the indication quickly became positive. If the door was then shut, and the window again opened, the natural effect was slowly recovered. A current of air, to feed the lecture-room fire, was found entering by either door or window when the other was shut. This alternate positive and negative electric ventilation may be repeated many times without renewing the positive electricity of the adjoining room by turning the machine afresh. * I find that steam from a kettle boiling briskly on a common fire is an excellent insulator. I allow it to blow for a quarter of an hour or more against an insulated electrified conductor, without discovering that it has any effect on the retention of the charge. The electricity of the steam itself, in such circumstances, as is to be expected from Faraday's investigation, is not considerable. Common air loses nearly all its resisting power at some temperature between that of boiling water and red-hot iron, and conducts continuously (not, as I believe is generally supposed to be the case, by dis- ruption) as glass does at some temperature below the boiling point, with so great ease as to discharge any common insulated conductor almost completely in a few seconds. 232 Atmospheric Electricity. [xvr. 300. The out of doors air potential, as tested by a portable electrometer in an open place, or even by a water dropping nozzle outside, two or three feet from the walls of the lecture room, was generally on these occasions positive, and the earth's surface itself, therefore, of course, negative ; the common fair weather condition, which I am forced to conclude is due to a paramount influence of positive electricity in higher regions of the air, notwithstanding the negative electricity of the air in the lower stratum near the earth's surface. On the two or three occasions when the in-door atmospheric electricity was found positive, and, therefore, the surface of the floor, walls, and ceil- ing negative, the potential outside was certainly positive, and the earth's surface out of doors negative, as usual in fair weather. 300'. Extract from letter addressed to General Sabine : " During my recent visit to Creuznach I became acquainted with Mr Dellman of that place, who makes meteorological, chiefly electrical, observations for the Prussian Government, and I had opportunities of witnessing his method of electrical observation. It consists- in using a copper ball about 6 inches diameter, to carry away an electrical effect from a position about two yards above the roof of his house, depending simply on the atmospheric 'potential' at the point to which the centre of the ball is sent; and it is exactly the method of the 'carrier ball' by which Faraday investigated the atmospheric potential in the neighbourhood of a rubbed stick of shell-lac, and other electrified bodies (Experimental Researches, Series xi. 1837). The whole process only differs from Faraday's in not employing the carrier ball directly, as the repeller in a Coulomb-electro- meter, but putting it into communication with the conductor of a separate electrometer of peculiar construction. The collecting- part of the apparatus is so simple and easily managed that an amateur could, for a few shillings, set one up on his own house, if at all suitable as regards roof and windows ; and, if provided with a suitable electrometer, could make observations in atmo- spheric electricity with as much ease as thermometric or baro- metric observations. The electrometer used by Mr Dellman is of his own construction (described in Poggendorffs Annalen, 1853, Vol. LXXXIX., also Vol. LXXXV.), and it appears to be very xvi.] Atmospheric Electricity. 283 satisfactory in its operation. It is, I believe, essentially more accurate and sensitive than Peltier's, and it has a great advan- tage in affording a very easy and exact method for reducing its indications to absolute measure. I was much struck with the simplicity and excellence of Mr Bellman's whole system of observation on atmospheric electricity; and it has occurred to me that the Kew Committee might be disposed to adopt it, if determined to carry out electrical observations. When I told Mr Dellman that I intended to make a suggestion to this effect, he at once offered to have an electrometer, if desired, made under his own care. I wish als"o to suggest two other modes of observing atmospheric electricity which have occurred to me, as possessing each of them some advantages over any of the systems hitherto followed. In one of these I propose to have an uninsulated cylindrical iron funnel, about 7 inches diameter, fixed to a height of two or three yards above the highest part of the building, and a light moveable continuation (like the telescope funnel of a steamer) of a yard and a half or two yards more, which can be let down or pushed up at pleasure. Insu- lated by supports at the top of the fixed part of the funnel, I would have a metal stem carrying a ball like Dellman's, stand- ing to such a height that it can be covered by a hinged lid on the top of the moveable joint of the funnel, when the latter is pushed up ; and a fine wire fixed to the lower end of the insu- lated stem, and hanging down, in the axis of the funnel to the electrometer. When the apparatus is not in use, the moveable joint would be kept at the highest, with its lid down, and the ball uninsulated. To make an observation, the ball would be insulated, the lid turned up rapidly, and the moveable joint carrying it let down, an operation which could be effected in a few seconds by a suitable mechanism. The electrometer would immediately indicate an inductive electrification simply propor- tional to the atmospheric potential at the position occupied by the centre of the ball, and would continue to indicate at each instant the actual atmospheric potential, however variable, as long as no sensible electrification or diselectrification has taken place through imperfect insulation or convection by particles of dust or currents of air (probably for a quarter or a half of an hour, when care is taken to keep the insulation in good order). 234 Atmospheric Electricity. [xvi. This might be the best form of apparatus for making observa- tions in the presence of thunder-clouds. But I think the , best possible plan in most respects, if it turns out to be practicable, of which I can have little doubt, will be to use, instead of the ordinary fixed insulated conductor with a point, a fixed conductor of similar form, but hollow, and containing within itself an apparatus for making hydrogen, and blowing small soap-bubbles of that gas from a fine tube terminating as nearly as may be in a point, at a height of a few yards in the air. With this arrangement the insulation would only need to be good enough to make the loss of a charge by conduction very slow in comparison with convective loss by the bubbles ; so that it would be easy to secure against any sensible error from defective insulation. If 100 or 200 bubbles, each -^ inch in diameter, are blown from the top of the conductor per minute, the electrical potential in its interior will very rapidly follow variations of the atmospheric potential, and would be at any instant the same as the mean for the atmosphere during some period of a few minutes preceding. The action of a simple point is (as, I suppose, is generally admitted) essentially unsa- tisfactory, and as nearly as possible nugatory in its results. I am not aware how flame has been found to succeed, but I should think not well in the circumstances of atmospheric observations, in which it is essentially closed in a lantern; and I cannot see on any theoretical ground how its action in these circumstances can be perfect, like that of the soap-bubbles. I intend to make a trial of the practicability of blowing the bubbles; and if it proves satisfactory, there cannot be a doubt of the availability of the system for atmospheric observations." [Addition, Feb. 1857.] The author has now made various trials on the last-mentioned part of his proposal, and he has not succeeded in finding any practicable self-regulating appa- ratus for blowing bubbles and detaching them one by one from the tube. He has seen reason to doubt whether it will be possible to get bubbles so small as those proposed above, to rise at all ; but he has not been led to believe that, if it is thought worth while to try, it will be found impracticable to construct a self-acting apparatus which will regularly blow and discharge separately, bubbles of considerably larger diameter, and so to Atmospheric Electricity. 235 secure the advantages mentioned, although with a proportion- ately larger consumption of the gas. On the other hand, he finds that, by the aid of an extremely sensitive electrometer which he has recently constructed, he will be able, in all probability with great ease and at very small cost, to bring into practice the first of his two plans, con- structed on a considerably smaller scale as regards height than proposed in the preceding statement. ON SOME REMARKABLE EFFECTS OF LIGHTNING OBSERVED IN A FARM-HOUSE NEAR MONIEMAIL, CUPAR-FIFE. (From Proceedings of the Philosophical Society of Glasgow.) 301. The following is an extract from a letter, addressed last autumn to me by Mr Leitch, minister of Moniemail parish: " MONIEMAIL MANSE, CUPAB-FIFE, 26th August, 1849. "... We were visited on the llth inst. with a violent thunder-storm, which did considerable damage to a farm-house in my immediate neighbourhood. I called shortly after- wards and brought away the wires and the paper which I enclose. . . . " I have some difficulty in accounting for the appearance of the wires. You will observe that they have been partially fused, and when I got them first they adhered closely to one another. You will find that the flat sides exactly fit. They were both attached to one crank, and ran parallel to one another. The question is, how were they attracted so power- fully as to be compressed together ? . . . " You will observe that the paper is discoloured. This has been done, not by scorching, but by having some substance deposited on it. There was painted wood also discoloured, on which the stratum was much thicker. It could easily be rubbed off, when you saw the paint quite fresh beneath. . . . " The farmer showed me a probang which hung on a nail. 236 Atmospheric Electricity. [xvi. The handle only was left. The rest, consisting of a twisted cane, had entirely disappeared. By minute examination I found a small fragment, which was not burnt, but broken off." [The copper wires and the stained paper, enclosed with Mr Leitch's letter, were laid before the Society.] The remarkable effects of lightning, described by Mr Leitch, are all extremely interesting. Those with reference to the copper wires are quite out of the common class of electrical phenomena; nothing of the kind having, so far as I am aware, been observed previously, either as resulting from natural dis- charges, or in experiments on electricity. It is not improbable that they are due to the electro-magnetic attraction which must have subsisted between the two wires during the dis- charge, it being a well-known fact that adjacent wires, with currents of electricity in similar directions along them, attract one another. It may certainly be doubted whether the in- appreciably short time occupied by the electrical discharge could have been sufficient to allow the wires, after having been drawn into contact, to be pressed with sufficient force to make them adhere together, and to produce the remarkable impres- sions which they still retain. On the other hand, the electro- magnetic force must have been very considerable, since the currents in the wires were strong enough nearly to melt them, and since they appear to have been softened, if not partially fused; the flattening and remarkable impressions might readily have been produced by even a slight force subsisting after the wires came in contact. The circumstances with reference to the probang, described by Mr Leitch, afford a remarkable illustration of the well- known fact, that an electrical discharge, when effected through the substance of a non-conducting (that is to say, a powerfully resisting) solid, shatters it, without producing any considerable elevation of its temperature; not leaving marks of combustion, if it be of an ordinary combustible material such as wood. Dr Robert Thomson, at my request, kindly undertook to examine the paper removed from the wall of the farm-house, and enclosed with his letter to me by Mr Leitch ; so as, if possible, by the application of chemical tests, to discover the staining substance deposited on its surface. Mr Leitch, in his xvi.] Atmospheric Electricity. 237 letter, had suggested that it would be worth while to try whether this case is an example of the deposition of sulphur, which Fusinieri believed he had discovered in similar circum- stances. Accordingly tests for sulphur were applied, but with entirely negative results. Stains presenting a similar appear- ance had been sometimes observed on paper in the neighbour- hood of copper- wires through which powerful discharges in experiments with the hydro-electric machine had been passed ; and from this it was suggested that the staining substance might have come from the bell-wires. Tests for copper were accordingly applied, and the results were most satisfactory. The front of the paper was scraped in different places, so as to remove some of the pigment in powder; and the powders from the stained, and from the not stained parts, were repeatedly examined. The presence of copper in the former was readily made manifest by the ordinary tests : in the latter, no traces of copper could be discovered. The back of the paper presented a green tint, having been torn from a wall which has probably been painted with Scheele's green ; and matter scraped away from any part of the back was found to contain copper. Since, however, the stains in front were manifestly superficial, the discolouration being entirely removed by scraping, and since there was no appearance whatever of staining at the back of the paper, nor of any effect of the electrical discharge, it was impossible to attribute the stains to copper produced from the Scheele's green on the wall below the paper. Dr Thomson, therefore, considered the most probable explanation to be, that the stains of oxide of copper must have come from the bell-wire. To ascertain how far this explanation could be supported by the circumstances of the case, I wrote to Mr Leitch asking him for further particulars, especially with re- ference to this point, and I received the following answer : " MONIEMAIL, CUFAE-FlFE, 30th Nov. 1849. ". . . . I received your letter to-day, and immediately called at Hall-hill, in the parish of Collessie, the farm-house which had been struck by the lightning. . . . 238 Atmospheric Electricity. [xvi. " I find that Dr Thomson's suggestion is fully borne out by the facts. I at first thought that the bell-wire did not run along the line of discolouration, but I now find that such was the case. . . . [From a drawing and explanation which Mr Leitch gives, it appears that the wire runs vertically along a corner of the room, from the floor, to about a yard from the ceiling, where it branches into two, connected with two cranks near one another, and close to the ceiling.] "The efflorescence [the stains previously adverted to] was on each side of this perpendicular wire. In some places it extended more than a foot from the wire. The deposit seemed to vary in thickness according to the surface on which it was deposited. There was none on the plaster on the roof. It was thinnest upon the wall-paper, and thickest upon the wood facing of the door*. This last exhibited various colours. On the thickest part it appeared quite black ; where there was only a slight film, it was green or yellow. . . . " I may mention that the thunder-storm was that of the llth of August last. It passed over most of Scotland, and has rarely been surpassed for terrific grandeur at least beyond the tropics. It commenced about nine o'clock P.M., and in the course of an hour it seemed to die away altogether. The peals became very faint, and the intervals between the flashes and the reports very great, when all at once a terrific crashing peal was heard, which did the damage. The storm ceased with this peal. " The electricity must have been conducted along the lead on the ridge of the house, and have diverged into three streams; one down through the roof, and the two others along the roof to the chimneys. One of these appears to have struck a large stone out from the chimney, and to have been conducted down the chimney to the kitchen, where it left traces upon the floor. It had been washed over before I saw it, but still the traces were visible on the Arbroath flags. The stains were of a lighter * These remarkable facts are probably connected with the conducting powers of the different surfaces. The plaster on the roof is not so good a conductor as the wall-paper, with its pigments ; and the painted wood is probably a better conductor than either. W. T. xvii.] Sound produced by the Discharge of a Condenser. 239 tint than the stone, and the general appearance was as if a pail of some light-coloured fluid had been dashed over the floor, so as to produce various distinct streams. All along the course of the discharge, and particularly in the neighbourhood of the bell- wires, there were small holes in the wall about an inch deep, like the marks that might be made by a finger in soft plaster. " Most of the windows were shattered, and all the fragments of glass were on the outside. I suppose this must be accounted for by the expansion of the air within the house. " The window-blind of the staircase, which was down at the time, was riddled, as if with small shot. The diameter of the space so riddled was about a foot. On minute examination I found that the holes were not such as could readily be made by a pointed instrument or a pellet. They were angular, the cloth being torn along both the warp and the woof. " The house was shattered from top to bottom. Two of the serving- maids received a positive shock, but soon recovered. A strong smell of what was supposed to be sulphur was per- ceived throughout the house, but particularly in the bed-room in which the effects I described before took place." XVII. SOUND PRODUCED BY THE DISCHARGE OF A CONDENSER. [LETTEE TO PROFESSOR TAIT.] KILMICHAEL, BRODICK, ISLE or ARRAN, Oct. 10, 1863. 302. Yesterday evening, when engaged in measuring the electrostatic capacities of some specimens of insulated wire designed for submarine telegraph cables, I had occasion fre- quently to discharge, through a galvanometer coil, a condenser consisting of two parallel plates of metal, separated by a space of air about '007 inch across, and charged to a difference of potentials equal to that of about 800 Daniell's elements. I remarked at an instant of discharge a sharp sound, with a very slight prolonged resonance, which seemed to come from 240 Discharge of a Condenser. [xvn. the interior of the case containing the condenser, and which struck me as resembling a sound I had repeatedly heard before when the condenser had been overcharged and a spark passed across its air-space. But I ascertained that this sound was distinctly audible when there was no spark within the con- denser, and the whole discharge took place fairly through the 200,0 yards of fine wire, constituting the galvanometer coil. I arranged the circuit so that the place where the contact was o J- made to produce the discharge was so far from my ear that the initiating spark was inaudible ; but still I heard distinctly the same sound as before from within the condenser. 303. Using instead of the galvanometer coil either a short wire or my own body (as in taking a shock from a Leyden phial), I still heard the sound within the condenser. The shock was imperceptible except by a very faint prick on the finger in the place of the spark, and (the direct sound of the spark being barely, if at all, sensible) there was still a very audible sound, always of the same character, within the condenser, which I heard at the same instant as 1 felt the spark on my finger. Mr Macfarlane could hear it distinctly standing at a distance of several yards. We watched for light within the condenser, but could see none. I have since ascertained that suddenly charging the condenser out of one of the specimens of cable charged for the purpose produces the same sound within the condenser; also that it is produced by suddenly reversing the charge of the condenser. 304. Thus it is distinctly proved that a plate of air emits a sound on being suddenly subjected to electric force, or on expe- riencing a sudden change of electric force through it. This seems a most natural result when viewed in connexion with the new theory put forward by Faraday in his series regarding the part played by air or other dielectric in manifestations of electric force. It also tends to confirm the hypothesis I suggested to account for the remarkable observation made regarding light- ning, when you told me of it about a year ago, and other similar observations which I believe have been reported, prov- ing a sound to be heard at the instant of a flash of lightning in localities at considerable distances from any part of the line of discharge, and which by some have been supposed to de- xvii.] Measurement of the Electrostatic -Force. 241 monstrate an error in the common theory of sound. I may add that Mr Macfarlane tells me he believes he has heard, at the instant of a flash of lightning, a sound as of a heavy body striking the earth, and imagined at first that something close to him had been struck, but heard the ordinary thunder at a sensible time later. XVIII. MEASUREMENT OF THE ELECTROSTATIC FORCE PRODUCED BY A DANIELL'S BATTERY. [Proceedings Royal Society, Feb. 23 and April 12, 1860, or Phil. Mag. 1860, second half-3'ear.] 305. In a paper "On Transient Electric Currents," published in the Philosophical Magazine for June 1853, [Mathematical and Physical Papers, Art. LXII.] I described a method for measuring differences of electric potential in absolute electro- static units, which seemed to me the best adapted for obtaining accurate results. The "absolute electrometer" which I ex- hibited to the British Association on the occasion of its meeting at Glasgow in 1855, was constructed for the purpose of putting this method into practice, and, as I then explained, was adapted to reduce the indications of an electroscopic* or of a torsion electrometer to absolute measure. 306. The want of sufficiently constant and accurate instru- ments of the latter class has long delayed my carrying out of the plans then set forth. Efforts which I have made to produce electrometers to fulfil certain conditions of sensibility, con- venience, and constancy, for various objects, especially the electrostatic measurement of galvanic forces, and of the differ- ences of potential required to produce sparks in air, under definite conditions, and the observation of natural atmospheric electricity, have enabled me now to make a beginning of abso- lute determinations, which I hope to be able to carry out soon in a much more accurate manner. In the meantime, I shall give a slight description of the chief instruments and processes * I have used the expression "electroscopic electrometer," to designate an electrometer of which the indications are merely read off in each instance by a single observation, without the necessity of applying any experimental process of weighing, or of balancing by torsion, or of otherwise modifying the conditions exhibited. T. E. 16 242 Measurement of the Electrostatic Force [xvm. followed, and state the approximate results already obtained, as these may be made the foundation of various important estimates in several departments of electrical science, 307. The absolute electrometer alluded to above (compare 358, below), consists of a plane metallic disc, insulated in a horizontal position, with a somewhat smaller plane metallic disc hung centrally over it, from one end of the beam of a balance. A metal case protects the suspended disc from currents of air, and from irregular electric influences, allowing a light vertical rod, rigidly connected with the disc at its lower end, and sus- pended from the balance above, to move up and down freely, through an aperture just wide enough not to touch it. In the side of the case there is another aperture, through which pro- jects an electrode rigidly connected with the lower insulated disc. The, upper disc is kept in metallic communication with the case. 308. In using this instrument to reduce the indications of an electroscopic or torsion electrometer to absolute electrostatic measure, the insulated part of the electrometer is kept in metallic communication with the insulated disc, while the cases enclosing the two instruments are also kept in metallic communication with one another. A charge, either positive or negative, is communicated to the insulated part of the double apparatus. The indication of the tested electrometer is read off, and at the same time the force required to keep the move- able disc at a stated distance from the fixed disc below it, is weighed by the balance. This part of the operation is, as I anticipated, somewhat troublesome, in consequence of the in- stability of the equilibrium, but with a little care it may be managed with considerable accuracy. The plan which I have hitherto followed, has been to limit the play of the arm of the balance to a very small arc, by means of firm stops suitably placed, thus allowing a range of motion to the upper disc through but a small part of its whole distance from the lower. A certain weight is put into the opposite scale of the balance, and the indications of the second electrometer are observed when the electric force is just sufficient to draw down the upper disc from resting in its upper position, and again when insufficient to keep it down with the beam pressed on its lower stop. This operation is repeated at different distances, xviii.] produced l>y a Daniell's Battery. 243 and thus no considerable error depending on a want of parallel- ism between the discs could remain imdetected. It may be remarked that the upper disc is carefully balanced by means of small weights attached to it, so as to make it hang as nearly as possible parallel to the lower disc. The stem carrying it is graduated to hundredths of an inch ('254 of a millimetre) ; and by watching it through a telescope at a short distance, it is easy to observe ^ of a millimetre of its vertical motion. 309. I have recently applied this method to reduce to ab- solute electrostatic measure the indications of an electrometer forming part of a portable apparatus for the observation of atmospheric electricity. In this instrument (compare 263) a very light bar of aluminium attached at right angles to the middle of a fine platinum wire, which is firmly stretched be- tween the inside coatings of two Leyden phials, one occupying an inverted position above the other, experiences and indicates the electrical force which is the subject of measurement, and which consists of repulsions in contrary directions on its two ends, produced by two short bars of metal fixed on the two sides of the top of a metal tube, supported by the inside coat- ing of the lower phial. 310. The amount of the electrical force (or rather, as it should be called in correct mechanical language, couple) is measured by the angle through which the upper Leyden phial must be turned round an axis coincident with the line of the wire, so as to bring the index to a marked position. An independently insulated metal case, bearing an electrode projecting outwards, to which the body to be tested is applied, surrounds the index and repelling bars, but leaves free apertures above and below, for the wire to pass through it without touching it; and by other apertures in its sides and top, it allows the motions of the index to be observed, and the Leyden phials to be charged or discharged at pleasure, by means of an electrode applied to one of the fixed bars described above. When by means of such an electrode the inside coatings of the Leyden phials are kept connected with the earth, this electrometer becomes a plain repulsion electrometer, on the same principle as Peltier's, with the exception that the index, supported by a platinum wire instead of on a pivot, is directed by elasticity of torsion instead 162 244 Measurement of the Electrostatic Force [xvni. of by magnetism ; and the electrical effect to be measured is produced by applying the electrified body to a conductor con- nected with a fixed metal case round the index and repelling bars, instead of with these conductors themselves. 311. This electrometer, being of suitable sensibility for direct comparison with the absolute electrometer according to the process described above, is not sufficiently sensitive to measure directly the electrostatic effect of any galvanic battery of fewer than two hundred cells with much accuracy. Not having at the time arrangements for working with a multiple battery of reliable character, I used a second torsion electrometer of a higher degree of sensibility as a medium for comparison, and determined the value of its indications by direct reference to a Daniell's battery of from six to twelve elements in good work- ing order. This electrometer, in which a light aluminium index, suspended by means of a fine glass fibre, kept constantly electrified by means of a light platinum wire hanging down from it and dipping into some sulphuric acid in the bottom of a charged Ley den jar, exhibits the effects of electric force due to a difference of potentials between two halves of a metallic ring separately insulated in its neighbourhood, will be suffici- ently described in another communication to the Royal Society. Slight descriptions of trial instruments of this kind have already been published in the Transactions of the Pontifical Academy of Rome*, and in the second edition of Nichol's Cyclopaedia (article Electricity, Atmospheric), 1860 ( 249, 266, above). 312. I hope soon to have another electrometer on the same general principle, but modified from those hitherto made, so as to be more convenient for accurate measurement in terms of constant units. In the meantime, I find that, by exercising sufficient care, I can obtain good measurements by means of the divided ring electrometer of the form described in Nichol's Cyclopaedia ( 263, above). 313. In the ordinary use of the portable electrometer, a con- siderable charge is communicated to the connected inside coat- ings of the Ley den phials, and the aluminium index is brought to an accurately marked position by torsion, while the insulated * Accademia Pontificia dei Nuovi Lyncei, February 1857, xvm.] produced by a DanieWs Battery. 245 metal case surrounding it is kept connected with the earth. The square root of the reading of the torsion-head thus ob- tained measures the potential, to which the inside coatings of the phials have been electrified. If, now, the metal case referred to is disconnected from the earth and put in con- nexion with a conductor whose potential is to be tested, the square root of the altered reading of the torsion-head required to bring the index to its marked position in the new circum- stances measures similarly the difference between this last potential and that of the inside coatings of the phials. Hence the excess of the latter square root above the former expresses in degree and in quality (positive or negative) the required potential. This plan has not only the merit of indicating the quality of the electricity to be tested, which is of great import- ance in atmospheric observation, but it also affords a much higher degree of sensibility than the instrument has when used as a plain repulsion electrometer ; and, on account of this last- mentioned advantage, it was adopted in the comparisons with the divided ring electrometer. On the other hand, the^"portable electrometer was used in its least sensitive state, that is to say, with its Leyden phials connected with the earth, when the comparisons with the absolute electrometer were made. 314. The general result of the weighings hitherto made, is that when the discs of the absolute electrometer were at a dis- tance of '5080 of a centimetre; the number of degrees of torsion in the portable electrometer was '20924 times the number of grammes' weight required to balance the attractive force ; and the number of degrees of torsion was "4983 times the number of grammes' weight found in other series of experiments in which the distance between the discs was 762 of a centimetre. According to the law of inverse squares of the distances to which the attraction between two parallel discs is subject when a constant difference of potentials is maintained between them*, the force at a distance of '254 of a centimetre would have been _.^_ j according to the first of the preceding results, or, accord- ing to the second, y^g f ^ e num ^er of degrees of torsion. The mean of these is -j^-, or '0777 ; and we may consider this * See 11 of Elements of Mathematical Theory of Electricity appended to the communication following this in the "Proceedings." 246 Measurement of the Electrostatic Force [xvm. number as representing approximately the value in grammes' weight at '254 of a centimetre distance between the discs of the absolute electrometer, corresponding to one degree of torsion of the portable electrometer. By comparing the indications of the portable electrometer with those of the divided ring electro- meter, and by evaluating those of the latter in terms of the electromotive force of a Daniell's battery charged in the usual manner, I find that 284 times the square root of the number of degrees of torsion in the portable electrometer is approxi- mately the number of cells of a Daniell's battery which would produce an electromotive force (or, which is the same thing, a difference of potentials) equal to that indicated. Hence the attraction between the discs of the portable electrometer, if at 254 of a centimetre distance, and maintained at a difference of potentials amounting to that produced by 284 cells, is "0777 of a gramme. The effect of 1000 cells would therefore be to give a force of *965 of a gramme, since the force of attraction is propor- tional to the square of the difference of potentials between the discs. The diameter of the opposed circular areas between which the attraction observed took place, was 14 '88 centi- metres. Its area was therefore 174*0 square centimetres, and therefore the amount of attraction per square decimetre, accord- ing to the preceding estimate for '254 of a centimetre distance and 1000 cells' difference of potential, is *554 of a gramme. Hence, with an electromotive force or difference of potentials produced by 1000 cells of DanielFs battery, the force of attrac- tion would be 3'57 grammes weight per square decimetre between discs separated to a distance of 1 millimetre. [The force in grammes weight is equal to '000357 x n 2 , if the area of each of the opposed surfaces is equal to a square whose side is n times the distance between them, provided n be a large number.] 315. This result differs very much from an estimate I have made according to Weber's comparison of electrostatic with elec- tro-magnetic units and my theoretical estimate of 2,500,000 British electro-magnetic units for the electromotive force of a single element of Daniell's. On the other hand, it agrees to a remarkable degree of accuracy with direct observations made for me, during my absence in Germany, by Mr Macfarlane, in xviii.] produced by a Daniell's Battery. 247 the months of June and July 1856, on the force of attraction produced by the direct application of a miniature Daniell's battery, of different numbers of elements, from 93 to 451, applied to the same absolute electrometer with its discs at 2006 of a centimetre asunder. These observations gave forces varying, on the whole, very closely according to the square of the number of cells used ; and the mean result re- duced according to this law to 1000 cells was 1*516 grammes. Reducing this to the distance of 1 millimetre, and dividing by 1'74, the area in square decimetres, we find 3'51 grammes per square decimetre at a distance of 1 millimetre. 316. Although the experiments leading to this result were executed with great care by Mr Macfarlane, I delayed publish- ing it because of the great discrepance it presented from the estimate which I deduced from Weber's measurement, pub- lished while my preparations were in progress. I cannot doubt its general correctness now, when it is so decidedly con- firmed by the electrometric experiments I have just described, which have been executed chiefly by Mr John Smith and Mr John Ferguson, working in my laboratory with much ability since the month of November. I am still unable to explain the discrepance, but it may possibly be owing to some miscalculation I have made in my deductions from Weber's result. GLASGOW COLLEGE, Jan. 18, 1860. [Addition, April 1870. From experiments of the present date, performed by Mr William Leitch and Mr Dugald M'Kichan, with the new Absolute Electrometer ( 364, below), it is deduced that with the difference of potentials produced by 1000 Daniell's cells in series, the force of attraction would be 57 grammes per square decimetre between discs separated to a distance of 1 millimetre, instead of 3'57 grammes as found in 314. This new measurement, with Maxwell's correction of Weber's number, which diminishes it by about 8 per cent. (Report of British Association for 1869, page 438 : Committee on Electrical Standards), seems to reduce to as nearly as may be nothing, the discrepance from my thermo-dynamic estimate of December 1851 (Philosophical Magazine) referred to in 318, 248 Measurement of the Electrostatic Force [XVIII. below. Calculating from it by 339, we find 374 for the dif- ference of potentials, or electromotive force in c. g. s. absolute electrostatic measure, produced by 1000 elements of Daniell's.] POSTSCRIPT, April 12, 1860. 317. I have since found that I had inadvertently misinter- preted Weber's statement in the ratio of 2 to 1. I had always, as it appears to me most natural to do, regarded the transference of negative electricity in one direction, and of positive elec- tricity in the other direction, as identical agencies, to which, in our ignorance as to the real nature of electricity, we may apply indiscriminately the one expression or the other, or a combina- tion of the two. Hence I have always regarded a current of unit strength as a current in which the positive or vitreous electricity flows in one direction at the rate of a unit of elec- tricity per unit of time ; or the negative or resinous electricity in the other direction at the same rate; or (according to the infinitely improbable hypothesis of two electric fluids) the vitreous electricity flows in one direction at any rate less than a unit per second, and the resinous in the opposite direction at a rate equal to the remainder of the unit per second. I have only recently remarked that Weber's expressions are not only adapted to the hypothesis of two electric fluids, but that they also reckon as a current of unit strength, what I should have called a current of strength 2, namely, a flow of vitreous electricity in one direction at the rate of a unit of vitreous electricity per unit of time, and of the resinous electricity in the other direction simultaneously, at the rate of a unit of resinous electricity per unit of time. 318. Weber's result as to the relation between electrostatic and electro-magnetic units, when correctly interpreted, I now find would be in perfect accordance with my own results given above, if the electromotive force of a single element of the Daniell's battery used were 2,140,000 British electro-magnetic units instead of 2,500,000, as according to my thermo-dynamic estimate. This is as good an agreement as could be ex- pected when the difficulties of the investigations, and the uncertainty which still exists as to the true measure of the xviii.] produced by a Daniell's Battery. 249 electromotive force of the Daniell's element are considered. It must indeed be remarked that the electromotive force of Daniell's battery varies by two or three or more per cent, with variations of the solutions used ; that it varies also very sensibly with temperature ; and that it seems also to be dependent, to some extent, on circumstances not hitherto elucidated. A thorough examination of the electromotive force of Daniell's and other forms of galvanic battery, is an object of high im- portance, which, it is to be hoped, will soon be attained. Until this has been done, at least for Daniell's battery, the results of the preceding paper may be regarded as having about as much accuracy as is desirable. 319. I may state, therefore, in conclusion, that the average electromotive force per cell of the Daniell's batteries which I have used, produces a difference of potentials amounting to 00296 [corrected to '00374, April 1870,] in [c. g. s.] absolute electrostatic measure. This statement is perfectly equivalent to the following in more familiar terms : One thousand cells of Daniell's battery, with its two poles connected by wires with two parallel plates of metal 1 millimetre apart, and each a square decimetre in area, produces an elec- trical attraction equal to the weight of 3*57 [corrected to 5*7] grammes. XIX. MEASUREMENT OF THE ELECTROMOTIVE FORCE REQUIRED TO PRODUCE A SPARK IN AIR BETWEEN PARALLEL METAL PLATES AT DIFFERENT DISTANCES. [Proceedings Eoyal Society, Feb. 23 and April 12, 1860, or Phil Nag., 1860, second half-year.] 320. THE electrometers used in this investigation were the absolute electrometer and the portable electrometer described in my last communication to the Royal Society, and the opera- tions, were executed by the same gentlemen, Mr Smith and Mr Ferguson. The conductors between which the sparks passed were two unvarnished plates of a condenser, of which one was moved by a micrometer screw, giving a motion of gig of an inch (about one millimetre) per turn, and having its head divided into 40 equal parts of circumference. The readings on the screw-head could be readily taken to tenth parts of a division, that is to say, to about ^^ of a millimetre on the distance to be measured. The point from which the spark would pass in successive trials being somewhat vari- able, and often near the edges of the discs, a thin flat piece of metal, made very slightly convex on its upper surface like an extremely flat watch-glass, was laid on the lower plate. It was then found that the spark always passed between the crown of this convex piece of metal and the flat upper plate. The curvature of the former was so small, that the physical circumstances of its own electrification near its crown, the opposite electrification of the opposed flat surface in the parts near the crown of the convex, and the electric pressure on or tension in the air between them could not, it was supposed, differ sensibly from those between two plane conducting surfaces at the same distance and maintained at the same difference of potentials. XIX.] Measurement of Electromotive Force. 251 321. The reading of the screw-head corresponding to the position of the moveable disc when touching the metal below, was always determined electrically by making a succession of sparks pass, and approaching the moveable disc gradually by the screw until all appearance of sparks ceased. Contact was thus pro- duced without any force of pressure between the two bodies capable of sensibly distorting their supports. With these arrangements several series of experiments were made, in which the differences of potentials producing sparks across different thicknesses of air were measured first by the absolute electrometer, and afterwards by the portable torsion electrometer. The following Tables exhibit the results hither- to obtained : 322. TABLE I. December 13, 1859. Measurements by absolute electrometer of maximum electrostatic forces* across a stra- tum of air of different thicknesses. Area of each plate of absolute electrometer = 174 square centimetres. Distance between plates of absolute electrometer= '508 of a centimetre. Length of spark in inches. Weight in grains required to balance in absolute elec- trometer. Electromotive force in units of the electrometer. Electrostatic force, or electromotive force per inch of air, in temporary units. S. W. Vw. ^/ w ^ s 007 6 2-4495 349-9 0105 9 3-0000 285-7 0115 10 3-1622 275-0 014 13 3-6055 257-5 017 16 4-0000 235-3 018 19 4-3589 242-2 024 30 5-4772 228-2 0295 40 6-3245 214-4 034 50 7-0710 208-0 0385 60 7-7459 201-2 041 70 8-3666 204-1 0445 80 8-9442 201-0 048 90 9-4868 197-6 052 100 10-0000 192-3 055 110 10-4880 190-7 058 120 10-9544 188-9 060 130 11-4017 190-0 323. These numbers demonstrate an unexpected and a very remarkable result, that greater electromotive force per unit See 331 below. 252 Measurement of Electromotive Force [xix. length of air is required to produce a spark at short distances than at long. When it is considered that the absolute electri- fication of each of the opposed surfaces* depends simply on the electromotive force per unit length of the space between them, or, which is the same thing, the resultant electrostatic force in the air occupying that space, it is difficult even to con- jecture an explanation. Without attempting to explain it, we are forced to recognise the fact that a thin stratum of air is stronger than a thick one against the same disruptive tension in the air, according to Faraday's view of its condition as tran- mitting electric force, or against the same lifting electric pres- sure from its bounding surfaces, according to the views of the eighteenth century school, as represented by Poisson. The same conclusion is established by a series of experiments with the previously-described portable torsion electrometer substi- tuted for the absolute electrometer, leading to results shown in the following Table : 324. TABLE II. January 17, 1860. Measurements by portable torsion electrometer of electromotive forces producing sparks across a stratum of air of different thicknesses. Length of spark in inches. S. Torsion in degrees required to balance in electrometer. 9. Electromotive force in units of the electrometer. V*. Electrostatic force, or electromotive force per inch of air, in temporary units. V0-5-*. 001 3 1-732 1732 002 7 2-646 1323 003 11 3-316 1105 004 14 3-742 935 005 18 4-243 849 006 22 4-690 782 007 27 5-196 742 008 30 5-477 685 009 33 5-744 638 010 38 6-164 616 Oil 43 6-557 596 012 48-5 6-964 580 013 54 7-348 565 014 59 7-681 549 015 66 8-124 542 016 73 8-544 534 017 79 8-888 523 018 85 9-219 512 * See 332 below. XIX.] required to produce a Spark. 253 325. The series of experiments here tabulated stops at the distance 18 thousandths of an inch, because it was found that the force in the electrometer corresponding to longer sparks than that, was too strong to be measured with certainty by the port- able electrometer, whether from the elasticity of the platinum wire, or from the rigidity of its connexion with the aluminium index being liable to fail when more than 85 or 90 of torsion were applied. So far as it goes, it agrees remarkably well with the other experiments exhibited in Table I., as is shown by the following comparative Table, in which, along with results of actual observation extracted from Table II., are placed results deduced from Table I. by interpolation for the same lengths of spark : TABLE III. ^Experiments of December 13, 1859, and January 17, 1860, compared. Col. 1. Col. 2. Col. 3. Col. 4. Electromotive force Electromotive force Length of spark in inches. per inch of air, Dec. 13, in temporary units of that day. per inch of air, Jan. 17, in temporary units of that day. Ratios of numbers in Col. 3 to num- bers in Col. 2. S. \/W V0 S S 007 349-3 742 2-13 0105 285-7 606 2-12 0115 275-0 588 2-14 014 257-5 549 2-14 017 235-3 523 2-22 018 242-2 512 2-11 Mean 2-14 The close agreement with one another of the numbers in Col. 4, derived from series differing so much as those in Cols. 2 and 3, and obtained by means of electrometers differing so much in construction, constitutes a very thorough confirmation of the remarkable result inferred above from the experiments of the first series, and shows that the law of variation of the electrostatic force in the air required to produce sparks of the different lengths, must be represented with some degree of accuracy by the numbers shown in the last column of either Table I. or Table III. The following additional series of experiments were made on precisely the same plan as those of Table II. : 254 Measurement of Electromotive Force [xix. TABLE IV. January 21, 1860. Measurements by portable torsion electrometer of electromotive forces producing sparks across a stratum of air of different thicknesses. Torsion in degrees Electromotive force Electrostatic force, or Length of spark in inches. required to balance in electrometer. in units of the electrometer. electromotive force per inch of air, in S. e. temporary units. 001 002 3-2 6-4 1-79 2-32 1790 1160 003 10-5 3-24 1080 004 13-2 3-63 907 005 14-2 3-77 754 006 18-2 4-27 712 007 21-7 4-66 666 012 41-2 6-42 535 013 46-7 6-83 525 014 53-2 7-29 521 015 57-2 7-56 504 016 63-2 7-95 497 017 68-2 8-26 486 018 78-2 8-84 491 TABLE V. January 23, 1860. Similar experiments repeated. s. * \/e. V-. 001 3-5 1-87 1870 002 6-5 2-55 1275 003 9-5 3-08 1027 004 12-7 3-56 890 005 15-5 3'94 788 006 18-5 4-30 716 007 23-0 4-80 686 008 25 62 5-06 632 009 30-5 5-52 613 010 35-0 5-92 592 on 39-5 6-28 571 012 44-0 6-63 553 013 50-0 7-07 544 014 54-0 7'35 525 015 59-0 7-68 512 016 63-5 7-97 498 017 69-5 8-34 490 018 74-5 8-63 479 The difference between the numbers shown in these two Tables and in Table II. above, are probably due in part to true differences in the resistance of the air to electrical disruption ; but variations in the electrometer, which was by no means of perfect construction, may have sensibly influenced the results, XIX.] required to produce a Spark. 255 especially as regards the differences between those shown in Table II. and those shown in Tables IV. and V., which, agree- ing on the whole closely with one another, fall considerably short of the former. 326. TABLE VI. Summary of results reduced to absolute measure. Col. 1. Length of spark in centimetres. Col. 2. Electrostatic forces according to simple determinations of Dec. 13, 1859. V M> X . 508 /981'4x87T* Col. 3. Electrostatic forces accord- ing to esti- mated average of various de- terminations. Col. 4. Differences. Col. 5. Pressures of electricity from either metallic surface balanced by air immediately before disruption, in grammes weight per square centimetref. S. S 1/4 -p j\ = R. XI. 8?r x 981-4 ' 00254 527-7 11-290 00508 .,. 367-8 5-484 00762 ..< 314-4 ... 4-007 01016 267-6 ... 2-903 01270 t 234-0 ... 2-220 01524 216-1 1-893 01778 211-4 208-2 + 3-2 1-757 02032 193-1 1-512 02286 183-4 ... 1-364 02540 177-5 1-277 02667 172-8 173-3 -0-5 1-217 02794 ... 171-0 1-185 02921 166-4 166-9 -0-5 1-129 03048 ... 163-2 ... 1-080 03302 159-4 ... 1-030 03556 155-8 155-8 '"0 984 03810 ... 152-6 ... 944 04064 ... 149-9 911 04318 142-5 144-4 -1-9 845 04572 146-7 145-7 + 1-0 860 06096 142-5 ... 823 07493 129-6 681 08636 126-0 >g 644 09779 121-8 601 10414 123-7 tf 620 11303 121-8 ... 601 12192 119-5 579 13208 116-3 548 13970 115-4 540 14732 114-5 it 531 15240 114-9 535 * Distance between discs of absolute electrometer = '508 of a centimetre. Area of each = 174 square centimetres. Force of gravity at Glasgow on unit mass = 981*4 dynamical units of force; that is to say, generates in one second a velocity of 981-4 centimetres per second. t This is most directly obtained by finding the force between the discs of the absolute electrometer per square centimetre, and reducing, according to the inverse proportion of squares of distances, to what it would have been if the distance between them had been equal to the length of the spark. 256 Measurement of Electromotive Force [xix. APPENDIX ( 327-338). 327. In order that the different expressions, " potential," " electromotive force," " electric force," or " electrostatic force," " pressure of electricity from a metallic surface balanced by air," used in the preceding statement, may be perfectly understood, I add the following explanations and definitions belonging to the ordinary elements of the mathematical theory of electricity: 328. Measurement of quantities of electricity. The unit quan- tity of electricity is such a quantity, that, if collected in a point, it will repel an equal quantity collected in a point at a unit distance with a force equal to unity. 329. [In absolute measurements the unit distance is one centimetre; and the unit force is that force which, acting on a gramme of matter during a second of time, generates a velocity of one centimetre per second. The weight of a gramme at Glasgow is 981 '4 of these units of force. The weight of a gramme in any part of the earth's surface may be estimated with about as much accuracy as it can be without a special experiment to determine it for the particular locality, by the following expression : In latitude X, average weight of a gramme = 978-024 x (1 + -005133 x sin 2 X) absolute kinetic units.] 330. Electric density. This term was introduced by Coulomb to designate the quantity of electricity per unit of area in any part of the surface of a conductor. He showed how to measure it, though not in absolute measure, by his proof plane. 331. Resultant electric force at any point in an insulating fluid [compare 65, above]. The resultant force at any point in air or other insulating fluid in the neighbourhood of an electrified body, is the force which a unit of electricity concentrated at that point would experience if it exercised no influence on the electric distributions in its neighbourhood. 332. Relation between electric density on the surface of a con- ductor, and electric force at points in the air close to it. Accord- ing to a proposition of Coulomb's, requiring, however, correction, and first correctly given by Laplace, the resultant force at any point in the air close to the surface of a conductor is perpendi- XEX.] required to produce a Spark. 257 cular to the surface and equal to 47T/9, if p denotes the electric density of the surface in the neighbourhood ( 87, Cor.). 333. Electric pressure from the surface of a conductor balanced by air. A thin metallic shell or liquid film, as for instance a soap-bubble, if electrified, experiences a real mechanical force in a direction perpendicular to the surface outwards, equal in amount per unit of area to 27rp z , p denoting, as before, the electric density at the part of the surface considered ( 88). This force may be called either a repulsion (as according to the views of the eighteenth century school) or an attraction effected by tension of air between the surface of the conductor and the conducting boundary of the air in which it is insu- lated, as it would probably be considered to be by Faraday ; but whatever may be the explanation of the modus operandi by which it is produced, it is a real mechanical force, and may be reckoned as in Col. 5 of the preceding Table, in grammes weight per square centimetre. In the case of the soap-bubble, for instance ; its effect will be to cause a slight enlargement of the bubble on electrification with either vitreous or resinous elec- tricity, and a corresponding collapse on being perfectly dis- charged. In every case we may regard it as constituting a deduction from the amount of air-pressure which the body experiences when unelectrified. The amount of this deduction being different in different parts according to the square of the electric density, its resultant action on the whole body disturbs its equilibrium, and constitutes in fact the resultant of the electric force experienced by the body. 334. Collected formulce of relation between electric density on the surface of a conductor, electric diminution of air-pressure upon it, and resultant force in the air close to the surface. Let, as before, p denote the first of these three elements, let p denote the second reckoned in units of force per unit of area, and let R denote the third. Then we have B = 4>7rp, 335. Electric potential [difference of potentials being what, after German usage, is still sometimes called "electromotive force." (Addition, April 1870.)] The amount of work required T. E. 17 258 Measurement of Electromotive Force [xix. to move a unit of electricity against electric repulsion from any one position to any other position, is equal to the excess of the electric potential of the second position above the electric potential of the first position. Cor. 1. The electric potential at all points close to the surface of an electrified metallic body has one value, since an electri- fied point, possessing so small a quantity of electricity as not sensibly to influence the electrification of the metallic surface, would, if held near the surface in any locality, experience a force perpendicular to the surface in its neighbourhood. Cor. 2. The electric potential throughout the interior of a hollow metallic body, electrified in any way by external influ- ence, or, if insulated, electrified either by influence or by com- munication of electricity to it, is constant, since there is no electric force in the interior in such circumstances. [It is easily shown by mathematical investigation, that the electric force experienced by an electric point containing an infinitely small quantity of electricity, when placed anywhere in the neighbourhood of a hollow electrified metallic shell, gradually diminishes to nothing if the electric point be moved gradually from the exterior through a small aperture in the shell into the interior. Hence the one value of the potential close to the surface outside, mentioned in Cor. 1, is equal to the constant value throughout the interior mentioned in Cor. 2.] 336. Interpretation of measurement by electrometer. Every kind of electrometer consists of a cage or case containing a move- able and a fixed conductor, of which one at least is insulated and put in metallic communication, by what I shall call the prin- cipal electrode passing through an aperture in the case or cage, with the conductor whose electricity is to be tested. In every properly constructed electrometer, the electric force experi- enced by the moveable part in a given position cannot be electrically influenced except by changing the difference of potentials between the principal electrode and the uninsulated conductor or conducting system in the electrometer. Even the best of ordinary electrometers hitherto constructed do not fulfil this condition, as the inner surface of the glass of which the whole or part of the enclosing case is generally made, is liable to become electrified, and inevitably does become so xix.] required to produce a Spark. 259 when any very high electrification is designedly or acciden- tally introduced, even for a very short time ; the consequence of which is that the moving body will generally not return to its zero position when the principal electrode is perfectly dis- insulated. Faraday long ago showed how to obviate this radi- cal defect by coating the interior of the glass case with a fine network of tinfoil ; and it seems strange that even at the pre- sent day electrometers for scientific research, as, for instance, for the investigation of atmospheric electricity, should be con- structed with so bad and obvious a defect uncured by so simple and perfect a remedy. When it is desired to leave the interior of the electrometer as much light as possible, and to allow it to be clearly seen from any external position with as little embarrassment as possible, a cage made like a bird's cage, with an extremely fine wire on a metal frame, inside the glass shade used to protect the instrument from currents of air, etc., may be substituted with advantage for the tinfoil network lining of the glass. It appears, therefore, that a properly constructed electrometer is an instrument for measuring, by means of the motions of a moveable conductor, the difference of potentials of two conducting systems insulated from one another, of one of which the case or cage of the apparatus forms part. It may be remarked in passing, that it is sometimes convenient in special researches to insulate the case or cage of the apparatus, and allow it to acquire a potential differing from that of the earth, and that then, as always, the subject of measurement is the difference of potentials between the principal electrode and the case or cage, while in the ordinary use of the instrument the potential of the latter is the same as that of the earth. Hence we may regard the electrometer merely as an instrument for measuring differences of potential between two conducting systems mutually insulated; and the object to be aimed at in perfecting any kind of electrometer (more or less sensitive as it may be, according to the subjects of investigation for which it is to be used), is, that accurate evaluations in absolute measure, of differences of potential, may be immediately derivable from its indications. 337. Relation between electrostatic force and variation of electric potential. 335, otherwise stated, is equivalent to this : The 172 260 Measurement of Electromotive Force [xix. average component electrostatic force in the straight line of air between two points in the neighbourhood of an electrified body is equal to their difference of potentials divided by their distance. In other words, the rate of variation of electric potential per unit of length in any direction is equal to the component of the electrostatic force in that direction. Since the average electrostatic force in the line joining two points at which the values of the potential are equal is nothing, the direction of the resultant electrostatic force at any point must be perpendicular to the equipotential surface passing through that point; or the lines of force (which are generally curves) cut the series of equipotential surfaces at right angles. The rate of variation of potential per unit of length along a line of force is therefore equal to the electrostatic force at any point. 338. Stratum of air between two parallel or nearly parallel plane or curved metallic surfaces maintained at different poten- tials-. Let a denote the distance between the metallic surfaces on each side of the stratum of air at any part, and V the differ- ence of potentials. It is easily shown that the resultant elec- trostatic force is sensibly constant through the whole distance, from the one surface to the other; and being in a direction V sensibly perpendicular to each, it must ( 337) be equal to . a Hence ( 332) the electric density on each of the opposed sur- y faces is equal to 7 . This is Green's theory of the Leyden phial. 339. Absolute Electrometer. As a particular case of 338, let the discs be plane and parallel : and let the distance be- tween them be small in comparison with their diameters, or with the distance of any part of either from any conductor differing from it in potential. The electric density will be uniform over the whole of each of the opposed surfaces and F equal to 7 , being positive on one and negative on the other; and in all other parts of the surface of each the electrification will be comparatively insensible. Hence the force of attraction F 2 between them per unit of area ( 333 and 334) will be - 5 ; XIX.] required to produce a Spark. 261 if A denote the area of either of the opposed surfaces, the F 2 whole force of attraction between them is therefore A - . STTO? Hence, if the observed force be equal to the weight of w grammes at Glasgow, we have F 2 981-4 xw = and therefore mi". -v- x 8-7T x w ADDITION, DATED APEIL 12, 1860. 340. Experiments on precisely the same plan as those of Table I. December 13, have been repeated by the same two ex- perimenters, with different distances from '75 to 1'5 of a centi- metre between the plates of the absolute electrometer, and results have been obtained confirming the general character of those shown in the preceding Tables. The absolute evaluations derived from these later series must be more accurate than those deduced above from the single series of December 13, when the distance between the plates in the absolute electrometer was only *5 of a centimetre. I therefore, by permission, add the following Table of absolute determinations : Length of spark in centimetres. Electrostatic forces according to estimated average of deter- minations of February 15, 23, 28, and 29, and March 2. s. R. 0086 267-1 0127 257-0 0127 262-2 0190 224-2 0281 200-6 0408 151-5 0563 144-1 0584 139-6 0688 140-8 0904 134-9 1056 132-1 1325 131-0 These results, as well as those shown in the preceding Tables, demonstrate a much less rapid variation with distance, of the 262 Measurement of Electromotive Force. [xix. electrostatic force preceding a spark, at the greater than at the smaller distances. It seems most probable that at still greater distances the electrostatic force will be found to be sensibly constant, as it was certainly expected to be at all distances. The limiting value to which the results shown in the last Table seem to point must be something not much less than 130. This corresponds to a pressure of 68 grammes weight per square decimetre. We may therefore conclude that the ordi- nary atmospheric pressure of 103,200 grammes per square deci- metre, is electrically relieved by the subtraction of not more than 68, on two very slightly convex metallic surfaces, before the air between them is cracked and a spark passes, provided the distance between them is not less than ^ of a centimetre. By taking into account the result of my preceding communica- tion to the Royal Society, we may also conclude that a Daniell's battery of 5510 elements can produce a spark between two slightly convex metallic surfaces at J of a centimetre asunder in ordinary atmospheric air. XX. ELECTEOMETERS AND ELECTROSTATIC MEASUREMENTS. [ 340' from British Association Eeport of Glasgow 1855 Meeting, 341 389 from Eeport of Dundee 1867 Meeting, being part of Report of Committee on Standards of Electrical Resistance."] 340'. IN this communication three instruments were de- scribed and exhibited to the Section : the first a standard electrometer, designed to measure, by a process of weighing the mutual attraction of two conducting discs, the difference of electrical potential between two bodies with which they are connected, an instrument which will be useful for determining the electromotive force of a galvanic battery in electrostatic measure, and for graduating electroscopic instruments so as to convert their scale indications into absolute measure; the second an electroscopic electrometer, which may be used for indicating electrical potentials in absolute measure, in ordinary experiments, and, probably with great advantage, in obser- vations of atmospheric electricity ; and the third, for which a scientific friend has suggested the name of Electroplatymeter, an instrument which may be applied either to measure the capacities of conducting surfaces for holding charges of elec- tricity, or to determine the electric inductive capacities of insu- lating media. 341. An electrometer is an instrument for measuring differ- ences of electric potential between two conductors through effects of electrostatic force, and is distinguished from the gal- vanometer, which, of whatever species, measures differences of electric potentials through electromagnetic effects of electric currents produced by them. When an electrometer merely indicates the existence of electric potential, without measuring its amount, it is commonly called an electroscope ; but the name electrometer is properly applied when greater or less degrees of difference are indicated on any scale of reckoning, 264 On Electrometers and Electrostatic Measurements, [xx. if approximately constant, even during a single series of experi- ments. The first step towards accurate electrometry in every case is to deduce from the scale-readings, numbers which shall be in simple proportion to the difference of potentials to be determined. The next and last step is to assign the corre- sponding values in absolute electrostatic measure. Thus, when for any electrometer the first step has been taken, it remains only to determine the single constant coefficient by which the numbers, deduced from its indications as simply proportional to differences of potential, must be multiplied to give differ- ences of potential in absolute electrostatic measure. This co- efficient will be called, for brevity, the absolute coefficient of the instrument in question. 342. Thus, for example, the gold-leaf electrometer indicates differences of potential between the gold leaves and the solid walls enclosing the air-space in which they move. If this solid -be of other than sufficiently perfect conducting material, of wood and glass, or of metal and glass, for instance, as in the instrument ordinarily made, it is quite imperfect and indefinite in its indications, and is not worthy of being even called an electroscope, as it may exhibit a divergence when the difference of potentials which the operator desires to discover is absolutely zero. It is interesting to remark ( 336) that Faraday first remedied this defect by coating the interior of the glass case with tinfoil, cut away to leave apertures proper and sufficient to allow indications to be seen, but not enough to cause these indications to differ sensibly from what they would be if the conducting envelope were completely closed around it; and that not till a long time after did any other naturalist, mathe- matician, or instrument-maker seem to have noticed the defect, or even to have unconsciously remedied it. 343. Electrometers may be classified in genera and species according to the shape and kinematic relations of their parts ; but as in plants and animals a perfect continuity of interme- diate species has been imagined between the rudimentary plant and the most perfect animal, so in electrometers we may actually construct species having intermediate qualities continuous between the most widely different genera. But, notwithstanding, some such classification as the following is xx.] On Electrometers and Electrostatic Measurements. 265 convenient with reference to the several instruments commonly in use and now to be described : I. Kepulsion electrometers. Pair of diverging straws as used by Beccaria, Volta, and others, last century. Pair of diverging gold leaves (Bennet). Peltier's electrometer. Delmann's electrometer. Old station-electrometer, described in lecture to the Royal Institution, May 1860 [ 274-275, above]; also in Nichol's Cyclopaedia, article " Electricity, Atmo- spheric" (edition, 1860) [ 263, above], and in Dr Everett's paper of 1867, " On Atmospheric Electricity" ( Philosophical Transactions) . II. Symmetrical electrometers. Bohnenberger's electrometer. Divided-ring electrometers. III. Attracted disc electrometers. Absolute electrometer. Long-range electrometer. Portable electrometer. Spring-standard electrometer. 344. Class I. is sufficiently illustrated by the examples referred to ; and it is not necessary to explain any of these instruments minutely at present, as they are, for the present at all events, superseded by the divided- ring electrometer and electrometers of the third class. There are at present only two known species of the second class ; but it is intended to include all electrometers in which a symmetrical field of electric force is constituted by two symmetrical fixed conductors at different electric potentials, and in which the indication of the force is produced by means of an electrified body moveable symmetrically in either direction from a middle position in this field. This definition is obviously fulfilled by Bohnenberger's well-known instrument*. * A single gold leaf hanging between the upper ends of two equal and similar dry piles standing vertically on a horizontal plate of metal, one with its positive and the other with its negative pole up. 2G6 On Electrometers and Electrostatic Measurements, [xx. / 345. My first published description of a divided-ring electro- meter is to be found in the Memoirs of the Roman Academy of Sciences* for February 1857; but since that time I have made great improvements in the instrument first, by applying a light mirror to indicate deflections of the moving body ; next, by substituting for two half rings four quadrants, and conse- quently for an electrified body projecting on one side only of the axis, an electrified body projecting symmetrically on the two sides, and moveable round an axis ; and lastly, by various mechanical improvements, and by the addition of a simple gauge to test the electrification of the moveable body, and of a replenish er to raise this electrification to any desired degree. 346. In the accompanying drawings, Plate I. fig. 1 repre- sents the front elevation of the instrument, of which the chief bulk consists of a jar of white glass (flint) supported on three legs by a brass mounting, cemented round the outside of its mouth, which is closed by a plate of stout sheet-brass, with a lantern-shaped cover standing over a wide aperture in its centre. For brevity, in what follows these three parts will be called the jar, the main cover, and the lantern. Fig. 5 represents the quadrants as seen from above ; they are shown in elevation at a and 6, fig. 1, and in section at c and d } fig. 2. They consist of four quarters of a flat circular box of brass, with circular apertures in the centres of its top and bottom. Their position in the instrument is shown in figs. 1, 2, and 6. Each of the four quadrants is supported on a glass stem passing downwards through a slot in the main cover of the jar, from a brass mounting on the outside of it, and admits of being drawn outwards for a space of about 1 centi- metre (| of an inch) from the positions they occupy when the instrument is in use, which are approximately those shown in the drawings. Three of them are secured in their proper posi- tions by nuts (e, e, e) on the outside of the chief flat lid of the jar shown in fig. 4. The upper end of the stem, carrying the fourth, is attached to a brass piece (/, fig. 6) resting on three short legs on the upper side of the main cover, two of these legs being guided by a straight V-groove at (g) to give them * Accademia Pontificia dei Nuovi Lincei. xx.] On Electrometers and Electrostatic Measurements. 267 freedom to move in a straight line inwards or outwards, and to prevent any other motion. This brass piece is pressed out- wards and downwards by a properly arranged spring (/&), and is kept from sliding out by a micrometer-screw (t) turning in a fixed nut. This simple kinematic arrangement gives great steadiness to the fourth quadrant when the screw is turned inwards or outwards, and then left in any position ; and at the same time produces but little friction against the sliding in either direction. The opposite quadrants are connected in two pairs by wires, as shown in fig. 5 ; and two stout vertical wires (I, m), called the chief electrodes, passing through holes in the roof of the lantern, are firmly supported by long perforated vulcanite columns passing through those holes, and serve to connect the pairs of quadrants with the external conductors whose difference of potentials is to be tested. Springs (n, o) at the lower ends of these columns, shown in figs. 1 and 2, main- tain metallic contact between the chief electrodes and the upper sides of two contiguous quadrants (a and b) when the lantern is set down in its proper position, but allow the lantern to be removed, carrying the chief electrodes with it, and to be replaced at pleasure without disturbing the quadrants. The lantern also carries an insulated charging-rod (p), or temporary electrode, for charging the inner coating of the jar ( 351) to a small degree, to be increased by the replenisher ( 352), or, it may be, for making special experiments in which the potential of the interior coating of the jar is to be measured by a separate electrometer, or kept at any stated amount of difference from that of the outer coating. When not in use this temporary electrode is secured in a position in which it is disconnected from the inner coating. 347. The main cover supports a glass column (q, fig. 2) projecting vertically upwards through its central aperture, to the upper end of which is attached a brass piece (r), which bears above it a fixed attracting disc (s), to be described later ( 353) ; and projecting down from it a fixed plate bearing the silk-fibre suspension of the mirror (t), needle (u), etc., seen in figs. 1 and 2, and fixed guard tubes (v, w), to be described presently. To the main cover also is attached the circular level (fig. 6), which is adjusted .to indicate the position of the 268 On Electrometers and Electrostatic Measurements, [xx. instrument in which the quadrants are level, and the guard- tubes just mentioned vertical. Its lower surface which rests on the cover is slightly rounded, like a convex lens, so as to admit of a slight further adjustment (see end of 348, Addition) by varying the relative pressure of the three screws by which it is fastened down to the cover. 348. The moveable conductor of the instrument consists of a stiff platinum wire (#), about 8 centimetres (3| inches) long, with the needle rigidly attached in a plane perpendicular to it, and connected with sulphuric acid in the bottom of the jar by a fine platinum wire hanging down from its lower end and kept stretched by a platinum weight under the level of the liquid. The upper end of the stiff platinum wire is supported by a single silk-fibre so that it hangs down vertically. The mirror is attached to it just below its upper end. Thus the mirror, the needle, and the stiff platinum stem constitute a rigid body having very perfect freedom to move round a vertical axis (the line of the bearing fibre), and yet practically prevented from any other motion in the regular use of the instrument by the weight of its own mass and that of the loose piece of platinum hanging from it below the surface of the liquid in the jar. A very small magnet is attached to the needle, which, by strong magnets fixed outside the jar, is directed to one position, about which it oscillates after it is turned through any angle round the vertical axis, and then left to itself. The external magnets are so placed that when there is magnetic equilibrium the needle is in the symmetrical position shown in figs. 5 and 6 with reference to the quadrants*. [Addition, April 1870. The success of the experiments re- ferred to in the footnote has led to the adoption of the bifilar suspension in all the Quadrant Electrometers now made. It is represented in the margin. The stiff platinum wire which carries the mirror and needle has a cross piece at its upper end, to which are attached the lower ends of the two suspending silk fibres ; the other ends being wound upon the two pins c, d, which may be turned in their sockets by a square-pointed key, to * Kecently I have made experiments ou a bifilar suspension with a view to superseding the magnetic adjustment, which promise well. xx.] On Electrometers and Electrostatic Measurements. 269 equalize the tensions of the fibres, and make the needle hang midway between the upper and under surfaces of the quadrants. The pins c, d, are pivoted in blocks carried by springs e, f, to allow them to be shifted horizontally when adjusting the position of the points of suspension. The screws a, 6, which traverse these blocks, have their points bearing against the fixed plate behind, so that when a or 6 is turned in the direction of the hands of a watch, the neighbouring point of sus- pension is brought forward, and con- versely. The needle may thus be made to turn through an angle, till it lies in the symmetrical position represented in fig. 5, Plate I., when all electrical disturbance has been guarded against by connecting the quadrants with the inside and out- side of the jar. The conical pin h passes between the two springs and screws into the plate behind; by screwing it inwards the points of suspension are made to recede from each other laterally, and the sensibility of the needle to a deflecting couple is diminished, and conversely. The method employed to test the symmetry of the suspen- sion is suggested by the consideration that if the tension be equally distributed between the two fibres, the sensibility of the needle to the same deflecting couple will be less than if the whole or the greater part of the weight were supported by one fibre ; also, the sensibility being a minimum, a small deviation from the conditions which make it so will produce the least change of sensibility, by the known property of a maximum or minimum. To test whether these conditions are attained, raise first one side of the instrument a little (one turn of the foot-screw on that side is usually sufficient), and then produce an equal deviation in the opposite direction from the position marked by the attached level ( 347) ; and in each 270 On Electrometers and Electrostatic Measurements, [xx. position of the instrument observe the deflection of the image on the scale produced by some constant difference of potentials, as that between the two poles of a Daniell's cell. This deflection ought to be very nearly equal in the three positions, but exactly equal in the two disturbed positions, and somewhat greater in these than in the middle or level position. When the instru- ment is far out of adjustment, the deviation will be greater in one of the disturbed positions and less in the other than in the middle position. When it is but slightly out of adjustment, the deflections in the disturbed positions may both somewhat exceed that in the middle position, but to different degrees. An approximation to symmetry thus far at least should be obtained by merely turning the pins (c, d) in their sockets as already directed, through the minutest angles sensible to the operator, without altering the adjustment of the spirit-level on the cover. When that has been done, the level on the cover ought to be adjusted ( 347) by successive trials to indicate the position of the instrument such that when equally dis- turbed from it in opposite directions, the deflections obtained are equally in excess of the deflection obtained in the indicated position.] 349. The needle (u) is of thin sheet aluminium cut to the shape seen in figs. 5 and 6 ; the very thinnest sheet that gives the requisite stiffness being chosen. Its area is 4-J square centi- metres, and weight '07 of a gramme. If the four quadrants are in a perfectly symmetrical position round it, and if they are kept at one electric potential by a metallic arc connecting the chief electrodes outside, the needle may be strongly electrified without being disturbed from its position of magnetic equili- brium ; but if it is electrified, and if the external electrodes be disconnected, and any difference of potentials established between them, the needle will clearly experience a couple turning it round its vertical axis, its two ends being driven from the positive quadrants towards the negative, if it is itself positively electrified. It is kept positive rather than negative in the ordinary use of the instrument, because I find that when a conductor with sharp edges or points is surrounded by another presenting everywhere a smooth surface, a much greater difference of potentials can be established between xx.] On Electrometers and Electrostatic Measurements. 271 them, without producing disruptive discharge, if the points and edges are positive than if they are negative. 350. The mirror (t) serves to indicate, by reflecting a ray of light from a lamp, small angular motions of the needle round the vertical axis. It is a very light, concave, silvered glass mirror, being of only 8 millimetres (-J of an inch) diameter, and 22 milligrammes (J of a grain) weight. I had for many years experienced great difficulty in getting suitable mirrors for my form of mirror galvanometer; but they are now supplied in very great perfection by Mr Becker, of Messrs Elliott Brothers, London. [Addition, May 1870. I have not succeeded in getting more of these light ground concave mirrors giving good images, after a few supplied by Mr Becker at the time when the report was written. The lightest ground mirrors that Mr Becker can guarantee to give good images, weigh ^ of a gramme ( T 7 ^ of a grain). These answer well enough for the electrometers, because the aluminium needle weighing -jL of a gramme (l^V g ram )> anc ^ being of much greater linear dimensions, its moment of inertia is not largely increased by the addition of a mirror of that weight ; and they are preferred for this purpose to the exquisite light mirrors supplied by Mr White, as being stronger and less liable to warp in being mounted. But for galvanometers, and especially telegraph - signal galvanometers, it is important that the mirrors be the very lightest possible. The only mirrors suitable for this purpose which I can now obtain are supplied by Mr White. They give very perfect images, and weigh -^ of a gramme (-$7 of a grain) without the magnets, and ^ of a gramme with the magnets attached. Mr White produces them by cutting out and silvering a large number of circles of the thinnest microscope glass, attaching the magnets (four on the back of each mirror), and finally testing for the image. Out of fifty tried, about ten or fifteen are generally found satisfactory. A mirror may give a good image before the magnets are attached, and become warped out of shape and give a bad image after the magnets have been cemented to it.] The focus for parallel rays is about 50 centimetres (20 inches) from the mirror, and thus the rays of the lamp placed at a distance of 1 metre (or 40 inches) are brought to a focus at 272 On Electrometers and Electrostatic Measurements, [xx. the same distance. The lamp is usually placed close behind the vertical screen a little below or above the normal line of the mirror, and the image is thrown on a graduated scale extending horizontally above or below the aperture in the screen through which the lamp sends its light. When the mirror is at its magnetic zero position, the lamp is so placed that its image is, as nearly as may be, in a vertical plane with itself, and not more than an inch above or below its level, so that there is as little obliquity as possible in the reflection, and the line traversed by the image on the screen during the deflection is, as nearly as may be, straight. The distance of the lamp and screen from the mirror is adjusted so as to give as perfect an image as possible of a fine wire which is stretched vertically in the plane of the screen across the aperture through which the lamp shines on the mirror ; and with Mr Becker's mirrors, as with Mr White's selected galvanometer mirrors, I find it easy to read the horizontal motions of the dark image to an accuracy of the tenth of a millimetre. In the ordinary use of the instrument a white paper screen, printed from a copper-plate, divided to fortieths of an inch, is employed, and the readings are commonly taken to about a quarter of a scale- division; but with a little practice they may, when so much accuracy is desired, be read with considerable accuracy to the tenth of a scale- division. Formerly a slit in front of the lamp was used, but the wire giving a dark line in the middle of the image of the flame is a very great improvement, first intro- duced by Dr. Everett (in consequence of a suggestion made by Professor P. G. Tait) in his experiments on the elasticity of solids made in the Natural Philosophy Laboratory of Glasgow University*. 351. The charge of the needle remains sensibly constant from hour to hour, and even from day to day, in virtue of the arrangement by which it is kept in communication with sulphuric acid in the bottom of the jar, the outside of the * A Drummond light placed about 70 centimetres from the mirror gives an image, on a screen about 3 metres distant, brilliant enough for lecture- illustrations, and with sufficient definition to allow accurate readings of the positions on a scale marked by the image of a fine vertical wire in front of the light. xx.] On Electrometers and Electrostatic Measurements. 273 jar being coated with tinfoil and connected with the earth, so that it is in reality a Leyden jar. The whole outside of the jar, even where not coated with tinfoil, is in the ordinary use of the instrument, especially in our moist climate, kept virtually at one potential through conduction along its surface. This potential is generally, by connecting wires or metal pieces, kept the same as that of the brass legs and framework of the instru- ment. To prevent disturbance in case of strongly electrified bodies being brought into the neighbourhood of the instrument, a wire is either wrapped round the jar from top to bottom, or a cage or network of wire, or any convenient metal case, is placed round it ; but this ought to be easily removed or opened at any time to permit the interior to be seen. When the instrument is left to itself from day to day in ordinary use, the needle, connected with the inner coating of the jar as just described, loses, of course, unless replenished, something of its charge; but not in general more than J percent, per day, when the jar is of flint-glass made in Glasgow. On trying similar jars of green glass I found that they lost their charge more rapidly per hour than the white glass jars per month, I have occa- sionally, but very rarely, found white glass jars to be as defec- tive as those green ones, and it is possible that the defect I found in the green jars may have been an accident to the jars tested, and not an essential property of that kind of glass. 352. I have recently made the very useful addition of a replenish er to restore electricity to the jar from time to time when required. It consists of (1) a turning vertical shaft of vulcanite bearing two metal pieces called carriers (b, b, figs. 17 and 18) ; (2) two springs (d, d, figs. 16 and 18), con- nected by a metallic arc, making contact with the carriers once every half turn of the shaft, and therefore called connectors ; and (3) two inductors (a, a) with receiving springs (c, c) attached to them, which make contact with the carriers once every half turn, shortly before the connecting contacts are made. The inductors (a, a, figs. 16 and 18) are pieces of sheet metal bent into circular cylindrical shapes of about 120 each ; they are placed so as to deviate in the manner shown in the drawing from parts of a cylindrical surface coaxial with the turning- shaft, leaving gaps of about 60 on each side. The diameter of T. E. 18 274 On Electrometers and Electrostatic Measurements, [xx. this cylindrical surface is about 15 millimeters (about f of an inch). The carriers (b, b, figs. 17 and 18) are also of sheet metal bent to cylindrical surfaces, but not exactly circular cylinders; and are so placed on the bearing vulcanite shaft that each is rubbed by the contact springs over a very short space, about 1 millimeter beyond its foremost edge, when turned in the proper direction for replenishing. The receiving springs (c, c, figs. 17 and 18) make their contacts with each carrier immediately after it has got fairly under cover, as it were, of the inductor. Each carrier subtends an angle of about 60 at the axis of the turning-shaft. The connecting contacts are completed just before the carriers commence emerging from being under cover of the inductors. The carriers may be said to be under cover of the inductors when they are within the angle of 120 subtended by the inductors on each side of the axis. One of the inductors is in metallic communication with the outside coating of the jar, the other with the inside. Figs. 16, 17, and 18 illustrate sufficiently the shape of carriers and the succession of the contacts. The arrow-head indicates the direction to turn for replenishing. When it is desired to diminish the charge, the replenisher is turned backwards. A small charge having been given to the jar from an independent source, the replenisher when turned forwards increases the dif- ference of potentials between the two inductors and therefore between the two coatings of the jar connected with them by a constant percentage per half turn, unless it is raised to so high a degree as to break down the air-insulation by disruptive dis- charge. The electric action is explained simply thus : The carriers, when connected by the connecting springs, receive op- posite charges by induction, of which they deposit large propor- tions the next time they touch receiving springs. Thus, for example, if the jar be charged positively, the carrier emerging from the inductor connected with the inner coating carries a negative charge round to the receiving spring connected with the outside coating, while the other carrier, emerging from the induc- tor connected with the outside coating, carries a positive charge round to the receiving spring connected with the inside coating. If the carriers are not sufficiently well under cover of the in- ductors during both the receiving contacts and the connecting xx.] On Electrometers and Electrostatic Measurements. 275 contacts to render the charges which they acquire by induction during the connecting contacts greater than that which they carry away with them from the receiving contacts, the rotation, even in the proper direction for replenishing, does not increase, but, on the contrary, diminishes the charge of the jar. The deviations of the inductors from the circular cylinder, referred to above, have been adopted to give greater security against this failure. A steel pivot fixed to the top of the vulcanite shaft, and passing through the main cover, carries a small milled head (y, fig. 1) above, on the outside, which is spun rapidly round in either direction by the finger, and thus in less than a minute a small charge in the jar may be doubled. The diminution of the charge, when the instrument is left to itself for twenty-four hours, is sometimes imperceptible ; but when any loss is discovered to have taken place, even if to the extent of 10 per cent., a few moments 1 use of the replenisher suffices to restore it, and to adjust it with minute accuracy to the required degree by aid of the gauge to be described pre- sently. The principle of 'the " replenish er " is identical with that of the " doubler " of Bennet. In the -essentials of its con- struction it is the same as Varley's improved form of Nichol- son's " revolving doubler." 353. The gauge consists of an electrometer of Class III. The moveable attracted disc is a square portion of a piece of very thin sheet aluminium of the shape shown at a in fig. 4. It is supported on a stretched platinum wire passing through two holes in the sheet, and over a very small projecting ridge of bent sheet aluminium placed in the manner shown in the magnified drawing, fig. 3. The ends of this wire are passed through holes in curved springs, shown in fig. 4, and are bent round them so as to give a secure attachment without solder, and without touching the straight stretched part of the wire. The ends of the platinum wire (/3, 0) are attached by cement to the springs, merely to prevent them from becoming loose, care being taken that the cement does not prevent metallic contact between some part of the platinum wire and one or both of the brass springs. I have constantly found fine platinum wire rendered brittle by ordinary solder applied to it. The use of these springs is to keep the platinum wire stretched 182 276 On Electrometers and Electrostatic Measurements, [xx. with an approximately constant tension from year to year, and at various temperatures. Their fixed ends are attached to round pins, which are held with their axes in a line with the fibre by friction, in bearings forming parts of two adjustable brass pieces (7, 7) indicated in fig. 4 ; these pieces are adjusted once for all to stretch the wire with sufficient force, and to keep the square attracted disc in its proper position. The round pins bearing the stretching springs are turned through very small angles by pressing on the projecting springs with the finger. They are set so as to give a proper amount of torsion tending to tilt the attracted disc (a) upwards, and the long end of the aluminium lever (), of which it forms a part, downwards. The downward motion of the long end is limited by a properly placed stop. Another stop (e) above limits the upward motion, which takes place under the influence of electrification in the use of the instrument. A very fine opaque black hair (that of a small black-and-tan terrier I have found much superior to any hitherto tried) is stretched across the forked portion of the sheet aluminium in which the long arm of the lever terminates. Looked at horizontally from the outside of the instrument it is seen, as shown in fig. 7, Plate I., against a white background, marked with two very fine black circles. These sight-plates in the instruments, as now made by Mr White, are of the same material as the ordinary enamel watch-dials, with black figures on a white ground. The white space between the two circles should be a very little less than the breadth of the hair. The sight-plate is set to be as near the hair as it can be without impeding its motion in any part of its range; it is slightly convex forwards, and is so placed that the hair is nearer to it when in the middle between the black circles than when in any other part of its range. It is thus made very easy, even without optical aid, to avoid any considerable error of parallax in estimating the position of the hair relatively to the two black circles. By a simple plano-convex lens (<, fig. 2), with the convex side turned inwards, it is easy, in the ordinary use of the instrument, to distinguish a motion up or down of the hair amounting to -^^ of an inch. With a little care I have ascertained, Dr Joule assisting, that a motion of no more than --i of an inch from one definite central position can be xx.] On Electrometers and Electrostatic Measurements. 277 securely tested without the aid of other magnifying power than that given by the simple lens. The lens during use is in a fixed position relatively to the framework bearing the needle, but it may be drawn out or pushed in to suit the focus of each observer. To give great magnification, it ought to be drawn out so far that the hair and sight-plate behind may be but little nearer to the lens than its principal focus, and the observer's eye ought to be at a very considerable distance from the instru- ment, no less than 20 centimetres (8 inches) to get good mag- nification ; and a short-sighted person should use his ordinary concave eye-lens close to his eye. The reason for turning the convexity of the small plano-convex lens inwards is, that with such a lens so placed, if the eye of the observer is too high or too low, the hair seems to him curved upwards or downwards, and he is thus guided to keep his eye on a level sufficiently constant to do away with all sensible effects of parallax on the position of the hair relatively to the black circles. The framework carry- ing the stretched platinum wire and moveable attracted disc is O -L above the brass roof of the lantern, in which a square aperture is cut to allow the square portion constituting the short arm of the aluminium balance to be attracted downwards by the fixed attracting disc ( 347), to be presently described. A side view of the attracting plate, the brass roof of the lantern, the alu- minium balance, the sight-plate, the hair, and the plano-convex lens is given in section (fig. 2) ; also a glass upper roof to pro- tect the gauge and the interior of the instrument below from dust and disturbance by currents of air, to which, without this upper roof, it would be exposed, through the small vacant space around the moveable aluminium square. The fixed attracting disc is borne by a vertical screw screwing into the upper brass mounting (z, fig. 2) ( 347), connected with the inner coating of the Leyden jar through the guard tubes, etc., and is secured in any position by 'the "jam nut/' shown in the drawing at z, fig. 2. This disc (s) is circular, and about 38 millimetres (1J inch) in diameter, and is placed horizontally with its centre under the centre of the square aperture in the roof of the lantern. Its distance from the lower surface of the roof and of the moveable attracted disc may be from 2} to 5 millimetres (from -^ to i of an inch), and is to be adjusted, along with the 278 On Electrometers and Electrostatic Measurements, [xx. amount of torsion in the platinum wire bearing the aluminium balance-arm, so as to give the proper sensibility to the gauge. The sensibility is increased by diminishing the distance from the attracting to the attracted plate, and increasing the amount of torsion. Or, again, the degree of the potential indicated by it when the hair is in the sighted position is increased by in- creasing the distance between the plates, or by increasing the amount of torsion. If the electrification of the needle is too great, its proper position of equilibrium becomes unstable ; or before this there is sometimes a liability to discharge by a spark across some of the air-spaces. The instrument works extremely well with the needle charged but little less than to give rise to one or both of these faults, and I adjust the gauge accordingly. 354. The strength of the fixed steel directing magnets is to be adjusted to give the desired amount of deflection with any stated difference of potentials maintained between the two chief electrodes, when the jar is charged to the degree which brings the hair of the gauge to its sighted position. In the instruments already made, the deflection* by a single cell of Daniell's amounts to about 100 scale-divisions (of ^ of an inch each and at a distance of 40 inches), if the magnetic directive force is such as to give a period of vibration equal to about T5 seconds, when the jar is discharged and the four quadrants are connected with one another and with the inner coating of the jar. Lower degrees of sensibility may be attained better by increasing the magnetic directing force than by diminishing the charge of the jar. Thus, for instance, when it is to be used for measuring and photographically recording the potential of atmospheric electricity at the point where the stream of the water-dropping collector*)- breaks into drops, the magnetic directing force may be made from 10 to 100 times greater than that just described. When this is to be done it may be con- venient to attach a somewhat more powerful magnetic needle than that which has been made in the most recent instruments where a high degree of sensibility has been provided for. But it * That is to say, the number of scale-divisions over which the luminous image moves when the chief electrodes are disconnected from one another and put in metallic connexion with the two plates of a Daniell's battery. t See Koyal Institution Lecture, May 18, 1860 ( 278, 279, above), or Nichol's Cyclopedia, article "Electricity, Atmospheric" (Edition 1860) ( 262, above). xx.] On Electrometers and Electrostatic Measurements. 270 is to be remarked that in general the directing-force of the ex- ternal steel magnets cannot be too strong, as the stronger it is the less is the disturbance produced by magnetic bodies moving in the neighbourhood of the instrument*. In laboratory work, where numerous magnetic experiments are being performed in the immediate neighbourhood, and in telegraph factories where there is constant disturbance by large moving masses of iron, the artificial magnetic field of the electrometer ought to be made very strong. To allow this, and yet leave sufficient sensibility to the instrument, the suspended magnetic needle has been made smaller and smaller, until it is now reduced to two small pieces of steel side by side, 6 millimetres (J of an inch) long. For a meteorological observatory all that is neces- sary is, that the directing magnetic force may be so great that the greatest disturbance experienced in magnetic storms shall not sensibly deflect the luminous image. 355. The sensibility of the gauge should be so adjusted that a variation in the charge of the jar, producing an easily per- ceived change in the position of the hair, shall produce no sensible change in the deflection of the luminous image pro- duced by the greatest difference of potentials between the quadrants, which is to be measured in the use of the instru- ment. I believe the instruments already made, when adjusted to fulfil these conditions, may be trusted to measure the dif- ference of potentials produced by a single cell of Daniell's to an accuracy of a quarter per cent. It must be remembered that the constancy of value of the unit of each instrument depends not only on the constancy of the potential indicated by the gauge, but also on the constancy of the magnetic force in the field traversed by the suspended magnet, and on the con- stancy of the magnetic moment of the latter. As each of these may be expected to decrease gradually from year to year (al- though very slowly after the first few hours or weeks), rigorous methods must be adopted to take such variations into account, if the instrument is to be trusted as giving accurately comparable indications at all times. The only method hitherto provided * All embarrassment from this source will be done away with if the bifilar plan be adopted (see 348, Addition). 280 On Electrometers and Electrostatic Measurements, [xx. for this most important object consists in the observation of the deflection produced by a measured motion of one of the quadrants by the micrometer screw (i) when the four quadrants are put in metallic communication with one another through the principal electrodes ; the jar being brought to one constant potential by aid of the gauge, and therefore the force producing the deflection being constant. The amount of the deflection will show whether or not the force of the magnetic field has changed, and will render it easy at any time to adjust the strength of the magnets, if necessary, to secure this constancy. But to attain this object by these means, the three quadrants not moved by the micrometer screw must be clamped by their fixing-screws so that they may be always in the same position. 356. The absolute constancy of the gauge cannot be altogether relied upon. It certainly changes to a sensible degree with tem- perature ; and in different instruments, to very different degrees, and even in different directions, as will be seen ( 377) in con- nexion with the description of the portable electrometer to be given later. But this temperature variation does not amount in ordinary cases probably to as much as one per cent. ; and it is probable- that after a year or two any continued secular variation of the platinum torsion spring will be quite insensible. It is to be remarked, however, that secular experiments on the elasticity of metals are wanting, and ought at least to be commenced in our generation. In the meantime it will be desirable, both on account of the temperature variation and of the possible secular variation in the couple of torsion, to check the gauge by accu- rate measurements of the time of oscillation of the needle with its appurtenances. The moment of inertia of this rigid body, except in so far as it may be influenced by oxidation of the metal, of which I have as yet discovered no signs, may be regarded as constant, and therefore the amount of the direct- ing couple due to the magnets may be- determined with great accuracy by finding the period of an oscillation when the four quadrants are put in connexion through the charging rod with the metal mounting bearing the guard plates, etc; I hare not as yet put into practice any of the obvious methods, founded on the general principle of coincidences used in pendulum observations, for determining the period of the oscillation ; but xx.] On Electrometers and Electrostatic Measurements. 281 although not more than twenty or thirty complete oscillations can be counted, it seems certain that with a little trouble the period of one of them may be easily determined to an accuracy of about y 1 ^ per cent. 357. [Addition, May 1870. The most direct and obvious method of using the Quadrant Electrometer is to connect the two chief electrodes, with the two bodies whose difference of potentials is to be measured, and one of them with the case of the instrument. With the instruments made at the present date, a difference of potentials equal to that of the opposite poles of a single Daniell's cell gives, when measured in this manner, a deflection of the image over about 60 scale-divisions, more or less according to the distance at which the points of sus- pension of the silk fibres have been adjusted ( 348, Addition). The difference of potentials due to six cells in series would thus deflect the image to the extremity of the scale, and be the greatest difference of potentials that could be measured by the electrometer, if these were the only connexions available for measurements. A second and much lower grade of sensibility is obtained by simply raising, so as to disconnect from the quadrant beneath it, the electrode connected with the case. This being done, it requires a battery of about 10 or 15 cells to produce the deflection pre- viously produced by a single cell. Several still lower grades of sen- sibility have been provided for in the instruments recently made, by the addition of an induction- plate, insulated directly over one of the quadrants behind the mirror. The sketch in the mar- gin represents a vertical section through the induction-plate (e), insulating glass stem (i) by which it is supported, its elec- trode (a), the quadrant (c), and A main glass stem (q). The line AB in the horizontal plan be- 282 On Electrometers and Electrostatic Measurements, [xx. low is the line of section, passing through the centres of the electrode and insulating stem of the induction-plate, and that of the main glass stem, which are in one straight line. The plan represents that part of the main cover as seen from above, when the lantern and upper works are removed. The plate (b) which supports the main stem (q) has been enlarged to bear also the insulating support (i) of the induction-plate. The outline of the induction-plate falls within that of the quadrant beneathUt by 16 of a centimetre (-^ of an inch) all round. It is distant '48 of a centimetre (-5^- of an inch) from the upper surface of the quadrant. The dimensions in the figure are half full size. With an electrometer fitted with the induction-plate, the usual connexions for the first or direct method of measure- ment are the same as above mentioned. The electrode of the induction-plate may be connected with that of the quadrant beneath it, or with the case, or it may be insulated, without sensibly affecting the indications of the instrument. For the second grade of sensibility the induction-plate is con- nected with the case, and the difference of potentials to be measured is established between it and the distant pair of quadrants, the nearer pair being insulated by raising their electrode. To free the latter from the induced charge which they commonly receive by the act of raising their electrode, a disinsulator is provided, consisting of a light arm or spring which may be turned so as to make contact with the quadrant by means of a small milled head projecting above the cover. For a certain lower grade the arrangement is the same, except that the distant pair of quadrants, instead of the induction-plate, is connected with the cover, and the difference of potentials to be measured is established between the cover and the induction- plate. With this arrangement the deflections measure about five times the difference of potentials producing the same deflections by the second grade. The connexions may be further varied so as to produce other degrees of sensibility giving indications perfectly trustworthy and available for comparative measurements. The different methods of forming the connexions, with or without an in- ductor, are indicated in the following table, where R means the xx.] On Electrometers and Electrostatic Measurements. 283 electrode of the pair of quadrants marked RR' in the figure, L that of the pair LL' y and / that of the induction-plate; G is the conductor led from one of the bodies experimented upon, the conductor led from the other and connected to the outer metallic case of the instrument, which may be insulated from the table if necessary by placing a small block or cake of clean paraffin under each of the three feet on which the instrument stands ; (R) or (L) means that the electrode of RR or LL' is to be raised so as to be disconnected from its pair of quadrants. Thus in the grade of diminished power or sensibility standing first in the table on the right, the electrode L is raised, one conductor is connected with R ; / and the other with the case of the instrument. The grade standing last in the table, in which L and R are both raised, is the least sensitive of all. In each of these methods the correctness of the indications has been verified by measurements taken simultaneously with the Standard Electrometer ( 379), the measured difference of potentials being that of the earth and of a Leyden jar fitted with a replenisher, by means of which its potential was varied so as to make the deflected image stand at all points between the extremity of the scale and the zero position. The working of the replenisher being suspended at intervals to allow an accurate reading to be taken of the position of the image and the indication of the Standard Electrometer, the subsistence of a correct proportion between the deflection and the measure- ment obtained from the Standard Electrometer was verified at all points of the range. WITHOUT INDUCTOE. WITH INDUCTOE. Full Power. Full Power. LC~\ [RC'] [LCI R0\ r \LO\ W r Diminished Power. Grades of Diminished Power. m o\ io io \IC~\ ilLO 0\ 284 On Electrometers and Electrostatic Measurements, [xx. Scale Ri'jhi The facility afforded by the num- ber of these arrangements for vary- ing the sensibility of the instn*- ment even to a moderate or slight degree without altering the adjust- ment of the fibres, will be found useful in some kinds of observa- tions. For instance^ if it be de- sired to observe the fluctuations of a varying potential, a degree of sensibility which throws the de- flected image nearly to the ex- tremity of the scale will cause the fluctuations to be twice as sensible and accurately read as if the de- flection were only half as much, as they will bear the same proportion to the whole deflection in the two cases. It is intended in future to make the induction-plate smaller and more distant from the quadrant, in order to diminish the inductive effect and permit of the measure- ment of from 100 to 5000 cells by the least sensitive method. In some electrometers also the first two grades of sensibility may be considered sufficient, and the induction-plate dispensed with.] ABSOLUTE ELECTROMETER. 358. The absolute electrometer (fig. 11, Plate II.) and the other instruments of Class III. are founded on a method of experimenting introduced by Sir William Snow Harris, and described in his first paper "On the Elementary Laws of Electricity*," thirty-four years ago. In these experiments a conductor, hung from one arm of a balance and kept in metallic communication with the earth, is attracted by a fixed insulated conductor, which is electrified, and, for the sake of keeping its electric potential constant, is connected with the * Philosophical Transactions, 1834. xx.] On Electrometers and Electrostatic Measurements. 285 inner coating of a Leyden battery. The first result which he announced is, that, when other circumstances remain the same, the attraction varies with the square of the quantity of electricity with which the insulated body is charged and is independent of the unopposed parts. " It is readily seen "that, in the case of Mr Harris's experiments, it will be " so slight on the unopposed portions that it could not be " perceived without experiments of a very refined nature, such " as might be made by the proof plane of Coulomb, which is, " in fact, with a slight modification, the instrument employed " by Mr Faraday in the investigation. Now to the degree of " approximation to which the electrification of the unopposed " parts may be neglected, the laws observed by Mr Harris when " the opposed surfaces are plane may be readily deduced from " the mathematical theory. Thus let v be the potential in the " interior of A, the charged body, a quantity which will depend " solely on the state of the interior coating of the battery with " which, in Mr Harris's experiments, A is connected, and will " therefore be sensibly constant for different positions of A "relative to the uninsulated opposed body B. Let a be the " distance between the plane opposed faces of A and B, and " let S be the area of the opposed parts of these faces, which " will in general be the area of the smaller, if they be unequal. " When the distance a is so small that we may entirely neglect " the intensity an all the unopposed parts of the bodies, it is " readily shown, from the mathematical theory, that (since the " difference of the potentials at the surfaces of A and B is v) " the intensity of the electricity produced by induction at any " point of the portion of the surface of B which is opposed to " A is , the intensity at any point which is not so situated 47T& " being insensible. Hence the attraction on any small element " w, of the portion S of the surface of B, will be in a direction " perpendicular to the plane and equal to 2?r I -^-\ o>*. Hence " the whole attraction on B is * See Mathematical Journal, vol. in. .p. 275 (VII. above, 146, 147). 286 On Electrometers and Electrostatic Measurements, [xx. "This formula expresses all the laws stated by Mr Harris " as results of his experiments in the case when the opposed " surfaces are plane*." 359. After many trials to make an absolute electrometer founded on the repulsion between two electrified spherical conductors for which I had given a convenient mathematical formula in 4 of the paper just quoted ( 30, above), it occurred to me to take advantage of the fact noticed by Harris, but easily seen as an immediate consequence of Green's mathematical theory, that the mutual attraction between two conductors used as in his experiments is but little influenced by the form of the unopposed parts; and in 1853, in a paper " On Transient Electric Currents-)-," I described a method for measuring differences of electric potential in absolute electrostatic measure founded on that idea. The " absolute electrometer," which I exhibited to the British Association at its Glasgow Meeting in 1855, was con- structed for the purpose of putting these methods into practice. This instrument consists of a plane metal disc insulated in a fixed horizontal position with a somewhat smaller fixed metal disc hung centrally over it, from one end of the beam of a balance. In two papers { entitled "Measurement of Electro- static Force produced by a Battery," and "Measurement of the Electromotive Force required to produce a Spark in Air between Parallel Metal Plates at Different Distances," published in the Proceedings of the Royal Society for February 1860, I described applications of this electrometer, in which, for the first time I believe, absolute electrostatic measurements were made. The calculations of differences of potential in absolute measure were made according to the formula quoted above ( 358) from my old paper on "The Elementary Laws of Statical Electricity." 360. This formula is rigorous only if the distance between the discs is infinitely small in comparison with their diameters; and therefore, in my earliest attempt to make absolute electro- static measurements, I used very small distances. I found * " On the Elementary Laws of Statical Electricity," Cambridge and Dublin Mathematical Journal, 1846 ; and Philosophical Magazine, July, 1854 (II. above, 27). t Philosophical Magazine, June, 1853. XVIII. and XIX. above, 310340. xx.] On Electrometers and Electrostatic Measurements. 287 great difficulty in securing that the distance should be nearly enough equal between different parts of the plates, and in measuring its absolute amount with sufficient accuracy; and found besides serious inconveniences in respect of sensibility and electric range : later I made a great improvement in the instrument by making only a small central area of one of the discs moveable. Thus the electric part of the instrument becomes two large parallel plates with a circular aperture in one of them, nearly filled up by a light circular disc supported properly to admit of its electrical attraction towards the other being accurately measured in absolute units of force. The disc and the perforated plate surrounding it will be called, for brevity, the disc and the guard-plate. The faces of these two next the other plate must be as nearly as possible in one plane when the disc is precisely in the position for measuring the electric force upon it, which, for brevity, will be called its sighted position. The space between the disc and the inner edge of its guard-ring must be a very small part of the diameter of the aperture, and must be very small in comparison with the distance between the plates; but the diameter of the disc may be greater than, equal to, or less than the distance between the plates. 361. Mathematical theory shows that the electric attraction experienced by the disc is the same as that experienced by a certain part of one of two infinite planes at the same distance, with the same difference of electric potentials, this area being very approximately the mean between the area of the aperture and the area of the disc, and that the approximation is very good, even should the distance between the plates be as much as a fourth or fifth, and the diameter of the disc as much as three-fourths of the diameter of the smaller of the two plates. This conclusion will be readily assented to when we consider that* the resultant electric force at any point in the air between the two plates is equal numerically to the rate of conduction of heat per unit area across the corresponding space in the follow- ing thermal analogue. Let a solid of uniform thermal conduc- tivity replace all the air between and around the plates; and in * " On the Uniform Conduction of Heat through Solid Bodies, and its con- nexion with the Mathematical Theory of Electricity," Cambridge Mathematical Journal, Feb. 1842; and Philosophical Magazine, July, 1854 (I. above, 16). 288 On Electrometers and Electrostatic Measurements, [xx. place of the plates let there be hollow spaces in this solid. Let these hollow spaces be kept at two uniform temperatures, differing by a number of degrees equal numerically to the difference of potentials in the electric system, the space corre- sponding to the disc and guard-ring being at one temperature, and that corresponding to the opposite plate at the other tem- perature; and let the thermal conductivity of the solid be unity. If we attempt to draw the isothermal surfaces between the hollow corresponding to the continuous plate on the one side, and that corresponding to the disc and guard-ring on the other, we see immediately that they must be very nearly plane, from very near the disc all the way across to the corre- sponding central portion of the opposite plate, but that there will be a convexity towards the annular space between the disc and guard-ring. 362. Thus we see that the resultant electric force will, to a y very close approximation, be equal to -^ for all points of the air between the plates at distances from the outer bounding edges exceeding two or three times the distance between the plates, and at distances from the interstice between the guard- ring and disc not less than the breadth of this interstice. Hence, if p denote th -electric density of any point of the plate or disc far -enough from the edges, we have V But the outward force experienced by the surface of the electrified conductor per unit of area at any point is 27r/o 2 , and therefore if F -denote the force experienced by any area A of the fixed plate, no part of which comes near its edge, we have which will clearly be equal to the attraction experienced by the moveable disc, if A be the mean area defined above. This /o ET gives FsD*7-~- , the formula by which difference of poten- tials in absolute electrostatic measure is calculated from the xx.] On Electrometers and Electrostatic Measurements. 289 result of a measurement of the force F, which, it must be remembered, is to be expressed in kinetic units. Thus if W be the mass in grammes to which the weight is equal, we have F=gW, where g is the force of gravity in centimetres per second per second. The difficulty which, in first applying this method about twelve years ago, I found in measuring accurately the distance D between the plates and in avoiding error from their not being rigorously parallel, I now elude by measuring only differ- ences of distance, and deducing the desired results from the difference of the corresponding differences of potentials. Thus let V be the difference of potentials between the plates re- quired to give the same force F\ when the difference of poten- tials is V instead of V, we have 363. The plan of proceeding which I now use is as follows : Each plate (fig. 11, Plate II.) is insulated; one of them, the continuous one, for instance, is kept at a potential differing from the earth by a fixed amount tested by aid of a separate idiostatic* electrometer (; the other plate (the guard -ring and moveable disc in metallic communication with one another) is alternately connected with the earth and with the body whose potential is to be measured. The lower plate is moved up or down by a micrometer screw until the moveable disc balances in a definite position, indicated by the hair (with background of white with black dots) seen through a lens, as shown in fig. 11. Before and after commencing each series of electrical experiments, a known weight is placed on the disc, and a small wire rider on the lever from which the disc hangs is adjusted to bring the hair to its sighted position when there is no electric force. This last condition is secured by putting the two plates * See 385, below. t [A Ley den jar with an idiostatic gauge and replenisher fitted to the cover by which it is closed has been found very suitable for this purpose. The gauge can be adjusted to a higher degree of sensibility than is attainable in an electrometer for general purposes, as the Standard or the Portable Electrometer, and the micrometer movements and graduations of these electrometers are not required. May, 1870.] T. E. 19 290 On Electrometers and Electrostatic Measurements, [xx. in metallic communication with one another. For the electric experiments the weight is removed, so that when the hair is in the sighted position the electric attraction on the moveahle disc is equal to the force of gravity on the weight. The electric connexions suitable in using this instrument for determining in absolute electrostatic measure the difference of potentials maintained by a galvanic battery between its two electrodes are indicated in fig. 11. No details as to the case for preventing disturbance by currents of air, and for maintaining a dry atmo- sphere, by aid of pumice impregnated with strong sulphuric acid, are shown, because they are by no means convenient in the instrument at present in use, which has undergone so many transformations that scarcely any part of the original structure remains. I hope soon to construct a compact instrument con- venient for general use. The amount of force which is constant in each series of experiments may be varied from one series to another by changing the position of the small wire rider on the lever. The electric system here described is heterostatic ( 385 below), there being an independent electrification besides that whose difference of potential is to be measured. NEW ABSOLUTE ELECTROMETER. [ 364... 367 added May, 1870.] 364. Plate III. is a sketch in perspective of this instru- ment, one-third of the full size. As in the Absolute Electro- meter just described, the electric system is heterostatic; with this addition, that the potential of the auxiliary charge is tested and maintained, not by a separate electrometer and electric machine, but by an idiostatic arrangement forming part of the instrument itself. This consists of a Leyden jar, forming the case of the instrument ; a gauge ; and a replenishes The Leyden jar is a white (flint) glass cylinder, coated inside and outside with tinfoil to nearly the height of the circular plate (A) ; apertures being left to admit the requisite light to the interior and allow the indications of the vertical scale (r) and divided circle (t) to be read. A brass mounting is cemented round the upper rim of the jar, to which is screwed the cover xx.] On Electrometers and Electrostatic Measurements. 291 of stout sheet-brass (C), which closes the jar at the top. By another brass mounting cemented round its lower rim, the jar is fastened down to the cast-iron sole -plate (D) which closes its lower end. The sole-plate is supported on three legs similar to those shown in fig. 13, Plate II. The cover (C) supports the replenisher (E), and the aluminium balance-lever of the idiostatic gauge, which are identical in construction with those described in 352, 353, but on a larger scale. The air inside is kept dry by aid of pumice soaked with strong sulphuric acid, contained in glass vessels placed in the bottom of the jar. The moveable disc or balance (c) hangs in a circular aperture in the plate (A\ which rests on three fixed supports (z t z, .) cemented to the interior surface of the jar, and in metallic con- nexion with the inside coating ; the manner of support is that of the hole, slot, and plane, described in 380, (2), below. This perforated plate or guard-plate supports on a brass pillar the attracting plate (F) of the idiostatic gauge, which thus tests the potential of the guard-plate, balance, and inside coat- ing. This potential is kept constant during any series of ex- periments by using the replenisher according to the indications of the gauge, which is made extremely sensitive by a proper adjustment of the distance from the attracting plate (F) to the balance-lever and of the torsion by which the electrical attraction is balanced (see end of 353). The replenisher has metallic contact with the guard- plate through the spring (e). The jar is charged by an insulated charging-rod let down for the occasion through a hole in the cover. 365. The balance (c) is a light aluminium disc, about 46 millimetres in diameter, strengthened by an elevated rim and radial ribs on its upper surface, but having its lower surface plane and smooth. It nearly fills the aperture in the guard- plate, sufficient clearance being left (75 of a millimetre all round) to allow it to move up and down without risk of fric- tion. It is supported by three delicate steel springs, each of which consists of two parts ; the upper end of the upper part is attached to the lower extremity of a vertical insulating stem (i) directly above the centre of the disc, where the cor- responding end of the lower part is fixed. The opposite ends, 192 292 On Electrometers and Electrostatic Measurements, [xx. which project considerably beyond the circumference of the disc, are riveted together. One of these springs (s) is shown in the figure. Their general form may be compared to that of coach-springs. The point of attachment of their upper parts is moved vertically by a kinematic arrangement precisely the same as that employed in the Portable Electrometer ( 369). The insulating stem (i) is attached to a brass tube (a), which slides up and down in V guides by the action of a micrometer screw. This micrometer screw is worked by means of the milled head (m) projecting above the cover (C) ; the guides for the tube (a) and index (x) which moves up and down with the tube, are similar to those represented more fully in fig. 10, Plate II., and are rigidly attached to a strong brass plate (b) lying across the mouth of the jar below the cover, and resting upon the flange of the brass mounting, to which it is fastened by screws. The plate (b) is so adjusted that the balance may hang concentric with the perforation in the guard-plate. The tube (a) is similar in construction to that represented in fig. 8, Plate II., and described in 369, below. The micrometer screw carries a horizontal circular disc (d) graduated by 100 equal angular divisions. An aperture is left in the cover through which its indications can be read off by reference to a fixed mark on the sloping edge of the aperture. This, together with the scale (jf ), each division of which corresponds to one full turn of the micrometer screw, measures the vertical distance through which the tube (a) and the points of attachment of the springs are moved. Metallic communication between the balance and the guard- plate is maintained by a light spiral wire attached to the pillar (g) and to the upper support of the springs, which is a brass piece cemented to the insulating stem. An arm, not seen in the figure, projects from the guard-plate over the disc so that its extremity is between the centre of the disc and the upper end, bent horizontally, of an upright fixed to the disc; thus serving as a stop to confine the motion of the disc between certain limits. A very fine opaque black hair ( 353) is stretched between two small uprights (one of which is seen in the figure) standing in the centre of the disc. An achromatic convex lens (h), fixed on the guard-plate, stands opposite, and pro- xx.] On Electrometers and Electrostatic Measurements. 293 duces an image of the hair in the conjugate focus, which is just over the outer edge of the guard-plate. The two opposed screw-points (&) are adjusted to touch each side of the image thus thrown by the lens, which, on the principle of the astro- nomical telescope, is observed through an eye-lens (I), attached outside of the jar to the upper brass mounting. By this arrangement the error of parallax in observing the position of the hair relatively to the two points is avoided; the position of the eye may be varied in any direction without causing any change in the apparent relative position of the hair (image) and points. In adjusting these different parts, it is arranged that when the image of the hair is exactly between the two points, or in what is called the sighted position, the under surfaces of the balance and guard-plate may be as nearly as possible in one horizontal plane. The balance and springs are protected, in the use of the instrument, from disturbing electrical forces, by a brass cover in two halves (y, y), one of which is represented displaced in the figure, to show the interior arrangements. The two halves, when placed together, form a circular box, with an aperture in front in which the lens (h) stands, and another aperture behind to admit light from the sky or from a lamp placed outside of the jar in the line of the hair, lens, and points. 366. The electrical part of the instrument is completed by the continuous attracting plate (B\ under and parallel to the guard-plate and spring-balance. This is a stiff circular brass plate with parts cut out to allow it to move freely past the fixed supports (z, z t .) of the guard-plate. An electrode (n) projecting through a hole in the sole-plate from an insulating stem (p) is kept in metallic communication by a spiral wire with an arm projecting from the centre of the continuous plate. The plate (B) is supported by a brass pillar (g), from which it is insulated by a short glass stern. It is moved vertically by the micrometer screw (w) (step -fo of an inch) ; and this motion is measured by a vertical scale (r) and horizontal graduated circle (t) attached to the screw. The screw projects below the sole- plate, and is worked by the milled head (u), the nut (v) being fixed in the centre of the sole-plate. The pillar (q) moves in 294 On Electrometers and Electrostatic Measurements, [xx. V or ring guides, and rests upon the upper end of the screw- in the manner represented in fig. 14, Plate II. 367. Before this instrument is available for absolute electro- static measurements, the force required to move the balance through any fixed vertical distance (the point of suspension being unmoved) must be known. This is ascertained by weighings conducted in the following manner : The cover (C) is removed, and all electrical force upon the balance is guarded against by putting the electrode (n) in metallic communication with the guard-plate. The balance is then brought, by turning the micrometer circle (d), to the sighted position ; and the reading on the scale (/) and graduated circle (d) is noted. A known weight is then distributed symmetrically over the disc ( T % of a gramme has been used hitherto), which displaces it below the sighted position. It is now raised to the sighted position by turning the disc (d), and the altered micrometer reading is noted. The difference between the two readings measures the distance through which the given weight displaces the balance in opposition to the tension of the springs ; and conversely, when the balance has been displaced through the same distance by electrical attraction between it and the continuous plate below it, this known weight is the measure of the force exerted upon it. It has been thus found by repeated weighings, that a weight of T 6 o of a gramme displaces the balance through a distance corresponding to two full turns of the micrometer screw and a fraction of one division of the circle, in the instru- ment belonging to the Laboratory of the Glasgow University. This distance having been ascertained with all possible care and at different temperatures, in view of the possible effect of temperature on the elasticity of the springs, the plan of pro- ceeding to absolute electrostatic measurements is as follows, the weights being removed and covers (y, y, C) replaced. All electrical influence having been removed by a wire led from the electrode (n) through the hole in the cover (C) to the guard-plate, the balance is brought to the sighted position. Starting from this point, it is raised by the micrometer screw through any distance which has been ascertained to correspond to a known weight, e.g. the distance just mentioned. This cor- XX.] On Electrometers and Electrostatic Measurements. 295 responds exactly to the removal of the weight ( 363) in the use of the Absolute Electrometer already described. The jar is then charged, and the potential is kept constant during the experiments by using the replenisher according to the indica- tions of the gauge, which, as already said, has been made extremely sensitive for the purpose. The attracting plate (B) is connected by its electrode (ri) alternately with the outside coating of the jar (which may be either connected with the earth or insulated) and with the body the difference of whose potential from that of the outside coating is to be measured. In each case the balance is brought to the sighted position by moving the plate (B) up or down by the micrometer screw (w), and the reading on the vertical scale (r) and graduated circle (t) is noted. The difference of the two readings gives the differ- ence of the two distances between balance and attracting plate, from which the difference of potentials is deduced by the formula at the end of 362. In measuring the difference of potentials between the poles of a voltaic battery, it is found very convenient to connect the poles, through a Steinheil (or double Bavarian) key, either with the outer coating of the jar (or earth), the other with the insulated electrode (n). The reading being taken and the key reversed, the difference of readings, it is evident, measures a difference of potentials double that of the poles of the battery. Two observers are convenient, one to watch the gauge and use the replenisher accordingly, the other to take the readings. PORTABLE ELECTROMETER. 368. In the ordinary use of the portable electrometer (figs. 8, 9, and 10, Plate II.), the electric system is heterostatic and quite similar to that of the absolute electrometer, when used in the manner described above in 363. But the balance is not adapted for absolute measure of the amount of force of attrac- tion experienced by the moveable disc ; on the contrary, it is precisely the same as that described for the gauge of the quad- rant electrometer in 353 above, only turned upside down. 296 On Electrometers and Electrostatic Measurements, [xx. Thus, in the portable instrument, the square disc (/) forming part of the lever of thin sheet aluminium is attracted upivards by a solid circular disc of sheet-brass ( DD', both roots are negative ; and the electrification comes to zero in time, whatever may be the initial charges. But when IV < DD', one root is positive and the other negative, and ultimately the charges augment in proportion to ept if p be the positive root. 323 ' Electrophoric Apparatus, [XXIII. FIG. 4. inductor be dry polished zinc, and the vessel of water above be copper, the receiver acquires a continually increasing charge of negative electricity. There is little or no effect, either posi- tive or negative, if the inductor present a surface of polished copper to the drops where they break from the continuous water above: but if the copper surface be oxidized by the heat of a lamp, until, instead of a bright metallic surface of copper, it presents a slate-coloured surface of oxide of copper to the drops, these become positively electrified, as is proved by a continually increasing positive charge exhi- bited by the electrometer. When the inner surface of the inductor is of bright metallic colour, either zinc or copper, there seems to be little difference in the effect whether it be wet with water or quite dry; alsa I have not found a considerable difference produced by lining the inner surface of the inductor with moist or dry paper. Copper filings falling from a copper funnel and breaking away from contact in the middle of a zinc inductor, in metallic commu- nication with a copper funnel, as shown in fig. 4, produce a rapidly increasing negative charge in a small insulated can catching them below. The quadrant divided-ring electrometer* in- dicating, by the image of a lamp on a scale, angular motions of a small concave mirror Q- of a grain in weight) such as I use in galvanometers, is very convenient for exhibiting these results. Its sensibility is such that it gives a deflection of 100 scale- divisions ( ? iy of an inch each) on either side of zero, as the effect of a single cell of Daniell's ; the focusing, by small con- cave mirrors supplied to me by Mr Becker, being so good that a deflection can easily be read with accuracy to a quarter of a scale-division. By adopting Peltier's method of a small mag- netic needle attached to the electric moveable body (or "needle"), and by using fixed steel magnets outside the instrument to give directing force (instead of the glass-fibre suspension of the * See Nichol's Encyclopedia. 1860, article "Electricity, Atmospheric;" or Proceedings of the Royal Institution, May 1860, Lecture on Atmospheric Electricity [ 249... 293, above]. Copper Filings. Induci 6 Inductor Zinc, c Receiver. xxiii.] and Illustrations of Voltaic Theory. 329 divided-ring electrometers described in the articles referred to), and by giving a measurable motion by means of a micrometer screw to one of the quadrants, I have a few weeks ago succeeded in making this instrument into an independent electrometer, instead of a mere electroscope, or an electrometer in virtue of a separate gauge electrometer, as in the Kew recording atmo- spheric electrometer, described in the Royal Institution lecture. 407. Reverting to the arrangement described above of a copper vessel of water discharging water in drops from a nozzle through an inductor of zinc in metallic connection with the copper, let the receiver be connected with a second inductor, this inductor insulated ; and let a second nozzle, from an uninsulated stream of water, discharge drops through it to a second receiver. Let this second receiver be connected with a third inductor used to electrify a third stream of water to be caught in a third receiver, and so on. We thus have an ascending scale of electrophorus action analogous to the beautiful mechanical electric multiplier of Mr. C. F. Yarley, with which, by purely electrostatic induc- tion, he obtained a rapid succession of sparks from an ordinary single voltaic element. This result is easily obtained by the self-acting arrangement now described, with the important modification in the voltaic element according to which no chemi- cal action is called into play, and work done by gravity is sub- stituted for work done by the combination of chemical elements. ON A UNIFORM ELECTEIC CURRENT ACCUMULATOR. [From the Philosophical Magazine, January 1868.] 408> CONCEIVE a. closed circuit, C T A B (7, according to the following description : One portion of it, T A, tangential to a circular disc of conducting material and somewhat longer than the radius; the continuation, A B, at right angles to this in the plane of the wheel, of a length equal to the radius ; and the completion of the circuit by a fork, B C, extending to an axle bearing the wheel. If all of the wheel were cut away except a portion, C T, from the axle to the point of contact at the circumference, the circuit would form a simple rectangle, C T A B, except the bifurcation of the side B C. Let a 330 On a Uniform Electric . [xxur. fixed magnet be placed so as to give lines of force perpen- dicular to the wheel, in the parts of it between G the centre and T the point of the circumference touched by the fixed conductor; and let power be applied to cause the wheel to rotate in the direction towards A. According to Faraday's well-known discovery, a current is induced in the circuit in such a direction that the mutual electromagnetic action between it and the fixed magnet resists the motion of the wheel. Now the mutual elec- tromagnetic force between the portions A B and G T of the circuit is repulsive, according to the well-known elemen- tary law of Ampere, and therefore resists the actual motion of the wheel; hence, if the magnet be removed, there will still be electromagnetic induction tending to maintain the current. Let us suppose the velocity of the wheel to have been at first no greater than that practically attained in ordinary ex- periments with Barlow's electromagnetic disc. As the magnet is gradually withdrawn let the velocity be gradually increased so as to keep the strength of the current constant, and, when the magnet is quite away, to maintain the current solely by electromagnetic induction between the fixed and moveable por- tions of the circuit. If, when the magnet is away, the wheel be forced to rotate faster than the limiting velocity of our pre- vious supposition, the current will be augmented according to the law of compound interest, and would go on thus increasing without limit were it not that the resistance of the circuit would become greater in virtue of the elevation of temperature pro- duced by the current. The velocity of rotation which gives by induction an electromotive force exactly equal to that required to maintain the current, is clearly independent of the strength of the current. The mathematical determination of it becomes complicated by the necessity of taking into account the diffusion of the current through portions of the disc not in a straight line between G and T; but it is very simple and easy if we prevent this diffusion by cutting the wheel into an infinite number of infinitely thin spokes, a great number of which are to be simul- xxin.] Current Accumulator. 331 taneously in contact with the fixed conductor at T. The linear velocity of the circumference of the wheel in the limiting case bears to the velocity which measures, in absolute measure, the resistance of the circuit, a ratio (determinable by the solu- tion of the mathematical problem) which depends on the pro- portions of the rectangle GTAB, and is independent of its absolute dimensions. 409. Lastly, suppose the wheel to be kept rotating at any constant velocity, whether above or below the velocity deter- mined by the preceding considerations ; and suppose the current to be temporarily excited in any way (for instance, by bringing a magnet into the neighbourhood and then withdrawing it); the strength of this current will diminish towards zero or will increase towards infinity, according as the velocity is below or above the critical velocity. The diminution or augmentation would follow the compound interest law if the resistance in the circuit remained constant. The conclusion presents us with this wonderful result : that if we commence with absolutely no electric current and give the wheel any velocity of rotation exceeding the critical velocity, the electric equilibrium is un- stable : an infinitesimal current in either direction would aug- ment until, by heating the circuit, the electric resistance be- comes increased to such an extent that the electromotive force of induction just suffices to keep the current constant. 410. It will be difficult, perhaps impossible, to realize this result in practice, because of the great velocity required, and the difficulty of maintaining good frictional contact at the circum- ference, without enormous friction, and consequently frictional generation of heat. 411. The electromagnetic augmentation and maintenance of a current discovered by Siemens, and put in practice by him, with the aid of soft iron, and proved by Maxwell to be theoreti- cally possible without soft iron, suggested the subject of this communication to the author, and led him to endeavour to arrive at a similar result with only a single circuit, and no making and breaking of contacts ; and it is only these characteristics that constitute the peculiarity of the arrangement which he now describes. 332 On Volta-Convection by Flame. [XXIII. FIG. 1. ON VOLTA-CONVECTION BY FLAME. [From the Philosophical Magazine, January 1868.] 412. IN Nichol's Cyclopaedia, article " Electricity, Atmo- spheric" (2d edition), and in the Proceedings of the Eoyal Institution May 1860 (Lecture on Atmospheric Electricity), [ 249... 293, above] the author had pointed out that the effect of the flame of an insulated lamp is to reduce the lamp and other conducting material connected with it to the same potential as that of the air in the neighbourhood of the flame, and that the effect of a fine jet of water from an insulated vessel is to bring the vessel and other conducting material connected with it to the same potential as that of the air at the point where the jet breaks into drops. In a recent communication to the Royal Society " On a Self-acting Ap- paratus for Multiplying and Maintaining Electric Charges, with applications to illustrate the Voltaic Theory," [ 401... 407, above,] an experiment was described in which a water- dropping apparatus was em- ployed to prove the difference of potential in the air, in the neighbourhood of bright metallic surfaces of zinc and copper metallically connected with one another, which is to be expected from Volta's discovery of contact- electricity. In the present com- munication a similar experiment is described, in which the flame of a spirit-lamp is used instead of a jet of water breaking into drops. 413. A spirit-lamp is placed on an insulated stand connected with a very delicate electrometer. Copper and zinc cylinders, in metallic connection with the metal case of the electrometer, are alternately held vertically in such a position that the XXIIL] On Volta-Convection by Flame. 333 flame burns nearly in the centre of the cylinder, which is open at both ends. If the electrometer reading, with the copper cylinder surrounding the flame, is called zero, the reading observed with the zinc cylinder surrounding the flame indicates positive electrification of the insulated stand bearing the lamp. 414. It is to be remarked that the differential method here followed eliminates the ambiguity involved in what is meant by the potential of a conducting system composed partly of flame, partly of alcohol, and partly of metal. In a merely illustrative experiment, which the author has already made, the amount of difference made by substituting the zinc cylinder for the copper cylinder round the flame was rather more than half the differ- ence of potential maintained by a single cell of Daniell's. Thus, when the sensibility of the quadrant divided-ring electrometer ( 406) was such that a single cell of Daniell's gave a deflection of 79 scale-divisions, the difference of the reading when the zinc cylinder was substituted for the copper cylinder round the in- sulated lamp was 39 scale-divisions. From other experiments on contact-electricity made seven years ago by the author, and agreeing with results which have been published by Hankel, it appears that the difference of potentials in the air in the neigh- bourhood of bright metallic surfaces of zinc and copper in metallic connexion with one another is about three-quarters of that of a single cell of Daniell's. It is -quite certain that the difference produced in the metal connected with the insulated lamp would be exactly equal to the true contact difference of the metals, if the interior surfaces of the metal cylinders were perfectly metallic (free from oxidation or any other tarnishing, such as by sulphur, iodine, or any other body) ; provided the distance of the inner surface of the cylinder from the flame were everywhere sufficient to prevent conduction by heated air be- tween them, and provided the length of the cylinder were infinite (or, practically, anything more than three or four times its diameter). 415. The author hopes before long to be able to publish a complete account of his old experiments on contact-electricity, of which a slight notice appeared in the Proceedings of the Literary and Philosophical Society of Manchester [ 400, above]. 334 Electric Replenisher. [XXIIT. ON ELECTKIC MACHINES FOUNDED ON INDUCTION AND CONVECTION. [From the Philosophical Magazine, January 1868.] 416. To facilitate the application of an instrument, which I have recently patented, for recording the signals of the Atlantic Cable, a small electric machine running easily enough to be driven by the wheel work of an ordinary Morse instrument was desired; and I have therefore designed a combination of the electrophorus principle with the system of reciprocal induction explained in [ 401... 407] a recent communication to the Royal Society (Proceedings, June 1867), which may be briefly described as follows : 417. A wheel of vulcanite, with a large number of pieces of metal (called carriers, for brevity) attached to its rim, is kept ro- tating rapidly round a fixed axis. The carriers are very lightly touched at opposite ends of a diameter by two fixed tangent springs. One of these springs (the earth-spring) is connected with the earth, and the other (the receiver-spring) with an in- sulated piece of metal called the receiver, which is analogous to the "prime conductor" of an ordinary electric machine. The point of contact of the earth-spring with the carriers is exposed to the influence of an electrified body (generally an in- sulated piece of metal) called the inductor. When this is negatively electrified, each carrier comes away from contact with the earth-spring, carrying positive electricity, which it gives up, through the receiver-spring, to the receiver. The receiver and inductor are each hollowed out to a proper shape, and are pro- perly placed to surround, each as nearly as may be, the point of contact of the corresponding spring. 418. The inductor, for the good working of the machine, should be kept electrified to a constant potential. This is effected by an adjunct called the replenisher, which may be applied to the main wheel, but which, for a large instrument, ought to be worked by a much smaller carrier- wheel, attached either to the same or to another turning-shaft. 419. The replenisher consists chiefly of two properly shaped pieces of metal called inductors, which are fixed in the neighbour- hood of a carrier-wheel, such as that described above, and four XXIII.] Electric Replenisher. 335 fixed springs touching the carriers at the ends of two diameters. Two of these springs (called receiver-springs) are connected respectively with the inductors; and the other two (called con- necting springs) are insulated and connected with one another (one of the inductors is generally connected with the earth, and the other insulated). They are so situated that they are touched by the carriers on emerging from the inductors, and shortly after FIG. 1. Section. Elevation. the contacts with the receiver-springs. If any difference of potential between the inductors is given to begin with, the action of the carriers, as is easily seen, increases it according to the compound-interest law as long as the insulation is perfect. Practically, in a few seconds after the machine is started running, bright flashes and sparks begin to fly about in various parts of the apparatus, even although the inductors and connectors have been kept for days as carefully discharged as possible. Forty elements of a dry pile (zinc, copper, paper), applied with one pole to one of the inductors, and the other for a moment to the connecting springs and the other inductor, may be used to de- termine, or to suddenly reverse, the character (vitreous or 336 Electric Replenisher. [xxm. resinous) of the electrification of the insulated inductor. The only instrument yet made is a very small one (with carrier- wheel 2 inches in diameter), constructed for the Atlantic FIG. 2. Telegraph application ; but its action has been so startlingly successful that good effect may be expected from larger machines on the same plan. 420. When this instrument is used to replenish the charge of the inductor in the constant electric machine, described above, one of its own inductors is connected with the earth, and the other with the inductor to be replenished. When accurate constancy XXIIL] Applications of the Electric Replenisher. 337 is desired, a gauge- electroscope is applied to break and make contact between the connector-springs of the replenisher when the potential to be maintained rises above or falls below a certain limit. 421. Several useful applications of the replenisher for scien- tific observation were shown by the author at the recent meeting of the British Association (Dundee), among others, to keep up the charge in the Ley den jar for the divided-ring mirror-elec- trometer, especially when this instrument is used for recording atmospheric electricity. A small replenisher, attached to the instrument within the jar, is worked by a little milled head on the outside, a few turns of which will suffice to replenish the loss of twenty-four hours. POSTSCRIPT, Nov. 23, 1867. 422. As has been stated, this machine was planned originally for recording the signals of the Atlantic Cable. The small " replenisher " represented in the diagrams has proved perfectly suitable for this purpose. The first experiments on the method for recording signals which I recently patented were made more than a year ago by aid of an ordinary plate- glass machine worked by hand. This day the small " replenisher " has been connected with the wheelwork drawing the Morse paper on which signals are recorded, and, with only the ordinary driving-weight as moving power, has proved quite successful. 423. The scientific applications indicated when the communi- cation was made to the British Association have been tested with- in the last few weeks, and especially to-day, with the assistance of Professor Tait. The small replenisher is now made as part of each quadrant electrometer. It is permanently placed in the interior of the glass Leyden jar ; and a few turns by the finger applied to a milled head on the outside of the lid are found sufficient to replenish the loss of twenty-four hours. A small instrument has also been made and tested for putting in prac- tice the plan of equalizing potentials, described verbally in the communication to the British Association, which consisted in a mechanical arrangement to produce effects of the same char- acter as those of the water-dropping system, described several T T? 22 338 Potential-Equalizer. [xxrii. years ago at the Royal Institution*. The instrument is repre- sented in the annexed sketch (fig. 3). AT and AT are two springs touching a circular row of small brass pegsf insulated from one another in a vulcanite disc. These springs are insu- lated, one or both, and are connected with the two electrodes of FIG. 3. the electrometer or one of them with the insulated part of the electrometer, and the other with the metal enclosing the case when there is only one insulated electrode. One application is to test the " pyro-electricity " of crystals ; thus a crystal of tour- maline, PN, by means of a metal arm holding its middle, is sup- ported symmetrically with reference to the disc in a position parallel to the line TT', and joining the lines of contact of the springs. When warmed (as is conveniently done by a metal plate at a considerable distance from it), it gives by ordinary tests, as is well known, indications of positive electrification to- * Lecture on Atmospheric Electricity, Proceedings of the Royal Institution, May 1860. See also Michel's Cyclopedia, article "Electricity, Atmospheric" [.249... 293]. t [I now find a smaller number of larger discs to be preferable, as consider- able disturbances are produced by the numerous breakings of contact unless the two springs are in precisely the same condition as to quality and clean- ness of metal surface. Thin stiff platinum pins attached to the discs, and very fine platinum springs touching them as they pass, will probably give good and steady results if the springs are kept very clean. The smallest quantity of the paraffin (with which, as usual in electric instruments, the vulcanite is coated), if getting on either spring, would probably produce im- mense disturbance. December 23, 1867.] xxiii.] Applications of Potential-Equalizer. 339 wards the one end P, and of negative electrification towards the other end N. The wheel in the arrangement now described is kept turning at a rapid rate ; and the effect of the carrier is to produce in the springs TA, T'A' the same potentials, approxi- mately, as those which would exist in the air at the points T, T' if the wheel and springs were removed. The springs being connected with the electrodes of the divided-ring quadrant electrometer, the spot of light is deflected to the right, let us say. After continuing the application of heat for some time the hot plate is removed, and a little later the spot of light goes to zero and passes to the left, remaining there for a long time, and indicating a difference of potentials between the springs, in the direction A'T positive and AT negative. The electrometer being of such sensibility as to give a deflection of about 100 scale-divisions to the right or left when tested by a single gal- vanic cell, and having a range of 300 scale-divisions on each side, it is necessary to place the tourmaline at a distance of several inches from the disc to keep the amount of the deflec- tion within the limits of the scale. 424. Another application of this instrument is for the experimental investigation of the voltaic theory, according to the general principle described [ 406] in the communi- cation to the Royal Society already referred to.* In it two inductors are placed as represented in fig. 4. The inner FIG. 4. surface of each of these is of smooth brass ; and one of them is lined wholly, or partially, with sheet zinc, copper, silver, or other metal to be tested. Thus, to experiment upon the contact difference of potentials between zinc and copper, Proceedings of the Royal Society, May 1867. 222 340 Applications of Potential-Equalizer. [xxiu. one of the inductors is wholly lined with sheet zinc or with sheet copper, and the two inductors are placed in me- tallic communication with one another. The springs are eacli in metallic communication with the electrodes of the quadrant mirror electrometer, and the wheel is kept turning. The spot of light is observed to take positions differing, according as the lining is zinc or copper, by 72J per cent, of the difference pro- duced by disconnecting the two inductors from one another and connecting them with the two plates of a single Daniell's cell, when either the zinc or the copper lining is left in one of them. These differences are very approximately in simple proportion to the differences of potentials between the pairs of the opposite quadrants of the electrometer in the different cases. The dif- ference between the effects of zinc and of copper in this arrange- ment is of course in the direction corresponding to the positive electrification of the quadrants connected with the spring whose point of contact is exposed to the zinc-lined inducing surface. It must be remembered, however, as is to be expected from Hankel's observations, that the difference measured will be much affected by a slight degree of tarnishing by oxidation, or other- wise, of the inner surface of either inductor. When the copper surface is brought to a slate-colour by oxidation under the influence of heat, the contact difference between it and polished zinc amounts sometimes, as I found in experiments made seven years ago [ 400, above], to 125, that of a single cell of Daniell's being called 100. 425. A useful application of the little instrument represented in fig. 4 is for testing insulation of insulated conductors of small capacity, as for instance, short lengths (2 or 3 feet) of submarine cable, when the electrometer used is such that its direct appli- cation to the conductor to be tested would produce a sensible disturbance in its charge, whether through the capacity of the elec- trometer being too great, or from inductive effects due to motion of the moveable part, or parts, especially if the electrometer is " heterostatic " [ 385]. In this application one of the induc- tors is kept in connection with a metal plate in the water sur- rounding the specimen of cable to be tested ; and the other is connected with the specimen, or is successively connected with the different specimens under examination. The springs are xxiii.] On the Reciprocal Electrophorus. 341 connected with the two electrodes of. the electrometer as usual. The small constant capacity of the insulated inductor, and the practically perfect insulation which may with ease be secured for the single glass or vulcanite stem bearing it, are such that the application of the testing apparatus to the body to be tested produces either no sensible change, or a small change which can be easily allowed for. It will be seen that the small metal pegs carried away by the turning-wheel from the point of the insulated spring, in the arrangement last described, correspond precisely to the drops of water breaking away from the nozzle in the water-dropping collector for atmospheric electricity. 426. A form bearing the same relation to that represented in the drawings that a glass-cylinder electric machine bears to a plate-glass machine of the ordinary kind will be more easily made, and will probably be found preferable, when the dimensions are not so great as to render it cumbrous. In it. it is proposed to make the carrier-wheel nearly after the pattern of a mouse- mill, with discs of vulcanite instead of wood for its ends. The inductor and receiver of the rotatory electrophorus or the two inductor -receivers of the replenisher, may, when this pattern is adopted, be mere tangent planes; but it will probably be found better to bend them somewhat to a curved cylindrical shape not differing very much from tangent planes. When, however, great intensity is desired, the best pattern will pro- bably be had by substituting for the carrier-wheel an endless rope ladder, as it were, with cross bars of metal and longitudinal cords of silk or other flexible insulating material. This, by an action analogous to that of the chain-pump, will be made to move with great rapidity, carrying electricity from a properly placed inductor to a properly shaped and properly placed re- ceiver at a distance from the inductor which may be as much as several feet. ON THE RECIPROCAL ELECTROPHORUS. [From the Philosophical Magazine, April 1868.] 427. Having been informed by Mr. Fleeming Jenkin that he had heard from Mr. Clerk Maxwell that the instrument which I described under the name " Replenisher," in the Philosophical 342 On the Reciprocal Electrophorus. [xxm. Magazine for January 1868, was founded on precisely the same principle as an instrument "for generating electricity" which had been patented some years ago by Mr. C. F. Varley, I was surprised ; for I remembered his inductive machine which had been so much admired at the Exhibition of 1862, and which certainly did not contain the peculiar principle of the "Replenisher." But I took the earliest opportunity of looking into Mr. Varley 's patent (1860), and found, as was to be ex- pected, that Mr. Maxwell was perfectly right. In that patent Mr. Varley describes an instrument agreeing in almost every detail with the general description of the "Replenisher" which 1 gave in the article of the Philosophical Magazine already referred to. The only essential difference is that no contacts are made in Mr. Varley's instrument, but, instead, the carriers pass, each at four points of its circular path, within such short distances of four metallic pieces that when a sufficient intensity of charge has been reached, sparks pass across the air-intervals. Hence to give a commencement of action to Mr. Varley's instru- ment, one of the inductors must be charged from an indepen- dent source to a considerable potential (that of several thousand cells for instance), to make sure that sparks will pass between the carriers and the metal piece (corresponding to one of my connecting springs) which it passes under the influence of that inductor. In my " Replenisher," however well discharged it may be to begin with, electrification enough is reached after a few seconds (on the compound interest principle, with an in- finitesimal capital to begin with) to produce sparks and flashes in various parts of the instrument. In Mr. Varley's instrument, what corresponds to my connector is described as being con- nected with the ground ; and the effect is to produce positive and negative electrification of the two inductors. In this re- spect it agrees with the self-acting apparatus for multiplying and maintaining electric charges, described in a communication to the Royal Society last May.* From this arrangement I passed to the "Replenisher" by using a wheel with carriers as a substitute for the water-droppers, and arranging that the connectors might be insulated and one of the inductors con- * Proceedings of the Royal Society, 1867; or, Phil. Nag., November 1867. xxin.] On the Reciprocal Electrophorus. 343 nected with the earth, which, of course, may be done in Mr. Yarley's instrument, and which renders it identical with mine, with the exception of the difference of spring-contacts instead of sparks. This difference is essential for some of the applica- tions of the " Replenishes" which I described, and have found very useful, especially the small internal replenish er, for reple- nishing, when needed, the charges of the Leyden jar of my heterostatic electrometers. But the reciprocal-electrophorus principle, which seemed to me a novelty in the communication to the Royal Society and in the Philosophical Magazine article of last January referred to, had, as I now find, been invented and published by Mr. Varley long before, in his patent of 1860, when it was, I believe, really new to science. 428. POSTSCRIPT. GLASGOW COLLEGE, March 20, 1868. In looking further into Mr. Varley 's patent, I find that he describes an arrangement for making spring-contacts instead of the narrow air-spaces for sparks, and that he uses the spring-contacts to enable him to commence with a very small difference of poten- tials, and to magnify on the compound interest principle. He even states that he can commence with such a difference of potentials as can be produced by a single thermo-electric element, and by the use of his inductive instrument can multiply this in a measured proportion until he reaches a difference of potentials measurable by an ordinary electrometer. Thus it appears that his anticipation of all that I have done in my " Replenish er" is even more complete than I supposed when writing the preceding. 429. SECOND POSTSCRIPT (1870). On having had my atten- tion called to Nicholson's "Revolving Doubler," I find in it the same compound interest principle of electrophone action. It seems certain that the discovery is Nicholson's, and about one hundred years old. Holtz's now celebrated electric machine, which is closely analogous in principle to Varley's of 1860, is, I believe, a descendant of Nicholson's. Its great power depends on the abolition by Holtz of metallic carriers, and of metallic make-and-break contacts. Its inductive principle is identical with that of Varley's earlier and my own later invention. It differs from Varley's and mine in leaving the inductors to them- selves, and using the current in the "connecting" arc ( 419), which, when sparks are to be produced, is broken. XXIV.-A MATHEMATICAL THEORY OF MAGNETISM. [Abstract from the Proceedings of the Royal Society, June 1849.] 430. THE theory of magnetism was first mathematically treated in a complete form by Poisson. Brief sketches of his theory, with some simplifications, have been given by Green and Murphy in their works on Electricity and Magnetism. In all these writings a hypothesis of two magnetic fluids has been adopted, and strictly adhered to throughout. -No physical evidence can be adduced in support of such a hypothesis ; but on the contrary, recent discoveries, especially in electro- magnetism, render it extremely improbable. Hence it is of importance that all reasoning with reference to magnetism should be conducted without assuming the existence of those hypothetical fluids. 431. The writer of the present paper endeavours to show that a complete mathematical theory of magnetism may be established upon the sole foundation of facts generally known, and Cou- lomb's special experimental researches. The positive parts of this theory agree with those of Poisson's mathematical theory, and consequently the elementary mathematical formulae coin- cide with those which have been previously given by Poisson. The paper at present laid before the Royal Society is re- stricted to the elements of the mathematical theory, exclusively of those parts in which the phenomena of magnetic induction are considered. The author hopes to have the honour of laying before the Society a continuation, containing some original mathematical investigations on magnetic distributions, and a theory of induc- tion, in ferromagnetic or diamagnetic substances. xxiv.] A Mathematical Theory of Magnetism. 345 [Transactions of the Roijal Society for June 1849, and June 1850.] Introduction. 432. THE existence of magnetism is recognised by certain phenomena of force which are attributed to it as their cause. Other physical effects are found to be produced by the same agency; as in the operation of magnetism with reference to polarized light, recently discovered by Mr Faraday ; but we must still regard magnetic force as the characteristic of mag- netism, and, however interesting such other phenomena may be in themselves, however essential a knowledge of them may be for enabling us to arrive at any satisfactory ideas regarding the physical nature of magnetism, and its connexion with the general properties of matter, we must still consider the investi- gation of the laws, according to which the development and the action of magnetic force are regulated, to be the primary object of a Mathematical Theory in this branch of Natural Philosophy. 433. Magnetic bodies, when put near one another, in general exert very sensible mutual forces; but a body which is not magnetic can experience no force in virtue of the magnetism of bodies in its neighbourhood. It may indeed be observed that a body, M, will exert a force upon another body A ; and again, on a third body J3; although when A and B are both removed to a considerable distance from M, no mutual action can be discovered between themselves ; but in all such cases A and B are, when in the neighbourhood of M, temporarily magnetic ; and when both are under the influence of M at the same time, they are found to act upon one another with a mutual force. All these phenomena are investigated in the mathematical theory of magnetism, which, therefore, comprehends two dis- tinct kinds of magnetic action the mutual forces exercised between bodies possessing magnetism, and the magnetization induced in other bodies through the influence of magnets. The First Part of this paper is confined to the more descriptive and positive details of the subject, with reference to the former class of phenomena. After a sufficient foundation has been laid in it, by the mathematical exposition of the distribution of magnetism in bodies, and by the determination and expression 346 A Mathematical Theory of Magnetism. [xxiv. of the general laws of magnetic force, a Second Part will be devoted to the theory of magnetization by influence, or magnetic induction. FIRST PART. ON MAGNETS, AND THE MUTUAL FORCES BETWEEN MAGNETS. CHAPTER I. Preliminary Definitions and Explanations. 434. A magnet is a substance which intrinsically possesses magnetic properties. A piece of loadstone, a piece of magnetized steel, a galvanic circuit, are examples of the varieties of natural and artificial magnets at present known ; but a piece of soft iron, or a piece of bismuth tem- porarily magnetized by induction, cannot, in unqualified terms, be called a magnet. A galvanic circuit is frequently, for the sake of distinction, called an " electro-magnet;" but, according to the preceding definition of a magnet, the simple term, without qualification, may be applied to such an arrangement. On the other hand, a piece of apparatus con- sisting of a galvanic coil, with a soft iron core, although often called simply "an electro-magnet," is in reality a complex arrangement involving an electro-magnet (which is intrinsically magnetic as long as the electric current is sustained) and a body transiently magnetized by induction. 435. In the following analysis of magnets, the magnetism of every magnetic substance considered will be regarded as ab- solutely permanent under all circumstances. This condition is not rigorously fulfilled either for magnetized steel or for load- stone, as the magnetism of any such substance is always liable to modification by induction, and may therefore be affected either by bringing another magnet into its neighbourhood, or by breaking the mass itself and separating the fragments. When, however, we consider the magnetism of any fragment taken from a steel or loadstone magnet, the hypothesis will be that it retains without any alteration the magnetic state which it actually had in its position in the body. The general theory of the distribution of magnetism founded upon conceptions of this kind, will be independent of the truth or falseness of any such hypothesis which may be made for the sake of conveni- xxiv.] A Mathematical Theory of Magnetism. 347 ence in studying the subject; but of course any actual experi- ments in illustration of the analysis or synthesis of a magnet would be affected by a want of rigidity in the magnetism of the matter operated on. For such illustrations electro-magnets [without iron or other magnetic substance] are extremely appropriate, as in them, except during the motion by which any alteration in their form or arrangement is effected, no appreciable inductive action can exist. 436. In selecting from the known phenomena of magnetism those elementary facts which are to serve for the foundation of the theory, all complex actions depending on the irregularities of the bodies made use of should be excluded. Thus if we were to attempt an experimental investigation of the action between two amorphous fragments of loadstone, or between two pieces of steel magnetized by ordinary processes, we should probably fail to recognise the simple laws on which the actions resulting from such complicated circumstances depend ; and we must look* for a simpler case of magnetic action before we can make an analysis which may lead to the establishment of the fundamental principles of the theory. Much complica- tion will be avoided if we take a case in which the irregularities of one, at least, of the bodies do not affect the phenomena to be considered. Now, the earth, as was first shown by Gilbert, is a magnet ; and its dimensions are so great that there is no sensible variation in its action on different parts of any ordinary magnet upon which we can experiment, and conse- quently, in the circumstances, no complicacy depending on the actual distribution of terrestrial magnetism. We may therefore, with advantage, commence by examining the action which the earth produces upon a magnet of any kind at its surface. 437. At a very early period in the history of magnetic dis- covery the remarkable property of " pointing north and south " was observed to be possessed by fragments of loadstone and magnetized steel needles. To form a clear conception of this phenomenon, we must consider the total action produced by the earth upon a magnet of any kind, and endeavour to dis- tinguish between the effects of gravitation which the earth exerts upon the body in virtue of its weight, and those which result from the magnetic agency. 348 A Mathematical Theory of Magnetism. [xxiv. 438. In the first place, it is to be remarked that the mag- netic agency of the earth gives rise to no resultant force of sensible magnitude, upon any magnet with reference to which we can perform experiments [that is to say, small enough to be a subject for laboratory experiments], as is proved by the following observed facts : (1.) A magnet placed in any manner, and allowed to move with perfect freedom in any horizontal direction (by being floated, for example, on the surface of a liquid), experiences no action which tends to set its centre of gravity in motion, and there is therefore no [directly observable] horizontal force upon the body. (2.) The magnetism of a body may be altered in any way, without affecting its weight as indicated by a balance. Hence there can be no [directly observable] vertical force upon it depending on its mag- netism. 439. It follows that any magnetic action which the earth can exert upon a magnet [of dimensions suitable for laboratory experiments] must be [sensibly] a couple. * To ascertain the manner in which this action takes place, let us conceive a magnet to be supported by its centre of gravity* and left per- fectly free to turn round this point, so that, without any con- straint being exerted which could balance the magnetic action, the body may be in circumstances the same as if it were with- out weight. The magnetic action of the earth upon the magnet gives rise to the following phenomena : (1 .) The magnet does not remain in equilibrium in every position in which it may be brought to rest, as it would do did it experience no action but that of gravitation. (2.) If the magnet be placed in a position of equilibrium there is a certain axis (which, for the present, we may conceive to be found by trial), such that if the magnet be turned round it, through any angle, and be brought to rest, it will remain in equilibrium. * The ordinary process for finding experimentally the centre of gravity of a body fails when there is any magnetic action to interfere with the effects of gravitaton. It is, however, for our present purpose, sufficient to know that the centre of gravity exists ; that is, that there is a point such that the vertical line of the resultant action of gravity passes through it, in whatever position the body be held. If it were of any consequence, a process some- what complicated by the magnetic action, for actually determining, by ex- periment, the centre of gravity of a magnet might be indicated, and thus the experimental treatment of the subject in the text would be completed. xxiv.] A Mathematical Theory of Magnetism. 349 (3.) If the magnet be turned through 180, about an axis perpen- dicular to this, it will again be in a position of equilibrium. (4.) Any motion of the magnet whatever, which is not of either of the kinds just described, nor compounded of the two, will bring it into a position in which it will not be in equilibrium. (5.) The directing couple experienced by the magnet in any posi- tion depends solely on the angle of inclination of the axis described in (1.) to the line along which it lies when the magnet is in equilibrium; being independent of the position of the plane of this angle, and of the different positions into which the magnet is brought by turning it round that axis. 440. From these observations we draw the conclusion that a magnet always experiences a directing couple from the earth unless a certain axis belonging to it is placed in a determinate position. This line of the magnet is called its magnetic axis.* 441. The direction towards which the magnetic axis of the magnet tends in virtue of the earth's action, is called " the line of dip," or "the direction of the total terrestrial magnetic force," at the locality of the observation. 442. No further explanation regarding phenomena which depend on terrestrial magnetism is required in the present chapter ; but, as the facts have been stated in part, it may be right to complete the statement, as far as regards the action experienced by a magnet of any kind when held in different positions in a given locality, by mentioning the following conclusions, deduced in a vory obvious manner from the general laws of magnetic action stated below, and verified fully by experiment : If a magnet be held with its magnetic axis inclined at any angle to the line of dip, it will experience a couple, the moment of which is proportional to the sine of the angle of inclination, acting in a plane containing the magnetic axis and the line of dip. The position of equilibrium towards which this couple tends to bring the magnetic axis is stable, and if the direc- tion of the magnetic axis be reversed, the magnet may be left balanced, but it will be in unstable equilibrium. * Any line in the body parallel to this might, with as good reason, be called a magnetic axis, but when we conceive the magnet to be supported by its centre of gravity, the magnetic axis is naturally taken as a line through this point. [See addition to 444.] 350 A Mathematical Theory of Magnetism. [xxiv. 443. The directive tendency observed in magnetic bodies being found to depend on their geographical position, and to be related, in some degree, to the terrestrial poles, received the name of polarity, probably on account of a false hypothesis of forces exercised by the pole-star* or by the earth's poles upon certain points of the loadstone or needle, thence called the " poles of the magnet." The terms " polarity " and " poles " are still retained, but the use of them, which has very generally been made, is nearly as vague as the ideas from which they had their origin. Thus, when the magnet is an elongated mass, its ends are called poles if its magnetic axis be in the direction of its length; no definite points, such as those in which the surface of the body is cut by the magnetic axis, being pre- cisely indicated by the term as it is generally used. If, how- ever, the body be symmetrical about its magnetic axis, and symmetrically magnetized, whether elongated in that direction or not, the poles might be definitely the ends of the magnetic acds (or the points in which the surface is cut by it), unless the magnet be annular and not cut by its magnetic axis (a ring electro-magnet, for instance), in which case the ordinary con- ception of poles fails. Notwithstanding this vagueness, how- ever, the terms poles and polarity are extremely convenient, and, with the following explanations, they will frequently be made use of in this paper : 444. Let be any point in a magnet, and let NOS be a straight line parallel to the line defined above as the magnetic axis through the centre of gravity. If the point 0, however it has been chosen, be called the centre of the magnet, the line NS, terminated either at the surface, on each side, or in any arbitrary manner, is called the magnetic axis, and the ends N, S, of the magnetic axis are called the poles of the magnet. -f * In the poem of Guiot de Provence (quoted in Whewell's History of the Inductive Sciences, vol. ii. p. 46), a needle is described as being magnetized and placed in or on a straw (floating on water it is to be presumed) " Puis se torne la pointe toute Centre 1'estoile sans doute." t A definition of poles at variance with this is adopted in some special cases, especially in that of the earth considered as a great magnet, but the manner in which the term will be used in this paper will be such as to produce no confusion on this account. xxiv.] A Mathematical Theory of Magnetism. 351 [Addition, 1871. Later, 494, a proper central axis, to be called the magnetic axis, and a point in it which may be called the magnetic centre, will be defined according to purely mag- netic conditions.] 445. That pole (marked N) which points, on the whole, from the north, and, in northern latitudes, upwards, is called the north pole, and the other ($), which points from the south, is called the south pole. 446. The sides of the body towards its north pole and south pole are said to possess " northern polarity " and " southern polarity " respectively, an expression obviously founded on the idea that the surface of a magnet may in general be contem- plated as a locus of poles. 447. If a magnetic body be broken up into any number of fragments, each morsel is found to be a complete magnet, presenting in itself all the phenomena of poles and polarity. This property is generally contemplated when, in modern writings on physical subjects, polarity is mentioned as a property belonging to a solid body ; and a corresponding idea is involved in the term when it is applied with reference to the electric state which Mr Faraday discovered to be induced in non-conductors of electricity ("dielectrics") when subjected to the influence of electrified bodies.* However different are the physical circumstances of. magnetic and electric polarity, it appears that the positive laws of the phenomena are the same,-)- and therefore the mathematical theories are identical. Either subject might be taken as an example of a very important branch of physical mathematics, which might be called "A Mathematical Theory of Polar Forces." 448. Although we have seen that any magnet, in general, experiences from the earth an action subject to certain very simple laws, yet the actual distribution of the magnetism which it possesses may be extremely irregular. We may certainly conceive that if the magnetized substance be a regular crystal of magnetic iron ore, the magnetism is distri- * Faraday's Experimental Researches in Electricity, Eleventh Series, t Sec a paper "On the Elementary Laws of Statical Electricity, " published in the Cambridge and Dublin Mathematical Journal (vol. i.) in December 1845. 352 A Mathematical Theory of Magnetism. [xxiv. buted through it according to some simple law; but by taking an amorphous and heterogeneous fragment of ore presenting magnetic properties, by magnetizing in any way an irregular mass of steel, by connecting any number of morsels of magnetic matter so as to make up a complex magnet, or by bending a galvanic wire into any form, we may obtain magnets in which the magnetic property is distributed in any arbitrary manner, however irregular. Excluding for the present the last-men- tioned case, let us endeavour to form a conception of the distribution of magnetism in actually magnetized matter, such as steel or loadstone, and to lay down the principles according to which it may in any instance be mathematically expressed. 449. In general we may consider a magnet as composed of matter which is magnetized throughout, since, in general, it is found that any fragment cut out of a magnetic mass is itself a magnet possessing properties entirely similar to those which have been described as possessed by any magnet whatever. It may be, however, that a small portion cut out of a certain position in a magnet, may present no magnetic phenomena; and if we cut equal and similar portions from different posi- tions, we may find them to possess magnetic properties differing to any extent both in intensity and in the directions of their magnetic axes. 450. If we find that equal and similar portions, cut in parallel directions, from any different positions in a given magnetic mass, possess equal and similar magnetic properties, the mass is said to be uniformly magnetized. 451. In general, however, the intensity of magnetization must be supposed to vary from one part to another, and the magnetic axes of the different parts to be not parallel to one another. Hence, to lay down deter minately a specification of the distribution of magnetism through a magnet of any kind, we must be able to express the intensity and the direction of magnetization at each point. Before attempting to define a standard for the numerical expression of intensity of magneti- zation, it will be convenient to examine the elementary laws upon whieh the phenomena of magnetic force depend, since it is by these effects that the nature and energy of the magnetism to which they are due must be estimated. xxiv.] A Mathematical Theory of Magnetism. 353 CHAPTER II. On the Laws of Magnetic Force, and on the Distribution of Magnetism in Magnetized Matter. 452. The object of the elementary magnetic researches of Coulomb was the determination of the mutual action between two infinitely thin, uniformly and longitudinally magnetized bars. The magnets which he used were in strictness neither uniformly nor longitudinally magnetized, such a state being unattainable by any actual process of magnetization; but, as the bars were very thin cylindrical steel wires, and were symmetrically magnetized, the resultant actions were sensibly the same as if they were in reality infinitely thin, and longi- tudinally magnetized ; and from experiments which he made, it appears that the intensity of the magnetization must have been very nearly constant .from the middle of each of the bars to within a short distance from either end, where a gradual decrease of intensity is sensible*. 453. These circumstances having been attended to, Coulomb was able to deduce from his experiments the true laws of the phenomena, and arrived at the following conclusions : (1) If two thin uniformly and longitudinally magnetized bars be held near one another, an action is exerted between them which consists of four distinct forces, along the four lines joining their extremities. (2) The forces between like ends of the two bars are re- pulsive-f. (3) The forces between unlike ends are attractive. (4) If the bars be held so that the four distances between their extremities, two and two, are equal, the four forces between them will be equal. (5) If the relative positions of the bars be altered, each force will vary inversely as the square of the mutual distance of the poles between which it acts. * See note on 469, below. t Hence we see the propriety of the terms north and south applied to the opposite polarities of a magnet, as explained above. Thus we designate the polarity, or the imaginary magnetic matter of the northern and southern magnetic hemispheres of the earth, as northern and southern respectively; and since the poles of ordinary magnets which are repelled by the earth's northern or southern polarity must be similar, these also are called northern or southern, as the case may be. T. E. 23 354 A Mathematical Theory of Magnetism. [xxiv. 454. To establish a standard for estimating the strength of a magnet, let us conceive two infinitely thin bars to be placed so that either end of one may be at unit of distance from an end of the other. Then, if the bars be equally magnetized, each uniformly and longitudinally, to such a degree that the force between those ends shall be unity, the strength of each bar- magnet is unity*. 455. If any number, m, of such unit bars, of equal length, be put with like ends together, so as to constitute a single complex bar, the strength of the magnet so formed is denoted by m. If there be any number of thin bar-magnets of equal length, and each of them of such a strength that q of them, with like ends together, would constitute a unit-bar ; and if p of those bars be put with like ends together, the strength of the complex magnet so formed will be - . 456. If a single infinitely thin bar be magnetized to such a degree that in the same positions it would produce the same effects as a complex bar of any strength m (an integer or fraction), the strength of this magnet is denoted by m. 457. If two complex bar-magnets, of the kind described above, be put near one another, each bar of one will act on each bar of the other with the same forces as if all the other bars were removed. Hence, if the distance between the two poles be unity, and if the strengths of the bars be respectively m and m' (whether these numbers be integral or fractional), the force between those poles will be mm'. If, now, the relative position of the magnets be altered, so that the distance between two poles may be /, the force between them will, according to Coulomb's law, be mm * The Koyal Society, in its Instructions for making observations on Terres- trial Magnetism, adopts one foot as the unit of length; and that force which, if acting on a grain of matter, would in one second of time generate one foot per second of velocity, as the unit of force ; which is consequently very nearly -1_ of the weight, in any part of Great Britain or Ireland, of one grain. [Note, 1871. The British Association's Committee on Electric Measurement have recently adopted the centimetre as unit of length, and the gramme as unit of mass, instead of the foot and grain.] xxiv.] A Mathematical Theory of Magnetism. 355 According to the definition given above of the strength of a simple bar-magnet, it follows that the same expression gives the force between two poles of any thin uniformly and longi- tudinally magnetized bars, of strengths m and m'. 458. The magnetic moment of an infinitely thin, uniformly and longitudinally magnetized bar, is the product of its length into its strength. 459. If any number of equally strong, uniformly and longi- tudinally magnetized rectangular bars of equal infinitely small sections, be put together with like ends towards the same parts, a complex uniformly magnetized solid of any form may be produced. The magnetic moment of such a magnet is equal to the sum of the magnetic moments of the bars of which it is composed. 460. The magnetic moment of any continuous solid, uni- formly magnetized in parallel lines, is equal to the sum of the magnetic moments of all the thin uniformly and longitudinally magnetized bars into which it may be divided. It follows that the magnetic moment of any part of a uni- formly magnetized mass is proportional to its volume. 461. The intensity of magnetization of a uniformly magnet- ized solid is the magnetic moment of a unit of its volume. It follows that the magnetic moment of a uniformly mag- netized solid, of any form and dimensions, is equal to the product of its volume into the intensity of its magnetization. 462. If a body be magnetized in any arbitrary regular or irregular manner, a portion may be taken in any position, so small in all its dimensions that the distribution of magnetism through it will be sensibly uniform. The quotient obtained by dividing the magnetic moment of such a portion, in any posi- tion P, by its volume, is the intensity of magnetization of the substance at the point P; and a line through P parallel to its lines of magnetization, is the direction of magnetization, at P. CHAPTER III. On the Imaginary Magnetic Matter ly means of which the Polarity of a Magnetized Body may be represented. 463. It will very often be convenient to refer the phenomena of magnetic force to attractions or repulsions mutually exerted 232 356 A Mathematical Theory of Magnetism. [xxiv. between portions of an imaginary magnetic matter, which, as we shall see, may be conceived to represent the polarity of a magnet of any kind. This imaginary substance possesses none of the primary qualities of ordinary matter, and it would be wrong to call it either a solid, or the " magnetic fluid " or "fluids"; but, without making any hypothesis whatever, we may call it " magnetic matter," on the understanding that it possesses only the property of attracting or repelling magnets, or other portions of "matter" of its own kind, according to certain determinate laws, which may be stated as follows : (1) There are two kinds of imaginary magnetic matter, northern and southern, to represent respectively the northern and southern magnetic polarities of the earth, or the similar polarities of any magnet whatever. (2) Like portions of magnetic matter repel, and unlike por- tions attract, mutually. (3) Any two small portions of magnetic matter exert a mutual force which varies inversely as the square of the dis- tance between them. (4) Two units of magnetic matter, at a unit of distance from one another, exert a unit of force, mutually. 464. If quantities of magnetic matter be measured numeri- cally in such units, and if the positive or negative sign be prefixed to denote the species of matter, whether northern (which, by convention, we may call positive) or southern, all the preceding laws are expressed in the following proposi- tion : If quantities m and m', of magnetic matter be concentrated respectively at points at a distance, f, from one another, they will repel with a force algebraically equal to mm' ~F\ 465. It appears from the explanations given above that the circumstances of a uniformly magnetized needle may be repre- sented if we imagine equal quantities of northern and southern magnetic matter to be concentrated at its two poles, the numerical measure of these equal quantities being the same as that of the " strength " of the magnet. The mutual action between two needles would thus be xxiv.] A Mathematical Theory of Magnetism. 357 reduced to forces of attraction and repulsion between the portions of magnetic matter by which their poles are represented. 466. Any magnetic mass whatever may, as we have seen, be regarded as composed of infinitely small bar-magnets put to- gether in such a way as to produce the distribution of mag- netism which it actually possesses ; and hence, by substituting imaginary magnetic matter for the poles of these magnets, we obtain a distribution of equal quantities of northern and southern magnetic matter through the magnetized substance, by which its actual magnetic condition may be represented. The distribution of this matter becomes very much simplified, from the circumstance that we have in general unlike poles of the elementary magnets in contact, by which the opposite kinds of magnetic matter are partially (or in a class of cases wholly*) destroyed through the interior of the body. The determination of the resulting distribution of magnetic matter, which represents in the simplest possible manner the polarity of any given magnet, is of much interest, and even importance, in the theory of magnetism, and we may therefore make this an object of investigation, before going further. 467. Let it be required to find the distribution of imaginary magnetic matter to represent the polarity of any number of uniformly magnetized needles, S^, S 9 N 9 ,...S n N n , of strengths fjL t) jui 2 , ... yu- n respectively, when they are placed together, end to end (not necessarily in the same straight line). If A denote the position occupied by S l when the bars are in their places ; if N t and $ 2 are placed in contact at K^ ; N 9 and $ 3 , at A" 2 ; and so on until we have the last magnet, with its end S n , in contact with N n _ lt at K n _ lt and its other end, N n , free, at a point B ; we shall have to imagine /jL t units of southern magnetic matter to be placed at A ; ^ units of northern, and fju 2 units of southern matter at K^ ; /* 2 units of northern, and ^ of southern matter at K z ; //, M _, units of northern, and p n of southern matter at K n _^ and lastly, fju n units of northern matter at B. * In all cases when the distribution is " solenoidal. " See below, Chap. v. 499 ; communicated to the Koyal Society, June 20, 1850. 358 A Mathematical Theory of Magnetism. [xxiv. Hence the final distribution of magnetic matter is as follows : . . . at A and fji n . . . . . -v . . . . B. 468. The complex magnet AKf^..R n _J$ consists of a number of parts, each of which is uniformly and longitudinally magnetized, and it will act in the same way as a simple bar of the same length, similarly magnetized ; and hence the magnetic matter which represents a bar-magnet AB of this kind is con- centrated in a series of points, at the ends of the whole bar, and at all the places where there is a variation in the strength* of its magnetization. 469. If the length of each part through which the strength of the magnetism is constant, be diminished without limit, and if the entire number of the parts be increased indefinitely, a straight or curved infinitely thin bar may be conceived to be produced, which shall possess a distribution of longitudinal magnetism varying continuously from one end to the other according to any arbitrary law. If the strength of the magnet- ism at any point P of this bar be denoted by JJL, and if [//-] and (//,) denote the values of p, at the points A and B, the investi- gation of 467, with the elementary principles and notation of the differential calculus, leads at once to the determination of the ultimate distribution of magnetic matter by which such a bar-magnet may be represented. Thus if AP be denoted by s ; fi will be a function of s, which may be supposed to be known, and its differential coefficient will express the continuous dis- tribution of magnetic matter which replaces the group of material points at K I} K 2 , etc.; so that the entire distribution of polarity in the bar and. at its ends will be as follows: in * This expression is equivalent to the product of the intensity of magnetiza- tion into the section of the bar; and by retaining it we are enabled to include cases in which the bar is not of uniform section. xxiv.] A Mathematical Theory of Magnetism. 359 any infinitely small length, ] at A, and (JJL) at B. It follows that if, through any part of the length of a bar, the strength of the magnetism is constant, there will be no magnetic matter to be distributed through this portion of the magnet ; but if the strength of the magnetism varies, then, according as it diminishes or increases from the north to the south pole of any small portion, there will be a distribution of northern or southern magnetic matter to represent the polarity which results from this variation. Corresponding inferences may be made conversely, with re- ference to the distribution of magnetism, when the distribution of the imaginary magnetic matter is known. Thus Coulomb found that his long thin cylindrical bar-magnets acted upon one another as if each had a symmetrical distribution of the two kinds of magnetic matter, northern within a limited space from one end, and southern within a limited space from the other, the intermediate space (constituting generally the greater part of the 'bar) being unoccupied ; from which we infer that no variation in the magnetism was sensible through the middle part of the bar, but that, through a limited space on each side, the intensity of the magnetization must have decreased gradu- ally towards the ends*. * This circumstance was alluded to above, in 452. Interesting views on the subject of the distribution of magnetism in bar-magnets are obtained by taking arbitrary examples to illustrate the investigation of the text. Thus we may either consider a uniform bar variably magnetized, or a thin bar of varying thickness, cut from a uniformly magnetized substance ; and, accord- ing to the arbitrary data assumed, various remarkable results may be ob- tained. We shall see afterwards that any such data, however arbitrary, may be actually produced in electro-magnets, and we have therefore the means of illustrating the subject experimentally, in as complete a manner as can be conceived, although from the practical non-rigidity of the magnetism of magnetized substances, ordinary steel or loadstone magnets would not afford such satisfactory illustrations of arbitrary cases as might be desired. The distribution of longitudinal magnetism in steel needles actually magnetized in different ways, and especially " magnetized to saturation," has been the 360 A Mathematical Theory of Magnetism. [xxiv. 470. The distribution of magnetic matter which represents the polarity of a uniformly magnetized body of any form, may be immediately determined if we imagine it divided into in- finitely thin bars, in the directions of its lines of magnetization; for each of these bars will be uniformly and longitudinally magnetized, and therefore there will be no distribution of matter except at their ends. Now the bars are all terminated on each side by the surface of the body, and consequently the whole magnetic effect is represented by a certain superficial distribution of northern and southern magnetic matter. It only remains to determine the actual form of this distribution; but, for the sake of simplicity in expression, it will be con- venient to state previously the following definition, borrowed from Coulomb's writings on electricity : 471. If any kind of matter be distributed over a surface, the superficial density at any point is the quotient obtained by dividing the quantity of matter" on an infinitely small element of the surface in the neighbourhood of that point, by the area of the element. 472. To determine the superficial density at any point in the case at present under consideration, let o> be the area of the perpendicular section of an infinitely thin uniform bar of the solid, with one end at that point. Then, if i be the intensity of magnetization of the solid, ico will be, as may be readily shown, the " strength " of the bar-magnet. Hence at the two ends of the bar we must suppose to be placed quantities of northern and southern imaginary magnetic matter each equal to io). In the distribution over the surface of the given magnet, these quantities of matter must be imagined to be spread over the oblique ends of the bar. Now if denote the inclination of the bar to a normal to the surface through one end, the area of that end will be ^ , and therefore in that part of the cos# surface we have a quantity of matter equal to ico spread over an area -^. Hence the superficial density is cos i cos 6. object of interesting experimental and theoretical investigations by Coulomb, Biot, Green, and Kiess. xxiv.] A Mathematical Theory of Magnetism. 36*1 This expression gives the superficial density at any point, P, of the surface, and its algebraic sign indicates the kind of matter, provided the angle denoted by 6 be taken between the external part of the normal, and a line drawn from P in the same direction as that of the motion of* a point carried from the south pole to the north pole, of a portion close to P, of the infinitely thin bar-magnet which we have been con- sidering. 473. Let it be required, in the last place, to determine the entire distribution of magnetic matter necessary to represent the polarity of any given magnet. We may conceive the whole magnetized mass to be divided into infinitely small parallelepipeds by planes parallel to three planes of rectangular co-ordinates. Let a, /3, 7 denote the three edges of one of these parallelepipeds having its centre at a point P (oc, y, z). Let i denote the given intensity, and I, m, n the given direction cosines of the magnetization at P. It will follow from the preceding investigation that the polarity of this infinitely small uniformly magnetized parallelepiped may be represented by imaginary magnetic matter distributed over its six faces in such a manner that the density will be uniform over each face, and that the quantities of matter on the six faces will be as follows : il . fiy, and il . /3y ; on the two faces parallel to YOZ ; im . 72, and im . 7 ; on the two faces parallel to ZOX ; in . a/3, and in . a/3 ; on the two faces parallel to XO Y. Now if we consider adjacent parallelepipeds of equal dimen- sions, touching the six faces of the one we have been consider- ing, we should find from each of them a second distribution of magnetic matter, to be placed upon that one of those six faces which it touches. Thus if we consider the first face 7, or that of which the distance from YOZ is x - Ja ; we shall have a second distribution upon it derived from a parallelepiped, the co-ordinates of the centre of which are x-a, y, z\ and the quantity of matter in this second distribution will be 362 A Mathematical Theory of Magnetism. [xxiv. This, added to that which was found above, gives for the total amount of matter upon this face. Again, the quantity in the second distribution on the other face, /5 7 , is equal to ( ., d (il) and therefore the total amount of matter on this face will be By determining in a similar way the final quantities of matter on the other faces of the parallelepiped, we find that the total amount of matter to be distributed over its surface is - 2 f^ + ^ + ^l a/37 - Now as the parallelepipeds into which we imagined the whole mass divided are infinitely small, we may substitute a con- tinuous distribution of matter through them, in place of the superficial distributions on their faces which have been de- termined ; and in making this substitution, the quantity of matter which we must suppose to be spread through the in- terior of any one of them must be half the total quantity on its surface, since each of its faces is common to it and another parallelepiped. Hence the quantity of matter to be distributed through the parallelepiped a/3 7 is equal to (d (il) d (im) d (in) \ dx dy dz Besides this continuous distribution through the interior of the magnet, there must be a superficial distribution to represent the un-neutralized polarity at its surface. If p denote the density of this distribution at any point ; [I], [m], [n] the direction- cosines, and [i] the intensity of the magnetization of the solid close to it ; and A, p, v the direction-cosines of a normal to the surface, we shall have, as in the case of the uniformly magnet- ized solid previously considered, p= [i] cos 0= \il\.\ + [im]./j,+ [in] . v (1). If according to the usual definition of " density," k denote the xxiv.] A Mathematical Theory of Magnetism. 363 density of the magnetic matter at P, in the continuous distri- bution through the interior, the expression found above for the quantity of matter in the element or, /3, 7, leads to the formula (d(il) d(im) d(in) ' ~~ ~ These two equations express respectively the superficial distri- bution, and the continuous distribution through the solid, of the magnetic matter which entirely represents the polarity of the given magnet. The fact that the quantity of northern matter is equal to the quantity of southern in the entire distri- bution, is readily verified by showing from these formula, as may readily be done by integration, that the total quantity of matter is algebraically equal to nothing. 474. If there be an abrupt change in the intensity or direc- tion of the magnetization from one part of the magnetized sub- stance to another, a slight modification in the formulae given above will be convenient. Thus we may take a case differing very little from a given case, but which, instead of presenting finite differences in the intensity or direction of magnetization on the two sides of any surface in the substance of the magnet, has merely very sudden continuous changes in the values of those elements : we may conceive the distribution to be made more and more nearly the same as the given distribution, with its abrupt transitions, and we may determine the limit towards which the value of the expression (2) approximates, and thus, although according to the ordinary rules of the differential calculus this formula fails in the limiting case, we may still derive the true result from it. It is very easily shown in this way, that, besides the continuous distribution given by the expression (2) applied to all points of the substance for which it does not fail, there will be a superficial distribution of mag- netic matter on any surface of discontinuity ; and that the density of this superficial distribution will be the difference between the products of the intensity of magnetization into the cosine of the inclination of its direction to the normal, on the two sides of the surface. 475. This result, obtained by the interpretation of formula (2) in the extreme case, might have been obtained directly from the original investigation, by taking into account the 364 A Mathematical Theory of Magnetism. [xxiv. abrupt variation of the magnetization at the surface of dis- continuity, as ( 472) we did the abrupt termination of the magnetized substance at the boundary of the magnet, and re- presenting the un-neutralized polarity which results, by a super- ficial distribution of magnetic matter. CHAPTEK IV. Determination of the Mutual Actions between any Given Portions of Magnetized Matter. 476. The synthetical part of the theory of magnetism has for its ultimate object the determination of the total action between two magnets, when the distribution of magnetism in each is given. The principles according to which the data of such a problem may be specified have been already laid down ( 459... 62), and we have seen that, with sufficient data in any case, Coulomb's laws of magnetic force are sufficient to enable us to apply ordinary statical principles to the solution of the problem. Hence the elements of this part of the theory may be regarded as complete, and we may proceed to the mathematical treatment of the subject. 477. The investigations of the preceding chapter, which show us how we may conventionally represent any given mag- net, in its agency upon other bodies, by an imaginary magnetic matter distributed on its surface and through its interior; enable us to reduce the problem of finding the action between any two magnets, to the known problem of determining the resultant of the attractions or repulsions exerted between the particles of two groups of matter, according to the law of force which is met with so universally in natural phenomena. The direct formulas applicable for this object are so readily obtained by means of the elementary principles of statics, and so well known, that it is unnecessary to cite them here, and we may regard equations (1) and (2) of the preceding chapter ( 473) as sufficient for indicating the manner in which the details of the problem may be worked out in any particular case. The expression for the "potential," and other formulas of importance in Laplace's method of treating this subject, are given below ( 482), as derived from the results expressed in equations (1) and (2). xxiv.] A Mathematical Theory of Magnetism. 365 478. The preceding solution of the problem, although ex- tremely simple and often convenient, must be regarded as very artificial, since in it the resultant action is found by the com- position of mutual actions between the particles of an imaginary magnetic matter, which are not the same as the real mutual actions between the different parts of the magnets themselves, although the resultant action between the entire groups of matter is necessarily the same as the real resultant action between the entire magnets. Hence it is very desirable to investigate another solution, of a less artificial form, in which the required resultant action may be obtained by compounding the real actions between the different parts into which we may conceive the magnets to be divided. The remainder of the chapter, after some preliminary explanations and definitions, will be devoted to this object. 479. The "resultant magnetic force at any point" is an expression which will very frequently be employed in what follows, and it is therefore of importance that its signification should be clearly defined. For this purpose, let us consider separately the cases of an external point in the neighbourhood of a magnet, and a point in space which is actually occupied by magnetic matter. (1) The resultant force at a point in space, void of magnet- ized matter, is the force that the north pole of a unit-bar (or a positive unit of imaginary magnetic matter), if placed at this point, would experience. (2) The resultant force at a point situated in space occupied by magnetized matter, is an expression the signification of which is somewhat arbitrary. If we conceive the magnetic substance to be removed from an infinitely small space round the point, the preceding definition would be applicable; since, if we imagine a very small barTinagnet to be placed in a definite position in this space, the force upon either end would be determinate. The circumstances of this case are made clear by considering the distribution of imaginary magnetic matter required to represent the given magnet, without the small portion we have conceived to be removed from its interior; which will differ from the distribution that represents the entire given magnet, in wanting the small portion of the 366 A Mathematical TJieory of Magnetism. [xxiv. continuous interior distribution corresponding to the removed portion, and in having instead a superficial distribution on the small internal surface bounding the hollow space. If we con- sider the portion removed to be infinitely small, the want of the small portion of the solid magnetic [imaginary] matter will produce no finite effect upon any point; but the superficial distribution at the boundary of the hollow space will produce a finite force upon any magnetic point within it. Hence the resultant force upon the given point round which the space was conceived to be hollowed, may be regarded as compounded of two forces, one due to the polarity of the complete magnet, and the other to the superficial polarity left free by the removal of the magnetized substance*. The former component is the force meant by the expression " the resultant force at a point within a magnetic substance," when employed in the present paperf. 480. The conventional language and ideas with reference to the imaginary magnetic matter, explained above ( 463... 75), enable us to give the following simple statement of the defini- tion, including both the cases which we have been considering. * If the portion removed be spherical and infinitely small, it may be proved that the force at any point within it, resulting from the free polarity of the solid at the surface bounding the hollow space, is in the direction of the lines of magnetization of the substance round it, and is equal to - . o This theorem (due to Poisson) will be demonstrated at the commencement of the Theory of Magnetic Induction, because we shall have to consider the "magnetizing force" upon any small portion of an inductively magnetized substance as the actual resultant force that would exist within the hollow space that would be left if the portion considered were removed, and the magnetism of the remainder constrained to remain unaltered. t If we imagine a magnet to be divided into two parts by any plane pass- ing through the line of magnetization at any internal point, P, and if we imagine the two parts to be separated by an infinitely small interval, and a unit north pole to be placed between them at P, the force which this pole would experience is " the resultant force at a point, P, of the magnetic sub- stance." This is the most direct definition of the expression that could have been given, and it agrees with the definition I have actually adopted; but I have preferred the explanation and statement in the text, as being practically more simple, and more directly connected with the various investigations in which the expression will be employed. [Note added June 15, 1850. Some subsequent investigations on the com- parison of common magnets and electro-magnets have altered my opinion, that the definition in the text is to be preferred; and I now believe the definition in the note to present the subject in the simplest possible manner, and in that which, for the applications to be made in the continuation of this Essay, is most convenient on the whole.] xxiv.] A Mathematical Theory of Magnetism. 367 The resultant magnetic force at any point, whether in the neighbourhood of a magnet or in its interior, is the force that a unit of northern magnetic matter would experience if it were placed at that point, and if all the magnetized substance were replaced by the corresponding distribution of imaginary mag- netic matter. 481. The determination of the resultant force at any point is, as we shall see, much facilitated by means of a method first introduced by Laplace in the mathematical treatment of the theory of attraction, and developed to a very remarkable extent by Green in his " Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" (Not- tingham, 1828), and in his other writings on the same and on allied subjects in the Cambridge Philosophical Transactions, and in the Transactions of the Royal Society of Edinburgh. Laplace's fundamental theorem is so well known that it is unnecessary to demonstrate it here ; but for the sake of re- ference, the following enunciation of it is given. The term " potential," defined in connexion with it, was first introduced by Green in his Essay (1828). It was at a later date intro- duced independently by Gauss, and is now in very, general use. Theorem (Laplace). The resultant force produced by a body, or a group of attracting or repelling particles, upon a unit particle placed at any point P, is such that the difference be- tween the values of a certain function, at any two points p and p' infinitely near P, divided by the distance pp, is equal to its component in the direction of the line joining p and p'. Definition (Green). This function, which, for a given mass, has a determinate value at any point P, of space, is called the potential of the mass, at the point P. It follows from Laplace's general demonstration, that, when the law of force, is that of the inverse square of the distance, the potential is found by dividing the quantity of matter in any infinitely small part of the mass, by its distance from P, and adding all the quotients so obtained. 482. The same demonstration is applicable to prove, in virtue of Coulomb's fundamental laws of magnetic force, the same theorem with reference to any kind of magnet that can 368 A Mathematical Theory of Magnetism. [xxiv, be conceived to be composed of uniformly magnetized bars, either finite or infinitely small, put together in any way, that is, of any magnet other than an electro-magnet ; and the in- vestigation, in the preceding chapter, of the resulting distribu- tion of magnetic matter that may be imagined as representing in the simplest possible way the polarity of such a magnet, enables us to determine at once, from equations (1) and (2) of 473, its potential at any point. Thus if V denote the poten- tial at a point P, whose co-ordinates are f, 77, f, and if dS denote an element of the surface of the magnet, situated at a point whose co-ordinates are [a?], [y], [z], we have, by the pro- position enunciated at the end of 480, d(im) d(in) + "" + ^ dlyd*...(3) 9 where A and [A] are respectively the distances of the points x, y, z and [x, y, z\ from the point P, and are given by the equations A 2 =(_) + (,-y)' + (-) [AP = (-M) 2 + (n- [2/]) 2 + denote the volume of an infinitely small element of uniformly magnetized matter, and i the intensity of its magnetization, the potential which it produces at any point P, at a finite distance from it, will be id), cos A* ' where A denotes the distance of P from any point, E, within the element, and 6 the angle between EP and a line drawn through E, in the direction of magnetization of the element, towards the side of it which has northern polarity. 486. Let us now suppose the element E to be a part of a magnet of finite dimensions, of which it is required to deter- mine the total potential at an external point, P. Let f , ij, be the co-ordinates of P, referred to a system of rectangular axes, and let x, y, z be those of E. We shall have xxiv.] A Mathematical Theory of Magnetism. 371 and, if I, m, n denote the direction-cosines of the magnetization . Hence the expression for the potential of the element E be- comes Now the potential of a whole is equal to the sum of the poten- tials of all its parts, and hence, if we take (ft = dxdydz, we have, by the integral calculus, the expression for the potential at the point P, due to the entire magnet.* [ 487... 494 added September, 1871.] [487. The expansion of this in ascending powers of - , - , - , where is necessarily convergent for all space outside the least spherical surface with the origin of co-ordinates for centre, enclosing the whole magnet. To find it, we have first to expand il ( #) + im (r) y) + in (% z) {(?-*)+ fo -y) f +(?-*)'}*- by Taylor's Theorem, in a series of ascending powers of #, y, z, which is necessarily convergent or divergent according as *J(a? + y*+s?) is less or greater than V(? 2 + *? 2 + f *) Tnus > for the part of V depending on il, we find where 222 denotes summation from to relatively to integers s, t, IL Hence, remarking that -^ -^ - , and putting * From the form of definition given in the second footnote on 479, for the magnetic force at an internal point, it may be shown that the expression (5), as well as the expression (3), is applicable to the potential at any point, whether internal or external. The same thing may be shown by proving, as may easily be done, that the investigation of 487 does not fail or become nugatory when (, 17, ) is included in the limits of integration. 242 372 A Mathematical Theory of Magnetism. [xxiv. -b f,] ....... (7) subject to the exception that terms of the first member involv- ing a?" 1 , or y 1 , or z~ l , are to be omitted, we have , c are constants (depending on the magnetism of the magnet, and the position relatively to it of the axis of co-ordinates) given by the equations L =fffildxdydz, M=fffimdxdydz, N=fffindxdydz .............................. (10), A=fffilxdxdydz, B=fffimydxdydz, C =fffinzdxdydz a =fff(imz + iny)dxdydz, 6 =fff(inx + ilz)dxdy.dz, c =fff(ily + imx}dxdydz 489. If we put (12), ~ L x My Nz N and eos = _- + _^ + _- : .................. (13), in the first term of (9) it becomes ; ^ ........................... (14), which is the first approximate expression for the potential of the magnet at a very distant point, and agrees with the rigorous expression ( 485) for the potential of an infinitely small uni- formly magnetized magnet at the origin of co-ordinates, having its magnetic moment equal to K, and its direction of magnetiza- tion specified by the direction-cosines ' .. K' K y K" Hence K, given by (12) and (10), is defined as the magnetic xxiv.] A Mathematical Theory of Magnetism. 373 moment of the given magnet ; and the direction (15) is readily proved to fulfil the condition stated in 439, 440 as the definition of a magnetic axis, determinate in direction but ( 444) left till now indeterminate as to its position in the magnet. It is to be remarked that the values of L, M, N given by (10) are independent of the position of the origin of co- ordinates, and depend only on the positions of the co-ordinate axes relatively to the magnet. 490. Let now the axes of co-ordinates be turned to bring one of the three into parallelism with the direction of the magnetic axis (15). Calling this OX, and using the same symbols, x, y, z, I, m t n, for co-ordinates and direction-cosines relatively to the new axes, we have, instead of (9) and (10), Kx K= fffildxdydz ; fffimdxdydz = ; fffindxdydz = ... (1 7), with equations (11) unchanged. 491. Secondly, let the axis of x be transferred from OX to the parallel line through any point for which b c The values of the integrals for the new axes corresponding to b and c are each zero, as is readily seen from (11) and (17). Hence, altering the notation y, z to correspond to the new axes, we have Kx with fff (inx + Hz} dxdydz = ; ///(% + imx) dxdydz = (20), and (11) in other respects unchanged. Now for 2i/ 2 - / - x\ and 2^ 2 - x 2 - y\ ] we may write > (20), -i(2a-jf -*)+! (y'-z*), and -\(^-f-f} -f (f-f)\ a transformation which, simple as it is, has an important signi- ficance in "spherical harmonics." Hence if we put a = ^ff/(2ilx - imy -inz) dxdydz, and (3=%f/f(imy - inz) dxdydz (21 ), (19) becomes 374 A Mathematical Theory of Magnetism. [xxiv. 492. Thirdly, shift the origin from to the point * = ! ................................ (23) in OX; that is to say, for x substitute x + j~. By (21) and (17) we have ffj(2ilx imy inz) dxdydz = ; /3 = $fff(*my~iiut) dxdydz (24) ; and (22) becomes Kx P(f-?) + *ay* . (a? +/ + /)* 493. Lastly, turn the axes OF, 0^, round OX through an angle equal to Jtan- 1 ^ ......................... (26). Relatively to OX, OF, OZ in this final position we have (17) and (24) unchanged, and fff(imz + iny) dxdydz = 0, fff(inx + ilz) dxdydz = 0, fff(ily + imx) dxdydz = (27) ; and (25) becomes reduced to 494. This is the simplest expression to the second degree of approximation for the distant potential of a magnet having any irregular distribution of magnetism. The axis determined by 489 (15) and 491 (18) is the magnetic axis, and the point in it determined by 492 (23) is the magnetic centre, of which definitions were promised in the addition to 444.] 495. The expression (5) of 486 is susceptible of a very remarkable modification, by integration by parts. Thus we may divide the second member into three terms, of which the following is one : fff - 1 jj - Integrating here by parts, with reference to x, we obtain d (il) where the brackets enclosing the double integral denote that xxiv.] A Mathematical Theory of Magnetism. 375 the variables in it must belong to some point of the surface. If X, //., v denote the direction-cosines of a normal to the surface at any point [f, ?;, f], and dS an element of the surface, we may take dydz = \. dS, and hence the double integral is reduced to IF and, as we readily see by tracing the limits of the first integral with reference to x, for all possible values of y and z this double integral must be extended over the entire surface of the mag- net. By treating in a similar manner the other two terms of the preceding expression for F, we obtain, finally, d(il) d(irn) d(in) " ~ The second member of this equation is the expression for the potential of a certain complex distribution of matter, consisting of a superficial distribution and a continuous internal distribu- tion. The superficial density of the distribution on the surface, and the density of the continuous distribution at any internal point, are expressed respectively by [il] \ + \im\ p + [in] v, and (d (il) d (im) d (in)) TT . ,> ,, c 1 -4^ H -- \ ' + -V 4 . Hence we infer that the action of doc dy dz J the complete magnet upon any external point is the same as would be produced by a certain distribution of imaginary mag- netic matter, determinable by means of these expressions, when the actual distribution of magnetism in the magnet is given.* The demonstration of the same theorem, given above ( 473), illustrates in a very interesting manner the process of integra- tion by parts applied to a triple integral. 496. The mutual action of any two magnets, considered as the resultant of the mutual actions between the infinitely small elements into which we may conceive them to be divided, con- sists of a force and a couple of which the components will be expressed by means of six triple integrals. Simpler expres- * This very remarkable theorem is due to Poisson, and the demonstration, as it has been just given in the text, is to be found in his first memoir on Magnet- ism. The demonstration which I have given in 473 may be regarded as exhibiting, by the theory of polarity, the physical principles expressed in the analytical formulae. 376 A Mathematical Theory of Magnetism. [xxiv. sions for the same results may be obtained by employing a notation for subsidiary results derived from triple integration with reference to one of the bodies, in the following manner : 497. Let us in the first place determine the action exerted by a given magnet upon an infinitely thin uniformly and longi- tudinally magnetized bar, placed in a given position in its neighbourhood. We may suppose the rectangular co-ordinates, , 77, f, of the north pole, and f ', ?/, f of the south pole of the bar to be given, and hence the components X, Y, Z and X' ', Y', Z ', of the re- sultant forces at those points due to the other given magnet may be regarded as known. Then, if p denote the " strength" of the bar-magnet, the components of the forces on its two poles will be respectively PX, PY, PZ, on the point ( 77, g), and -PX', -PY', -PZ', on the point (', if t O- The resultant action due to this system of forces may be deter- mined by means of the elementary principles of statics. Thus if we conceive the forces to be transferred to the middle of the bar by the introduction of couples, the system will be reduced to a force, on this point, whose components are and a couple, whose components are {/? (Z+Z') . i(, - r,") - ft (F+ F) . i (C- r)}, {/3(x+x').i(r- n -0(z+z') . H?-r n, {/3( Y+ F) . Kf - r) -/3 (X+X') . fo - r,'}}. 498. Let I, m, n denote the direction-cosines of a line drawn along the bar, from it's middle towards its north pole, and if a be the length of the bar, we shall have f;' al, rj rj'am, ' = an. Hence, if the bar be infinitely short, and if x, y, z denote the co-ordinates of its middle point, we have ^ ._, dX 7 dX dX X-^X = -j- . at -f -j- . am + j . an, doc dy dz v , dY . dY dY Y Y = -j .al + ^ . am + -j- . an, dx dy dz r ., dZ 7 dZ dZ and Zi Zi = = . al + -r~ . ow + -* an. dx dy dz xxiv.] A Mathematical Theory of Magnetism. 377 Multiplying each member of these equations by /3, we obtain the expressions for the components of the force in this case ; and the expressions for the components of the couples are found in their simpler forms, by substituting for f f , etc., their values given above ; and, on account of the infinitely small factor which each term contains, taking 2JT, 2F, and 2Z, in place of X + X', F+ F', and Z+Z'. 499. Let us now suppose an infinite number of such infinitely small bar-magnets to be put together so as to constitute a mass, infinitely small in all its dimensions, uniformly magnetized in the direction (I, m, n) to such an intensity that its magnetic moment is p. We infer, from the preceding investigation, that the total action on this body, when placed at the point x } y> z, will be composed of a force whose components are fdX, dX dX \ AM-J- z + -y- m+ -T- ra], \dx dy dz J fdY, dT dY \ u, { -y- I -H -j m + -T- n } , \dx dy dz J (dZ . dZ dZ \ /* ( -j- * + j- m + -j~ n ) > \dx dy dz J acting at the centre of gravity of the solid supposed homo- geneous ; and a couple of which the components are li,(Zm-Yn), jj, (Xn - Zl), 500. The preceding investigation enables us, by means of the integral calculus, to determine the total mutual action between any two given magnets. For, if we take X, Y, Z to denote the components of the resultant force due to one of the magnets, at any point (x, y, z) of the other, and if i denote the intensity and (I, m, n) the direction of magnetization of the substance of the second magnet at this point, we may take p = i.dxdydz in the expressions which were obtained, and they wiM then express the action which one of the magnets exerts upon an element dxdydz of the other. To determine the total resultant action, we may transfer all the forces to the origin of co-ordi- nates, by introducing additional couples ; and, by the usual pro- cess, we find, for the mutual action between the two magnets, 378 A Mathematical Theory of Magnetism. [xxiv. a force in a line through this point, and a couple, of which the components, F, G, H, and L, M, JY, are given by the equa- tions dX , dX\, I ^+ m ^r d y dz dY . _ f[[f.,dZ . dZ H= w P fo + im dy Zi rr TT I ? **" \A*-4 . \AJ \ = HI \ imZinY+y(il-j- -\-irn -^ + in -=- JJJ ( ^V (to ^ rf^/ /. 7 ^F, . ^F . dY\] , 7 , z (il -j + im-j h *ft -j- 1 r dxdydz \ dx dy dz ) } 9 x(il dZ . dZ . dZ\} -T- +im- r +m-j- )[ dx dy dz)\ fff f -7T7- V f'J dY ' dY . dY\ =lll \ ilYimX+x[il r +im- r --}-m T- JJJ [ \ dx dy dz J dX . dX\] , , , -j-+m r > dxdydz dx dy L 7 j 501. If, in the second members of these equations, we em- ploy for X } Y, Z respectively their values obtained, as indicated in equations (4) of 483, by the differentiation of the expres- sion (5) for V in 486, we obtain expressions for F, 6r, H, L, M, N, which may readily be put under symmetrical forms with reference to the two magnets, exhibiting the parts of those quantities depending on the mutual action between an element of one of the magnets, and an element of the other. Again, expressions exhibiting the mutual action between any element of the imaginary magnetic matter of one magnet, and any element of the imaginary magnetic matter of the other, may be found by first modifying by integration by parts, as in 495, from the expressions which we have actually obtained for F t G, H, L,M,N; and then substituting for X, Y, and Z their values obtained by the differentiation of the expression (3) of 482, for F. xxiv.] A Mathematical Theory of Magnetism. 379 It is unnecessary here to do more than indicate how such other foramlse may be derived from those given above; for whenever it may be required, there can be no difficulty in applying the principles which have been established in this paper to obtain any desired form of expression for the mutual action between two given magnets. 502 and 503.* On the Expression of Mutual Action between two Magnets by means of the Differential Coefficients of a Function of their relative Position. 502. By a simple application of the theory of the potential, it may be shown that the amount of mechanical work spent or gained in any motion of a permanent magnet, effected under the action of another permanent magnet in a fixed position, depends solely on the initial and final positions, and not at all upon the positions successively occupied by the magnet in passing from one to the other. Hence the amount of work requisite to bring a given magnet from being infinitely distant from all magnetic bodies into a certain position in the neigh- bourhood of a given fixed magnet, depends solely upon the dis- tributions of magnetism in the two magnets, and on the relative position which they have acquired. Denoting this amount by Q, we may consider Q as a function of co-ordinates which fix the relative position of the two magnets; and the variation which Q experiences when this is altered in any way will be the amount of work spent or lost, as the case may be, in effect- ing the alteration. This enables us to express completely the mutual action between the two magnets, by means of dif- ferential coefficients of Q, in the following manner : If we suppose one of the magnets to remain fixed during the alterations of relative position conceived to take place, the quantity Q will be a function of the linear and angular co-ordinates by which the variable position of the other is expressed. Without specifying any particular system of co- ordinates to be adopted, we may denote by d^Q the augmenta- * Communicated June 20, 1850. 380 A Mathematical Theory of Magnetism. [xxiv. tion of Q when the moveable magnet is pushed through an infinitely small space dg in any given direction, and by d$Q the augmentation of Q when it is turned round any given axis, through an infinitely small angle d. Then, if F denote the force upon the magnet in the direction of d, and L the moment round^the fixed axis of all the forces acting upon it (or the component, round the fixed axis, of the resultant couple ob- tained when all the forces on the different parts of the magnet are transferred to any point on this axis), we shall have -Fd = d 6 Q, and - Ld = d^Q, since a force equal to F is overcome through the space dg in the first case, and a couple, of which the moment is equal to L, is overcome through an angle d(f> in the second case of motion. Hence we have d ' 503. It only remains to show how the function Q may be determined when the distributions of magnetism in the two magnets and the relative positions of the bodies are given. For this purpose, let us consider points P and P', in the two magnets respectively, and let their co-ordinates with refer- ence to three fixed rectangular axes be denoted by x, y, z and #', y' } z\ let also the intensity of magnetization at P be denoted by i t and its direction-cosines by l,m,n; and let the correspond- ing quantities, with reference to P', be denoted by i, I', ra', n'. Then it may be demonstrated without difficulty that , . . , + mn -, , , dydx dydy dydz jt dzdy xxiv.] A Mathematical Theory of Magnetism. 381 where, for brevity, A is taken to denote {(x x'f + (y y')* + (z z) 2 }%, and the differentiations upon -^ are merely indicated. Now, by any of the ordinary formulae for the transformation of co-ordinates, the values of x, y, z, and x, y, z', may be expressed in terms of co-ordinates of the point P with reference to axes fixed in the magnet to which it belongs, of the co-ordinates of the point P' with reference to axes fixed in the other, and of the co-ordinates adopted to express the relative position of the two magnets : and so the preceding expression for Q may be trans- formed into an expression involving explicitly the relative co-ordinates, and containing the co-ordinates of the points P and P' in the two bodies only as variables in integrations, the limits of which, depending only on the forms and dimensions of the two bodies, are absolutely constant. Thus Q is obtained as a function of the relative co-ordinates of the bodies, and the solution of the problem is complete. There is no difficulty in working out the result by this method, so as actually to obtain either the expressions of 500, or the expressions indicated in 501, although the process is somewhat long. [Addition, Dec. 11, 1871. If in the formula for Q we suppose the integration with respect to x, y, z to be performed, we have Q=-f* ^ I" dxdydz (&' + &' + '} ...... (5). 47TJ -a) J -oo J -co This is a very important result, as we shall see in Chapter VII. Compare 561.] The method just explained for expressing the mutual action between two magnets in terms of a function of their relative position, has been added to this chapter rather for the sake of completing the mathematical theory of the division of the subject to which it is devoted, than for its practical usefulness in actual problems regarding magnetic force, for which the most convenient solutions may generally be obtained by some of the more synthetical methods explained in the preceding parts of the chapter. There is, however, a far more important application of the principles upon which this last method is founded which remains to be made. The mechanical value of a distribution of magnetism, although it has not, I believe, been noticed in any writings hitherto published on the mathe- matical theory of magnetism, is a subject of investigation of great interest, and, as I hope on a later occasion* to have an opportunity of showing, of much consequence, on account of its maximum and minimum problems, which lead to demonstra- tions of important theorems in the solutions of inverse problems regarding magnetic distribution. CHAPTER V. On Solenoidal and Lamellar Distributions of Magnetism.-^ 504. In the course of some researches upon inverse problems regarding distributions of magnetism, and upon the comparison * [Chap. VII.. ..X. below; Dec. 1871.] f Communicated to the Royal Society June 20, 1850. xxiv.] Solenoidal and Lamellar Distributions. 383 of electro-magnets and common magnets, I have found it extremely convenient to make use of definite terms to express certain distributions of magnetism and forms of magnetized matter possessing remarkable properties. The use of such terms will be of still greater consequence in describing the results of these researches, and therefore, before proceeding to do so, I shall give definitions of the terms which I have adopted, and explain briefly the principal properties of the magnetic distributions to which they are applied. The remainder of this chapter will be devoted to three new methods of analysing the expressions for the resultant force of a magnet at any point, suggested by the consideration of these special forms of mag- netic distribution. A Mathematical Theory of Electro-Magnets, and Inverse Problems regarding magnetic distributions, are the subjects of papers which I hope to be able to lay before the Royal Society on a subsequent occasion. [They are published for the first time in this volume : Chaps. VI.... X.] 505. Definitions and explanations regarding Magnetic Sole- noids. (1) A magnetic solenoid* is an infinitely thin bar of any form, longitudinally magnetized with an intensity varying in- versely as the area of the normal section in different parts, The constant product of the intensity of magnetization into the area of the normal section, is called the magnetic strength, or sometimes simply the strength of the solenoid. Hence the magnetic moment of any straight portion, or of an infinitely small portion of a curved solenoid, is equal to the product of the magnetic strength into the length of the portion. (2) A number of magnetic solenoids of different lengths may be put together so as to constitute what is, as far as regards magnetic action, equivalent to a single infinitely thin bar of any form, longitudinally magnetized with an intensity varying * This term (from o-wX^, a tube) is suggested by the term "electro-dynamic solenoid" applied by Ampere to a certain tube-like arrangement of galvanic circuits which produces precisely the same external magnetic effect as is pro- duced by ordinary magnetism distributed in the manner defined in the text. The especial appropriateness of the term to the magnetic distribution is mani- fest from the relation indicated in the footnote on 513 below, between the intensity and direction of magnetization in a solenoid, and the velocity and direction of motion of a liquid flowing through a tube of constant or varying section. 384 A Mathematical Theory of Magnetism. [xxiv. arbitrarily from one end of the bar to the other. Hence such a magnet may be called a complex magnetic solenoid. The magnetic strength of a complex solenoid is not uniform, but varies from one part to another. (3) An infinitely thin closed ring, magnetized in the manner described in (1), is called a closed magnetic solenoid. 506. Definitions and explanations regarding Magnetic Shells. (1) A magnetic shell is an infinitely thin sheet of any form, normally magnetized with an intensity varying inversely as the thickness in different parts. The constant product of the intensity of magnetization into the thickness is called the magnetic strength, or sometimes simply the strength of the shell. Hence the magnetic moment of any plane portion, or of an infinitely small portion of a curved magnetic shell, is equal to the product of the magnetic strength into the area of the portion. (2) A number of magnetic shells of different areas may be put together so as to constitute what is, as far as regards mag- netic action, equivalent to a single infinitely thin sheet of any form, normally magnetized with an intensity varying arbitrarily over the whole sheet. Hence such a magnet may be called a complex magnetic shell. The magnetic strength of a complex shell is not uniform, but varies from one part to another. (3) An infinitely thin sheet, of which the two sides are closed surfaces, is called a closed magnetic shell. 507. Solenoidal and Lamellar Distributions of Magnetism. If a finite magnet of any form be capable of division into an infinite number of solenoids which are either closed or have their ends in the bounding surface, the distribution of magnet- ism in it is said to be solenoidal, and the substance is said to be solenoidally magnetized. If a finite magnet of any form be capable of division into an infinite number of magnetic shells which are either closed or have their edges in the bounding surface, the distribution of magnetism in it is said to be lamellar,* and the substance is said to be lamellarly magnetized. * The term lamellar, adopted for want of a better, is preferred to "lami- nated"; since this might be objected to as rather meaning composed of plane xxiv.] Solenoidal and Lamellar Distributions. 385 508. Complex Lamellar Distributions of Magnetism. If a finite magnet of any form be capable of division into an infinite number of complex magnetic shells, it is said to possess a com- plex lamellar distribution of magnetism. 509. Complex Solenoidal Distributions of Magnetism. Since, by cutting it along lines of magnetization, every magnet of finite dimensions may be divided into an infinite number of longitu- dinally magnetized infinitely thin bars or rings, any distribu- tion of magnetism which is not solenoidal might be called a complex solenoidal distribution ; but no advantage is obtained by the use of this expression, which is only alluded to here, on account of the analogy with the subject of the preceding definition. 510. PJROP. The action of a magnetic solenoid is the same as if a quantity of positive or northern imaginary magnetic matter numerically equal to its magnetic strength were placed at one end, and an equal absolute quantity of negative or southern matter at the other end. The truth of this proposition follows at once from the in- vestigation of Chap. III. 467, 468, 469. Cor. 1. The action of a magnetic solenoid is independent of its form, and depends solely on its strength and the positions of its extremities. Cor. 2. A closed solenoid exerts no action on any other magnet. Cor. 3. The "resultant force" (defined in Chap. IV. 480) at any point in the substance of a closed magnetic solenoid vanishes. 511. PEOP. If\ be the intensity of magnetization, and co the area of the normal section at any point P, at a distance sfrom one extremity of a complex solenoid, and if [i} denote the values of the product of these quantities at the extremity from which s is measured, and at the other extremity respectively ; the magnetic action will be the same as if there were a distribution of imaginary magnetic matter, through the length of the bar of which the quantity is an infinitely small portion ds, of the length at the plates, than composed of shells whether plane or curve, and is besides too much associated with a mechanical structure such as that of slate or mica, to be a convenient term for the magnetic distributions denned in the text. T. E. 25 386 A Mathematical Theory of Magnetism. [xxiv. point P, would be \ - ds, and accumulations of quantities equal to [io>] and {io>} respectively at the two extremities. The truth of this proposition follows immediately from the conclusions of Chap. III. 469. 512. PROP. The potential of a magnetic shell at any point is equal to the solid angle which it subtends at that point multiplied by its magnetic strength*. Let dS denote the area of an infinitely small element of the shell, A the distance of this element from the point P, at which the potential is considered, and 6 the angle between this line, and a normal to the shell drawn through the north polar side of dS. Then if \ denote the magnetic strength of the shell, the magnetic moment of the element dS will be \dS, and ( 485) the potential due to it at P will be \dS.cos0 A 2 Now ^T^ is the solid angle subtended at P by the element dS, and therefore the potential due to any infinitely small element, is equal to the product of its magnetic strength into the solid angle which its area subtends at P. But the poten- tial due to the whole is equal to the sum of the potentials due to the parts, and the strength is the same for all the parts. Hence the potential due to the whole shell is equal to the pro- duct of its strength into the sum of the solid angles which all its parts, or the solid angle which the whole, subtends at P. n 1 rrn dS . COS 6 , . , . , , Cor. 1. The expression -^ , which occurs in the pre- ceding demonstration, being positive or negative according as 6 is acute or obtuse, it appears that the solid angle subtended by different parts of the shell at P must be considered as posi- tive or negative according as their north polar or their south polar sides are towards this point. * This theorem is due to Gauss (see his paper "On the General Theory of Terrestrial Magnetism," 38 ; of which a translation is published in Taylor's Scientific Memoirs, vol. n.). Ampere's well-known theorem, referred to by Gauss, that a closed galvanic circuit produces the same magnetic effect as a magnetic shell of any form having the circuit for its edge, implies obviously the truth of the first part of Cor. 2 below. XXIV.] Lamellar Distributions. 387 Cor. 2. The potential at any point due to a magnetic shell is independent of the form of the shell itself, and depends solely on its bounding line or edge, subject to an ambiguity, the nature of which is made clear by the following statement : If two shells of equal magnetic strength, X, have a common boundary, and if the north polar side of one, and the south polar side of the other be towards the enclosed space, the potentials due to them at any external point will be equal ; and the potential at any point in the enclosed space, due to that one of which the northern polarity is on the inside, will exceed the potential due to the other by the constant 4-TrX. Cor. 3. Of two points infinitely near one another on the two sides of a magnetic she]!, but not infinitely near its edge, the potential at that one which is on the north polar side exceeds the potential at the other by the constant 4?rX. Cor. 4. The potential of a closed magnetic shell of strength X, with its northern polarity on the inside, is 47rX, for all points in the enclosed space, and for all external points ; and for points in the magnetized substance it varies continuously from the inside, where it is 4?rX to the outside, where it is 0. Cor. 5. A closed magnetic shell exerts no force on any other magnet. Cor. 6. The "resultant force" as defined at 479, 480 [polar definition], is equal to - , at any point in the sub- . ' T stance of a closed magnetic shell, if r be the thickness, or to 4?', if i be the intensity of magnetization of the shell in the neighbourhood of the point, and is in the direction of a normal drawn from the point through the south polar side of the shell. [The " resultant force " as defined below in 517, by the electro- magnetic definition, is zero at any point in the substance of a closed magnetic shell, or of a lamellar distribution consisting of closed shells.] Cor. 7. If the intensity of magnetization of an open shell be finite, the resultant force at any external point not infinitely near the edge is infinitely small ; but the force at any point in the substance not infinitely near the edge is finite, and is equal to 4?', if i be the intensity of the magnetization in the neigh- 252 388 A Mathematical Theory of Magnetism. [xxiv. bourhood of the point, and is in the direction of a normal through the south polar side. 513. PEOP. A distribution of magnetism expressed by i( a j A 7) at (x, y, z)}* is solenoidal if, and is not solenoidal da d d? unless, j- + -f- + -^ = 0. dx dy dz The condition that a given distribution of magnetism, in a substance of finite dimensions, may be solenoidal, is readily deduced from the investigations of 473, by means of the pro- positions of 510 and 511. For, if the distribution of mag- netism be solenoidal, the imaginary magnetic matter by which the polarity of the whole magnet may be represented will be situated at the ends of the solenoids, according to 510, and therefore ( 507) will be spread over the bounding surface. On the other hand, if the distribution be not solenoidal, that is, if the magnet be divisible into solenoids, of which some, if not all, are complex; there will, according to 511, be an internal distribution of imaginary magnetic matter in the representa- tion of the polarity of the whole magnet. Hence it follows from 473 that if a, ft 7 denote the components of the intensity of magnetization at any internal point (sc, y, z), the equation d* d/3 dy_ (I) dx + dy + dz~ " ( ' } expresses that the distribution of magnetism is solenoidarf . * Where a, p, 7, which may be called the components, parallel to the axes of co-ordinates, of the magnetization at (x, y, z), denote respectively the products of the intensity into the direction cosines of the magnetization. t The analogy between the circumstances of this expression and those of the cinematical condition expressed by "the equation of continuity" to which the motion of a homogeneous incompressible fluid is subject, is so obvious that it is scarcely necessary to point it out. When an incompressible fluid flows through a tube of variable infinitely small section, the velocity (or rather the mean velocity) in any part is inversely proportional to the area of the section. Hence the intensity and direction of magnetization, in a solenoid, according to the definition, are subject to the same law as the mean fluid velocity in a tube with an incompressible fluid flowing through it. Again, if any finite portion of a mass of incompressible fluid in motion be at any instant divided into an infinite number of solenoids (that is, tube-like parts), by following the lines of motion, the velocity in any one of these parts will, at different points of it, be inversely proportional to the area of its section. Hence the intensity and direction of magnetization in a solenoidal distribution of magnetism, according to the definition, are subject to the same condition as the fluid-velocity and its direc- tion, at any point in an incompressible fluid in motion. It may be remarked, that by making an investigation on the plan of 473 to express merely the condition that there may be no internal distribution of imaginary magnetic xxiv.] Lamellar Distributions. 389 514. PROP. A distribution of magnetism { (a, /3, 7) at (x, y,z] } is lamellar if, and is not lamellar unless, adx + /3dy 4- 7dz is the differential of a function of three independent variables. Let ty be a variable which has a certain value for each of the series of surfaces, by which the magnet may be divided into magnetic shells ; so that, if ty be considered as a function of x, y, z, any one of these surfaces will be represented by the equation ^(x^y.z) = II ........................ (a) ; and the entire series will be obtained by giving the parameter IT, successively a series of values each greater than that which precedes it by an infinitely small amount. According to the definition of a magnetic shell ( 506), the lines of magnetiza- tion must cut these surfaces orthogonally; and hence, since a, /3, 7 denote quantities proportional to the direction cosines of the magnetization at any point, we must have JL---, JL (b\ dx dy dz Let us consider the magnetic shell between two of the con- secutive surfaces corresponding to values of the parameter of which the infinitely small difference is r. The thickness of this shell at any point (x, y, z) will be (dy dy dy\* \dx* + cty a+ dz*J Now the product of the intensity of magnetization, into the thickness of the shell, must be constant for all points of the matter, the equation +^ + ^=Ois obtained in a manner precisely similar dx dy dz to a mode of investigating the equation of continuity for an incompressible fluid, now well known, which is given in Duhamel's Cours de Mecanique, and in the Cambridge and Dublin Mathematical Journal, vol. n. p. 282. The following very remarkable 'proposition is an immediate consequence of the proposition that "& closed solenoid exerts no action on any other magnet" ( 510, Cor. 2 above), m virtue of the analogy here indicated. "If a closed vessel, of any internal shape, be completely filled with an in- compressible fluid, the fluid set into any possible state of motion, and the vessel held at rest ; and if a solid mass of steel of the same shape as the space within the vessel be magnetized at each point with an intensity proportional and ma direction corresponding to the velocity and direction of the motion at the corresponding point of the fluid at any instant ; the magnet thus formed will exercise no force on any external magnet." 390 A Mathematical Theory of Magnetism. [xxiv. same shell; and hence, since & is constant, and since a, /3, 7 denote quantities such that (a 2 + /3 2 + 7*)^ is the intensity of magnetization at any point, we must have where F(^) denotes a quantity which is constant when ifr is constant. This equation, and the two equations (6), express all the conditions required to make the given distribution lamellar. By combining them we obtain the following three, which are equivalent to them : and hence, \i$F(fy)dty be denoted by <, we have where < is some function of x, y, and z. Hence the condition that a magnetic distribution (a, /3, 7) may be lamellar, is simply that adx + jBdy + at any point in the surface of the magnet, which, as appears from the preceding investigations, is all that is necessary for determining the potential due to a lamellar magnet at any point not contained in the magnetized substance, may, according to well-known principles, be determined by integration, if the tangential component of the magnetization at every point of the magnet infinitely near its surface be given. It appears therefore that, if it be known that a magnet is lamellarly magnetized throughout its interior, it is sufficient to know the tangential component of its magnetization at every point infinitely near the surface, or to have enough of data for determining it, without any further specification re- garding the interior distribution than that it is lamellar, to enable us to determine completely its external magnetic action. This conclusion is analogous to a conclusion which may be drawn, for the case of a solenoidal distribution, from the ex- pression obtained in 482, for the potential of a magnet of any kind. For, from this expression, we have, according to 513, the following in the case of a solenoidal distribution : m/3 + ny) dSl -If (VIIL); from which we conclude, that without further data regarding the interior distribution than that it is solenoidal, it is sufficient to know the normal component of the magnetization at every point infinitely near the surface to enable us to determine the external magnetic action. Yet, although analogous con- clusions are thus drawn from these two formulae, the formulae themselves are not analogous, as the former (that of 482) is applicable to all distributions, whether solenoidal or not, and shows precisely how the resultant magnetic action will in general depend on the interior distribution besides the normal expression (VII.) (1) as for external points, is, for any internal point, the force at a point within an infinitely small crevass perpendicular to the lines of magnetization ; as it is easily shown that the differential coefficients of 4?r(0) are the rectangular components of the force at such a point due [ 7 (5)] to the free contrary polarities on the two sides of the crevass. xxiv.] A Mathematical Theory of Magnetism. 397 magnetization near the surface, according to the deviation from being solenoidal which it presents; while the formula of 515 merely express a fact with reference to lamellar distributions, and being only applicable to lamellar distributions, do not indicate the effect of a deviation from being lamellar, in a distribution of general form. Certain considerations regard- ing the comparison between common magnets and electro- magnets, suggested by Ampere's theorem that the magnetic action of a closed galvanic circuit is the same as that of a "magnetic shell" (as denned in 506) of any figure having its edge coincident with the circuit, led me to a synthetical in- vestigation [ 554 below] of a distribution of galvanism through the interior and at the surface of a magnet magnetized in any arbitrary manner, from which I deduced formula for the resultant force at any external or internal point, giving the desired indication regarding effect of a deviation from being lamellar, on expressions which, for lamellar distributions, de- pend solely on the tangential component of magnetization at points infinitely near the surface. These galvanic elements throughout the body, from the action of which the resultant force at any external point is compounded, produce effects which are not separately expressible by means of a potential, and therefore, although of course when the three components X, Y, Z of the total resultant force have been obtained, they will be found to be such that Xdoc + Ydy +Zdz is a complete differential, the separate infinitely small elements of which these forces are compounded by integration with reference to the elements of the magnet, do not separately satisfy such a con- dition. Hence the investigation does not lead to an expression for the potential ; but by means of it the following expressions for the three components of the force at any external point, or at a point within any infinitely small crevass perpendicular to the lines of magnetization, have been obtained*: * The expression Xdx + Ydy + Zdz will not be a complete differential for internal points unless the distribution of magnetism be lamellar, since, for any internal point, X, F, Z differ from the rectangular components of the " resultant force," as defined in 479, by the quantities 4, 4ir, 4iry, respectively, and since ( 483) the "resultant force," for all points, whether internal or external, is derivable from a potential. (See Postscript to 517.) 398 A Mathematical Theory of Magnetism. [xxiv. v ftfj j j (vi-yfd* d/3\ Z-zfdy dz\} X=\\\ dxdydz \ L - j rf- \-j - ^ry- -^ - -j- )[ JJJ \ A 3 \dy dx) A 3 \efo? dz) } dS\ (IX.). [Postscript to 517, Nov. 17, 1871. These expressions, to be proved in 518 for external points, may be taken as a definition for "resultant force" at points in the magnetized substance. They are simplified by putting ~7 r = u ) "i T~ = v > i ?~ = w \ / \ dy dz dz dx dx dy > (a), and nft my = U, ly nx s= F, wa I ft = W J which, with a/, y', / substituted for x, y, z ; w', v', w/ for w, v, w; and a?, 2/, z for f, 97, f; reduces them to with the symmetrical forms for F and Z. Now observe that is the 7/-component of the resultant force at (so, y, z) due to a distribution of imaginary matter through the magnet and over its surface, having w for density at any interior point (a?, y, z}, and W for surface density at [x, y, z] ; and for the other terms of (6), etc., consider corresponding distributions (v, F), and (u, U)] and therefore instead of (b), etc., write _ dM v _dL dN 7 _dM dL . - j ~~ ^ JL - ~~j~ ~~ ~~^ , JU j ~~ j ...... ( C ) dy dz dz doc dx dy denoting * by * This notation has been introduced to agree with that used by Helmholtz in corresponding formulae with reference to Vortex Motion. It is to be remarked xxiv.] A Mathematical Theory of Magnetism. 399 L the potential of distribution (u, U) M (v, V) ...... (d) so that L-IIJ<*$ + [jj] , if=etc., *=etc. (.); and, by Poisson's theorem, where, as is now usual, -7-5 + -^ + -j- is denoted bv X7 2 . The //''/* f/?7 CM P second members of (/) vanish for all points external to the magnet, because there u = 0, v = 0, w = 0. Now for simplicity suppose the magnetization to diminish gradually, not abruptly, to zero at the boundary of the magnet. The second terms of the expressions (d) for L, M, N will disappear, and by diffe- rentiations and summation we have fdu' dv' dw^ dL dM dN _ f[f\dx' dy di dx dy dz ~ jjj D But (a) show that ...................... dx dy dz , ,, dL dM dN ,, x and therefore -y- + -= + -y- =0 ...................... (h). dx dy dz However quick the gradation from finite values of u, v, w within the magnet, to zero through external space, this equa- tion holds, and therefore it holds in the limit, when the mag- netization comes to an end abruptly at the boundary. To prove (h) directly from the expressions (e\ with the surface - terms included, will be found a good exercise for the student. From (c) by differentiations, and application of (/) and (h}, we find dX dY dZ_ k] ~~ + ~ " /C dZ dY dX dZ ' dY dX - --- = 4t7ru, -^ -- -j- = 47TV, -j -- -T- dy dz dz dx dx dy or in virtue of (a) dZ^_dY_. (dy_<\ ^_^_4 (^L ^I\ ^T_^ = 4 /^_^\ dy dz ~ \dy dz) ' dz dx~ \dz~ dx)' dx dy \dx dy) j that the quantities u, v, w, U, V, W thus introduced fulfil the equations (1) and (2) of 539. They represent the components of the internal and superficial distributions of electric currents, in the electro-magnetic representative ( 554) of the given polar magnet. 400 A Mathematical Theory of Magnetism. [xxiv. The corresponding properties of X, , 52, if these denote the components of the " resultant force " as denned in 479, 480, are [see 473 (2) and 483] dX c d& (dz , , =-4-7r [-J- +-j- + -r) ......... (m), -T- -T- -j- - -J -j- -r dx dy dz \dx dy dz dX ^52 d dX ,; j U ......... ( n ) . 7 ; f 7 , y j dy dz dz dx dx dy These equations, as well as (k) and (I), hold through all space, the values of a, ft, y being zero in every part of space not con- taining magnetized matter. Some if not all of the differential coefficients appearing in them become infinite when the mag- netization varies abruptly from one side to the other of any surface, but the interpretation presents no difficulty. Taking for instance the case when the magnetization, finite up to the boundary of the magnet, comes to an end abruptly there, let X, and X,, denote the values of X at points infinitely near one another outside and inside the boundary; and similarly for Y 9 Z, X, Y, 5S. We have by 7 (5), 517 (c) and (e), and 473 (1), X t -X lt = 4:Tr(nV-mW), Y t - Y lt = iir(lW-nU), Z t - Z it = Tr (mU -IV) (o) and - = , , where p = lu + mv + nw j ^ By (a) we have nV-m W= I (m/3 -f ny) (m 2 + n 2 ) a = I (la. -f m/3 + ny) - a. Hence, with the notation of (p), (o) becomes -Y t = TT (nip - 0), Z, - Z u = TT (np - y) (q). In a foot-note to 517 above it was stated that the values of X, Y, Z differ from what in this postscript I call X, , 52 by quantities equal to 4?ra, 47T/5, 4?T7, respectively; a statement which is no doubt to be proved directly by carefully examining the meaning of the integrals of 518 for internal points. We may now verify it by taking the difference between (k) and (m), and the differences between (Q and (n). If in these we put dP dQ dR theyg,ve _ + + _ xxiv.] A Mathematical Theory of Magnetism. 401 ^_^Q = Q dP__dR = Q dQ_dP = dy dz ' dz dx ' ' dx dy These last three equations show that where ty if not zero is a function of x, y, z\ and the first then becomes _ ~ This equation must hold through all space when there is no abrupt variation of magnetization ; and, as ty must vanish at an infinite distance from the magnet in any direction, we must ( 206 above) therefore (whether there are abrupt variations or not) have ty = 0. The proof may be illustrated for abrupt variations, by taking the differences of equations (q) and (p), which show that (-T- X - 4wa), -(X-%- 4),, = 0; (F etc., Z etc.) ; or P,-P,, = 0, -& = <), 12, -.8,, = 0; which prove that ^ ^,,= 0, the suffixed accents denoting values for infinitely near points on the two sides of the surface of abrupt change. We conclude that through all space X = X + 4>7ra, F = g? + 47r/3, Z=<% + 4>7ry ......... (r); which, for space unoccupied by magnetized matter, give (what we knew before) x=s, r=u, z=&. For space within the magnet, it was shown in 479 that the force (, |, 5&) is the resultant force experienced by a unit pole in a crevass tangential to the lines of magnetization. From this, and (r), it follows that, as was asserted in 517, the force (X } Y, Z] is the resultant force experienced by a unit pole in a crevass perpendicular to the lines of magnetization. Of these two definitions of " resultant force " for space within a magnet, the former, as suitable to a polar magnet ( 549), will some- times be called the " polar definition," and the latter, as suit- T. E. 26 402 A Mathematical Theory of Magnetism. [xxiv. able for an electromagnet, the " electromagnetic definition," for the sake of brevity.] 518. The investigation by which I originally obtained the expressions (IX. of 517) is, with reference to galvanism, precisely analogous to the investigation in 473 with refer- ence to imaginary magnetic matter. It cannot be given with- out explanations regarding the elements of electro -magnetism which would exceed the limits of the present communication*; but when I had once discovered the formulas I had no diffi- culty in working out the subjoined analytical demonstration of them for the case of an external point, which is precisely analo- gous to Poisson's original investigation (given in 495 above) of the formula of 482. Equations (3) and (4) of 482 and 483 lead to expres- sions for the components of the resultant force at any point in the neighbourhood of a magnet. Taking X only (since the expressions for the three components are symmetrical), we have 1 ,1 ,1 7 . dy ' dz] Now if the factor of dxdydz in the second member of this equation be differentiated with reference to f, an expression is obtained which does not become infinitely great for any values of a?, y, z included within the limits of integration, since the point (f, ?;, f) is considered to be external in the present investigation. Hence the differentiation with reference to f may be performed under the integral sign; and, since ^! ,7* d-r- d-r A_ A lf = ~~' we thus obtain * [Note, Nov. 1871. It is given in 554, below.] t If the point (, 97, f) be either within the magnet or infinitely near it, the factor of dxdydz in this integral is infinitely great for values of (#, y, z) included xxiv.] A Mathematical Theory of Magnetism. 403 Now, for all points included within the limits of integration, we have, from Laplace's well-known equation, and therefore Dividing the second member into four terms, and applying an obvious process of integration by parts, we deduce i , ? l i A ff.iirl. ?.^-rj~* dy X=\ 1 1 I a -T-cMz-a -7- dxdy+fi dydz -f y -7- dydz fff JJJ [ ,!_ ,1^ ,1 doi A <** * ^ S ^- T - + - J - - -- -f -3 -- -rT- ajf cZy dz dz dx dy dec dz Modifying the double integral by assuming, in its different terms, dydz Id 8" dzdx = mdS' } dxdy = ndS, and altering the order of all the terms, we obtain within the limits of integration ; and it may be demonstrated that the value of a part of the integral corresponding to any infinitely small portion of the magnet infinitely near the point (, 17, f) is in general finite, and that it depends on the form of this portion, on its position with reference to the line of magnetization through (, 17, f ), and on the proportions of the distances of its different parts from this point. It follows that if the point ( 77, f) he internal, and if a portion of the magnet round it be omitted from the integral, the value of the integral will be affected by the form of the omitted portion, however small its dimensions may be, and consequently the complete integral has no determinate value if the point (, 17, f) be internal. Hence although, as we have seen above ( 482, 483), has in all cases a determinate value, which, by the definition ( 479), is called the component parallel to OX of the resultant force at (, 77, ), the expression -III has no meaning when (, 77, f ) is in the substance of the magnet. 262 404 A Mathematical Theory of Magnetism. [xxiv. = 1 1 1 dxdydi dy \dy dx) dz a! Afdy_d*\ ,dx dz) This expression, when the indicated differentiations are actually performed upon , becomes identical with the expression for X at the end of 517, and the formulae which it was required to prove are therefore established. 519. The triple integrals in these expressions vanish in the case of a lamellar distribution, in virtue of the equations (III.) of 514; and we have simply dS To interpret these expressions, let us assume, for brevity, From these we deduce mW nV= a. I (la. + mfi + ny) = a, (XII); Z F m ?7= yn (lot. + mfi + ny) =y where a /; /3 t , y t denote the rectangular components of the tangential component of the magnetization at a point infinitely near the surface. Conversely, from these equations we deduce U=n/3 t -my,', V=ly,-nz l ; W = ma, - //3, . . . (XIII.) Now the direct data required for obtaining the values of X, Y, and Z, by means of formulae (X.), are simply the values of U, F, W at all points of its surface. Equations (XII.) show that with these data the values of a,, ft,, y / may be calculated; and again, equations (XIII.) show conversely that if a y , /3 t , y, xxiv.] A Mathematical Theory of Magnetism. 405 be given the required data for the problem may be immediately deduced. We infer that the necessary and sufficient data for determining the resultant force of a lamellar magnet, at any external point, by means of formulae (X.), are equivalent to a specification of the direction and magnitude of the tangential component of the intensity of magnetization at every point infinitely near the surface of the magnet ; and we conclude, as we did in 517 from a very different process of reasoning, that besides these data, nothing but that it is lamellar through- out need be known of the interior distribution. 520. The close analogy which exists between solenoidal and lamellar distributions of magnetism having led me to the new formulae which have just been given, it occurred to me that a formula (or formulae, if it were necessary here to separate the cases of internal and external points), for solenoidal distribu- tions analogous to the formulae (VII.) of 516 for lamellar distributions might be discovered. Taking an analytical view of the problem (the synthetical view, although itself much more obvious, not showing any very obvious way of arriving at a formula of the desired kind), I observed that the formula eft . costidS . g deduced from the g enera l expression for the potential by a partial integration performed upon factors in- volving a, /3, 7, and depending on the integrability of the function ados + fidy + ^dz, insured by the equations M_^-0 ^- = -^? = ~dz dy~~ ' dx dz ' dy dx for a lamellar distribution ; and I endeavoured to find a corre- sponding mode of treatment for solenoidal distributions, to consist of a partial integration, commencing still with factors involving a, /3, 7, but depending no\v upon the single equation ^ + f + J/ = (a), dx dy dz instead of the three equations required in the former process. After some fruitless attempts to connect this equation with the integrability of some function of two independent variables, I fell upon the following investigation, which exactly answered my expectations : 406 A Mathematical Theory of Magnetism. [xxiv. 521. In virtue of the preceding equation (a), we may assume dH dG dF dH dG dF where F, G, H are three functions to a certain extent arbitrary. These functions I have since found, have for their most general expressions iff 7 7 (&$ F = 4/1 dydz ( -y- - 3 JJ ' \dy + ~- dz) dx where i/r denotes an absolutely arbitrary function; and the indicated integrations are indefinite, with the arbitrages which they introduce subject to the equations (XI V.). The demonstration of these equations follows immediately from the results obtained by differentiating the three equations (XIV.) with reference to x, y, and z respectively. The simplest final forms for F 9 6r, and H are the following, which are de- duced from the preceding by integration : .(XVI). az j Making substitutions according to the formulae (XIV.) for a, @, 7 in the general expression for the potential, we have ( ,1 a ,1) jr [ffi j A( dH d&\ A (dF dH\*A (dG dF\*&l. V=\\\dxdydz\(-j - U-+ - _-+ _f JjJ [\dy dz J dx \dz dx J dy \dx dyjdz] Dividing the second member into six terms, and integrating each by parts, commencing upon the factors such as -^ dy, ay we obtain an expression, with a triple integral involving six terms which destroy one another two and two because of xxiv.] A Mathematical Theory of Magnetism. 407 properties such as ,1 1 dy "dx" ~ dx ~dy ' and besides, a double integral, which may be reduced in the usual manner to a form involving dS, an element of the surface. We thus obtain, finally, 1 ,1 ,1 (XVII.). 522. The second member of this equation expresses the potential of a certain distribution of magnetism in an infinitely thin sheet coinciding with the surface of the body ; the total magnetic moment of the magnetism in the area dS being {(mH- nGf + (nF - IH) 2 + (IG- mF)*}* dS, and its direction cosines proportional to mH-nG, nF-lH, IG-mF. Now we have identically, I (mH- nG)+m (nF- IE) + n (IG - mF] = ; and hence the direction of this imaginary magnetization at every point of the surface is perpendicular to the normal. It follows that we have found a distribution of tangential mag- netism in an infinitely thin sheet coinciding with the bounding surface which produces the same potential at any point, in- ternal or external, as the given solenoidal magnet. [It is re- markable that the imaginary tangential magnetization thus found [depends ( 523) upon the normal component of the actual magnetization infinitely near the surface ; so that, besides this normal component, nothing need be known of the actual magnetization except that it is solenoidal. Compare conclu- sion of 519.] 523. The conclusion of 522 may be arrived at syntheti- cally in a very obvious manner, by taking into account the property of a solenoid stated in 510, according to which it appears that any two solenoids of equal strength, with the same ends, produce the same force at any point whether in the magnetized substance of either, or not. For it follows from 408 A Mathematical Theory of Magnetism. [xxiv. this, that when a magnet is divisible into solenoids with their ends on its surface, we may by joining the two ends of each solenoid by any arbitrary curve on this surface, and laying a solenoid of equal strength along this curve, obtain a series of solenoids, constituting by their superposition, a tangential distribution of magnetism in an infinitely thin sheet coinciding with the bounding surface, which produces the same resultant force at any internal or external point as the given magnet. It is not, however, easy to deduce from this synthesis a formula involving the requisite arbitrary functions to express a super- ficial distribution satisfying the existing conditions in the most general manner. The analytical investigation given above, supplies, in reality, a complete solution of this problem. It may be remarked that the sole condition which F, G and H considered as functions of the co-ordinates, x, y, z, of some point in the surface of the magnet, and therefore functions of two independent variables, must satisfy in order that (XVII.) may express correctly the potential at any point, is ,/dH dG\ fdF dH\ /dG dF\ 1 t j j- + m hr T- )+ n \j -r-) = loL+m0+ny (XVIII.) , \dy dz) \dz dx J \dx dy J x y y, and z of course being supposed to satisfy the equation to the surface ; and it may be proved, by a demonstration inde- pendent of the investigation which has been given, that the second member of (XVII.) has the same value for any func- tions F, G, H whatever, which are subject to this relation. {Postscript, Dec. 7, 1871, and Jan. 6, 1872. Inasmuch as the second member of (XVIII.) is ( 473 (1)), the surface density of the imaginary magnetic matter, representing the polarity of the given solenoidal magnet, we may eliminate the idea of magne- tization, and so arrive at the following remarkable theorem : Let p be the (density at any point of a superficial distribution of matter on a surface $, which may be either a closed surface or an open shell, there being as much negative matter as posi- tive in the whole distribution, and let F, G, H be any three quantities such that dH dG\ , (dF dH\ /dG dF\ -j -7-1 -H&l-j -j-)+n (j -T- } = p> (XIX.): dy dz J \dz dx J \dx dy J ^ xxiv.] Electromagnets. 409 the potential of this distribution, that is to say // , is correctly expressed by the formula (XVII.). When S is a closed surface this expression holds for the space within S, as well as for external space. From the remark with reference to (XVII.) and (XVIII.) at the conclusion of the section in the original now numbered 523, it appears that the values of F, G, H given by (XVI.), although expressing the most general solution of (XIV.), are not the most general expressions for functions F, G, H to satisfy (XVII.); and that instead of F t G, H in (XVII.) we may substitute F+F' 9 G + G f , H + H' where F, G, H are given by (XVI.) and F' t Q', H' are any three functions of x 9 y> z, which, over the whole surface S, satisfy the equation H' cZ6A (dF* dET\ /dG' dF'\ -j -y- 1 + (-j 7 )+ n (-J r~ ) = 0... dy dzj \dz dx ) \dx dy ) The surface distribution of tangential magnetization specified by F', G' } H' in accordance with the explanations of 522, consists of closed solenoids lying on the surface S.] CHAPTER VI.* On Electromagnets. 524. Oersted's discovery of the mutual forces between magnets and conductors containing electric currents gave rise to the science of electromagnetism. It was soon found that there are also mutual forces between different conductors and between different parts of the same conductor conveying electric currents : and various very remarkable electro-magenetic phenomena were observed by different experimenters, of which the most remarkable are the continuous rotations of portions of conductors round magnets and of magnets round conductors, * [Note, October 1871. This chapter was written twenty-two years ago, and has lain in manuscript ever since, because I had not succeeded in finding time to write a sequel on inverse problems. It is now printed from the original manuscript with only a few verbal alterations, and it will be followed in this volume (Chap. IX.) by the long-projected article on inverse problems, of which something was communicated to the British Association at its Oxford Meeting of 1847, but not published except in the very short abstract contained in the Report of that meeting,] 410 A Mathematical Theory of Magnetism. [xxiv. discovered by Faraday to result, in certain circumstances, from their mutual actions. The laws to which all these actions are subject were first completely investigated by Ampere. His experiments are the foundation, and the conclusions which he deduces from them constitute the elements, of the Mathematical Theory of Electromagnetism. As a complete and satisfactory account of these researches is to be found in Ampere's original papers*, and a succinct exposition of the mathema- tical part of the investigations, in Murphy's Treatise on Elec- tricity, the results will be considered as fully established, and those of them which are required in the present essay will be quoted. 525. Let P and P be points in two conductors, of which the lateral dimensions are very small compared with the distance PP \ let a- and or' be the length of infinitely small elements of these conductors, with their centres at the points P and P' respectively, and terminated by planes perpendicular to the directions of the conductors; let PP be denoted by r\ let and 6' denote the angles at which the directions of the con- ductors at P and P' are inclined to the line PP' ; and let

respectively. There will be such a mutual action between the magnet and the galvanic arc a that each will experience a force, the resultant of two forces through P per- pendicular respectively to the planes of NP and er, and of SP and tr, given in amount by the following expressions respec- tively : in . *vcr . -.m. 70" . . / '-^ sm<, and-JpsmcJ.. The directions of these forces, upon the element, if the direction of the current be from east to west, and if N and 8 be each north of P, will be; the former obliquely or directly down- wards, and the latter, upwards. [For mnemonic principle see below 547.] The magnet will be acted upon as if a point in the position of P, rigidly connected with it, experienced two forces equal and opposite to the forces of which the action on A sin < A sin $ 529. Hence the components of the force experienced by the element of electric current are given in magnitude and direc- tion by the following expressions : A 3 A s A 3 If the axes of co-ordinates be so chosen that when OX is from south to north, and OY from east to west, OZ will be vertically upwards, these expressions will be applicable, as far as regards signs, to the direction of the action which the electric arc experiences ; and it would be necessary to change the sign of each, to make them applicable to the direction of the force upon the pole. 530. These expressions, since they involve I, m, n only line- arly, show that a galvanic arc cr, of strength 7, in the direction I, m, n, produces the same effect either upon another arc, or upon a magnet, as three arcs parallel to the axes of co-ordinates, each of the same strength, 7, and of lengths respectively equal to alt <* m > ' n - 531. The factor 7 being taken as the numerical measure of the strength of the current in the circuit of which cr is an arc, the unit of strength for an electric current may be defined in the following manner : If a galvanic current, in a conductor of infinitely small section, be such that the mutual action between any infinitely small arc of it, and a unit magnetic pole held in a direc- tion perpendicular to the length of the arc, at a unit of dis- tance, is numerically equal to cr the infinitely small length of the element, the strength of the current is unity. xxiv.] Electromagnets. 413 Or in the following manner : If a galvanic current in a conductor of infinitely small section be such that the action between two infinitely small portions of it in line with one another and at a distance unity from one another, is numerically equal to the product of the length of the elements, the strength of the current is unity. 532. If what is called "an electric current" be in reality the transference of matter along the conductor in which it exists, the "strength of the current" numerically measured in the manner which has been explained, will depend upon the quantity of this matter transmitted in a given time; and a unit of time may be chosen, according to the unit of electrical quantity which is adopted, so that the quantity 7, measured as above explained by the electro-magnetic action of the conductor, may be numerically the quantity of electricity which flows across any section of it in a unit of time. 533. In a continuous current, this quantity is of course the same for every section; and, as it is impossible that a continu- ous stream of electricity can emanate from one body, and be dis- charged into another, the current must be re-entering, or every continuous current must form what is termed " a closed circuit." It is found by experiment that whatever be the dimensions or material of the different parts of the conductor along which the current flows, provided always the dimensions of the section be small compared with the distances through which the electro-magnetic action is observed, the quantity 7 has the same value for all parts of it; and even in the places where the electro-motive force operates, as has been shown by Faraday, as in the liquid of any ordinary galvanic battery, or in a con- ductor in motion in the neighbourhood of a magnet, the electro- magnetic effects are observable and probably to exactly the same degree; so that it would probably be found that a gal- vanic circuit consisting of a battery of small cells arranged in a circular arc, and a wire completing the circuit by joining the poles, would produce the same electro-magnetic effects at all points symmetrically situated with reference to the circle, irrespectively of the part of the circuit, whether the cells or the wire; provided always that the distances considered be great compared with either the dimensions of a section of the 414 A Mathematical Theory of Magnetism. [xxiv. wire, or of any of the cells made by planes perpendicular to the plane of the circle, through its centre. 534. Hypothesis of Matter flowing . In the theory of electro- magnetism it is quite unnecessary to adopt any such hypothesis as this, however probable or improbable it maybe as an ulterior theory; and all that we could introduce as depending upon it is that, for a linear circuit of varying section or material, the quantity 7 is the same throughout the circuit, and that all finite circuits possessing continuous currents are necessarily closed ; two facts which cannot be assumed a priori, but which are in reality established by satisfactory experimental evidence. 535. Division of Electromagnets into three Classes Linear, Superficial, and Solid. If all the dimensions of any section of the conductor along which the current is communicated be infinitely small, the complete circuit constitutes what will be called a linear electromagnet. When the electric currents are confined to a shell of which the thickness is infinitely small, and when they are continu- ously distributed through it, or distributed through it in such a manner as not to satisfy the condition by which a linear electromagnet is defined, the entire group of the complete circuits constitutes what is called a superficial electromagnet [or surface-electromagnet]. When electric currents are so arranged as to fill any solid portion of space, the group of the complete circuits constitutes a solid electromagnet. It is clear that, in practice, electromagnets may be treated as linear, or superficial if the quantities which ought to be in- finitely small, are merely very small compared with the dimen- sions of the magnets, and with the distances at which the electro-magnetic action are to be observed; and again, if wires, or linear currents of any kind, be disposed upon any surface or through any space, so that the distances between those which are adjacent are small compared with the dimensions of the circuits, or of the curves, or with the distances at which the magnetic actions are to be observed, the group may be considered as constituting practically a superficial electro- magnet; and a solid electromagnet may be composed of a group of galvanic wires similarly arranged through a solid space. xxiv.] Electromagnets. 415 536. Linear Electromagnets. A linear electromagnet is com- pletely specified when the form of the closed curve of the current, and 7, the strength, are given. Irrespectively of any theory, the term " electric current " will often be made use of; but as the terms, literally interpreted, imply a theory which, to say the least, is doubtful, it must be borne in mind that they are not to be interpreted literally, and that they are only used in this essay occasionally for conveni- ence ; and especially because of the almost universal use which is made of them by writers on the same subject. The term " galvanism " will often be used to denote the agency to which the phenomena presented by continuous electric currents are due, and quantity of galvanism in a linear conductor will be measured according to the following standard : The strength of the current in a linear electromagnet into the length of any part of the conductor in which it exists, is the quantity of galvanism in that portion. The term intensity will be used with reference to linear currents, according to the following definition : The intensity of the galvanism in any part of a linear electro- magnet is equal to the strength of the current, divided by the area of the section of the conductor. Hence in a linear conductor of which the section is not uniform throughout, the intensity of the galvanism will vary inversely as the section from one part to another of the conductor*. 537. Superficial Electromagnets. Def. The quantity of gal- vanism on any small portion of the surface, divided by its area, is the superficial intensity of the galvanism at that point. If the superficial intensity f and the direction of the galvanism is given at every point of a given surface, the specification of the superficial electromagnet is complete. There are, however, certain conditions to which such a specification is subject, and an arbitrary specification, not satisfying them, will not corre- spond to any possible superficial electromagnet. The founda- * [Note, Oct. 25, 1871. I leave this section exactly as I find it in the old manuscript, under protest that I do not now approve of the mode m which the word "galvanism" is used in the terms which it proposes. Where these terms occur henceforth it is because I have not invariably altered the manuscript to substitute more convenient modes of expression.] t [In 1871 we should rather say surface-intensity than superficial intensity.] 416 A Mathematical Theory of Magnetism. [xxiv. tion of these conditions is the fact that no incomplete circuit can exist permanently, from which it follows that all the currents, or the continuous superficial flux of electricity con- stituting a superficial electromagnet, must be resolvable into a group of closed galvanic currents. This will lead to a condition which must be satisfied at every point of a purely superficial electromagnet, and again, a con- dition which must be satisfied at the boundary, if the surface be not closed. The mathematical expression of these conditions will be given later. 538. Solid Electromagnets. Def. The intensity of the gal- vanism at any point within a solid electromagnet is the quan- tity of galvanism in a space of infinitely small dimensions round that point, divided by the volume of the space. The complete specification of a -solid electromagnet will be the expression of the intensity and direction of the galvanism at every point of it. Here again there will be conditions to be satisfied by the specification, to express the fact that all the galvanism consists of a group of closed circuits. 539. After these preliminary explanations we may enter upon a regular analytical treatment of the subject ; commencing with investigations of the conditions to which the distribution of galvanism in solid and in superficial electromagnets is subject. Let u, v, w denote the components of the flux at any point (x } y } z) within a solid electromagnet; and, if there be besides a superficial distribution of galvanism on the bounding surface, let U, Vj W be the components of the superficial flux at the point (x, y, z) when this point belongs to the surface. These quantities must satisfy the following conditions, in order that the galvanism expressed by u, v, w, U, V, W may consist of a group of closed circuits :- dx dy dz for every point (x, y, z) of the magnet, and dU dV dW fdn_dm dx dy dz \dy < xxiv.] Electromagnets. 417 for every point (a?, y, z) of the surface of the magnet, the direc- tion-cosines of a normal to the surface being denoted by I, m, n. 540. To demonstrate these conditions, let us consider an in- finitesimal tubular portion of the magnet bounded by stream- lines*, and by the surface of the magnet, if any of these lines cut it. Let the stream-lines thus considered be infinitely near one another, so that the portion of the magnet contained by them may be a ring of infinitely small section, cut or not as the case may be, by the surface of the magnet. The conditions to be satisfied with reference to this portion of the magnet are, that the intensity of the galvanic stream at each point must be inversely proportional to the area of section perpendicular to the stream-lines of galvanism ; and that if the ring be cut by the surface of the magnet, the incomplete galvanic arc thus existing within the magnet must be completed along the sur- face. Since the whole body may be divided into portions of this kind, we have a condition for every internal point, and by ex- pressing that the superficial distribution U, Y, W must be such as to complete circuits for the galvanic arcs, of which the ends are in the surface, the condition io which U, V, W are subject is obtained. To investigate the condition for u, v, w, consider an infinitely small parallelepiped afiy, of which the centre is at (x, y, z), and the edges respectively parallel to OX, OY, OZ, and let the galvanic arcs into which the whole magnet is divided be supposed to be of sections so small that an infinite number of them will pass through this parallelepiped. The condition to be expressed will be that the sum of the products of the intensities into the sections at one set of the ends of these arcs shall be equal to the sum of the corresponding products at the other set of ends. The sums of these products for all the ends which lie on the two faces /3, 7, of which the distances from YOZ are x , x -f a are respectively equal to ( u - ^ ' and w + ^ * [Note, Oct. 25, 1871. This term (its introduction is I believe due to Rankine) is now much used in writings on hydrokinetics. It is substituted for "lines of galvanism," which I find in my old manuscript.] T V 27 418 A Mathematical Theory of Magnetism. [xxiv. and similarly, for the faces 7, a, we obtain the sums and, fora, /3, (w- i7:r) a A and \. CLZ / Now when w, t>, and w are all positive, one set of the ends of the galvanic arcs will lie on the three faces of the parallele- piped of which the distances from the co-ordinate planes are respectively x J a, y ^/3, -iy; and the other three faces will contain the other set of ends, and we must therefore have whence* + ! + = o. [(1) of 539]. dx dy dz 541. To investigate the conditions for the surface of the body, it may be remarked that if there were no galvanic arcs from within, terminated at the surface, there might be no superficial galvanism, and that any superficial galvanism there could be must constitute a group of closed circuits ; but that when there are interior galvanic arcs of which the ends lie on the surface, the superficial distribution must complete the circuits for them, besides containing any arbitrary distribution of closed circuits. Hence, if P and P' be two points on a band of the surface between two lines of superficial galvanism infinitely near one another, /3 and @' the breadths of the band at these points, and / and /' the superficial intensities of the * It is scarcely necessary to remark that this is the same as the " equation of continuity," for the motion of an incompressible fluid, of which the velocity at any point (x, y, z) is the resultant of w, v, w. The condition that as much fluid leaves the parallelepiped a.py as enters it, in a unit of time would lead to pre- cisely the same investigation as that of the text (see Duhamel's Cours de Mecanique, or Cambridge and Dublin Mathematical Journal, 1847, p. 282). The electrical matter which may be imagined to be flowing through the body, must not become accumulated, nor leave a deficiency hi any part. [Note, Jan. 1872. When this was written, upwards of twenty years ago, the investigation of the "equation of continuity" here referred to, adapted from Fourier, was but little known.] xxiv.] Electromagnets. 419 galvanism ; the values of the products I.j3 and 1'.$' must differ by an amount equal to the sum of the strengths of the interior arcs of which the ends lie on the band between P and P'. Now if ds denote the length of an element of the band, the sum of the strengths of all the interior arcs having their ends on this part of the band will be and therefore if P and P' be situated at the two extremities of ds, we must have I'ff-I0 = (lu + mv + nw)l3d8 ............. (1); or, if the symbol ft denote differentiation performed with refer- ence to variations along the superficial stream-line through P t %(I@) = (lu + mv + nw)@ds ............... (2). Now let (f) be such a function of x, y, z that the equation = & .............................. (3), with different constant values given to k, shall represent any set of surfaces cutting the surface of the magnet along the stream-lines; that is to say (as the direction cosines of the stream-line are proportional to U, F, W), let be any function satisfying the equation U^+V^ + W^ = ............. (4). dx dy dz And, because the stream-line lies on the surface, we have lU+mV+nW = ..................... (5). If K be the difference of the values of k for the two bounding stream-lines on the two sides of the band through P which we have been considering, we readily obtain, for the breadth of the band, the following expression : (6). (/ d$ d\* , / dj> $Y/* m ! ( w* j- w 7 ) +{n-i I ~j + ( ^ ^r ~~ m ~j~] |\ dz dy) \ doc dz) \ dy dx) Equations (4) and (5) with U*+V* + W> = F ....................... (7), resolved for U, V, W, give 420 A Mathematical Theory of Magnetism. [xxiv. TT \ dy dz) U j_j F= - dz ax W dx dy (8), where, in virtue of (6) i i K- .(9). And from (8) we have ',d(b TF-?iF)=(m 2 -Mi 2 )-- ' dx and therefore, if we put d(b ,/ dd> d --M m-r- + n-^-} = -^- dx \ dy dz) dx dx dy dd> , d\ 5 : -f n^ d dz) we have d ff -j-H-my^-f ?i-j^=II dx dy dz dy .(10). Differentiating (9) along the stream-line, we have *(/)_ ** 7 --- f~*1 7 as K ds Hence Now &ds _ ds ds & dx ds dy ds'*' dz "' -F, I=F..... .......... (12). ds ds Using these in (11) and then putting for U-r- etc., the equi- ax valent formulae B -r- , etc., and making use of equations xxiv.] Electromagnets. 421 (8) we find = dV dW fids dx dy dz 4.1_J^^_^4-^^_^ . N + V[l-j- + 'm>-j- + n-j-}+W{l-j- + mj- + n -j- I. ..(16). V dx dy dzj \ dx dy dz) 544. The mutual actions between electromagnets and com- mon magnets, or between any part of an electromagnet and other partial or complete electromagnets or common magnets, may be determined by means of the expressions of 525 529 ; and when the data are sufficient, the application of elementary statical principles leads to the solution of any problem that can be proposed. The mode of specifying the distribution of galvanism in an electromagnet, explained in 531 539, leads immediately, by means of Ampere's formula given above, 525, 527, to proper expressions for the mutual action between any two solid electromagnets by means of four definite integrals representing the parts of that component due to the mutual actions of the solid and superficial parts of their distributions of electric current. 545. A similar synthetical solution of the problem of deter- mining the mutual action between an electromagnet and a common magnet, is obtained by first investigating a formula for the mutual action between an element of a galvanic circuit, and an infinitely small magnet, which may be done at once by means of the formulae of 520, 529, and the synthesis of a magnet explained in 461, 462, and then applying statical principles to derive formulas for the components (both of force and couple) of the mutual action. It is sufficient here to indicate the method of proceeding, for such problems ; and unnecessary to write down the formulae, which, in fact, may always, when wanted, be written down at once from the formulse of the preceding chapters, according to the principles which have been now explained. Thus, write down the formulae for the rectangular components of the force exerted by the electromagnet (u, v, w, U, V, W, 539), upon a positive xxiv.] Electromagnets. 423 unit pole, according to the formulae of 529 ; and for the com- ponents of couple which would be given by transferring the con- stituent forces from their supposed lines through the elements of electric current, to parallel lines through the magnetic pole. It will be found that in the integrals the components of couple disappear, and thus is proved Cor. 5 of 549 ; that the resultant force is in a line through the pole. The expressions for the components of this force are, as mere inspection of the formulae of 529 proves, identical with those of 517, (fr). I proceed to propositions regarding electromagnetic force, the importance of which will appear from the application made of them in subsequent investigations. 546. Proposition. The action of an infinitely small plane closed circuit on an element of another circuit, or on another complete electromagnet or magnet of any kind, is the same as would be produced by an infinitely small magnet, in the same position, with its axis perpendicular to the plane of the circuit*. [The proof is easily worked out from the formulae of 483, 485, 529.] * [Note added Jan. 1872.] Hence Ampfere's theory of magnetism, according to which, magnetization of ^steel or load-stone, or soft iron, or any other polar magnet ( 549) consists of electric currents circulating round the molecules of the magnetized substance in planes perpendicular to the directions of magneti- zation. From twenty to five-and-twenty years ago, when the materials of the present compilation were worked out, I had no belief in the reality of this theory (compare 602) ; but I did not then know that motion is the very essence of what has been hitherto called matter. At the 1847 meeting of the British Association in Oxford, I learned from Joule the dynamical theory of heat, and was forced to abandon at once many, and gradually from year to year all other, statical preconceptions regarding the ultimate causes of apparently statical phenomena. In a paper communicated to the Koyal Society of London, 10th May 1856, under the title "Dynamical Illustrations of the Magnetic and the Heli9oidal Kotatory effects of Transparent Bodies on Polarized Light," [Art. xcui. of Eeprint of Mathematical and Physical Papers (Vol. n.)] after proving that the helicoidal property shown by syrup, oil of turpentine, quartz crystals, etc., is due to a right or left-handed asymmetry in the constituent molecules, I made the follow- ing statement regarding the nature of magnetism : "The magnetic influence on light discovered by Faraday depends on the 'direction of motion of moving particles. For instance, in a medium possess- ing it, particles in a straight line parallel to the lines of magnetic force, dis- ' placed to a helix round this line as axis, and then projected tangentially with ' such velocities as to describe circles, will have different velocities according as 'their motions are round in one direction (the same as the nominal direction of 'the galvanic current in the magnetizing coil), or in the contrary direction. But 'the elastic reaction of the medium must be the same for the same displace- 'ments, whatever be the velocities and directions of the particles ; that is to say, 'the forces which are balanced by centrifugal force of the circular motions are 'equal, while the luminiferous motions are unequal. The absolute circular 'motions being therefore either equal or such as to transmit equal centrifugal 424 A Mathematical Theory of Magnetism. [xxiv. Cor. 1. The magnetic moment of the infinitely small magnet which produces the same magnetic effects as an infinitely small plane closed circuit is equal to the galvanic strength of the circuit, multiplied into the plane area which it encloses. 547. Rule for Directions. The magnet must be so held relatively to the current which it represents, that if the circuit and it be placed at the centre of the earth, with its plane in the earth's equator, and with the current going round from east to west, the north polar side of the magnet shall be towards the earth's North Pole. [Mnemonic principle : Remember that if terrestrial magnetism were due to currents in the earth's crust, their general direction would be "the way of the sun;" that is to say, from east to west.] 548. Cor. 2. The magnetic action of a linear electromagnet ( 535) [that is to say, a galvanic circuit in an infinitely thin conducting ring] of any form is the same as that of a uniform magnetic shell ( 506) of any shape having its edge coincident with the circuit, and having its magnetic strength numerically equal to the galvanic strength of the circuit. The rule for "forces to the particles initially considered, it follows that the luminiferous "motions are only components of the whole motion; and that a less lumi- "niferous component in one direction, compounded with a motion existing in "the medium when transmitting no light, gives an equal resultant to that of a "greater luminiferous motion in the contrary direction compounded with the "same non-luminous motion. I think it is not only impossible to conceive any "other than this dynamical explanation of the fact that circularly polaiized light "transmitted through magnetized glass parallel to the lines of magnetizing "force, with the same quality, right-handed always, or left-handed always, is "propagated at different rates according as its course is in the direction or is "contrary to the direction in which a north magnetic pole is drawn; but I "believe it can be demonstrated that no other explanation of that fact is possible. "Hence it appears that Faraday's optical discovery affords a demonstration of "the reality of Ampere's explanation of the ultimate nature of magnetism ; and "gives a definition of magnetization in the dynamical theory of heat. The "introduction of the principle of moments of momenta ('the conservation of "areas') into the mechanical treatment of Mr Rankine's hypothesis of 'molecular "vortices,' appears to indicate a line perpendicular to the plane of resultant "rotatory momentum ('the invariable plane') of the thermal motions as the "magnetic axis of a magnetized body, and suggests the resultant moment of "momenta of these motions as the definite measure of the 'magnetic moment.' "The explanation of all phenomena of electro-magnetic attraction or repulsion, "and of electro-magnetic induction, is to be looked for simply in the inertia and "pressure of the matter of which the motions constitute heat. Whether this "matter is or is not electricity, whether it is a continuous fluid interpermeating "the spaces between molecular nuclei, or is itself molecularly grouped; or "whether all matter is continuous, and molecular heterogeneousness consists in "finite vortical or other relative motions of contiguous parts of a body; it is ' ' impossible to decide, and perhaps in vain to speculate, in the present state of "science." xxiv.] Electromagnets. 425 directions is, that if the circuit be held so that in any part of it the current is from east to west, then a point carried in a circle round that part of the galvanic arc northwards above it and southwards below it, will cut the shell through from its north polar to its south polar side. 549. Cor. 3. A common magnet [or a polar magnet as I shall henceforth call anything magnetized after the manner of a load- stone or a steel magnet] may be found which shall produce the same action as any given complete electromagnet, upon other magnets of either kind, or upon any portion of an electromagnet [or arc of an electric circuit]. Cor. 4. The distribution of ordinary [or polar] magnetism which produces the same force, according to the "electro- magnetic definition" ( 517), as a given electromagnet is indeterminate. [Because any lamellar distribution consist- ing of closed shells may ( 512, Cor. 6) be superimposed on a distribution of magnetism without altering the resultant force electromagnetically defined in 517. Compare below 584588.] Cor. 5. The mutual action between a magnetic point or pole, that is, an end of an infinitely thin uniformly and longitudin- ally magnetized bar, and a complete electromagnet, is in a line through that point. [Compare 526, 545.] 550. Cor. 6. The definition (1) of 479 and the definition of the potential with the propositions on which it is founded, as set forth in 481, 483 may be applied without alteration to an electromagnet, as far as regards points external to the conduct- ing matter through which the electric currents pass. 551. With regard to internal points, the definition given in 517 for the resultant force requires no conventional under- standing of an analogous character to that which was made in the case of points in the substance of common magnets, and set forth in the text and in the second foot-note of 479. We can- not, as in the case of a common magnet, suppose a portion to be cut from the substance of an electromagnet, without deranging the magnetic condition of the remainder. If we imagine a space hollowed out in the substance of an electromagnet, we must suppose such arrangements made that the vacancy 426 A Mathematical Theory of Magnetism. [xxiv. shall only deflect, not interrupt the electric currents. If a small spherical portion, for example, be cut from an electro- magnet, there may be either a gradual deflection of the current through some space round the part cut out ; or the interrupted circuits may be completed by a condensation of electric cur- rent on the surface bounding the hollow. But it is satisfac- tory to know that the resultant magnetic force at any point within such a hollow space is infinitely little affected by the supposed deflection of the currents, when the space is infinitely small. This follows from the comparison of similar circum- stances for similar hollows of different dimensions, which shows that the disturbing influence is in simple proportion to the linear dimensions of the hollow. Or, simply taking the triple integrals of 545, 517 (b) or (c), and using them for a point, P, within the conducting substance, we see in a moment that the part of each integral belonging to any small space round P diminishes in proportion to the linear dimensions of this space when made infinitely small without change of shape or of position relatively to P. Hence there is no necessity for hollow- ing out a space in the electromagnet or of further considering the complicated circumstances referred to above, and the re- sultant force at any point within or without an electromagnet is the force which may be simply defined as the force expressed by the formulae of 528, according to the modes of specification and principles explained in 536, 537, 538, 545 ; [that is to say, simply the formulae (b} of 517]. 552. If an electromagnet consist of a number of conductors which when put together fit close to one another, without touching, or of a single wire of a rectangular or hexagonal section, rolled up with the different parts of the wire not touch- ing one another, but lying close together so as to be separated by spaces infinitely small compared with the lateral dimensions of the wire ; the preceding definition of the resultant force at any point of the magnet considered as a single solid electro- magnet will give sensibly the same resultant force at neighbour- ing points whether in the substance of the conductor or in the interstitial space. [Addition and correction, Oct. 27, 1871. But even if the spaces between the different circuits, or the neighbouring por- xxiv.] Electromagnets. 427 tions of one circuit constituting an ordinary artificial electro- magnet, be not infinitely small or be infinitely great compared with the sections of the conductors, the variation of force from point to point between two neighbouring portions of circuit will be small in comparison with the whole force generally, pro- vided that the ratio of space occupied to whole space within the bounds of the electromagnet be great in comparison with the ratio of the diameter of the wire to the diameter of a section of the electromagnet across all the circuits or wires. This is easily proved from (c) of 517. Consideration of the corre- sponding gravitational case is instructive. In the first place for simplicity ; consider a great spherical space, 8, of radius, R, with a great number, n, of equal homogeneous spheres of very small radius, r, and density, p, distributed with average homo- geneousness through it, so as to give an average density equal tp !!, At the boundary of 8 the resultant force will be Jt approximately towards the centre and equal to 47r nr 3 p ~3~T?~^ ; and at distance x from the centre, it will be approximately towards the centre and equal to 4?r nr s p Tl?~ a The greatest deviation from these approximations would be produced by taking one of the small constituent spheres from a great distance, and bringing it into contact with the point attracted, which would introduce a force amounting to 4?r IT^ and therefore would produce but a small difference on either T the magnitude or the direction of the resultant force if - is small in comparison with ^ . Generally, for any group of molecules attracting according to the Newtonian law, if the pro- duct of the density into the diameter of a molecule be very small in comparison with the product of mean density into diameter of the whole ; the masses of the molecules might be expanded 428 A Mathematical Theory of Magnetism. f xxiv. into the interstices so as to continuously occupy the whole volume of the whole group, without producing anywhere more than a very small change in the resultant force.] 553. A superficial distribution of electric currents gives the same normal component, but different tangential components, for the resultant magnetic force at points infinitely near it on its two sides. The tangential component at one side is found by compounding with a force equal and parallel to the tangen- tial component force at the other side, a force perpendicular to the stream-lines and equal to 47r/, if / denote the surface intensity of the electric stream. [These propositions are easily proved from the surface term of the expression (6) of 517, applied to the present subject according to 551. They are in fact proved by equations (o) of 517. Equations (p) of the same section express in symbols the well-known corresponding proposition in respect to a superficial distribution of matter acting according to the inverse square of the distance, which in words is, that the tangential component is the same, for points infinitely near one another on the two sides of the sur- face, but the normal components differ by ^irp, if p denote the surface density.] 554. Original investigation of 517 (IX.) referred to in 518. " Glasgow College, 7th November, 1849. Yesterday I fell upon " a train of synthesis and analysis of galvanic distributions "which I think will add much consistence and symmetry to " the whole first part of my paper on magnetism (a portion of " the first part was communicated on the 21st of June last, by " Colonel Sabine, to the Royal Society), and it will help me in "getting to work to write out the matter I have had so long "in hand. It occurred to me to treat galvanic distributions "according to the analogy of Chapter III., 'On the imaginary " magnetic matter by which the polarity of a magnet may be "represented [ 463 475 above];' thus, a, /3, 7, being the "components of the intensities of magnetization at (x, y, z), " consider Ampere's imaginary currents round dxdydz. We "have strength of current round OX, along faces dxdy, dxdz, " dxdy, and dxdz, = adx. " Consider all the partial currents parallel to OX. We have xxiv.] Electromagnets. 429 "-ftdydx, along one of the dydz faces (that which corresponds " to x, y t z + dz], and ydzdx along one of the dzdx faces (that " which corresponds to x, y + dy, z). "The coincident face, dydx, of a contiguous elementary " parallelepiped has "and the coincident face dzdx of another contiguous parallel- " epiped has "Hence (as in Chapter III.) the share for the element " dxdydz, of galvanism parallel to OX, is, d0 " So for shares parallel to Y and OZ we find dy dx t( But at the surface of the magnet there is unneutralized " galvanism. Hence, besides the internal distribution we have " a superficial distribution ; and the share to a superficial ele- "ment ds has, I find, for its components parallel to OX, OY t " OZ, the following : ((Sn 2 d*dyd* -.(I) -co./ co J co O7T where E denotes the resultant force at x, y, z. This is an obvious conclusion from the following investiga- tion for the mutual potential energy ( 503, Addition of date llth December, 1871) of two distributions of matter; or, as for brevity we may call them, two bodies. * "Matter" is here used conventionally and merely for brevity, to denote a substance fulfilling the conditions by which "imaginary magnetic matter" ( 463) is defined; that is, substance of which any two, small portions repel one another mutually with a force equal to the product of their quantities divided by the square of the distance between them. Either or both quan- tities may be negative, and the negative product of unlike masses indicates attraction. Not being in any way occupied with Kinetics at present, we suppose this imaginary matter to remain where it is placed until we please to move it; so that a "distribution" of it maybe supposed to be either a rigid body or a flexible body, or a flexible and compressible body, held at rest by the necessary force, except when we suppose it to move; and then we per- form work, positive or negative, upon it to whatever amount is necessary to produce, irrespectively of inertia, the supposed motion against or with the forces resulting from attraction or repulsion, which the portions of the matter moved experiences. All the formulae and conclusions are applicable to real matter, gravitating according to the Newtonian law, if we substitute attrac- tion for repulsion, that is to say, change the signs of each formula for force or work, and exclude negative matter. In applications of gravity, therefore, instead of the "mechanical value" or "potential energy" of a distribution of the imaginary magnetic matter, we have an "exhaustion of energy" (Thomson and Tait's Natural Philosophy, 549) in a distribution of real matter. 438 A Mathematical Theory of Magnetism. [xxvi. Let p be the density at any point (#, y, z) of one of these bodies M ; and let V be the potential at the same point, due to the other body M'. Then denoting by Q the mutual poten- tial energy of the two, we have r r ooJ OOJ CO We have by Poisson's theorem, dY dZ\ dy + dz ) where X, Y, Z denote the components of the force at (#, y, z,) due to the body M. This equation (as it also expresses Laplace's theorem for space containing none of the matter of M, since there p = ;) holds throughout space. Hence for (2) we may write !//. p fdx d y dZ Q = -T-| (T" + ^- + T~ 4<7rJ-J-. x> J-. 00 \dx dy dz Hence by integration by parts Q i r r r (xx , + Yy/+ 4*7T J ooJ 00^' GO where X' Y'Z' denote the components of the force at (#, y, z,) due to M'. Let now the second body consist of a distribution of matter coincident with the first and similar to it throughout, but let the whole quantity of matter in the second body be infinitely small and be denoted by dm, that of the first being denoted by m : we shall have v , dm v v , dm , dm A = - A, i = - I , Zi = - . m m m Instead of Q write now dE. We have This formula expresses the quantity of work required to add dm similarly distributed to a distribution m already made. Our supposed matter being not subject to the law of impenetra- bility, we might simply suppose the distribution of dm, precisely similar to that of m, to be given at an infinite distance and to be moved against the repulsion of m into coincidence : the work xxvi.] Mechanical Values of Distributions of Matter. 439 required is that which is denoted by dE. So far it is not necessary to suppose dm infinitely small. But if dm be in- finitely small, the work required to bring it in infinitely smaller parts from infinite mutual distances into the supposed position of coincidence with the distribution of m, would involve only an infinitely small amount of the second degree of infinitesimals, on account of the mutual influences of the different parts of dm. Hence the formula (5) represents the work required to augment the supposed distribution from m io m + dm, by bringing altogether from a state of infinite diffusion the in- finitesimal portion of matter dm ; and therefore the integral of this formula from to m is the whole work required to build up the distribution m from infinitely diffused matter. Now, with reference to the variation of m, each of X, Y, Z varies in simple proportion to m, and therefore the triple integral may be denoted by Cm 2 , so that we have dE=-r- Gmdm, 47T which gives E- 07T Finally eliminating C we have E=-F r r dxdydz(X*+Y*+Z*) (6). 07TJ a,./ ooJ oo The preceding deduction of the formula (4) from (2) mutatis mutandis allows us to come back to the following important alternative formula p .. J caJ aoJ oo The direct proof of this formula by integration with reference to m, commencing with an expression for dE derived from (2) is obvious. 562. The forces at points similarly situated relatively to similar bodies, are proportional to the linear dimensions of the bodies, and to their densities in corresponding places. The values of (1) for similar bodies are therefore as the fifth powers of the linear dimensions, and as the squares of the densities. Hence if a homogeneous rectangular parallelepiped 440 . A Mathematical Theory of Magnetism. [xxvi. be divided into i 3 equal and similar parts, and these parts be separated to infinite distances from one another, the whole value of the integral (1) for the scattered parts is equal to 2 of its value for the undivided body. It follows that if a finite body be divided into an infinite number of infinitely small parts, and these parts be separated to infinite distances from one another, the value of the integral (1) for all the parts be- comes an infinitely small quantity of the same order as the square of the diameter of one of the parts. Hence the integral (1) relatively to a finite body or distribution of matter, composed of ultimately homogeneous continuous substance, expresses the work required to build it up out of infinitely small parts having the same density (or any other density not too infinitely great) and given at infinitely great distances from one another. 563. A complete analytical view of the circumstances con- templated in 562 is, as is generally the case, easier than the quasi- elementary method, involving intricacies of language and perplexities of "compound proportion," to which, as the only alternative to utter vagueness, " popular " expositions are com- monly restricted. At any point (#, y, ,) let V be the potential and X, Y t Zihe components of force due to a body M\ and let m be its mass. Consider a similar distribution of matter of <7-fold density at corresponding points, and of p-fold linear dimensions. The mass of this body will be p s qm, and its potential and force-components at the point corresponding to (a?, y, z t ) will be P*$V* P<][X> pq.Y y pqZ. Hence if we put J Too Too /*co 8?rJ -ooJ-ooJ-oo that is to say, if E denote the mechanical value of the distri- bution M } the mechanical value of the supposed similar distri- bution of altered dimensions will be 564. Considering now similar magnets of different dimen- sions, whether polar magnets or electro-magnets, we see from the fundamental formulse ( 482, 483, 486, 544) that the forces at corresponding points are independent of the linear dimensions, xxvi.] Mechanical Values of Polar Magnets. 441 and are equal, with equal intensities of magnetization, when polar magnets are compared, or with intensities of electric currents inversely proportional to the linear dimensions of the bodies when electro-magnets are compared. Hence the values of the integral (1) of 561 for similar magnets are simply pro- portional to their volumes ; provided that, when polar magnets are compared their intensities of magnetization are equal, and when electro-magnets, the intensities of their electric currents are inversely proportional to their linear dimensions. Farther when polar magnets are compared, the proposition holds whether the polar or the electro-magnetic definition ( 517) of resultant force through interiors is adopted. But an electro-magnet can- not be simply divided into parts infinitely small in all their dimensions each of which is an independent electro-magnet ; and therefore the further consideration of electro-magnets must be deferred, while we use the divisibility of a polar magnet asserted in 447, to investigate the mechanical value of a distribution of polar magnetism, after the manner of 562. 565. At any point (x, y, 2,) let 3ft denote the resultant force due to a polar magnet ; the definition of 480 being adopted when (#, y, 2,} is in the substance of the magnet. The prelimi- nary proposition ( 561) is immediately applicable, and shows that the work required to produce any change in the relative position of a set of magnets is equal to the augmentation of /oo /*ao /*oo 3fJ 2 f-dxdydz (1). J-aoJ-aoJ-to 07T Hence ( 564) when a uniformly magnetized magnet is of such a shape that it can be divided into similar parts, the mechanical value of the whole is simply equal to the sum of the mechanical values of the parts ; [a remarkable contrast to the corresponding proposition ( 562) relative to a homo- geneous distribution of matter]. In other words, the work required to separate to infinitely great mutual distances any number of parts, each similar to the whole, of a uniformly mag- netized magnet, is zero. It follows that if an infinite number of infinitely small magnets, each distributed through a finite volume of space, with their magnetic axes parallel and with equal sums of magnetic moments in equal finite portions of 442 A Mathematical Theory of Magnetism. [xxvi. that space, no work will be required to condense or rarefy the distribution without altering the proportions of mutual dis- tances, or the direction of the magnetic axes relatively to the lines of these distances ; provided that the condensation is never pushed so far as to bring the constituents within dis- tances not infinitely great in comparison with the linear dimen- sions of the constituent magnets. This last proviso is unne- cessary when the constituents are uniformly magnetized, all with the same intensity of magnetization, and are so shaped that when brought into contact in the supposed condensation they fit together and form a whole, similar in shape to each part. 566. Consider now a bar or cylinder of uniformly and longi- tudinally magnetized substance, terminated by planes perpen- dicular to its length; and let i denote the intensity of the magnetization. This limit is approximately reached when the length of the bar is very great in comparison with its greatest transverse diameter. The corresponding distribution of imagi- nary magnetic matter consists ( 473) of distributions of positive and negative matter, of surface density i on the two terminal planes. The resultant force at points infinitely near the edge of either of these planes is infinite; but notwith- standing this, it is easily proved that the value of the integral (1) is finite. If we suppose the bar to be at first infinitely short and to be gradually increased in length, the value of the integral (1), expressing the work required to draw the two terminal planes asunder against their mutual attraction, increases continuously from zero to a limiting value equal to twice the value of the corresponding integral for either of the terminal planes alone. Hence, because for similar bars the values of the integral are ( 565) as the volumes of the bars, it follows that for bars of similar cross sections the integral has values proportional to the cubes of transverse dimensions and independent of the lengths, provided only that the length of each bar considered is very great in comparison with its greatest transverse diameter. Hence, if any polar magnet be divided into infinitely thin bars* along its lines of magnetiza- * By an infinitely thin bar, I mean a bar of which the transverse diameters are all infinitely small in comparison with the length. xxvi.] Mechanical Values of Polar Magnets. 443 tion, and if these bars be separated to infinite distances from one another, the whole value of the integral (1) becomes in- finitely small*. 567. Hence if magnetized substance given in infinitely thin bars at infinitely great distances from one another be put to- gether so as to form a polar magnet, the value of integral (1) for this magnet expresses the amount of work which was spent in thus building it up. Neglecting then the (unknown) mecha- nical value of the material, supposed given in infinitely thin permanently magnetized bars at infinitely great distances from one another, and defining the mechanical value of a magnet as the amount of work required to build it up of such materials, we see that this is expressed by the integral (1) of 565. 568. The value of the integral (1) ( 565) is zero, when the magnet consists of closed solenoids; because, in this case ( 510 Cors. 2 and 3) 1& = for every point. This result might at first sight appear erroneous, because a finite positive amount of work is required to cut up a finite closed solenoid into bars and separate them to infinite distances from one another. But it is verified by remarking that if each such bar, being of finite transverse dimensions, is split up into infinitely thin bars, work is gained by allowing these infinitely thin bars to repel one another to infinite mutual distances; and that the whole amount of work thus gained is exactly equal to what was spent in reducing the solenoid to separate finite bars. Or vary the process by supposing a finite solenoid to be first split up into an infinite number of infinitely thin solenoids; then the sum of the infinitely great number of infinitely small amounts of work required to break these infinitely thin solenoids into bars and separate the bars to infinite mutual distances, is infinitely small. In short the explanation of the apparent difficulty is contained in 566. 569. It is only for a magnet consisting of closed solenoids that 3ft, is everywhere zero. For every other magnet, the * But if each of these bars be divided into lengths comparable with its transverse dimensions, and if these parts be separated to distances from one another infinitely great in comparison with their dimensions, the integral (1) acquires a finite value which is equal to the amount of work necessary to produce this separation. 444 A Mathematical Theory of Magnetism. [xxvi. integral (1) of 565 has consequently a finite positive value. This I shall now prove to be always less than / 00 / GO p 00 l / / i*dxdydz J J GoJ X /oo /.x /oo and therefore <& -f E= 2?r i*dxdydz ....... . . (8). J -oo J x J oo Now E has always a positive finite value except for the extreme case of a magnet consisting of closed shells, when it is zero, because (512 cor. 6), R = in this case for every point whether in the substance of the magnet or not. Hence the proposition is proved. 570. For X* + P + Z* take, in virtue of (c), 517, ~ } Y(- } \dz dy ) \dz dx ) \dx dy and integrate by parts after the manner of 518, but with infinities for limits. We thus find or by 517 (0 E=%r ^ I" J x J coJ co This, which is the analogue to (7) of 569, was discovered for fluid motion by Helmholtz, and given in his paper on Yortex Motion (Crelle's Journal, 1858, or, translation by Tait, Philo- sophical Magazine, 1867, second half year). Lastly, substitut- ing for u, v, w their values by (a) of 517, and integrating again by parts as before, we find J E=|f f f" J -x-' X^ 00 The analogue to this is [compare 503 (2)], = -if f f dxdydz.(<& + J _55V -J co The addition of these two formulae verifies (8) of 569. 571. In a memorandum-book under date Oct. 16th, 1851, I find the following statement: "I concluded that the value of 446 A Mathematical TJieory of Magnetism. [xxvi. "a current in a closed conductor, left without electromotive " force, is the quantity of work that would be got by letting " all the infinitely small currents into which it may be divided "along the lines of motion of the electricity come together "from an infinite distance, and make it up. Each of these " ' infinitely small currents ' is of course in a circuit which is "generally of finite length. It is the section of each partial " conductor and the strength of the current in it that must be " infinitely small." A memorandum of principles and formulae proving this statement had been written a few days previously (Oct. 13th, 1851). A somewhat amplified statement of the principle was first published, but without the formulae, in I860, in the second edition of Nichol's Cyclopaedia (Article " Magnet- ism, Dynamical Relations of"). Though the subject does not belong properly to the present volume, I append in foot- notes the original memorandum*, and an extract from Nichol's * Memorandum, Oct. 13, 1851. Eefers first to an erroneous temporary conclusion which led me to think "that the value of a current in a closed "conductor will be effected by steel magnets in its neighbourhood." "From "this I was shaken a little by Faraday's finding (Exp. Res. 1100) 'that steel "does not do so well as soft iron," etc. [in respect to electro -magnetic induc- tion], "and I soon saw that I must have fallen into some mistake. . . . "I made out the true state of the case. This is the explanation. Let "-T- -j-dt.y be the quantity of work done in time dt, by bringing a steel "magnet towards a galvanic current, kept up, say, by a battery. Then (7, "the electromotive force due to the chemical action, will be increased by -j--. Hence if k be the resistance in absolute measure as dt "so that if wdt denote the work dEdsf dEds\ ds dt( + ds dt) wdt = ^ ' dt, rC "and if Mdt be the mechanical equivalent of the chemical action (increased "on account of the increased current), we have Mdt = Cydt v dt "Lastly, if Hdt be the heat developed, we have xxvi.] Mechanical Values of Electro-Magnets. 447 Cyclopaedia*, containing the amplified statement. Denning then the dynamical value of an electro-magnet as the quantity of ' We ^ conclude that the work actually spent, together with the mechanical 'equivalent of the chemical action, together produce exactly an equivalent 'of heat, and therefore no other effect. Hence the mechanical values of the 'current and of the magnet together are not altered. On the other hand, 'let two_ pure electro-magnets be brought towards one another. Adopting 'a notation corresponding to the former we have dE_ds^ ds ~ "Hence [J" denoting Joule's equivalent] there is more heat evolved than " (M+M' + w) by -p-w, and therefore the mechanical value of two cur- u J "rents is diminished by -^ wdt in the time dt." J * "Electricity in motion. If an electric current be excited in a conductor, 'and then left without electro-motive force, it retains energy to produce heat, 'light, and other kinds of mechanical effect, and it gradually falls in strength 'until it becomes insensible, as is amply demonstrated by the initial experi- 'ments of Faraday and Henry, on the spark which takes place when a gal- 'vanic circuit is opened at any point, and by those of Weber, Helmholtz, and 'others on the electro-magnetic effects of varying currents. Professor W. ' Thomson has shown how the mechanical value of all the effects that a cur- 'rent in a closed circuit can produce after the electro-motive force ceases, 'may be ascertained by a determination, founded on the known laws of 'electro-dynamic induction, of the mechanical value of the energy of a cur- 'rent of given strength, circulating in a linear conductor (a bent wire, for 'instance) of any form. To do this, it may be remarked, in the first place, 'that a current, once instituted in a conductor, and circulating in it after 'the electro-motive force ceases, does so just as if the electricity had inertia, 'and will diminish in strength according to the same, or nearly the same, 'laws as a current of water or other fluid, once set in motion and left with- 'out moving force, in a pipe forming a closed circuit. But according to 'Faraday, who found that an electric circuit consisting of a wire doubled on 'itself, with the two parts close together, gives no sensible spark when 'suddenly broken, in comparison with that given by an equal length of wire 'bent into a coil, it appears that the effects of ordinary inertia either do not 'exist for electricity in motion, or are but small compared with those which, 'in a suitable arrangement, are produced by the 'induction of the current "upon itself.' In the present state of science it is only these effects that 'can be determined by a mathematical investigation ; but the effects of elec- 'trical inertia, should it be found to exist, will be taken into account by 'adding a term of determinate form to the fully^ determined result of the 'present investigation which expresses the mechanical value of a current in 'a linear conductor as far as it depends on the induction of the current on 'itself. "The general principle of the investigation is this If two conductors, 'with a current sustained in each by a constant electro-motive force, be 448 A Mathematical Theory of Magnetism. [xxvi. work specified in the statement quoted above in the text, we have in equation (5) a proof the first hitherto published, of the assertion in the extract from Nichol's Cyclopaedia quoted in the foot-note, that the dynamical value of a current in a closed circuit may be calculated by the formula (4). For let open magnetic shells ( 506, 548) be substituted for the " infinitely small currents " referred to in the preceding statement, sup- posed first to be in their actual positions in the electro-magnet composed of them ; and let these shells be separated to infinite distances from one another. It is easily proved by considera- tions of infinitesimals analogous to those fully set forth in 566, that when the shells are brought to infinite distances from one another, the value of E becomes zero ; and, therefore, as the second member of (5) remains constant, the value of E before the circuits were separated, is equal to the addition of value which ( experiences during the process of separation, "slowly moved towards one another, and there be a certain gain of work on "the whole, by electro-dynamic force operating during the motion, there 'will be twice as much as this of work spent by the electro-motive forces ' (for instance, twice the equivalent of chemical action in the batteries, should 'the electro-motive forces be chemical) over and above that which they ' would have had to spend in the same time, merely to keep up the currents, 'if the conductors had been at rest, because the electro-dynamic induction 'produced by the motion will augment the currents; while on the other 'hand, if the motion be such as to require the expenditure of work against 'electro-dynamic forces to produce it, there will be twice as much work ' saved off the action of the electro-motive forces by the currents being dimin- ' ished during the motion. Hence the aggregate mechanical value of the 'currents in the two conductors, when brought to rest, will be increased in "the one case by an amount equal to the work done by mutual electro- "dynamic forces in the motion, and will be diminished by the corresponding "amount in the other case. The same considerations are applicable to "relative motions of two portions of the same linear conductor (supposed "perfectly flexible). Hence it is concluded that the mechanical value of a "current of given strength in a linear conductor of any form, is determined "by calculating the amount of work against electro-dynamic forces, required "to double it upon itself, while a current of constant strength is sustained in "it. The mathematical problem thus presented leads to an expression for "the required mechanical value consisting of two factors, of which, one is "determined according to the form and dimensions of the line of the con- ductor in any case, irrespectively of its section, and the other is the square "of the strength of the current. The mechanical value of a current in a "closed circuit, determined on these principles, may be calculated by means " of the following simple formula, not hitherto published : ^]]jP?dxdydz, "where E denotes the resultant electro-magnetic force at any point (x, y, z). "This expression is very useful in the dynamical theory of magneto-electric "machines and electro-magnetic engines." From Article "Magnetism, "Dynamical Kelations of," Nichol's Cyclopcedia, edit. 1860. XXVII.] Hydro-kinetic Analogy. 449 that is to say, is equal to the work spent in effecting this process. 572. Equation (5) expresses the following very remarkable proposition. The sum of the dynamical values of an electro- magnet and of any corresponding lamellar polar magnet is equal to 2?r multiplied into the sum of the squares of the intensities of magnetization of all parts of the latter ; the two species of dynamical value understood, being those defined in 571 and 567. XXVII. [Jan. 1872.] CHAPTER VIII. Hydro-kinetic Analogy. 573. The hydro-kinetic analogy for the force of a polar magnet seems to have been first perceived by Euler. It re- quires the supposition of generation and annihilation of fluid in places of positive and negative magnetic polarity, if we adopt for "the resultant force" in the magnetic substance the definition proper for a polar magnet laid down in 479 ; unless we limit the field of force considered, to places void of mag- netized matter, whether external to the magnet or in hollows within it. Thus, if we consider all space as filled with an incompressible frictionless liquid initially at rest, and if at certain points, lines, surfaces, or volumes, we assume more liquid of the same density to be continuously generated, and at the same time in other places liquid in equal quantity to be continuously annihilated, the velocity of the resulting fluid motion would be the same in direction and magnitude as the resultant magnetic force due to a distribution of magnetism presenting unneutralized polarity, positive (or northern) in the places of the fluid analogue where there is generation, and negative (or southern) in the places where there is annihilation. There is, however, no interest in pursuing the consideration of this extension of the hydro-kinetic analogy through spaces occupied by magnetized matter, involving as it does the strained supposition of the generation and annihilation of matter in spaces through which the liquid is perfectly free to move. 574. On the other hand, the hydro-kinetic analogy limited to spaces unoccupied by magnetized matter is perfectly satisfactory, T. E. 29 450 A Mathematical Theory of Magnetism. [xxvrr; as far as it goes. Let all these spaces be occupied by incom- pressible liquid, and let the magnetized matter be replaced by a rigid body perforated so as to constitute an infinitely numer- ous group of infinitely fine tubes fulfilling the following con- ditions : Divide the whole surface of the magnet into infinitely small areas inversely proportional to the magnitudes of the normal component forces across them whether outwards or inwards. Because the surface integral of the normal com- ponent force for the whole surface of the magnet is zero, the number of these infinitesimal areas in that part of the surface where the normal component force is outwards must be equal to the number in the remainder of the surface. Now to pass to the fluid analogue; instead of the magnet substitute a rigid body perforated from each of the infinitesimal areas in the part of the surface where the normal component force is positive, by a single tunnel through to one of the areas in the other part of the surface. Let there be in the first place a piston in each of these tunnels or tubes, and apply force to it until it moves with such a velocity that the velocity of efflux at one end and influx at the other is numerically equal to the normal component of the magnetic force to be represented : and when this condition has been once reached let the pistons become dissolved into perfect liquid homogeneous with the rest. The solid with its perforations remaining a rigid tubular system, the liquid will continue for ever circulating through the tubes and the free external space : and its motion through all external space will be such that the velocity is everywhere of the same magnitude and in the same direction as the re- sultant magnetic force in the corresponding position relatively to the magnet. The proof of this proposition * is ; that accord- ing to a well-known hydro-kinetic theorem, the motion of the liquid must be everywhere " irrotational " [Vortex Motion, 59 (e)], and that if the normal component fluid velocity, or normal component force in the magnetic analogue, be given * All the hydro-kinetic terminology and propositions used in the remainder of this volume are fully explained, with demonstrations when necessary, in the portion already published (in the Transactions of the Royal Society of Edinburgh, April 1867 and Dec. 1869) of a paper on "Vortex Motion," with the continuation of which I am at present occupied. Eeferences to it are given when necessary to justify any of the assertions in hydro-kinetic subjects made henceforward. xxvil.] Hydro-kinetic Analogy. 451 over the whole surface, the fluid motion or magnetic force is determinate through all external space ( 591, Theorems 1 and 2). The permanence of the fluid motion fulfilling the same condition follows at once from the constancy of the circulation through each perforation [Vortex Motion, 59 (d)], consequent upon the frictionless character which we assume the fluid to possess. 575. In the preceding statement no condition has been imposed as to the pairs of apertures in the surface of the rigid body substituted for a magnet, which are to be connected through the internal tubes; no such condition having been necessary, because we supposed the apertures over the whole surface to be inversely proportional to the magnitude of the normal component force. The statement may be varied thus : take all that part of the surface for which the normal component force is outwards, and divide it in any manner into infinitesimal areas. From each point in the boundary of any one of these areas, draw a line through external space till it meets again, as it will meet again, the surface of the magnet. By doing this for every infinitesimal area of the boundary traversed outwards, a corresponding area, where the normal component force is inwards, is found, 'and the whole remainder of the surface is thus divided into areas corresponding to those chosen in the first part. Let the pairs of corresponding areas be con- nected by internal tubes. The remainder of the statement may be applied without alteration to this tubular arrange- ment. The fluid analogue thus constructed, will have the peculiarity, that each portion of fluid circulates for ever along one circuit (that is to say, closed curve). 576. The hydro-kinetic analogy is both more complete and more simple, it is in fact perfectly complete, and therefore per- fectly simple, if instead of as in 479 adopting the definition proper for a polar magnet ( 549), we adopt the "electro- magnetic definition" (517 and postscript to 517), for the resultant force at any point in the substance of the magnet, whether it be a polar magnet or an electro-magnet. The resultant force defined " electromagnetically" for the space occupied by the magnet, and the resultant magnetic force accord- ing to the unambiguous definition for space not occupied by the 292 452 A Mathematical Theory of Magnetism. [xxvu. magnet, agree everywhere in magnitude and direction with the velocity in a possible case of motion of an incompressible liquid filling all space. To prove this it is only necessary to remark that the sole condition that X, Y, Z, may be the velocity-com- ponents in a possible case of motion of an incompressible fluid, is that they fulfil the equation of continuity dX dY dZ_ ^ + dy + dz~ and we have seen ( 517) that _ dx dy dz throughout the substance of the magnet as well as through external space, if X, Y, Z denote components of the magnetic force. The component intensities of electric current in the electro-magnet producing this force are [ 517 (a), (I)] JL (*%_ dY\ _(dX_dZ\ J_ (dY_dX\ 4>7r\dy dz)' 4>7r\dz dx)' 4ar\dx dy)' 577. This proposition, which I found more than twenty years ago as an obvious deduction from my formulae for electro- magnetic force, published in the -Transactions of the Royal Society for June 1850 ( 515518 above), is purely kine- matical. Since that time it has acquired an interest which it did not then possess for me, in virtue of Helmholtz's splendid discovery of the dynamical laws of vortex motion*. I had not known more than that the distribution of " electro-magnetic " force through the substance of the magnet, as well as through external space, corresponded to a possible distribution of motion in a continuous incompressible fluid filling all space, and had no clue to the consequences of leaving a frictionless liquid to itself, with such a motion once established in it. By Helm- holtz's theory, it is demonstrated that the fluid motion alters so as to always remain the representative of the electromotive force due to an electro-magnet continuously varied according to the following law. Lines of fluid matter which initially coincided with the lines of electric current in the electro-magnet initially * Crelle's Journal, 1858, and (Tait's translation) Philosophical Magazine, July 1867. XXVII.] Hydro-kinetic Analogy. 453 replaced by the fluid, however they change in the subsequent motion, always mark the lines of electric current which must be constituted to produce the altered electro- magnet ; and the whole amount of the intensity of the electric current crossing any area bounded by any closed curve passing always through the same fluid particles remains constant. It is unnecessary, however, to enter now on the wide hydro-kinetic subject thus indicated ; although I cannot but refer to Helmholtz's theorem of vortex motion, not merely on account of its intrinsic beauty, but because I have found it of great value in assisting me to realize the purely kinematic representation of electro-magnetic force which fluid motion affords. The general hydro-kinematic analogy, and the dynamics of the irrotationally moving portions of the fluid, as they served me primarily twenty-four years ago in investigating the inverse problems, will be further considered in the following chapter. 578. The hydro-kinetic analogy is valuable in the mathe- matical theory of electro-magnetism as leading to a set of theorems respecting magnetic forces produced by electric cur- rents, precisely analogous to those theorems of Green's respect- ing forces due to centres acting according to the Newtonian law, which I deduced in 1841 from an analogy with the "Uniform motion of heat in homogeneous bodies," by the investigation forming the first part of this volume ( 1 4 above). The following theorems I III. are particular cases of the general proposition of 576, and require no further demonstration. 579. Theorem I. (Compare 594 below.) Considering all space as occupied by an incompressible frictionless liquid, let S be a closed surface, which (to facilitate conceptions) may be supposed to be constituted of a perfectly flexible and extensible membrane. At first let there be no motion of the liquid in any part of space, and then let any motion whatever be arbi- trarily given to S, subject only to the condition of not altering the volume enclosed by it. The motion which is given to the liquid will be everywhere irrotational .("Vortex Motion," 16 and 60), and will therefore be continuously expressible throughout external space by a potential; and continuously expressible, likewise, through the internal space: but there 454 A Mathematical Theory of Magnetism. [xxvu. will be a discontinuity at S ; on the two sides of which the velocity-potential must differ by an amount equal to P, the impulsive pressure which would have to be applied to S to pro- duce the actual motion instantaneously from rest. Divide S into infinitely narrow bands by lines corresponding to equal values of P, and in each of these bands let an electric current g73 circulate of strength equal to - where BP denotes the differ- TbTT ence of the values of P at its two boundaries. The magnetic force produced by the distribution of electric currents thus con- stituted, will agree in magnitude and direction with the fluid velocity in the hydro-kinetic analogue. This proposition I used in a communication to the British Association at Oxford, in June 1847, " On the Electric Currents by which the Phenomena of Terrestrial Magnetism may be produced ; " and it is referred to in the abstract of that communication (now reprinted in 602, 603 below), which appeared in the yearly volume. It was probably one of five propositions which I wrote to Liou- ville in the September following (see 589 below). 580. Corollary. In the electro-magnetic analogue the direc- tion of the electric current is perpendicular to the relative tangential motion of the liquid on the two sides of '8, and the surface intensity of the electric current is equal to the relative tangential velocity divided by 4?r. 581. Example. Let S be kept of constant figure, and let the motion given to it be purely translatory. The liquid within it will move as if it were a rigid body. Hence the interior velocity-potential will be Ux, if U be the velocity, and if its direction be parallel to the axis of x. Hence if we consider a solid carried along through a frictionless liquid ; determine the velocity and direction, relatively to the solid, of the liquid gliding along each part of its surface ; and construct the ana- logous surface electro-magnet according to the rule of 579 ; this distribution of electric currents will produce a uniform field of force, of intensity U throughout the space enclosed by the surface on which they are distributed, and will produce a resultant force at every external point, agreeing in magnitude and direction with the absolute velocity which the liquid is compelled to take in making way for the solid. The analytical xxvii.] Hydro-kinetic Analogy. 455 expression of this very interesting theorem is contained in (IX.) of 517, applied to the case in which 582. Theorem II. (Includes the case 581 of Theorem I.) Let any motion of rotation be given to a rigid body in an in- finite incompressible liquid. The magnetic analogue consists of a uniform current traversing the volume of the rigid body in lines parallel to the axis of rotation, and of intensity equal to twice the angular velocity; with the circuit completed superficially by the surface distribution constructed according to the rule of 581. The resultant force of the completed solid and superficial electro-magnet ( 535) thus formed will agree everywhere in magnitude and direction with the absolute velo- city of the matter, whether solid or liquid, in the kinematic analogue. The analytical expression of this theorem (if we take the axis of the solid's rotation for the axis of x) is had by putting in (IX.) of 517 583. Theorem III. Consider a fixed rigid ring, having, for simplicity, but one perforation, and therefore giving duplex continuity to the space external to it. Let the whole of the external space be occupied by an incompressible frictionless liquid in a state of cyclic motion, with the ring for core. Take any surface S bounded by stream lines. This is necessarily a surface of duplex continuity enclosing the ring. On one of the stream lines forming a circuit of $, take i points corre- sponding to infinitely small differences of the velocity potential, each an exact sub-multiple - of the " cyclic constant," or "whole circulation" (/c). Through these points draw equi- potential lines on 8, which therefore will each cut perpendicu- larly all the stream lines on 8. In each of the infinitely narrow bands into which 8 is thus divided (constituting a geometrical circuit which crosses all the stream line circuits), let an electric current of strength 7 circulate. The resulting electro-magnetic force will be zero at every point within 8, and will be equal to, and in the same direction as, the fluid velocity 456 A Mathematical Theory of Magnetism. [xxvm. in the space external to S. This interesting and important proposition is perfectly analogous to that which is given by Green for surface distribution of electricity and the resulting electric force in Article 12 of his Essay (to which reference is made in Thomson and Tait's Natural Philosophy, 507, under the designation "reducible case of Green's problem"). XXVIII. {Nov. 1871.] CHAPTER IX. Inverse Problems. 584. Inverse problems of magnetism are problems in which the data are of magnetic force, and it is required to find distri- butions of magnetism or of electric currents by which the given force can be produced. They fall under two classes : I. Those in which the force is gi venjbr every point of space: and II. Those in which the force or some component of the force is given through some portion of space, whether volume, surface, or line; and it is required, under certain limitations or condi- tions, to find distributions of magnetism or of electric currents by which the given force can be produced. A complete and unconditional solution of every problem of Class I. is, as we shall immediately see, always easily found. 585. Class I. First case, polar definition ( 479 and Post- script to 517) of resultant force adopted. In this case the magnetic force is expressible by means of a potential, and therefore the most general form of data is; given the potential at every point of space. Let V be its value at (x, y, z), so that if K, J, 5% denote the components of the magnetic force, dx dy dz If a, & 7 denote the rectangular components of the required magnetization, we have da d$ , dy 1 ApF d"F , d 2 F\ -T- + -T- + T* = 7- ( -7T + -7-5- + -TT K 517 (m) repeated], dx dy dz 4m\dx * dy* dz* J L * and a, ft 7 may be any functions whatever which fulfil this equation. Then as a particular solution we have xxvin.] Inverse Problems. 457 Let now a", ft", 7" denote any three functions whatever fulfilling the following equation : .. ... dx dy dz The complete solution of the problem is, <* = '+", /3=/3' + /9", 7 = 7 ' + 7 " ............... (4). The arbitrary part a", /3", 7", of this solution consists of any distribution of magnetization agreeing everywhere in intensity and direction with the velocity and direction of a possible motion of an incompressible fluid through all space. When the given function F is such that its first and second diffe- rential coefficients dV tfV dV d'V dV d*V dx ' dx* ' dy ' dy* ' dz ' dt* are everywhere finite, there is nothing more to be said in respect to the preceding solution ; but when the first differential co- dV efficients -r- , etc., though themselves everywhere finite, vary anywhere abruptly in their values, an interpretation of a suffi- ciently obvious character becomes necessary to deduce the solution from the preceding formulae. Or the form of solution may be varied by introducing the proper formulae [ 473 (1)] for surface-distributions of the imaginary magnetic matter at the surfaces of discontinuity. 586. Class I. Second case, electro-magnetic definition adopted. In this case the force, though expressible by mean's of a poten- tial throughout every portion of space free from magnetized matter, is not so expressible through the substance of the magnet. Hence the data must be the intensity and direction of the re- sultant force at every point of space ; but these data are not altogether arbitrary inasmuch as if X, F, Z denote the three rectangular components of the force, + + = t "7 W repeated]. Hence the problem is; given X, F, Z, each any function of (#, y, z\ but subject to equation (&) of 517 ; it is required 458 A Mathematical Theory of Magnetism. [xxvm. to find three quantities a, ft, 7 such that fdy d&\_dZ dY (cal portion of it tested by any physical agency exhibits no difference in quality however it is turned. Or, which amounts to the same, a cubical portion cut from any position in an isotropic body exhibits the same qualities relatively to each pair of parallel faces. Or two equal and similar portions ' cut from any positions in the body not subject to the condition of parallelism ' ( 675) are undistinguishable from one another. A substance which is not 'isotropic but exhibits differences of quality in different directions is called 4 zeolotropic." Thompson and Tait's Natural Philosophy, 676.] f " Memoire sur le Magnelisme en Mouvement." (Mem. de VInstitut, 1823, vol. vi. Paris, 1827.) For quotations from this and the two preceding memoirs of Poisson, showing his theoretical anticipation of the "discovery of magne- crystallic action, see the Appendix to this article. 472 A Mathematical Theory of Magnetism. [xxx. theory of magnetic induction is obvious. On the other hand, in the present state of science, no theory founded on Poisson's hypothesis of "two magnetic fluids" moveable in the "mag- netic elements " could be satisfactory, as it is generally admitted that the truth of any such hypothesis is extremely improbable. Hence it is at present desirable that a complete theory of mag- netic induction in crystalline or non-crystalline matter should be established independently of any hypothesis of magnetic fluids, and, if possible, upon a purely experimental foundation. With this object, I have endeavoured to detach the hypothesis of magnetic fluids from Poisson's theory, and to substitute elementary principles deducible from it as the foundation of a mathematical theory identical with Poisson's in all substantial conclusions. In the present communication I shall state these principles, and point out what modifications of them may be required by a more complete experimental investigation of the subject than has yet been made; and, adopting them tem- porarily as axioms of magnetic induction, I shall give an account of some important practical conclusions deduced from them, by mathematical reasoning which I propose to publish on a future occasion. Some explanations and definitions are prefixed to show the signification in which certain extremely convenient terms and expressions, occasionally employed by Faraday and other writers, will be used in what follows. 605. Definition. The force at any point due to a magnet is the force which it would exert on the north pole of an infinitely thin, uniformly and longitudinally magnetized bar of unit strength placed at that point*, if it experienced no inductive action from the latter magnet. Definition. The total magnetic force at any point is the force * "If two infinitely thin bars be equally, and each uniformly and longi- tudinally, magnetized, and if, when an end of one is placed at a unit of dis- tance from an end of the other, the mutual force between these ends is unity, the magnetic strength of each is unity." (Philosophical Magazine, Oct. 1850, pp. 241, 242.) The definition of magnetic force in the text will agree pre- cisely with the definition of "magnetic force in absolute measure" adopted by the Koyal Society, in its "Instructions for making observations on terrestrial magnetism," if, in the definition of a unit bar, the unit of length understood be one foot, and the unit of force, a force which, if acting on a grain of matter, would in one second of time generate one foot per second of velocity. (See Admiralty Manual of Scientific Inquiry, pp. 16, 33, 37.) xxx.] Magnetic Induction. 473 which the north pole of a unit bar-magnet would experience from all magnets which exert any sensible action on it, if it produced no inductive action on any magnet or other body. Or, The total mag netic force at any point is the quotient obtained by dividing the force experienced by either pole, placed at that point, of an infinitely thin bar, uniformly and longitudinally magnetized to a finite degree of intensity, by the infinitely small numerical measure of the magnetic strength of the bar; and its direction is that of the force experienced by the north pole of the bar. Definition. Any space at every point of which there is a finite magnetic force is called ' ; a field of magnetic force;" or, magnetic being understood, simply "a field of force ;" or, some- times, "a magnetic field." Definition. A "line of force" is a line drawn through a mag- netic field in the direction of the force at each point through which it passes ; or a line touched at each point of itself by the direction of the magnetic force. Definition. A "uniform field of magnetic force" is a space throughout which the lines of force are parallel straight lines, and the intensity of the force is uniform. Definition. A substance magnetized so that the intensity and direction of magnetization at each point ( 462) are repre- sented by the diagonal of a parallelogram, of which the sides represent the intensities and directions at the same point in two other distributions, is said to possess a distribution of magnetism which is the resultant of these two superimposed, one on the other. It is demonstrated by Poisson, that the force at any point due to a resultant distribution of m&gnetism is the resultant of It may be remarked, that this unit of force will be the fraction - of the weight, in any locality, of one grain of matter, if g denote the velocity acquired in one second by a falling body in that locality; and that it is therefore very nearly ^. of tlie wei 8 bt > in anv P art of Great Britain or Ireland, of a grain. [Addition, May 30, 1872. The units of mass and length now adopted are the gramme and the centimetre. As 32-2 feet is equal to 981-6 centimetres, we may take 982 as the number of absolute kinetic units of force, in the apparent force of gravity on one gramme of matter in these latitudes.] 474 A Mathematical Theory of Magnetism. [xxx. the forces that would be produced at the same point if the component distributions existed separately. 606. Axioms of Magnetic Force. I. All mechanical action which a magnet experiences in virtue of its magnetism is due to other magnets*. II. The action between any two magnets is mutual. III. The whole action experienced by any magnet is the mechanical resultant of the actions which it would experience from all the magnets in its neighbourhood, if each acted on it as if the others were removed, the distributions of magnetism in the two remaining unaltered. 607. Laws of Magnetic Induction according to Poisson's Theory. I. When a given body, susceptible of inductive magnetization (whether it be ferromagnetic or diamagnetic), is placed in the neighbourhood of a magnet, it becomes magnetized in a manner dependent solely on the field of force which it is made to occupy. II. Superposition of Magnetic Inductions. Different magnets placed simultaneously in the neighbourhood of an inductively magnetizable (ferromagnetic or diamagnetic) body induce in it a distribution of magnetism which is the resultant of the different distributions that would be induced by the separate influences of the different magnets, each in its own position, with the others removed. 608. The first of these two propositions merely implies that any magnet, whether an electro-magnet, or a magnet consisting of magnetized substance, which produces at each point of a certain space the same " force " as another magnet of any kind, would produce the same inductive effect on a magnetizable substance occupying that space. Everything that is known of inductive action is consistent with it ; and it is, I believe, universally admitted as an axiomatic principle. 609. The second proposition, which asserts the mutual inde- pendence of superimposed magnetic inductions, is equivalent to an assertion that, if the force at every point of a magnetic field be altered in a certain ratio, the magnetization of a substance placed in it will be altered proportionately. This is undoubtedly * This principle appears, from his discovery that the phenomena of terres- trial magnetism are produced by the earth acting as a great magnet, to have been first recognised by Gilbert. xxx.] Magnetic Induction. 475 not a principle of universal application. It is not applicable to steel, nor to the substances of which natural magnets are composed ; nor, in general, to substances possessing in any degree that property of resisting magnetization or demagnetiza- tion, called by Poisson " coercive force," in virtue of which they can permanently retain magnetism. Neither is it, as Joule's experiments, and the more recent experiments of Gartenhauser and Muller demonstrate, applicable to soft iron, except as an approximate law of the magnetization when the magnetizing force does not exceed certain limits of intensity. But, that it is very approximately, if not rigorously, fulfilled in the magneti- zation of all homogeneous substances of very feeble inductive capacity, and destitute of "coercive force" (as all known diamag- netics and all ferromagnetics which contain no iron or nickel, or only very small proportions in chemical combination, appear to be), is, I think, extremely probable. The foundation of a complete theory of magnetic induction requires an experimental investigation of the laws according to which the {C coercive force " acts in various substances, and of the variation of induc- tive capacity produced in soft iron, and it may be in other sub- stances, by actual magnetization. The following conclusions, being mathematical deductions from the laws stated above, are liable to modification, according to the deviations from those laws which actual experiments may point out : 610. 1. The determination of the conditions of magnetic induction in a body of any kind in any circumstances may be made to depend on a knowledge of the state of magnetization induced in a homogeneous sphere of the same substance, placed in a uniform field of magnetic force. 2. A homogeneous sphere of any substance placed in a uniform field of force becomes uniformly magnetized in parallel lines with an intensity which is independent of the radius of the sphere. [To prove this, imagine a uniformly magnetized sphere of substance having infinite "coercive power." Let a spherical portion be removed from its interior. The resultant force at any point in the hollow will be ( 479, 473) that due to " imaginary magnetic matter " or free polarity, as it may be properly called, on the outer and inner spherical surface bound- 476 A Mathematical Theory of Magnetism. [xxx. ing the magnetized matter which is left. The surface density of the polarity at any point of either surface will be equal to % cos 0, if i denote the intensity of the magnetization and the angle between the direction of magnetization and the radius through the point considered. The distribution on one alone of the spherical surfaces, according to a very elementary result of spherical analysis stated above in a foot-note on 479 (and proved in the appended foot-note*), is parallel to the direction of mag- * To find the resultant due to one such distribution of matter on a spheri- cal surface, imagine first a solid material globe of uniform volume-density p throughout. By Newton's theorems for the attraction of a uniform spheri- cal mass, acting according to his law of the inverse square of the dis- tance, the resultant force at any point within the substance will be towards the centre, and equal to ~- multiplied by the distance of the attracted point O from the centre of the globe. Consider now two equal globes, one of uni- form positive matter and the other of uniform negative matter of the same density, the former repelling and the latter attracting a unit of positive matter (as in the electric and magnetic applications of the Newtonian law). Let them be placed with their centres C and C', at any distance apart less than the sum of their radii, and first imagine their materials to co-exist in the space common to the two spherical volumes, each acting as if the other were away. The resultant force at any point P within this space will be found by compounding a force equal to -^ CP with a force -~ C"P, in O O the direction from P towards C", and therefore, according to the parallelo- gram of forces, will be in the direction PD parallel to CC', and will be equal to -J^ CC'. This (as the positive and negative matters in the space common O to the two spheres neutralize one another) is therefore the resultant force at P, due to uniform distribution of positive and negative matter in the two meniscuses formed by the non-coincident portions of the two spheres. Now let CC' become infinitely small, and p infinitely large, and denote by i the product pCC', which we may suppose to have any value we please. The two meniscuses become a continuous superficial distribution of matter over a single spherical surface, having for surface-density tcostf, at any point where the inclination of the normal to the diameter through CC' is 0. The re- 4iri sultant force is parallel to this diameter and of constant value equal to O throughout the entire spherical space. A similar investigation gives the resultant magnetic force at any point in the interior of a uniformly magnetized ellipsoid ; but in this case it is con- venient to consider components of magnetization and of force in the direc- tions of the three principal axes. Thus if a, , 7 be the components of magnetization, and , f|, Z the components of the magnetic force according xxx.] Magnetic Susceptibility different in different directions. 477 netization, and equal to ^- ; and therefore the two balance one o another for every point within the supposed hollow space. The resultant force is therefore zero throughout this space. Replacing now the magnetized material in the hollow space, let the uniformly magnetized hollow sphere be placed in a uniform field of force, and instead of " coercive power," let its substance be endowed with such inductive susceptibility in each part of it, that by induction it shall remain uniformly magnetized. The magnetizing force actually experienced by any spherical portion of it is the same as if the surrounding substance were removed. Hence different equal spherical portions of the whole require equal inductive susceptibilities to keep them equally magnetized; and as we may suppose these spherical portions to be as small as we please, it follows that the induc- tive susceptibility must be equal throughout, and that if the substance be seolotropic its quality must be throughout similarly related to the force of the field. Conversely, the inductive magnetization experienced by a globe of homogeneous substance devoid of " coercive power " when placed in a uniform field of force, must be uniform and in parallel lines.] 3. If the sphere be of isotropic substance, the lines of its to the polar definition, we find 47r%a M _47rB/3 _ir(&y *~ 3 ' g ~"T~' 3 ' where J&, 13, -1C denote the three elliptic integrals which appear in (6) of 23, above, each with the factor V(l - ^} sA 1 - e 2 In one respect the equilibrium might be said to be neutral rather than stable, since every position into which the body may be turned round ihe stable axis is a position of equilibrium. J In two respects the equilibrium might be said to be neutral ; since every position into which the body may be turned round the direction of the lines of force is a position of equilibrium, and every position into which it may be turned in the plane of the stable principal axes is a position of equilibrium. 480 A Mathematical Theory of Magnetism. [xxx. 614. 9. If a spherical portion, of volume a, of a substance of which the three principal inductive capacities are A, B, and C, be held in a uniform magnetic field where the intensity of the force in absolute measure is R, with the three principal axes of in- duction inclined to the direction of the force at angles of which the cosines are respectively I, m, n, it will receive a state of magnetization which is the resultant of three states of uniform magnetization ; one of intensity A . El, in the direction of the first principal axis ; a second of intensity B . Rm, in the direction of the second principal axis ; and a third, of intensity C . Rn, in the direction of the third principal axis ; and it will experience a turning action, of which the mechanical definition is a couple, of moment a-. R\ (mV (B - C) 2 + n*P(C- A) 2 +l 2 m 2 (A-B) 2 }*...(1\ in a plane of which the direction cosines* with reference to the three principal axes are respectively mn (B - C) nl(C-A) lm(A-B) D ' D D * '' where D denotes the square root of the sum of the squares of the numerators of these three fractions, or the third factor of the preceding expression. 615. 10. If the sphere be infinitely small, and if it be put into a uniform or non-uniform field of force, the entire action which it experiences, whether directive tendency or tendency to move from one part of the field to another, is defined by the following proposition : The quantity of mechanical work which is required to bring the body from a position where the intensity of the force is R, and its direction cosines with reference to the three principal inductive axes I, m, n, to a position where the intensity of the force is R', and its direction cosines with reference to the three principal inductive axes in their new positions I', m, ri, is equal to Ja {(Al* + Bm' 2 + Cn 2 ) R'* - (Al 2 + Bm 2 + Cn 2 ) R 2 } (3). 11. If A=B = C, this expression becomes simply \vA (R' 2 R 2 ), and the proposition is equivalent to the mathe- * Or the cosines of the inclinations of a perpendicular to the plane, to the three axes. xxx.] Ferromagnetics and Diamagnetics. 481 matical expression of Faraday's law regarding the tendency to places of stronger or of weaker force, of ferromagnetic or dia- magnetic non-crystalline substances, on which some remarks [reprinted, 647 668 below] are published in the Philoso- phical Magazine for October 1850. 616. 12. If, without moving its centre, the ball be turned so that its three principal axes shall successively be in the direction of the lines of force (the field being non-uniform, but the body infinitely small), it will in each position experience a force in the line of most rapid variation of the " force of the field ;" but the magnitude of the force will in general differ in the three posi- tions, being proportional to A, B, and C respectively*. If * Thus a ball cut out of a crystal of pure calcareous spar, which tends to turn with its optic axis perpendicular to the lines of force, and which tends as a whole from places of stronger towards places of weaker force, would experience this latter tendency less strongly when the optic axis is perpen- dicular to the lines of force than when it is parallel to them ; since, accord- ing to 612 of the text, the crystal must have greatest inductive capacity or (the language in the text being strictly algebraic when negative quantities are concerned) least capacity for diamagnetic induction perpendicular to the optic axis. I am not aware that this particular conclusion has been verified by any experimenter; but I am informed (Oct. 25, 1850) by Mr Faraday, that he finds a piece of crystalline bismuth to experience a different " repul- sion" according as it is held with its naagnecrystallic axis along or perpen- dicular to the lines of force in a non-uniform field ; the repulsion being less in the former case than in the latter, which agrees perfectly with the conclu- sions of the text, since, as a ball of bismuth would tend to place its magne- crystallic axis along the lines of force, that axis must, according to 612, be the principal axis of greatest inductive capacity, or, bismuth being diamag- netic, the axis of least diamagnetic capacity. It is right to add, that what, according to the theory explained in the text, must be the correct explanation of the peculiar phenomena of magnetic induction depending on magnecrystallic properties, was clearly stated in the form of a conjecture by Faraday in his 22d Series (2588) in the following terms: "Or we might suppose that the crystal is a little more apt for mag- ' netic induction, or a little less apt for diamagnetic induction, in the direc- ' tion of the magnecrystallic axis than in other directions. But, ^ if so, it ' should surely show * * * in the case of diamagnetic bodies, as bismuth, a ' difference in the degree of repulsion when presented with the ^ magne- 'crystallic axis parallel and perpendicular to the lines of magnetic force ' (2552) ; which it does not do." (Read before the Royal Society, December 7, 1848.) The failure of the first experiment (2552) to detect this difference of action need not be wondered at, when we consider how minute it must probably be; and the conjecture, apparently abandoned at the time by the author for want of experimental support, may be considered as fully established by his own subsequent experimental researches. [The following appeared in the Philosophical Magazine for 1851, second half- year, under the title "Magnecrystallic Property of Calcareous Spar":] Extract from letter to the Editors. Glasgow College, Nov. 7, 1851. * * * * In the passage, as originally published (line 4 from beginning of foot-note), the word "more" occurred m the place of " less." The mistake was pointed out to me last April by Professor T. E. 31 482 A Mathematical Theory of Magnetism. [xxx. each of these quantities be positive, the force on the ball in each position will be in the direction in which the force of the field increases ; if any one of these quantities be negative, the force on the ball when the corresponding principal axis is in the direction of the lines of force, will be in the contrary direction, or that in which the force of the field decreases most rapidly. 617. 13. If A, B, and G be all positive, the body is called ferro- magnetic ; if they be all negative, it is ^called diamagnetic. No substance has as yet been found to have some of the quantities A, B, G positive, and others negative. 618. 14. If the inductive capacities be very small, all the pre- ceding conclusions will be applicable to the actions experienced by bodies in air (ferromagnetic), or in any magnetizable fluid of either ferromagnetic or diamagnetic inductive capacity, pro- vided, instead of the absolute inductive capacities of the sub- stance in each case, we use for A, B, and (7, or for the " principal inductive capacities " in the verbal enunciations, the excesses of the absolute principal inductive capacities of the substance, above the inductive capacity of the fluid. 619. Curious experiments might be made by means of a vary- ing field of force occupied by a magnetizable fluid, and a ball of crystalline substance allowed to move freely in the line of most rapid variation of the force. If the inductive capacity (whether positive or negative) of the fluid be intermediate between the Stokes, and I immediately requested you to correct it, which you accordingly did by an intimation in the "Errata." When the perplexity occasioned by the mistake is removed, it is obvious to any one reading the passage carefully, that the mistake itself was only a slip of the pen, as at the conclusion of the sentence it is asserted that a crystal of pure calcareous spar must have the " least capacity for diamagnetic induction, perpendicular to the optic axis." This conclusion is verified by Dr Tyndall, who describes experiments, in a paper published in your September Number, by which it appears that the dia- magnetic inductive capacity of calcareous spar in a direction parallel to the optic axis is to its diamagnetic inductive capacity perpendicular to the optic axis as 57 to 51. I remain, gentlemen, your obedient servant, WILLIAM THOMSON. [We have also received a communication on this subject from Mr Tyndall, who in reference to a note received by him from Prof. Thomson, writes as follows : "I have only to say that the facts are precisely what they are here stated to be. Previous to writing the remarks in question, I looked to the Errata, but not it seems with sufficient attention, for Professor Thomson's cor- rection escaped me. Not only do our results agree in principle, but the same substance and form of substance which Professor Thomson had referred to in illustration of his theory was unwittingly examined by me in Berlin, and the exact result which he had theoretically predicted arrived at by way of experi- ment." EDIT,] xxx.] Poisson's Anticipation of Magnecrystallic Quality. 483 greatest and the least of the absolute principal inductive capa- cities of the substances, the ball will be urged from places of weaker towards places of stronger force when its axis of greatest inductive capacity is placed along the lines of force, and in the contrary direction when the axis of least inductive capacity is placed in the same direction. It would be easy to adjust the strength of a solution of sul- phate of iron so as to satisfy this condition for a ferromagnetic crystalline substance; but there might be great difficulty in demonstrating by experiment the existence of the forces, on account of their feebleness. APPENDIX. Quotations from Poisson regarding Magnecrystallic Action. 620. " la forme des e'le'mens pourra aussi influer " sur cette intensite' ; et cette influence aura cela de particulier, " qu'elle ne sera pas la meme en des sens diffe'rens. Supposons, " par exemple, que les Clemens magnetiques sont des ellipsoides "dont les axes ont la m&ne direction dans toute I'e'tendue " d'un meme corps, et que ce corps est une sphere aimantee par "influence, dans laquelle la force coercitive est nulle; les "attractions ou repulsions qu'elle exercera au-dehors seront " diff^rentes dans le sens des axes de ses elemens et dans tout " autre sens ; en sorte que, si Ton fait tourner cette sphere sur " elle-meme, son action sur un meme point changera, en g^neVal, " en grandeur et en direction : mais, si les elemens magne'tiques " sont des spheres de diametres ^gaux ou in^gaux, ou bien s'ils " s'^cartent de la forme spheVique, mais qu'ils soient disposes " sans aucune regularite dans 1'interieur d'un corps aimante par " influence, leurs formes n'influeront plus sur les resultats qui "d^pendront seulement de lasomme de leurs volumes, comparee " au volume entier de ce corps*, et qui seront alors les memes en " tout sens. Ce dernier cas est celui du fer forge*, et sans doute "aussi des autres corps non cristallises dans lesquels on a " observ^ le magn^tisme : mais il serait curieux de chercher si "le premier cas n'aurait pas lieu lorsque ces substances sont * [This error was corrected by Poisson himself in a subsequent memoir.] 312 484 A Mathematical Theory of Magnetism. [xxx. " cristallisees ; on pourrait s'en assurer par 1'experience, soit en " approchant un cristal d'une aiguille aimantee, librement sus- "pendue, soit en faisant osciller de petites aiguilles taille'es "dans des cristaux en toute sorte de sens et soumises a 1'action "d'un tres fort aimant." Pp. 258, 259, Memoire sur la TMorie du Magnetisme, par M. Poisson. Lu a 1' Academic des Sciences le 2 Fe'vrier, 1824. Mem. de I'lnst. 1821-22. Paris, 1826. " la forme des elemens et leurs positions par rapport "aux plans fixes des coordonne'es x, y y z, peuvent influer sur "rdtat magnetique de A, et sur les attractions ou repulsions " qu'il exerce au dehors. II pourrait meme arriver que cette " influence ne fut pas la meme en tout sens, en sorte que, si A " ^tait une sphere homogene, et qu'on fit tourner ce corps sans " d^placer son centre et sans rien changer aux forces exte'rieures " ou a la fonction V, les actions magnetiques de A changeraient "neanmoins en grandeur et en direction. Ce cas singulier, u que nous avons deja indiquedans le preambule de ce Memoire, " ne s'&ant pas encore presente a 1'observation, nous 1'exclurons "de nos recherches, quant a present, et nous allons, en conse- "quence, determiner les relations qui doivent exister entre a, "ft, 7'*, et les quantites a y , /8 /5 7^, pour qu'il n'ait pas lieu." Ibid. p. 278. 621. The following explanation may serve to give an idea of Poisson's mode of treating the subject of the last quotation, and to show the relation it bears to the theory of which an outline has been given above. A sphere of any homogeneous magnetizable substance being placed in a uniform field of force, intensity R, let the direction of the force make angles whose cosines are l } m, n with three rectangular axes fixed relatively to the substance ; and let a, /3, 7 be the components of the induced magnetization. Poisson deduces, from his hypothesis of magnetic fluids, equations^ * Component intensities of magnetization. t Components of the magnetizing force. $ The products of the first members of Poisson's three equations in p. 278 of his first Memoire, into k, the ratio of the sum of the volumes of the magnetic elements to the whole volume of the body, are respectively equal to the three components of the intensity of magnetization (a, @, 7); and ii A, B, etc., be taken to denote the values of the products of Tc into Poisson's coefficients P, Q, etc., respectively, the equations in the text coincide with those of Poisson, xxx.] Theorem of Principal Axes Demonstrated. 485 which are equivalent to the following: a = (Al +B'm +C"n)R\ p = (A"l + Bm +C f n)R\ (4), 7 = (Al +B"m+ Gn)R) where A, B, etc., are coefficients depending solely on the nature of the substance. These equations are deducible from the axioms and the hypothetical principle of the superposition of magnetic inductions, stated above, without the necessity of referring at all to the hypothesis of " fluids." All that remains of Poisson's theory is confined to the case of non- crystalline matter, with reference to which it is proved that A, B, and G must be equal to one another, and that each of the other six coefficients must vanish; and there is nothing to indicate the possibility of establishing any relations among the nine co- efficients which must hold for matter in general. I have found that the following relations, reducing the number of independent coefficients from nine to six, must be fulfilled, whatever be the nature of the substance : " = C', C" = A', A" = B' (5), the demonstration [added below, 622] being founded on no uncertain or special hypothesis, but on the principle that a sphere of matter of any kind, placed in a uniform field of force, and made to turn round an axis fixed perpendicular to the lines of force, cannot be an inexhaustible source of mechanical effect. All the conclusions with reference to magnecrystallic action enun- ciated in the preceding abstract are founded on these relations. [622. Demonstration: January 1872. Because the field of force is uniform the dynamical action experienced by the mag- netized sphere if of unit volume consists simply of a couple ( 499) whose components are (pn-ym)R, (', and let < denote the angle between the plane YOX and the plane of OX and (I, m, n). 486 A Mathematical Theory of Magnetism. [xxx. The work done by the magnetized substance during this motion will be (my - n0) Rd ........................ (7), which, if we put I = cos 0, m = sin 6 cos c/>, n = sin 9 sin $, and use (4), becomes E { sin e cos 6 (A' cos - A" sin 0) + sin 2 6 [(C - B) sin cos + B" cos 2 -C'sin 2 0]}d0...(8). Integrating this expression from = to < = 2?r, we find ^ sin 2 0(5"- (7')7r> for the integral amount of work done during a revolution round OX. But this must be zero, for avoidance of the " perpetual motion*," since the body is brought back to its primitive position and physical condition at the end of the motion; and therefore B" C'. Similarly, by turning the body once round the axis Y, we prove that C" = A', and by turning it round OZ we prove that A" B'. Thus are established the three relations between the co- efficients expressed by equations (5) above. 623. To find a symmetrical expression for the work done in any infinitesimal rotation, remark that when I is constant we have , . dm dn d6 = -- = . n m Hence (my n/3) d$ = ydn + /3dm. Hence by (7) and corresponding expressions for the work done in infinitesimal rotations, round Y and OZ, we find for the whole work, dQ, done by any infinitesimal rotation whatever dQ = R(adl+/3dn + ydm) ............... (9). Using in this for a, /3, 7, their expressions by (4), as linear functions of /, m, n, and looking to the relations (5) established between the coefficients, we see that d Q is a complete differ- ential of a quadratic function of I, m, n, as if these were three independent variables ; and therefore by integration Q = } (Al 2 + Bm 2 + Cri* + 2amn + 2bnl + 2clm) R\ . .(10), where a, b, c denote respectively the value of either members of the three equations (5). Hence by differentiation and compari- son with (4), IdQ IdQ IdQ and Q = %(a.l + /3m+yn)R ............... (12). See below, 670, footnote. xxxi.] Magnetic Permeability and Analogues. 487 This is necessarily equal to the exhaustion of energy (Thomson and Tait's Natural Philosophy, 549) in letting the globule come from any place of zero magnetic force, into its', actual position in the supposed magnetic field. Compare 732, and 722 (70) bis, and 503 (2). 624. The elementary theory of the transformation of quad- ratic functions shows how, when A, B, 0, a, b, c are known for any one set of three rectangular axes in the substance, we can find determinately by aid of the solution of a cubic equation, a set of three rectangular axes such that if we take them for axes of X, Y, Z, the coefficients of mn, nl, Im will vanish in the transformed quadratic function, and we should have simply Q = %(AP + Bm* + Cn*)IP ............ (13), and a = AlR, /3 = BmR, y=CnR ...... ..... (14). Hence the propositions of 612, 613, 614.] XXXI. Magnetic Permeability, and Analogues in Electro-static Induction, Conduction of Heat, and Fluid Motion. March 1872. 625. Supposing the coefficients A, B, G, and a, b, c of 621624, (5) and (10), to be known for a particular set of axes in a substance susceptible of magnetic induction, let it be required to find its susceptibility for magnetization in any given direction. Let a sphere of the substance be placed in a uniform field of force having components F, Q, H parallel to the axes of co-ordinates. By 623 (11) we have for the com- ponents of magnetization a = .(1); and denoting by^'-the intensity of the resultant magnetization, and I, m, n its direction-cosines, Conceive now an infinitely thin bar of the substance, of any length along the lines of magnetization, to be removed. The magnetic force in the hollow space will be compounded of the 488 A Mathematical Theory of Magnetism. [xxxi. force of the field (F, G, H) and the force due to the free surface- polarity of the sphere ; and therefore ( 610, 2, foot-note) if we denote "by X, Y, Zits components, we have 4-Tra 4-7T/3 4^7 JL = ^- -|p, r = (r -- -, 4 = H- ^ ...(4). It is this which is the magnetizing force actually experienced by the bar in its position as part of the sphere. The magnet- ization induced by it is of intensity \/(a 2 +/3 2 + 7 2 ), and is in the direction of the bar's length. Hence the magnetic suscep- tibility of the substance in the direction (3) of this bar is To find the magnetic susceptibility in any direction (I, m, ri) explicitly in terms of I, m, n and the co-efficients A, B, C, a, 6, c, all that is necessary is to eliminate a, j3, 7, X, Y, Z, F, 0, H from (5) by means of the nine equations (1), (3), (4). The algebraic process required involves only the solution of the three linear equations (1) for F, G-, H. The simplified solution given in the following section may be regarded as algebraically equivalent to an expression of the preceding direct solution in terms of symmetrical functions of the roots of a cubic equation. 626. To simplify let the axes of co-ordinates be chosen in the direction of the three principal axes ( 611) of magnetic susceptibility. This makes a = 0, 6 = 0, c = 0, and we have (6), =I-H (7). Hence by (5), (3), and (2) we have, for the magnetic suscepti- bility in the direction I, m, n, ,.-(8), , A B where X = .,.- r _ T_^?' 3 " 3 3 627. The coefficients denoted in (9) by X, /*, v are the three principal magnetic susceptibilities, as we see by considering the cases in which (I, m, n) coincides with the axes of co- xxxi.] Magnetic Permeability and Analogues. 489 ordinates. By equations (9), conversely, for the inductive mag- netization of a sphere when its principal susceptibilities X, /*, v are given, we find ~ - . 4-7T. ' ~~ 4-7T ' ~ 4-7T ""^ '' 628. In the exposition of Faraday's great electro -static discovery, given above ( 36 50), I pointed out a perfectly close analogy between the mathematical theories of the electro- polar induction which he found to be experienced by insulators in a field of electric force, of the inductive magnetization of ferromagnetics, air, and diamagnetics, and of the conduction of heat through a heterogeneous solid. This volume will end with a fourth analogy ( 751 763, below), in which it will be shown that precisely the same laws and mathematical ex- pressions are applicable to the flow of a frictionless incom- pressible liquid, through a porous solid of infinitely fine texture, when the motion of the liquid is throughout irrotational (or such as may be produced from rest by any motion given to the boundary of the liquid). The singular combination of mathe- matical acuteness, with experimental research and profound physical speculation, which Faraday, though not a "mathe- matician," presented, is remarkably illustrated by his use of the expression, conducting power of a magnetic medium for lines of force, referred to in the foot-note to 44, above. The ana- logue corresponding to conducting power of a solid for heat, or, as it is shortly called, "thermal conductivity," is, in electro- static induction, the " specific inductive capacity " of the di-electric; in magnetism it is not what has hitherto been called magnetic inductive capacity, a quality which is negative in diamagnetics, but it is Faraday's " conducting power for lines of force;" and in hydrokinetics it is ( 753, below) flux per unit area, per unit intensity of energy. The common word "permeability" seems well adapted to express the specific quality in each of the four analogous subjects. Adopting it we have thermal permeability, a synonym for thermal con- ductivity; permeability for lines of electric force, a synonym for the electro-static inductive capacity of an insulator; mag- netic permeability, a synonym for conducting power for lines 490 A Mathematical Theory of Magnetism. [xxxi. of magnetic force ; and hydrokinetic permeability, a name for the specific quality of a porous solid, according to which, when placed in a moving frictionless liquid, it modifies the flow. 629. To find the relation between what has been called above magnetic susceptibility and magnetic permeability, consider a body with no intrinsic magnetization ( 698, below) surrounded by air in a magnetic field. Let A be any infinitesimal area of its surface cutting perpendicularly one of the three principal inductive axes of the substance in its neighbourhood. Let S- be the normal component of the magnetization induced in the substance infinitely near A ; and let N, N' be the values of the normal component force at external and internal points infinitely near A, the latter according to the polar definition ( 517, Postscript). We have [ 473 (1), and 7] N' = N-47T* ....................... (11). Let now /i- be the magnetic susceptibility in the direction of the normal, so that ( 610, 3, definition 2) we have * = pN' .............................. (12). Eliminating S- from this by (11), we have N' = N 4 < 7r//JV v ', and therefore N ON (13). Hence (compare 44, above) 1 + 4?r//, is the magnetic permea- bility of the substance in the direction of its principal axis perpendicular to A. Thus we see that if //,, //, /A" denote the three principal magnetic susceptibilities of a substance, and -or, OT', iff" its principal magnetic permeabilities, we have 57 = 1 +4?r^, r' = 1 + 4w/, iff" = 1 + 47T//," ...... (14). 630. Experiment has hitherto given but little accurate know- ledge of the magnetic susceptibilities of different substances. Comparisons of the susceptibilities of diamagnetics and feeble ferromagnetics with one another and with that of iron have been attempted ; but the only determination in absolute measure hitherto made or even attempted is that of Thalen* for iron. He found the magnetic susceptibilities of different specimens to be very different. The greatest susceptibility which he found * "Beeherches sur les propriety magnetises du fer." Par T. E. Thalen. Extrait des actes de la Socie'te Boyale des Sciences d'Upsal. Serie iii e . T, iv. Upsal, 1861. xxxi.] Magnetic Permeability and Analogues. 491 was in some specimens of the best soft iron, and amounted to about 45. " Coercive force," the laws of which are at present wholly unknown, exists to a great degree in all varieties of iron and steel, including the softest iron ; and varies very much in the same specimen with its state of temper. It complicates excessively every investigation regarding the inductive quali- ties of iron and steel. On the other hand (and particularly now that the British Association has given to experimenters standards of electric resistance in absolute electro-magnetic measure*, and important contributions towards the general practice of the absolute system) it is a very easy thing to measure, with some degree of accuracy, the absolute value of the inductive quality of substances destitute of coercive force. (All fluids are necessarily so ; and, as stated in 609, it is pro- bable that all diamagnetics, and all homogeneous substances of feeble ferromagnetic quality, are nearly so.) As yet no such measurement has been made, but it is to be hoped that before long some experimenter will take up the subject. 631. ThaleVs number, 45, gives, according to (14), 1 -f 4?r x 45, or about 566 for the permeability of the best soft iron. It has been stated that the inductive susceptibility of cobalt is greater than that of soft iron, but this seems to be by no means certain; and I believe it is certain that all other substances hitherto experimented on are less susceptible than iron. The permea- bilities of all ferromagnetics exceed unity, but only by very small fractions, except the few so-called magnetic metals, or substances containing them in large proportion. It is also remarkable that no substance has been discovered for which the permeability falls short of unity by more than a very minute fraction, as is shown by the extreme feebleness of the forces due to diamagnetic induction in all cases which have been hitherto observed. If we knew something instead of nothing of the molecular theory of magnetic induction, we should probably see that the permeability of every substance must be positive. * British Association Committee on Electric Measurement, appointed first in the year 1860, and reappointed after that from year to year. A reprint of its successive Keports collected is being made by the Committee, with permission of the Council of the British Association, and will soon be ready for publication in a separate form. [Published in 1873 by E. and F. N. Spon, London, under the title of "Keports of Electrical Standards," edited by Prof, F. Jenkin, F.K.S.. LL.D.l 492 A Mathematical Theory of Magnetism. [xxxii. XXXII. Diagrams of Lines of Force ; to illustrate Magnetic Permeability. [May 29, 1872.] 632. The differential equation for lines of force in void space resulting from the Newtonian law is always integrable when the distribution is symmetrical round an axis, as was first shown in an article "On the Equations of Motion of Heat referred to Curvilinear Co-ordinates " in the Cambridge Mathe- matical Journal, Nov. 1843 [Art. ix. of my "Reprint of Mathe- matical and Physical Papers," Vol. I. University Press, Cam- bridge, 1882.] thus : In the case of symmetry round an axis, take for co-ordinates x along the axis of symmetry, and y per- pendicular to it in any plane through it. Laplace and Poisson's equation becomes d*7 d z V ldV_ Therefore through void space, + i-o (i). The differential equation of the lines of. force is dV, dV. -T~ dx = ay = 0. dy dx This, in virtue of (1), is rendered integrable by the factor y, and therefore the integral equation of the lines of force is T/T = const., j .here " - dV > dV ^ * (2) ' For example let V= . - Fx . , . . (3), (^ + 0T so that the distribution of force is that of a uniform field, of intensity F, disturbed by the presence of an infinitesimal magnet, of magnetic moment //., placed with its magnetic axis parallel to the lines of the undisturbed force. We find which, if we put - = a 3 , and - = b 2 .................. (5), or, resolved for at, x= U--tf ..................... (7). XXXII.] Examples of Lines of Force. 493 On account of the double sign of the radical in (6) we may, without loss of generality, suppose a always positive; and the branches of the curves corresponding to negative values of the radical will then correspond to the case in which the magnet is placed in the position in which, if it were rigidly magnetized, and free to turn, its equilibrium would be unstable. In these branches, which for brevity will be called exflected, f is every- where greater than 6 2 ; while in the branches corresponding to a magnet placed in position of stable equilibrium, which will be called inflected, f is everywhere less than 6 a . Of the Eadius of Circle = a. Y FIG. 1. annexed woodcuts*, fig. 1 represents the entire series of both sets of branches for all positive values of 6 2 ; fig. 2 the whole series of inflected branches ; fig. 3 the whole series of exflected branches; and figs. 4, 5, 6, 7 selections from the two sets to illustrate inductive influences of spherical bodies of various qualities, placed in a uniform current of incompressible friction- less liquid, or in uniform fields of electric or magnetic force. * From photographs of large-scale diagrams calculated from equation (7), and drawn for the Natural Philosophy Class in the University of Glasgow about twenty-three years ago by Mr. D. Macfarlane, to illustrate fluid motion a.nd the allied subjects of physical mathematics, 494 A Mathematical Theory of Magnetism. [xxxn. The two double points shown in figs. 1, 2, and 4 correspond to Fm. 2. the pairs of equal roots y = 3-7^ , y = -^ , which the two quintics Kadius of Circle = a. Fm. 3. /q have when b = ^r = 1*375. A circle (fig. 4) described from the v^ origin as centre through these double points, and therefore XXXII.] Examples of Lines of Force. 495 a having - for radius, cuts perpendicularly each of the inflected FIG. 4. in curves, except the one given by b = -7^- , which it cuts through the double points at angles of + tan" 1 j- . 496 A Mathematical Theory of Magnetism. [xxxn. Of the exflected curves (fig. 3), that given by b = consists of a circle of radius a, having its centre at the origin, together Radius of Circle =a. a FIG. 6. with the parts of the axis of x external to that circle, each doubled ; or the same circle, together with the part of the axis of x within it, doubled. FIG. 7. xxxii.] Examples of Lines of Force. 497 Fig. 4 represents the lines of electric force in the neighbour- hood of an uncharged insulated metal globe placed in a uniform field of electric force. It also represents (631) without sensible distinction the lines of magnetic force in the neighbourhood of a globe of soft iron in a uniform magnetic field. Fig. 6 repre- sents the stream lines of a frictionless incompressible liquid passing a fixed spherical obstacle. 633. To investigate the relation of the lines of force in the neighbourhood of a solid globe of any ferromagnetic or dia- magnetic homogeneous material destitute of intrinsic magnet- ism, put into a uniform magnetic field, with one of the three principal axes ( 611) if the substance be not isotropic, placed parallel to the lines of force : Let -cr be the permeability of the substance ( 629) and r the radius of the globe. The induced magnetization being ( 610) uniform, and parallel to the lines of force of the field, its action through external space will ( 610, foot-note) be the same as that of an infinitely small magnet at its centre. Hence, using the notation of (5) in (3), and instead of admitting the negative sign for the radical, taking the proper diamagnetic formula by itself, we have a?x *) - ...... (ferromagnetic) -* for the potential in external space due to the magnetism of the globe and the uniform force of the field. Throughout the in- ternal space the force is ( 610, foot-note) uniform, and its potential must be of the form Cx. Choosing C so that at the surface of the sphere (radius r) the external and internal poten- tials shall be equal, we find y = ipfe -2) x ...... (ferromagnetic) V , a * , F= - \F ( + 2 x. . .(diamagnetic) From this and (8) we find, for the force at any point in the 82 498 A Mathematical Theory of Magnetism. [xxxn. axis of x, = F (l + ^) ...... (ferromagnetic) (external) JJ ...(10); (diamagnetic) and - ...... (ferromagnetic) (internal) / .(ll). = ^ (l + J ^ J ...... (diamagnetic) For points in the axis of x infinitely near one another x = r, and ( 629) we have X (external) _ X (internal) Hence, by (10) and (11), cr= 3 ' (ferromagnetic) Co .(4 5- (diamagnetic) Cb or (resolving for r) r = a V 2(1^1) ...... (ferromagnetic^ (13). For great values of -cr we have r=^r(l + ) approximately ........... (14). Z L \ -CT/ Hence for such values of r as those discovered in soft iron by Thaldn ( 631 above) the value of r would be only greater by about g^ part than that shown in fig. 4. The circles shown in figs. 5 and 7 were described with radii chosen at random. By measuring them in proportion to a in each case, I find the permeabilities of the inductively magnetized globes, whose influence on the lines of magnetic force is represented in those diagrams, to be respectively 2*8 and '48. xxxiii.] Attraction of Ferromagnetics. 499 XXXIII. On the Forces experienced by Small Spheres under Magnetic Influence; and on some of the Phenomena pre- sented by Diamagnetic Substances. [From the Cambridge and Dublin Mathematical Journal, May 1847.] 634. THE circumstance that a magnet* attracts small pieces of iron, is the phenomenon of magnetism which was first ob- served ; and an analogous action, presented by rubbed amber, first drew attention to the phenomena of electricity. Now it has since been discovered that no mutual attraction or repulsion between two bodies can result from magnetism in one, unless the other be also magnetized, and that- no electric force can exist unless each body be electrically excited. Hence it ap- pears that the forces originally observed are the consequences of a temporary magnetic or electric state induced in a neutral body, when placed in the neighbourhood of a magnet or of an electrified body. In the following paper the law of such phenomena with reference to magnetism t is considered. It is easily shown however that, by taking i = 1 in the formulae obtained below, the corresponding results for small insulated conductors, elec- trified by influence, may be obtained, although the physical problems are entirely distinct. 635. We may commence by considering the case of a small sphere of soft iron, or of any other substance susceptible of magnetic induction; and it is easily shown that the formulae expressing the results may be applied to the case of a small cube by merely altering the value of a certain coefficient ; and in general to the case of a small portion of matter of any form, such that in whatever way it be turned, the resultant axis of magnetization, for the whole mass, shall coincide with the direction of the magnetizing force. * Originally a piece of magnetic iron-ore or loadstone. The term may now be applied to any mass possessing permanent magnetism, and may even be extended to a galvanic wire of any form. + This has not been made the subject of a special investigation by any writer, so far as I am aware, although the nature of the result, in the ease of magnetism, appears to be entirely understood by Mr Faraday. Thus, from 2418 of his Experimental Researches [quoted below, in the text ( 646)] we might infer that a small sphere or cube of soft iron would in some cases be " urged along, and in others obliquely or directly across the lines of magnetic force;" and that all the phenomena would resolve themselves into this, that such a portion of matter, when under magnetic action, tends to move from places of weaker to places of stronger force. 322 500 A Mathematical Theory of Magnetism. [xxxm. 636. It is well -known [and proved in 609 above] that if a small homogeneous sphere of soft iron, or of any other substance susceptible of magnetic induction, be placed in the neighbourhood of a magnet, it will become uniformly magnet- ized, throughout its mass, with an intensity numerically ex- pressed by multiplying the magnetizing force, by a coefficient independent of the dimensions of the sphere. Thus if E denote the resultant force of the magnet, or the force that it would exert upon an imaginary unit of magnetism, at the position occupied by the sphere, of which we suppose the dimensions to be so small that H has sensibly the same value and direction throughout; and if K be the intensity of the induced magnetism; we have where i is a proper fraction (nearly equal to unity for soft iron) depending on the capacity of the substance for magnetic in- duction. 637. If the force E were rigorously constant in magnitude and direction throughout the whole space S occupied by the sphere, then there would be no resulting force tending to move the sphere ; as, for example, we may conceive it to be, without committing an appreciable error, in the case of a ball of iron of any ordinary dimensions magnetized by the terrestrial force. In the investigation which follows we shall therefore have to consider the small variation of R through the space S, but although considering the effect of this small variation in caus- ing a moving force upon the magnetized sphere, we may neglect the deviation from rigorous uniformity of magnetization which it will produce. 638. Let X, F, zf be the components of R at the point (x,y,z), which may be taken as the centre of the small sphere. At any point (#+/), (# + #), (z +~h), in the sphere, we shall have, for the components of the resultant force due to the magnet, dX ,dX dX, dY, dY dY, -T-f+-j-g + -j-h dx j dy * dz dZ dZ dZ ' - T -f+-j-q + - 1 h. dx j d y dz XXXIII.] Attraction of Ferromagnetics. 501 By considering the effects of these forces upon the elements (as for instance thin bars, in the direction of magnetization) into which the magnetized sphere may be supposed to be divided, it is easily shown [ 500 above], as has also been done by Poisson, that the components of the resulting force on the sphere are given by the equations dX dx dY H = dX J- dy dX KG . m H 7- .Ko-.n, dz dY -J- . Kir dX v dY^ 7 dZ\_Si ~j 1" * ~7 i ^ ~~i ) T~~ 0" dz dz dz J 4-7T R dR dx R dR dy dR U dz~ ..(4). 502 A Mathematical Theory of Magnetism. [xxxm. From these we deduce Fdx + Gdy + Hdz = er . EdE = d(o which expresses fully the result of equations (4). 640. The interpretation of this result shows that a sphere of soft iron is urged in the direction in which the magnetizing force increases most rapidly; the components of the force in different directions being expressible by the differential coeffi- g cients of the function o-jR 2 . Thus in some cases it may O7T actually be urged across the direction of the magnetizing force. For instance, if a ball of soft iron be placed symmetrically with respect to the two poles of a horse-shoe magnet, and at some distance from the line joining them, it will be urged towards this line in a direction perpendicular to it, although the magnetizing force is parallel to it ; or if the magnetizing force be due to a straight galvanic wire, a ball of soft iron will be attracted to- wards the wire, although the force on an imaginary " magnetic point " is perpendicular to a plane through it and the wire. 641. The positions of equilibrium of a small sphere acted upon by the magnetic forces alone, will be points in the neigh- bourhood of which R* is stationary in value, or points where d (R 2 ) = 0. This condition is satisfied by either R = 0, or dR=0. Hence the sphere will be in equilibrium at points where the resultant magnetizing force vanishes ; where it is a maximum or minimum ; or where it is stationary in value. 642. A position of stable equilibrium will be such that jR 2 diminishes in every direction from it ; and hence, if there be any point, external to the magnet, at which the resultant force has a maximum value, it would be a position of stable equi- librium for a small ball of soft iron, and any other position of equilibrium is essentially unstable. 643. According to Mr Faraday's recent researches, it ap- pears that there are a great many substances susceptible of magnetic induction, of such a kind that for them the value of the coefficient i is negative. These he calls diamagnetic sub- stances, and, in describing the remarkable results to which his experiments conducted him with reference to induction in diamagnetic matter, he says : " all the phenomena resolve xxxiii.] Repulsion of Diamagnetics. 503 themselves into this, that a portion of such matter, when under magnetic action, tends to move from stronger to weaker places or points of force*." This is entirely in accordance with the result obtained above ; and it appears that the law of all the phenomena of induction discovered by Faraday with reference to diamagnetics may be expressed in the same terms as in the case of ordinary magnetic induction, by merely supposing the coefficient i to have a negative value f. 644. In the case of a diamagnetic sphere, the consideration of the stability or instability of equilibrium in different posi- tions, is extremely interesting. Thus, at a point where E 2 is a minimum, a small sphere of diamagnetic matter will be in stable equilibrium ; and this is actually the case at any point for which the force vanishes ; even if we take into account the weight of the sphere, it is readily shown that stable positions of equilibrium may exist. Thus a hollow cylindrical bar- magnet (if sufficiently powerful), held with its axis vertical, would support a small diamagnetic sphere in a position of stable equilibrium at a point in the axis, a little below the lower end of the magnet. For, considering different points in the axis, we perceive that there is one below the lower end (at a distance = -j= , if a, the radius of the cylinder, be very great Y* compared with its thickness, and very small compared with its length, and if the distribution of magnetism be uniform) at which the resultant force is a maximum. If, on moving a small diamagnetic sphere upwards from this position, we arrive at a point where the force urging it upwards is greater than the weight, and then let it move freely from rest, it will oscillate about a position of stable equilibrium. It will probably be impossible ever to observe this phenomenon, on account of the difficulty of getting a magnet strong enough, and a diamagnetic substance sufficiently light, as the forces manifested in all cases of diamagnetic induction hitherto examined are excessively feeble. * Experimental Researches, 2418. t The law of induction in a mass of any form, -whether of magnetic or diamagnetic matter, may be stated as follows: Let R be the magnetic force upon a point within an infinitely small spherical surface, described round a point P in the mass, resulting from the magnetism of all the matter external to this surface. The intensity of the magnetism at P is equal to f TriR, and its direction is that of the resultant force JR. 504 A Mathematical Theory of Magnetism. [xxxni. 645. A very curious phenomenon might readily be observed, according to the results given above, by placing two bar-mag- nets, with similar poles, in the neighbourhood of a ball of soft iron allowed to move in a horizontal straight line (or suspended in such a manner that any motion which can take place is in a circle of considerable radius). Thus if a pole, S, of a bar- magnet which we may regard for simplicity as very long and thin, be held in the neighbourhood, the ball will be drawn towards the point A, in which a perpendicular from $ meets the line of motion, and A will therefore be a position of stable equilibrium. If now a pole S', of an equally powerful magnet, be presented and held at an equal distance in SA produced, A will become an unstable position ; and if the ball be placed in its line of motion, at any distance from A less than ^ , it will \f 2 be repelled from A, although either magnet alone would cause it to move towards this point. 646. The result obtained above affords the true explanation of the phenomenon observed by Faraday, that a thin bar or needle of a diamagnetic substance, when suspended between the poles of a magnet, assumes a position across the line join- ing them. For such a needle has no tendency to arrange itself across the lines of magnetic force ; but, as will be shown [ 684, below] in a future paper, if it be very small compared with the dimensions and distance of the magnet (as is the case, for instance, with a bar of any ordinary dimensions, subject only to the earth's influence), the direction it will assume, when allowed to turn freely about its centre of gravity, will be that of the lines of force, whether the material of which it consists be diamagnetic, or magnetic matter such as soft iron : but Faraday's result is due to the rapid decrease of magnetic intensity round the poles of the magnet, and to the length of the needle, which is con- siderable compared with the distance between the poles of the magnet ; and is thus explained by the discoverer himself. ( 2269 of his Experimental Researches.) " The cause of the " pointing of the bar, or any oblong arrangement of the heavy " glass is now evident. It is merely a result of the tendency of " the particles to move outwards, or into the positions of weakest xxxiv.] Attractions and Repulsions. 505 "magnetic action*. The joint exertion of the action of all the " particles brings the mass into the position which, by experiment, " is found to belong to it." ST PETER'S COLLEGE, May 13, 1847. XXXIV. Remarks on the Forces experienced by Inductively Magnetized Ferromagnetic or Diamagnetic Non-Crystalline Substances. [From the Philosophical Magazine, October 1850.] THE remarkable law laid down by Faraday in [ 2418 of his Experimental Researches} his Memoir on the Magnetic Condition of all Matter [Transactions Royal Society, 1846, p. 21, or Phil. Mag. Vol. xxviil., 1846], that a small portion of diamagnetic matter placed in the neighbourhood of a magnet experiences a pressure urging it from places of stronger towards places of weaker force, is a simple conclusion, derived from the mathematical solution of the problem of determining the action experienced by a small sphere of matter magnetized inductively, and acted upon in virtue of its induced magnetism. Without entering upon the analytical investigation, which will be found in [ 634 646 above] a paper "On the Forces experienced by small Spheres under Magnetic Influence ; and on some of the Phe- nomena presented by Diamagnetic Substances -f," I shall, in the present communication, state and explain briefly the result, and point out some remarkable inferences which may be drawn from it. 647. Let P be any point in the neighbourhood of a magnet, and let P be a point at an infinitely small distance, which may be denoted by a, from P. Let R denote the force which a "unit north pole^:" if placed at P would experience, or, as it is called, "the resultant magnetic force at P;" and let R f * The extreme feebleness of the diamagnetic action on account of which any small sphere or cube of the matter will experience very nearly the same force as if all the rest were removed, seems fully to justify this explanation. t Cambridge and Dublin Mathematical Journal, May 1847. J That is, the end of an infinitely thin uniformly and longitudinally magnetized bar of "unit strength" which is repelled on the whole from the north by the magnetism of the earth; "unit strength" being denned by the following statement : If two infinitely thin bars be equally, and each uniformly and longitu- dinally, magnetized, and if, when an end of one is placed at a unit (an inch, for example) of distance from an end of the other, the mutual force between these ends is unity; the magnetic strength of each is unity. The force E, defined in the text, is of course equal and opposite to the force that a "unit south pole" would experience if placed at P. 506 A Mathematical Theory of Magnetism. [xxxiv. denote the same with reference to P'. Then, if a small sphere of any kind of non-crystalline homogeneous matter, naturally unmagnetic, but susceptible of magnetization by influence, be placed at P, it will experience a force of which the component along PP' is where e the magnetized bar, and ST, NT' straight lines touching the line of force in which, by hypothesis, its ex- tremities lie, and P a point on it, midway between them. The resultant force on the bar will be the resultant of two forces pulling its ends in the lines ST, NT'. If these two forces were equal (as they would be if the intensity of the field did not vary at all along a line of force, as for instance when the lines of force are concentric circles, as they are when simply due to a current of electricity passing along a straight conductor ; or if P were in a situation between two dissimilar poles symme- trically placed on each side of it), the resultant force would clearly bisect the angle between the lines TS, T'N, and would therefore be perpendicular to the bar and to the lines of force in the direction towards which they are curved ; that is (Prop. IV.), would be from places of weaker to places of stronger force, perpendicularly across the lines of force. On the other hand, if the line of force through P has no curvature at this point, or no sensible curvature as far from it as N and 8, the XXXVII.] Faraday's Law. 535 lines NT and ST' will be in the same straight line, and the resultant force on the bar will be simply the excess of the force on one end above that on the other acting in the direc- tion of the greater ; and since in this case (Prop. IV.) there is p' FIG. 2. no variation of the intensity of the force in the field in &, direction perpendicular to the lines of force, the resultant force experienced by the bar is still simply in the direction in which the intensity of the field increases, this being now a direc- tion coincident with a line of force. Lastly, if the intensity increases most rapidly in an oblique direction in the field, from P in some direction between PS and PP' } there must clearly be an augmentation (a "component" augmentation) from P towards P' ; and therefore (Prop. IV.) the line through P must be curved, with its concavity towards P', and also a " com- ponent" augmentation from JV towards 8, and therefore the end S must experience a greater force than the end N. It follows that the magnet will experience a resultant force along some line in the angle SNP', that is, on the whole from places of weaker towards places of stronger force, obliquely across the lines of force. 677. Prop. V. (Mechanical Lemma.) Two forces infinitely nearly equal to one another, acting tangentially in opposed direc- tions on the extremities of an infinitely small chord of a circle, are equivalent to two forces respectively along the chord and perpendicular to it through its point of bisection, of which the former is equal to the difference between the two given forces and acts on the side of the greater ; and the latter, acting towards the centre of the circle, bears to either of the given forces the ratio of the length of the arc to the radius. 536 A Mathematical Theory of Magnetism. [xxxvu. The truth of this proposition is so obvious a consequence of " the parallelogram of forces," that it is not necessary to give a formal demonstration of it here. 678. Prop. VI. A very short, infinitely thin, uniformly and longitudinally magnetized needle, placed with its two ends in one line offeree in any part of a magnetic field, experiences a force which is the resultant of a longitudinal force equal to the difference of the forces experienced by its ends, and another force perpendicular to it through its middle point equal to the difference between the force actually experienced by either end, and that which it would experience if removed, in the plane of curvature of the line of force, to a distance equal to the length of the needle, on one side or the other of its given position. NS being the bar as before, let I denote the intensity of the force in the field at the point occupied by N t I the intensity at S t J"the intensity at P on the line of force midway between S and N t and J' the intensity P at a point P', at a distance p PP' equal to the length of /^ the bar, in a direction per- pendicular to the line of force. Then if m denote the strength of magnetism of the bar, ml and ml' will be the forces on its two FIG. 3. extremities respectively. Hence by the mechanical lemma, the resultant of these forces will be the same as the resultant of a force m(I I') acting along the bar in the direction SN, and a force perpendicular to it towards the centre of curvature, bearing the same ratio to either ml or ml', or to mJ (which is their mean, and is infinitely nearly equal to each of them), as NS to the radius of curvature, or (by Prop. II.) the ratio of the excess of the intensity at P' above that at P to the inten- sity at either, that is the ratio of J 7 J to J, and therefore itself equal to m(J' J). The bar therefore experiences a force the same as the resultant of m(II') acting along it from $ towards N, and m(J' J) perpendicularly across it towards P f , through its middle point. 679. Cor. The direction of the resultant force on the bar is xxxvu.] Faraday s Law. 537 that in which the total intensity of the field increases most rapidly ; or, which is the same, it is perpendicular to the sur- face of no variation of the total intensity. Prop. VII. The resultant force on an infinitely small magnet of any kind placed in a magnetic field, with its magnetic axis along the lines of force, is in the line of most rapid variation of the total intensity of the field, and is equal to the magnetic moment of the magnet multiplied by the rate of variation of the total intensity per unit of distance ; being in the direction in which the force increases when the magnetic axis is " direct," (that is, in the position it would rest in if the magnet were free to turn about its centre of gravity). Cor. 1. The resultant force experienced by the magnet will be in the contrary direction, that is, the direction in which the total intensity of the field diminishes most rapidly, when it is held with its magnetic axis reverse along the lines of force of the field. 680. Cor. 2. A ball of soft iron, or of any non-crystalline paramagnetic substance, held anyhow in a non-uniform magnetic field, or a ball or small fragment of any shape, of any kind of paramagnetic substance whether crystalline or not, left free to turn about its centre of gravity, will experience a resultant force in the direction in which the total intensity of the field increases most rapidly, and in magnitude equal to the magnetic moment, of the magnetization induced in the mass multiplied by the rate of variation of the total intensity per unit distance in the line of greatest variation in the field. For such a body in such a position is known to be a magnet by induction, with its magnetic axis direct along the lines of force. 681. Cor. 3. A ball of non-crystalline diamagnetic substance held anyhow in a magnetic field, or a small bar or fragment of any shape of any kind of diamagnetic substance, crystalline or non-crystalline, held by its centre of gravity, but left free to turn about this point, experiences the same resultant force as a small steel or other permanent magnet substituted for it, and held with its magnetic axis reverse along the lines of force. For Faraday has discovered, that a large class of natural substances in the stated conditions experience no other action than a 538 A Mathematical Theory of Magnetism. [xxxvu. tendency from places of stronger towards places of weaker force, quite irrespective of the directions the lines of force may have, and he has called such substances diamagnetics. 682. Cor. 4. A diamagnetic, held by its centre of gravity but free to tarn about this point, must react upon other magnets with the same forces as a steel or other magnet substituted in its place, and held with its magnetic axis reverse along the lines of force due to all the magnets in its neighbourhood. 683. Cor. 5. Any one of a row of balls or cubes of diamag- netic substance held in a magnetic field with the line joining their centres along a line of force, is in a locality of less intense force than it would be if the others were removed ; but any one ball or cube of the row, if held with the line joining their centres perpendicularly across the line of force, is in a locality of more intense force than it would be if the others were removed. 684. Cor. 6. When a row of balls or cubes, or a bar, of per- fectly non-crystalline diamagnetic substance, is held obliquely across the lines of force in a magnetic field, the magnetic axis of each ball or cube, or of every small part of the substance, is nearly in the direction of the lines of force, but slightly inclined from this direction towards the direction perpendicular to the length of the row or bar. Hence, since the magnetic axis of every part differs only a little from being exactly reverse along the lines of force, the direction of the resultant of the couples with which the magnets, to which the field is due, act on the parts of the row or bar must be such as to turn its length along the lines of force. 685. Cor. 7. The positions of equilibrium of a row of balls or cubes rigidly connected, or of a bar of perfectly non-crystalline diamagnetic substance, free to move about its centre of gravity in a perfectly uniform field of force, are either with the length along or with the length perpendicularly across the lines of force : positions with the length along the lines of force are stable; positions with the length perpendicularly across the lines of force are unstable. 686. Cor. 8. The mutual influence and its effects, referred to in Cors. 5, 6, 7, is so excessively minute, that it cannot possibly have been sensibly concerned in any phenomena that have yet XXXVIL] Faraday's Law. 539 been observed ; and it is probable that it may always remain insensible, even to experiments especially directed to test it. For the influence of the most powerful electro-magnets induces the peculiar magnetic condition of which diamagnetics are capable, to so slight a degree as to give rise to only very feeble, scarcely sensible, mutual force between the diamagnetic and the magnet ; and therefore the magnetizing influence of a neigh- bouring diamagnetic, which could scarcely, if at all, be observed on a piece of soft iron, must be insensibly small on another diamagnetic. 687. Cor. 9. All phenomena of motion that have been ob- served as produced in a diamagnetic body of any form or sub- stance by the action of fixed magnets or electro-magnets, are due to the resultant of forces urging all parts of it, and couples tending to turn them ; the force and couple acting on each small part being sensibly the same as it would be if all the other parts were removed. 688. Cor. 10. The deflecting power (observed and measured by Weber) with which a bar of non-crystalline bismuth, placed vertically as core in a cylinder electro-magnet (a helix conveying an electric current), urges a magnetized needle on a level with either of its ends, is the reaction of a tendency of all parts of the bar itself from places of stronger towards places of weaker force in its actual field. The preceding investigation, leading to Props. VI. and VII., is the same (only expressed in non-analytical language) as one which was first published in the Cambridge and Dublin Mathe- matical Journal, May 1846 [ 638640 above]. The chief conclusions now drawn from it, with particulars not repeated, were stated in a paper entitled " Remarks on the Forces experi- enced by inductively magnetized Ferromagnetic or Diamagnetic Substances," in the Philosophical Magazine for October 1850 [Article XXXIV, above]. GLASGOW COLLEGE, March 15, 1855. 540 A Mathematical Theory of Magnetism, [xxxvm. XXXVIII. CORRESPONDENCE WITH PROFESSOR TYNDALL. Letter to Professor Tyndall on the "Magnetic Medium" and on the Effects of Compression. [From the Philosophical Magazine, April 1855.] [Editorial.] The following letter was received a few days ago. It was not written for publication, but the subject to which it refers being of general interest at present, I ven- tured to suggest to Professor Thomson the desirableness of having the letter printed. This he at once agreed to. With the exception of a paragraph relating to matters of a purely private nature, the letter appears as I received it. JOHN TYNDALL. March 24, 1855. 2 COLLEGE, GLASGOW, March 12, 1855. 689. MY DEAR SBR, Allow me to thank you for the abstract of your letter on magnetism, and the copy of your letter to Mr Faraday, which I have recently received from you, and have read with much interest. I am still strongly disposed to believe in the magnetic character of the medium occupying space, and I am not sure but that your last argument in favour of the reverse bodily polarity of diamagnetics may be turned to support the theory of universally direct polarity. There is no doubt but that the medium occupying interplanetary space, and the best approximations to vacuum which we can make, have perfectly decided mechanical qualities, and among others, that of being able to transmit mechanical energy in enormous quantities (a platinum wire, for instance, kept incandescent by a galvanic current in the receiver of an air-pump, emits to the glass and external bodies the whole mechanical value of the energy of current spent in overcoming its galvanic resistance). Some of these properties differ but little from those of air or oxygen at an ordinary barometric pressure. Why not, then, the magnetic property? (of which we know so little that we have no right to pronounce a negative). Displace the inter- planetary medium by oxygen, and you have a slight increase of magnetic polarity in the locality with a drawing in of the lines of force. Displace it with a piece of bismuth or a piece of wood, and a slight decrease of magnetic polarity through the xxxvin.] Correspondence with Professor Tyndall 541 locality takes place, accompanied by a pushing out of the lines of force. A state of strain by compression may enhance, in the direction of the strain, that quality of the substance by which it lessens the magnetizability of the space from which it displaces air or "ether;" just as a similar state may enhance, in the direction of compression, the augmenting power of a paramagnetic substance. 690. By the bye, a long time ago (rather more than a year after the Edinburgh meeting of the British Association) I re- peated with much pleasure some of your compression experi- ments, and found a piece of fresh bread instantly affected by pressure, so as always to turn the compressed line perpendicular to the lines of force, to whatever form the fragment was reduced. A very slight squeeze between the fingers was quite enough to produce this property, or again to alter it so as to make a new line of compression set equatorially. I repeated it a few days ago with the same results, and got a ball of bismuth, too, to act similarly. I remember formerly finding the bread attracted as a whole, instead of being repelled, as I expected from your results. I suppose, however, this must have resulted from some ferruginous impurities, which it may readily have got either in the course of the experiments with it, or in the baking. I mean to try this again*. 691. I do not quite admit the argument you draw from your compression experiments regarding the effect of contiguity of particles, because in fact we know nothing of the actual state of the molecules of a strained solid. You have made out a most interesting fact regarding their magnetic bearings ; but experi- ments are neither wanted, nor can be made, to show any sensible effect whatever of the mutual influence of a row of small pieces of bismuth placed near one another, or touching one another. It is perfectly easy to demonstrate that it must be such as to impair the " diamagnetization " of each piece when the line of the row is parallel to the lines of force, and to enhance it when that line is perpendicular to the lines of force, but in each case to so infinitesimally minute a degree, as to be * Prof. Thomson's supposition is correct; pure bread is repelled by a magnetic pole. I may remark that I am at present engaged in the further examination of the influence of compression, and have already obtained numerous instructive results. J. T. 542 A Mathematical Theory of Magnetism, [xxxviii. wholly inappreciable to the most refined tests that have ever been applied. For let the lines of force be parallel to the line shown in the figure, and act on a steel needle in the manner there represented. Then, whatever hypothesis be true for diamagnetism, there is not a doubt but that each piece is acted on, and consequently reacts, precisely as a piece of steel very feebly magnetized, with its magnetic axis reverse to that of a steel needle free to turn, substituted for it, would do. Each piece of bismuth therefore acts as a little magnet, having its polarity as marked in the diagram, would do. Hence the magnetizing force by which the middle fragment is influenced is less than if the two others were away (this being such a force as would be produced by a north pole on the left-hand side of the diagram, and a south pole on the right). It is easily seen, similarly, that if the line joining the centres be perpen- dicular to the lines of force, the magnetizing force on the space occupied by the middle fragment is increased. Corresponding assertions are true for the terminal fragments, although the disturbing effect will be less on them in each case than in the middle one. Hence the dia- magnetization of each will be en- feebled in the former case and enhanced in the latter, by the pre- sence of the others. It follows, according to the principle of su- perposition of magnetizations, that if the line of the row be placed obliquely across the lines of force, the magnetic axis of each particle, instead of being exactly parallel to the lines of force, will be a little inclined to them, in the angle between their direction and the direction transverse to the bar. The magnets causing the force of the field must act on the little dia- magnets, each with its axis thus rendered somewhat oblique, so as to produce on it a statical couple (as shown by the arrow- heads), and the resultant of the couples thus acting on the frag- xxxviii.] Correspondence with Professor Tyndall. 543 ments will, when all these are placed on a frame, or rigidly connected, tend to turn the whole mass in such a direction as to place the length of the bar along the lines of force. Still, I repeat, this action, although demonstrated with as much cer- tainty as the parallelogram of forces, is so excessively feeble as to be absolutely inappreciable. A fragment of bismuth, of any shape whatever, held in any position whatever in any kind of magnetic field, uniform or varying most intensely, only exhibits the resultant action of couples on all its small parts if crystal- line, and of forces acting always according to Faraday's law on them if the field in which it is placed be non-uniform. Some phenomena that have been observed are to be explained by the resultant of forces from places of stronger to places of weaker intensity in the field, others by the resultant of couples depend- ing on crystalline structure, and others by the resultant of such forces and couples co-existing; and none observed depend at all on any other cause. 692. I gave a very brief summary of these views (which I had explained somewhat fully and illustrated by experiments on paramagnetics of sufficient inductive capacity to manifest the effects of mutual influence, at the meeting at Belfast) as an abstract of my communication, for publication in the Keport of the Belfast meeting of the British Association, where you may see them [669 above] stated, I hope intelligibly. The experi- ments on the paramagnetics are very easy, and certainly exhibit some very curious phsenomena, illustrative of the resultant effects due to the attractions experienced by the parts in virtue of a variation of the intensity of the field, and to the couples they experience when their axes are diverted from parallelism to the lines of force by mutual influence of the magnetized parts. 693. I had no intention of entering on this long disquisition when I commenced, but merely wished to try and briefly point out, that the assertions I have made regarding mutual influence are demonstrable in every case without special experiment, are confirmed amply by experiment for paramagnetics, and are absolutely incontrovertible, as well as incapable of verification by experiment or observation on diamagnetics. Believe me, yours very truly, WILLIAM THOMSON. PROF. TYNDALL. 544 A Mathematical Theory of Magnetism, [xxxvni. On Reciprocal Molecular Induction: Letter from Professor Tyndall to Professor W. Thomson, F.B.S. [From the Philosophical Magazine, December 1855.] ROYAL INSTITUTION, Nov. 26, 1855. 694. MY DEAR SIR, The communication from Professor Weber which appears in the present number of the Philoso- phical Magazine, has reminded me, almost too late, of your own interesting letter on the same subject published in the April number of this Journal. A desire to finish all I have to say upon this question at present induces me to make the following remarks, which, had it not been for the circumstance just alluded to, might have been indefinitely deferred. "With reference to the mutual action of a row of bismuth par- ticles, you say that "it is perfectly easy to demonstrate that " it must be such as to impair the ' diamagnetization ' when the " line of the row is parallel to the lines of force " (the " must," you will remember, is put in italics by yourself). From this you infer, that in a uniform field of force a bar of bismuth would set its length along the lines of force. Further on it is stated that this action is " demonstrated with as much certainty " as the parallelogram of forces ;" and you conclude your letter by observing that " the assertions which I [yourself] have made " are demonstrable in every case without special experiment, . . . " and are absolutely incontrovertible, as well as incapable of " verification by experiment or observation on diamagnetics." Most of what I have to say upon this subject condenses itself into one question. Supposing a cylinder of bismuth to be placed within a helix, and surrounded by an electric current of sufficient intensity; can you say, with certainty, what the action of either end of that cylinder would be on an external fragment of bismuth presented to it ? If you can, I, for my part, shall rejoice to learn the process by which such certainty is attained : but if you cannot, it will, I think, be evident to you that the verb "must" is logically " defective." We know that magnetized iron attracts iron : we know that xxxviii.] Reciprocal Action of Diamagnetic Particles. 545 magnetized iron repels bismuth : this, so far as I can see, is your only experimental ground for assuming that magnetized bismuth repels bismuth, and yet you affirm that an action deduced from this assumption " is demonstrated with as much " certainty as the parallelogram of forces." Do I not state the question fairly ? I can, at all events, answer for my earnest wish to do so. It is needless to remind one so well acquainted with the mental experience of the scientific inquirer, that the very letters which you attach to your sketch, page 291 [of Philosophical Magazine, 691 above], may tempt us to an act of abstraction a forgetfulness of a possible physical difference between the n of iron and the n of bismuth which may lead us very wide of the truth. The very term " pole " often pledges us to a theoretic conception without our being conscious of it. You are also well aware of the danger of shutting the door against experimental inquiry on an unpromising subject; and when you apparently do this in your concluding paragraph, I simply accept it as a strong way of expressing your personal conviction, that the action referred to is too feeble to be rendered sensible by experiment. Believe me, dear Sir, most truly yours, JOHN TYNDALL. On the Reciprocal Action of Diamagnetic Particles: Letter from Prof essor Thomson to Professor Tyndall. [From the Philosophical Magazine, January 1856.] GLASGOW COLLEGE, Dec. 24, 1855. 695. MY DEAR SIR, I have been prevented until to-day, by a pressure of business, from replying to the letter you addressed to me in the number of the Philosophical Magazine published at the beginning of this month. You ask me the question, " Supposing a cylinder of bismuth " to be placed within a helix, and surrounded by an electric " current of sufficient intensity ; can you say, with certainty, " what the action of either end of that cylinder would be on an " external fragment of bismuth presented to it ?" 696. In answer, I say that the fragment of bismuth will be re- pelled from either end of the bar provided the helix be infinitely T. E. 35 546 A Mathematical Theory of Magnetism, [xxxvin. long, or long enough to exercise no sensible direct magnetic action in the locality of the bismuth fragment. I can only say this with the same kind of confidence that I can say the different parts of the earth's atmosphere attract one another. The con- fidence amounts in my own mind to a feeling of certainty. In every case in which the forces experienced by a little magnetized steel needle held with its axis reverse along the lines of force, and a fragment of bismuth substituted for it in the same locality of a magnetic field, have been compared, they have been found to agree. In a vast variety of cases, a fragment of bismuth has been found to experience the opposite force to that experienced by a little ball of iron, that is, the same force as a little steel magnet held with its axis reverse to the lines of force ; and in no case has a discrepance, or have any indica- tions of a discrepance, from this law been observed. I feel, therefore, in my own mind a certain conviction, that even when the action is so feeble that no force can be discovered at all on the bismuth by experimental tests, such in regard to sensi- bility as have been hitherto applied, the bismuth is really acted on by the same force as that which a little reverse magnet, if only feeble enough, would experience when substituted in its place. Now there is no doubt of the nature of the force experienced by the steel magnet, or by a little ball of soft iron, in the locality in which you put the fragment of bismuth. One end of a magnetized needle will be attracted, and the other end repelled by the neighbouring end of the bismuth bar ; and the attraction or the repulsion will preponderate according as the attracted or the repelled part is nearer. There is then certainly repulsion when the steel magnet is held in the reverse direc- tion to that in which it would settle if balanced on i'ts centre of gravity. In every case in which any magnetic force at all can be observed on a fragment of bismuth, it is such as the steel magnet thus held experiences. Therefore I say it is in this case repulsion. But it will be as much smaller in proportion to the force experienced by the steel magnet, as it would be if an iron wire were substituted for the bismuth core. Yet in this case the repulsion on the bismuth is very slight, barely sensible, or perhaps not at all sensible when the needle exhibits most energetic signs of the forces it experiences. You know xxxviii.] Reciprocal Action of Diamagnetic Particles. 547 yourself, by your own experiments, how very small is even the directive agency experienced by a steel magnet placed across the lines of force due to the bismuth core. You may judge how much less sensible would be the attraction or repulsion it would experience as a whole, if held along the lines of force; and then think if the corresponding force experienced by a fragment of bismuth substituted for it, is likely to be verified by direct experiment or observation. I think you will admit that it is "incapable of verification," as well as "incontro- vertible " by any collation of the results of experiments hitherto made on diamagnetics. As to the concluding paragraph of my letter which you quote, you do me justice when you say you accept it as an expression of my " personal conviction that the "action referred to is too feeble to be rendered sensible by " experiment." I will not maintain its unqualified application to all that can possibly be done in future in the way of experi- mental research to test the mutual action of diamagnetics under magnetic influence. On the contrary, I admit that no real physical agency can be rightly said to be " incapable of " verification by experiment or observation ; " and I will ask you to limit that expression to experiments and observations hitherto made, and to substitute for the concluding paragraph of my letter the following statement [ 686 above], written for publi- cation three days later, and published in the same number of the Magazine as that to which you communicated my letter (Phil. Mag., April 1855, p. 247). "The mutual influence" between rows of balls or cubes of bismuth in a magnetic field, " and its effects " in giving a tendency to a bar of the substance to assume a position along the lines of force, " are so excessively "minute, that they cannot possibly have been sensibly con- " cerned in any phenomena that have yet been observed ; and " it is probable that they may always remain insensible, even "to experiments especially directed to test them." I remain, my dear Sir, yours very truly, WILLIAM THOMSON. DR TYNDALL. 35 548 A Mathematical Theory of Magnetism. [xxxix. XXXIX. Inductive Susceptibility of a Polar Magnet. [March 1872. Not hitherto published.] 697. It is probable that every loadstone or steel magnet, or polar magnet of any kind, whatever degree of intrinsic mag- netization it may possess, has also a susceptibility for magnetic induction, according to which, under the influence of other magnets brought into its neighbourhood, it will experience inductive magnetization temporarily superimposed upon its in- trinsic magnetization. Hitherto experiment has given us little or no definite knowledge on this subject, or indeed generally on the relation between magnetic retentiveness and magnetic susceptibility. Waiting for more complete experimental in- vestigation of the magnetic properties of matter, I shall assume as a typical magnetic solid, a rigid body possessing any degree of intrinsic magnetization in any direction, with perfect re- tentiveness; and having inductive quality denned by three principal magnetic susceptibilities along three principal rect- angular axes of inductive capacity, in any given directions through it. The " rigid polar magnets " which we have hitherto considered are intrinsic magnets of zero susceptibility ; and it now becomes necessary to define intrinsic magnetization for a substance of which the susceptibility is not zero. 698. Def. The intrinsic magnetization of a body is the re- sultant ( 605) of the three intensities of magnetization found by cutting three infinitely thin bars from directions in it agree- ing with its principal inductive axes, and testing them in a uniform magnetic field of air by measuring the couples which they experience when held at right angles to the lines of force. Before going on with the general problem of magnetic induc- tion, we may consider the following particular case of it, merely as an illustration of this definition : 699. Problem. A solid sphere of uniform material, having IJL, jj,', fj," for its three principal magnetic susceptibilities, and possessing intrinsic magnetization of intensity i in the direc- tions specified with reference to the principal inductive axes by the direction-cosines, I, I', I", is placed in air with no dis- turbing body in its neighbourhood : it is required to find its XL.] General Problem of Magnetic Induction. 549 actual magnetization. Let f , f ', f ", be the components of induced magnetization in the directions of the three principal axes ; the required magnetization will be the resultant of il-t, U'-Z, tT-r .................. (1); and therefore the problem is solved when f, f ', f" are deter- mined. From the footnote to 609, it follows immediately that the resultant force at any point within the sphere has for its components, in the directions of the principal axes, Now f , ', f " are the intensities of induced magnetiza- tion due separately to these three components of magnetizing force, and therefore ( 610, Def. 2) Solving these for f, (', f ", we have 4V q lt > * , 4 ' and therefore (components of the whole magnetization) XL. General Problem of Magnetic Induction. [March 1872. Not hitherto published.] 700. This problem is ( 628) identical with the three general problems electro-static induction through a heterogeneous in- sulating solid, thermal or electric conduction through a hetero- geneous conducting solid, and (proved below, 751 759) the flow of a frictionless incompressible liquid through a hetero- geneous porous solid. 701. Let all space be occupied with matter of given permea- bilities, w, -or', r", along three principal inductive axes (I, m, n) } (l' t m, n'), (I", m", n"), ( 611) through any point (a?, y, z). 550 A Mathematical Theory of Magnetism. [XL. Let there be intrinsic magnetization (a, ft, 7) at (x, y, z) ; and let constant electric currents be maintained having u, v, w for components of intensity at (x, y, z} ; subject to the condition (540) du^dv + dw = Q (1). dx~* dy dz Let f, 77, be the components of induced magnetization at (x, y, z). Then r, r', tar", (Z, m, n), (Z', m', w'), (Z", m", ri'), a, ft, 7, u, v, w, being given for every point (x, y, z}, it is re- quired to find f, 77, f. This is the general problem of magnetic induction. In it a., ft, 7 are absolutely arbitrary functions of (x, y, z) ; their values being zero in any part of space destitute of intrinsic magnetization : and u, v, w are arbitrary functions of (x, y, z), subject only to the condition (1); their values being zero throughout any portion of space through which there is no electric current. 702. Let jff , (?Ur, pj be the components of the resultant mag- netic force according to the polar definition ( 517, Postscript), calculated from the given intrinsic magnetization on the sup- position of no induced magnetism ; and F, G, H the components of the unambiguous resultant force ( 551) calculated from the given electric currents. By 545 and 517 (m), (ri), and (k), (I), we have 4f7rp dx dy dz dy dz < dz dx ' dx dy i dft = dx dy dz dff dG dH dF dF dG ~j --- 7- = 4f7TU, -7 --- j- = &7TV, -j -- -j- dy dz dx dz dy dx Equations (2) suffice to determine Jf, (, |^ from the data > A 7, by expressing that they are the differential coefficients of a function, and that that function is the potential of a distri- bution of imaginary magnetic matter having ~^~+"T~ + ^~ for its density at (x, y, z), which we denote by p. Similarly XL.] General Problem of Magnetic Induction. 551 equations (3) determine F, G, H by virtually expressing that they are the components of the resultant magnetic force due to the given distribution of electric currents (u, v, w), and are therefore directly calculable from the data by the formulas (6) of 517 with F, G, ^instead of X, F, Z. 703. Let now F = Jf+F, G = doc dy dz I ,-, dH dG dF dH dG dF {'" ( h -f --- ~ = 4?, -p -- f=- =4 TTV, -f= -,= = 4nrw dy dz dz dx dx dy ) and these equations suffice to determine F, G, H fully, by virtually expressing that they are the sums of the two sets of components explicitly expressed in terms of the data, by the formulae referred to in the preceding section. As we shall see immediately that we require from the data respecting intrinsic magnetization and electric currents nothing but the values f > G } H, we may simply regard these quantities as express- ing the necessary data in this respect ; and it is important to remark that they are unconditionally arbitrary for every point (x, y } z}. 704. Let now the potential of the distribution of imaginary magnetic matter corresponding to the induced magnetism (? V> ?) b e denoted by 17; that is to say, let Iff be the function of (x } y, z} which through all space satisfies the equation andlet ^ = -^=-^ = - ......... (7). We shall see immediately that our problem is reduced to the determination of the single function 17; and we shall have simple equations [ 705 (10)] giving explicitly the required components of induced magnetization f , 77, f, in terms of the differential coefficients of this function. 705. Let I } /', /" denote the components of the resultant of F, G, H, and jfe, gb', b", the components of the resultant of K, , 3&, along the principal inductive axes. We have 552 A Mathematical Theory of Magnetism. [XL. I=lF+mG + nH, I' = l'F + m'G + n'H t r = l"F + m" F=lI+l'r + l"I", G=mI + m'I' + m"I, H = nI+nT The three principal magnetic susceptibilities ( 629) being 1r i rr i 07 1 -OF 1 4-7T ' 4?T ' 4?T ' the component intensities of induced magnetization along the principal inductive axes (to be denoted, 712 below, by ^ y *" are Hence taking components along the axes of (#, y, z\ and multi- plying by 4-7T, we have 706. These three equations, together with the three equations by which , |9, SB might, according to 518, 482, 483, be expressed in terms of f, 97, f, suffice to determine the six unknown quantities f, 77, , g^, 5^; but, by (7) and (6) intro- ducing 17, we may eliminate those six unknown quantities, and obtain a single equation for the one unknown quantity IT, thus : Taking ^- of the first of the three equations (10), -y- of the second, and -y- of the third, adding and using (6) and (7), we find dx dy dz d(F- wll - Tz'I'l'- w'TT] d(G- talm - is'l'm'- >a"I"m") d(H - dx dy dz Substituting in this for gb, ^', gb" their values by (8), then for , ^, 5^ by (7), and for /, /', I" their values by (8), we have explicitly a linear differential equation of the second order with second member a known function of (a?, y y z) to determine the unknown function J7. XL.] General Problem of Magnetic Induction. 553 707. The coefficients of , , - under the symbols dx dy dz :=-.- are related in the ordinary symmetrical manner dx dy dz to the coefficients which appear in the quadratic function ^] .......... (12) when expanded; and it is unnecessary to write them out ex- plicitly. A similar remark is applicable to the coefficients of F, G, H under differentiation in the second member. Denot- ing (12) by 4R, and the same function of F, G, H by Q, so that using again the notation of (8) for brevity, we have n ...... (is) and Q = - - ?&) + *> (&' - m? + " (&"- m". . .(IT), and the triple integral j0=r r r ^dxdydz ............ as). J aoj oo J oo (Compare 503, 561, 206, 732, and 753763). The function ^ is necessarily positive, except in the particular case of 3 = 3Bt, Sb' = ISt', >" = 3St", when it is zero. Remembering that Sb, Sb', o r 4. ,. , .., . - are linear functions of ^ , rf- , T~ > W1 ^h given func- ax ay az tions of (x t y, z) for their coefficients, apply the calculus of variations to assign F, so that E may be a minimum. Using for brevity the notation (7) of 704, we have Hence, following the usual process of integration by parts, ac- cording to the calculus of variations, we find for the condition that E may be a minimum, Now if we put which imply that > ...(20) and look to equations (13) and (8) of 707, we see that -p is the same quadratic function of K - it, | - J% 5& - M> that <& is of X, 9, 5& Hence , 1 , are linear functions of _ ^ ^ _ J^t, SB - ift ; and if we denote by $ the same quadratic function of '&, J^l, JJ that ^ is of , ^, jg, that is to say, if we put s=-l( OT m 2 +^m"+ w "ia" 2 ) ............ (21), O7T XL.] Determinacy and Singleness proved. 555 we have ^l-^.^L rfg_d__dX d^_m_d^ , dX~dX d%' djj>~d^ dJJV dZ," d%> d&'" Hence (19) becomes d d& d d% ( . " ^ __ dx dX dyd^ dz d%>~ 'dx dH. dy dffl dz which, expanded in terms of 17, is a linear partial differential equation of the second order, with right-hand member a given function of (a?, y, z\ The fulfilment of this equation through all space is the sole condition which 17 must fulfil to make E a minimum. Now it is possible to assign 17 so as to make E a minimum, and therefore there exists a function 17 which satisfies equation (23) through all space. This is an obvious extension of Theorem 1, 206. Demonstration 2 of 206 extended in an obvious manner proves that no function differing at any point from one function which satisfies (23) through all space, can satisfy (23) through all space. Hence the solution of this equation is determinate and free from all ambiguity or multiplicity of values. 710. The extension of 206, 2, gives the following useful theorems : Let 17 be a function of (#, y, z) satisfying (23) through all space ; let A17 be any function whatever of (x, y } z}\ let Agfc, A&', ASb", ^(A) be the values of b, g>', Sb", E, when AIT is substituted for F; and let E+&E be the value of E when 17 + A17 is substituted for IT. Then^ Theorem L f f ccJ aoJ _o ")=0 (24); proved by the ordinary integration by parts of 199, (a), (6), as extended in 206, Demonstrations 1 and 2, and now further extended. Theorem II. . A^=#(A) ........................ (25). This very important theorem is an instant consequence of Theorem I. As E (A) is necessarily positive, a function J?, which satisfies (23), has the unique characteristic that every function differing from it gives a larger value to E. 711. The first member of (23) is identical with the first 556 A Mathematical Theory of Magnetism. [XL. member of (15). We may make the second member of (23) equal to the second member of (15), by taking it =-/+-(/+ Ml + bm + fon), &' = -/' + , &' = -/' + , (/' + til' + bW + foV), ) ...... "+&"ro" + fo"n") J TV where u, fo, fo are any three quantities such that du db dto This we see at once by remarking that 47r = ^m? + w'mr + *rWl\ etc. etc, and 47r 3 = vll + sr'/T 4- w'TT, etc. etc, u/J^ and taking account of (8) and (5). Hence 709, 710, with the values (26) for 3S, 38t', 3K", prove that there exists a function F satisfying the inductive equation (16) through all space ; that this solution makes the triple integral E (18) a minimum ; that if 17 be a function satisfying (15), and AU any function whatever, U + AU substituted for 17 augments the value of E by the necessarily positive value of the triple integral found by substi- tuting A17 for 17 ; and, therefore, that no function differing from one which satisfies (15) can also satisfy it. 712. Preliminary to Second Demonstration of Deter minacy and Singleness. First, it will be convenient to put the inductive equations (11) and (16) into a different form, a form suitable to the uniform reckoning of "resultant magnetic force," accord- ing to the "electro-magnetic definition" ( 517, Postscript}. Eemembering ( 702, 704) that jf , $f, ffi and K, |B, S& are the components of the resultant forces calculated separately, ac- cording to the polar definition, from the intrinsic and induced magnetizations respectively, we see [517 (r)] that are the components of the resultant force of intrinsic and in- duced magnetizations together, according to the electro-magnetic definition. To these we must add F, 6r, H to find for the whole system (of inducing intrinsic magnetization and electric currents, and induced magnetization) the components of the resultant magnetic force, according to the electro-magnetic XL.] Electromagnetic Formula?. 557 definition. Calling these X, Y, Z, and taking advantage of the short notation (4), we have X=F+ +47r(a+a r=g+|)+47r(/3+7;), Z=H+&+4nr(, o', >" have still the same significance as that indicated in (8), 705, above. Now by (9) we have ( W -l)(/+g>), 47ry=( W '-l)(r+g>')> 4^"=( W "-l)(/ /f +Sb") (34). Hence eliminating S-, &' W from (30), ^(7+b)+47r^, S'=v'(r-%')+47rA', S"=^"(r^"}+^A" (35). Put now and let Cl+at+Cri''^, Cm+C'm'+C"m"=G, Cn+C'n'+C"n"=S,'\ implying I (37). C=lF+mG+nH, C'=l'F+m'~G+n'H, C"=l"F+m"G+ri'B) By (35) we have & = -4 S>' = J-C', " = ^-C"... (38). Hence /7 2 713. Now let Q = . tT + . ............... (40). [Compare (13) of 707.] Substituting for S, S', S" their values by (31), we have in Q a quadratic function of JT, Y", >f (corre- sponding in the electro-magnetic formulae to the function (Hi of , |9, SS in the polar formulae). Now (39) becomes 558 A Mathematical Theory of Magnetism. [XL. Eliminating K, |J, 5? by the condition that Xdx + T&dy + %dz is a complete differential, we have d_dQ_d_dQ,_ _(dH_dG\ d dQ_d^dQ_^fdF_dJ^\ dy dZ dzdY~TT\dy dz/' dz dX dxdZ~4ir\dz dx ) d_dQ L _d_dQ_ fdG _ dF\ dxdY dydZ~ir\dx dy) three linear partial differential equations in X, Y, Z, equivalent to two independent equations, because -y- of the first added to -y- of the second and -y- of the third constitutes an equation in dy dz which each member is identically zero. Also, by (29), (5), (7), and (6), we have dX dY dZ _ n , , 7 I -- 7 I -- T~~ " ............... . ..... C^^/' ax ay az These four, (42) and (43,) equivalent to three independent equa- tions, in which F, 6r, H are arbitrarily given functions of (x, y, z), determine fully and unambiguously the unknown X, Y, Z through all space, as will be proved immediately by the pro- mised fresh demonstration. But first it may be remarked that one obvious way of dealing with them leads us back to our former analysis, thus : The three equations (42) simply express that (dQ 1 -jfr^.fdQ 1 W\ , , fdQ 1 w TV ~~ T~ * i ax + ( T\r"~ A~~ "" ) dy + T^ -7 \dX 4?r I \dY 4?r / \dZ 4?r is a complete differential. Hence their most general integral is , dQ -p dV dQ T dV dQ = - ^ = ~ ^ = where 17 so far denotes an arbitrary function of (x, y, z). The first members here are merely short expressions for the linear functions of X, Y } Z which appear in (39) with S, S', 8" elimi- nated by (31). Solved for X } Y, Z, equations (44) give expres- sions which are the same as (29) with f, TJ, eliminated by (10), and , |9, % by (7); and eliminating by them X, 7, Z from (43) we have an equation for Jff identical with (11), which ( 708) determines 17 unambiguously through all space. 714. Second Proof of Determinateness and Singleness. Let .K", K', K" be any three arbitrarily given functions of (#, y, z)-, and put [where the surfix is appended to distinguish from the <2E of 729 ...731 below.] XL.] Second Proof of Determinacy and Singleness. 559 Consider the problem of finding X, Y, Z so as to make (, a minimum, subject to (43). Denoting by X an indeterminate multiplier, according to the ordinary method of the calculus of variations, make unconditionally a minimum. The resultant equations are dQ(S-K)_ 1 d\ dQ(S-K}_ 1 d\ dQ(S-K)_ 1 d\ dX ~47rdx'~ dY ~47r J < r r dxdydz(^^C+^'& f C' + ^"&"0" ............ (56). For C, C', C", taking their values by (36), and attending to (8), (28), and (32), we have sr(7 + rar'&'C" + w"&"C" = wl& + -nfl'S)' + &"!"&" + v (S>A +%'A' + %"A ") Putting in the second member for K, |9, 5S their values, XL.] Polar and Electromagnetic Formulae. 561 dlrT (da. d/3 dy\ ~^, etc., remembermg that p = -( + ^ + sJ' and m ' tegrating by parts as usual, we find dxdydz&gC + Tz'&'C' + &"&"") (If J -00 J -03 7 -t / the last step being simply an introduction of the notation of (15). Using this in (56), attending to (55) and (50), and transposing, we find -| f 00 y00 /* CO 8?rJ ooJaoJoo Compare 569 (7), (8) ; 717 (55), (58) ; 731 (99), (100). 718. The triple integral (53) denoted by E is of great import- ance, as being the expression for the whole kinetic energy in the hydro-kinetic analogue (Chapter XI. below). On account of the correspondence by opposites, which I perceived some years ago ( 733 739, below) between the forces experienced by solids held at rest in a moving liquid, and the forces experienced by magnetized matter in the corresponding cases of the magnetic analogue, I conclude that the diminution of the value of E produced by motion of any portion of matter, surrounded by space of uniform and isotropic permeability and not traversed by electric currents, is equal to the work required to effect the motion. Before proceeding to prove this proposition it is con- venient to notice that the triple integral may be put into se.veral other forms, each having a characteristic quality suitable for a class of applications. 719. These transformations will be simplified by, in the first place, substituting for electric currents, if there are any, distribu- tions of intrinsic magnetization giving the same contributions to the values of S, S', S" ; which may be done in an infinite variety of ways, as we see by the following considerations : For every closed circuit substitute ( 548) an open mag- netic shell producing the same potential as the circuit through- out space, except the portion occupied by the magnetized substance of the shell. The resultant force of the shell, T. E. 36 562 A Mathematical Theory of Magnetism. [XL. reckoned in the magnetized substance according to the electro- magnetic definition ( 517, Postscript), will throughout space be the same as that of the circuit. The values of S, S' S", will be everywhere unchanged if the whole magnetized substance thus introduced be placed in space of zero susceptibility (or unit permeability), and be itself of zero susceptibility. But this cannot be if there are circuits completely imbedded in matter of other than zero susceptibility; if, for instance, part of the given system consists of an electric circuit through the aperture of a soft iron ring. Hence to avoid loss of gener- ality we must suppose some part, if not the whole, of the intrinsic magnetization, which we are now introducing, to be placed in portions of space having in the original data, sus- ceptibility different from zero. The magnetizing force in these portions of space will be altered by the substitution of mag- netization for electric current, but to make the whole external effect the same, we have only to add in them an intrinsic mag- netization equal to the inductive magnetization lost by the change. 720. As an illustration we may consider the familiar case of Ampere's electro-dynamic solenoid (505, foot-note), with a soft iron core; what is commonly called a bar electro -magnet. First, suppose there to be no soft iron core. We may do away with the current and substitute a uniformly and longitudinally magnetized bar of steel, with flat .ends, occupying the whole internal space of the cylinder. This will, at every external point, give the same resultant force as the solenoid; and its resultant force, according to the electro-magnetic definition, will throughout its substance be the same as the resultant force of the solenoid throughout the cylindrical space between planes cutting it perpendicularly through its ends. In the substance of the steel magnet, the resultant force, according to the polar definition, will ( 479) be merely the resultant of the force calculable from positive and negative planes of imaginary mag- netic matter coincident with its two ends ; and this is what would be the magnetizing force due to the intrinsic magnetiza- tion of the steel if ( 697) we attribute magnetic susceptibility to its substance ; without depriving it of its intrinsic magnetiza- tion. It is of very small amount except very near the ends of XL.] Intrinsic Magnetization substituted for Currents. 563 the bar, and is, throughout the interior, opposite in direction to the resultant force of the solenoid. To pass then from the case of a bar electro-magnet with core of soft iron or other substance susceptible of magnetic induction^ to an arrangement producing the same external effects with intrinsic magnetization of the core instead of electric currents round it ; we may first give to the core the intrinsic magnetization of the steel magnet we have just been considering, and superimpose upon this so much more of intrinsic magnetization as shall bring the whole magnetiza- tion of the core up to the resultant of the inductive magnetiza- tion which it has from the electric currents, and the uniform longitudinal magnetization which we attributed to the steel magnet. The core thus intrinsically magnetized and still retain- ing its magnetic susceptibility, will act the same upon all other magnets, and experience the same action from them, as the given electro- magnet. The same result may be also attained without attributing intrinsic magnetization to the core, in any case in which it is completely surrounded by matter of zero susceptibility ; as is the case with an ordinary bar electro- magnet or horse- shoe electro-magnet, unless its ends be con- nected by an armature of soft iron or other susceptible substance (the substance of the electric conductor being supposed to be of zero -magnetic susceptibility). For in any such case the substance of the magnetic shells may be placed altogether outside the core of the electro-magnet, by hollowing them so that they may pass clear of the core round either end of it ; or some of them round one end and some round the other so as to enclose the core among them. Then by supposing the sub- stance of the shells to be of zero inductive susceptibility, we have a system in which the core is inductively magnetized in virtue of the intrinsic magnetization of the shells, to precisely the same degree as it was under the influence of the electric currents. The external resultant force is the same as that of the electro-magnet, being composed of a constituent due to the shells which is the same as that due to the electric currents, and a constituent due to the magnetization of the core, identical in the two cases. 721. Supposing then electric currents done away with by the process of 719, we may simply take the data to be; at any 362 564 A Mathematical Theory of Magnetism. [XL. point (x, y, z), intrinsic magnetization (2, /3, 7), and inductive permeabilities CT, OT', TH" along principal inductive axes (I, m, ?i), (l r , m, n'), (I", m", w"). Thus (35) becomes where J, 3T, 5" denote the components along the principal inductive axes, of the resultant of Jf, C5r, pj. Hence for - in (40) we may put ( $ + ^ + 4?r ) S, and so for the other OT \ t7/ terms. Now by the elementary formula for transformation of rectangular components, we have and because (jf + X) dx ((ffif +^) c?2/ + (|^ + 5&) dz is a complete T . /v , i i - d Y dZ , dinerential and -^ h -^- + -7- = 0, we have dx dy dz r r r J coJ oo J Thus (53) becomes ' r r -nJnJ-ao This is one of the transformed expressions promised in 718. 722. To find the others, substitute for S, S f , S" their values by (59); and then remarking that, by the transformation of rectangular components, we find /co rco f J J Remarking that (Jf + X) dx + (^r + ^)% + (|^4-5S) dz is a complete differential, put Then integrating by parts in (64) as usual, we find i- J2 J'2 /d'^x"! ~F /3 + 47r(-H-^ 7 - + Ar (66); V OT CT W /J XL.] Formulae for Exhaustion of Energy. 565 where [as in 702 (2)] . /*f , <*0 dy\_ 1 (dff +a)v=r r r d^dzv^+a)..^^. J J QOJ -CO J CO./ -CO./ -CO Hence instead of (73) we may write ATF=r ^ \" dxdydzV^p + v) ......... (76). J aoJ coj -co 726. Consider now (part of the second member of this equa- tion) /oo /co /co III dxdydzVkv, Put in it [ 722 (71)] td* d&r, d^\ AJ = : - J ...(oU). , A , 47T 47T 4?r 727. Remembering ( 725) that A prefixed to any function of (x, y, z) denotes the augmentation which the function experi- ences when B is moved in any manner as a rigid body with its magnetization unchanged, while (80) expresses the actually varying inductive magnetization, we see that, throughout the volume of B, A/". . .(81), 47T where A x denotes augmentation produced by giving the actual motion to 5, and moving all other magnetized matter as if 568 A Mathematical Theory of Magmtism. [XL. rigidly connected with it, the axes of (x, y t z) being held fixed. Hence (79) is equal to _ _L f " p [ dxdydz 4<7T J _cc J _cc J oo I"" d*dydzl(ia-l)JW+(w'-l)J'* l J' + (rf'-l)J''A l J^ ...... (82), J W where CT must be regarded as equal to unity through all space except that occupied by B. Now using the notation of 730 (93), we have -j- -r- dx dy dz and rectangular transformation gives TA r r/A r 7//A r/' A A A /0/(N JA/+ J'A t J + J A/ = ^ A, Tx + ^ A, ^ + 2- A, -g. (84). Using these in the second term of (82) and performing integra- tions by parts, we reduce that term to m" , . r d dP d dP d dP 1 d*V &V . dxd y dz iT x ^ + ^^ + d- z ^- d ~z it ~ v ~~z dx dy dz By (94) and (74) this becomes simply f f f dxdydza\V. (86), J ccJ co J oo where a must be regarded as zero through all space except that occupied by B. 728. Now from the definitions of A and A, it follows that A 1 o- = AJCD J ao J co J co Substituting the second member of this equation for the second term of (82), and going back through (79) to (78) : then transposing and halving, we find r r r dxdydzv^=-~r r J tt> J tt> J oo J on J oc J XL.] Force experienced by any Part. 569 Finally, using this in (76), we find AJP= I" j^ r clxdydz^Ap-^^^w + J'^^' + J^^iff")]...^^. 729. Now to prove 718: let 8 denote variation due to any motion of B as a rigid body, the magnetization of every portion of matter varying (according to its actual susceptibility) with the varying magnetizing force to which it is subjected. The work required to effect the motion of B, being infinitesimal, will be the same as if (according to the hypothesis of 725) the actual magnetization were everywhere rigid. Hence if (& c denote the work undone in removing B to an infinite distance from all other bodies possessing either intrinsic mag- netization or magnetic susceptibility different from zero (that is to say, permeability differing from unity), and c a constant so far as the present variation is concerned [to be arbitrarily assigned later (731)], we have SeB = AJF. .......................... (91). 730. Taking the variation of (66), 722, we have I" f dxdydz(VSp + P SV)....(92), -.00 J 00 J 00 A 9 A' z A"* as the term of the triple integral depending on -- 1 -- r H -- 7r does not vary. Now putting p = -L ( CT JT 2 + v'J'* + v"J"*) ......... (93), OTT we have, by (15), d dP d dP d dP - d -j- dx dP d dP \ -^V + dz7dT] d-^r- d-r- I dy dz / Hence ddv dp r r r . r f 05 r , /dp dsv dp LLL y p LLL y fe^" + ^r \ dx dy dz As P is a quadratic function of -7- , -j~ , -v- , the expression under the integral sign here is clearly a symmetrical function . dV dV dV , dSV dSV dSV , of -j- , -j- , -T- , and -y , y , -y J and we may write dx dy' dz dx dy d z it thus : 570 A Mathematical Theory of Magnetism. [XL. dP dSV dP dSV dP dSV dx dV -j dj- -j dx dy dz P dV.dP dx dy 731. Taking the first triple term alone and performing inte- grations by parts, we have dx dy dz ^dx dy dz Hence (92) becomes = - j f f dxdydzVSp + ^f j j dxdydz(J*5iff + J'*l J-mJ-nJ-ta J -=Sp, ACT = CT, ACT' = SCT', and ACT" = SCT", we see that -SE=AW. (98), which proves 718. In virtue of (98), (66), and (91) we may put <& = r r p dxdydzVp (99). J -CO/ CO./ -00 By 566 we see that this implies assigning to c of 729 a value equal to the work which, after B has been removed to an infinite distance, must be undone to divide into infinitely thin bars every part of the system* possessing intrinsic magnetization and separate these bars to infinite distances from one another ; their directions having been so chosen that when uninfluenced the magnetism of each is longitudinal. Thus we see that the function expressed by (99) is the "mechanical value" of the given magnetic system, according to the definition of 567 extended to include material susceptible of magnetic induction along with intrinsically magnetized matter. It is essentially positive. Were there no magnetic susceptibility in any of the * Not omitting B though infinitely distant, if it has intrinsic magnetization. XL.] Force experienced by any Part. 571 material concerned, it would be identical with the <2B of 569, 570. By (66) we have /co /.co / QO . ,J2 A / 2 J"2\ t7^^(- + ^ 7 + ^ 77 )...(100). -00^-00^-00 \W Of ^ / Compare 717 (55), (58). For the particular case of zero sus- ceptibility (or unit permeability) throughout the system, (& and E have the same significations as in 569 above. 732. The expressions (62), (64), (66), (69), (70), (70) bis, for E, and (99) for (, depend on the exclusion of electric currents by which ( 721) we simplified the formula for magnetic induction; but as ( 719) this simplification did not involve any loss of generality, it is in reality proved that the con- figurational function E, expressed by the formula g 2 c,, 2 cw 2 . + f + ^) [(53> f 716 repeated] not involving the exclusion of electric currents, represents by its variations the forces experienced by detached portions of any system composed of intrinsically magnetized polar mag- nets, electromagnets, and inductively magnetized matter; thus: The augmentation of this function produced by any motion of a rigid portion or portions of such a system, through space occupied by matter of zero susceptibility, is equal to the work gained by permitting the motion. [Addition of date March 5th, 1884. The student is re- commended to exercise .himself by going through the whole investigation of 700 732, for the simple case of equal permeability in all directions. It will then be seen that the seeming difficulties of the investigation as given above, are merely mathematical complexities essential to the expression of the formulae concerned, when the matter is aeolotropic. In respect to "mechanical values" of magnetic and electro- magnetic systems, the investigation for the case of isotropic matter is to be found in Article LXI. ("On the Mechanical Values of Distributions of Electricity, Magnetism, and Gal- vanism ") of Vol. I., of my " Mathematical and Physical Papers." W. T.] XLI. HYDKOKINETIC ANALOGY FOB THE MAGNETIC INFLUENCE OF AN IDEAL EXTREME DIAMAGNETIC. On the Forces experienced by Solids immersed in a Moving Liquid. [From the Proceedings of the Royal Society of Edinburgh for Feb. 1870.] 733. Cyclic irrotational motion*, [V. M. 60 (#)] once esta- blished through an aperture or apertures, in a moveable solid immersed in a liquid, continues for ever after with circulation or circulations unchanged, [V. M. 60 (a)] however the solid he moved, or bent, and whatever influences there may be from other bodies. The solid, if rigid and left at rest, must clearly continue at rest relatively to the fluid surrounding it to an infinite dis- tance, provided there be no other solid within an infinite distance from it. But if there be any other solid or solids at rest within any finite distance from the first, there will be mutual forces between them, which, if not balanced by proper application of force, will cause them to move. The theory of the equilibrium of rigid bodies in these circumstances might be called Kinetico- statics; but it is in reality a branch of physical statics simply. For we know of no case of true statics in which some if not all of the forces are not due to motion; whether, as in the case of the hydrostatics of gases, thanks to Clausius and Maxwell, we perfectly understand the character of the motion, or, as in the statics of liquids and elastic solids, we only know that * The references [V. M. ] are to the author's paper on Vortex Motion, recently published in the Transactions of the Eoyal Society of Edinburgh (1869), which contains definitions of all the new terms used in the present article. Proofs of such of the propositions now enunciated as require proof are to be found in a continuation of that paper. [They are found in 759 763, below.] XLI.] Hydrokinetic Analogy for Extreme Diamagnetic. 573 some kind of molecular motion is essentially concerned. The theorems which I now propose to bring before the Royal So- ciety regarding the forces experienced by bodies mutually in- fluencing one another through the mediation of a moving liquid, though they are but theorems of abstract hydrokinetics, are of some interest in physics as illustrating the great question of the 18th and 19th centuries: Is action at a distance a reality, or is gravitation to be explained, as we now believe magnetic and electric forces must be, by action of intervening matter? 734. I. (Proposition.) Consider first a single fixed body with one or more apertures through it; as a particular example, a piece of straight tube open at each end. Let there be irrotational circulation of the fluid through one or more such apertures. It is readily proved [from V. M. 63, Exam. (2.)]* that the velocity of the fluid at any point in the neighbourhood agrees in magni- tude and direction with the resultant electro-magnetic force, at the corresponding point in the neighbourhood of an electro- magnet replacing the solid, constructed according to the fol- lowing specification. The "core" on which the conductor is wound, is to be of any material having extreme diamagnetic inductive capacity }-, and is to be of the same size and shape as the solid immersed in the fluid. The conductor is to form an infinitely thin layer or layers, with one circuit going round each aperture. The whole strength of current in each circuit reckoned in absolute electro-magnetic measure, is to be equal to the circulation of the fluid through that aperture divided by 4-7T. The resultant electro-magnetic force at any point will be numerically equal to the resultant fluid velocity at the cor- responding point in the hydrokinetic system. 735. Thus, considering, for example, the particular case of a straight tube open at each end, let the diameter be infinitely small in comparison with the length. The "circulation" will exceed by but an infinitely small quantity the product of the velocity within the tube into the length. In the neighbour- * Or from Helmholtz's original integration of the hydrokinetic equations. t Keal diamagnetic substances are, according to Faraday's very expressive language, relatively to lines of magnetic force, worse conductors than air. The ideal substance of extreme diamagnetic inductive capacity is a substance which completely sheds off lines of magnetic force, or which is perfectly imper- vious to magnetic force [or of zero "permeability," ( 629)]. 574 .4 Mathematical Theory of Magnetism. [XLL* hood of each end, at distances from it great in comparison with the diameter of the tube and short in comparison with the length, the stream lines will be straight lines radiating from the end. The velocity, outwards from one end and inwards towards the other, will therefore be inversely as the square of the distance from the end. Generally at all considerable distances from the ends, the distribution of fluid velocity will be the same as that of the magnetic force in the neighbourhood of an infinitely thin bar longitudinally magnetized uniformly from end to end. 736. Merely as regards the comparison between fluid velocity and resultant magnetic forces, Euler's fanciful theory of magnet- ism ( 573) is thus curiously illustrated. This comparison, which has been long known as part of the correlation between the mathematical theories of electricity, magnetism, conduction of heat, and hydrokinetics, is merely kinematical, not dynamical. When we pass, as we presently shall, to a strictly dynamical comparison relatively to the mutual force between two hard steel magnets, we shall find the same law of mutual action between two tubes, with liquid flowing through each, but with this remarkable difference, that the forces are opposite in the two cases; unlike poles attracting and like poles repelling in the magnetic system, while in the hydrokinetic analogue there is attraction between like ends and repulsion between unlike ends. 737. II. (Proposition.) Consider two or more fixed bodies, such as the one described in Prop. I. [ 734]. The mutual actions of two of these bodies are equal, but in opposite direction, to those between the corresponding electro-magnets. The particular instance referred to above shows us the remarkable result, that through fluid pressure we can have a system of mutual action, in which like attracts like with force varying inversely as the square of the distance. Thus, considering tubes open at each end, with fluid flowing through them, if the exit ends be placed in the neighbourhood of one another, and the entering ends be at infinite distances, the mutual forces resulting will be simply attractions according to this law. The lengths of the tubes on this supposition are infinitely great, and therefore, as is easily proved from the conservation of energy, the quantities flowing out per unit of time are but infinitesimally affected by the mutual influence. [When any change is allowed in the relative XLL] Hydrokinetio Analogy for Extreme Diamagnetic. 575 positions of two tubes by which work is done, a diminution of kinetic energy of the fluid is produced within the tubes, and at the same time an augmentation of its kinetic energy in the external space. The former is equal to double the work done ; the latter is equal to the work done ; and so the loss of kinetic energy from the whole liquid is simply equal to the work done.] 738. III. (Proposition.) Proposition II. holds, even if one of the bodies considered be merely a solid, with or without apertures; if with apertures, having no circulation through them. In such a case as this, the corresponding magnetic system consists of a magnet or electro-magnet, and a merely diamagnetic body, not itself a magnet, but disturbing the distribution of magnetic force around it by its diamagnetic influence. Thus, for example, a spherical solid at rest in the field of motion due to a fixed body through apertures in which there is cyclic irrotational motion, will experience from fluid pressure a resultant force through its centre equal and opposite to that experienced by a sphere of infinite diamagnetic capacity, similarly situated in the neigh- bourhood of the corresponding electro- magnet. Therefore, ac- cording to Faraday's law for the latter, and the comparison asserted in Prop. I. [ 734], it would experience a force from places of less towards places of greater fluid velocity, irrespectively of the direction of the stream lines in its neighbourhood; a result easily deduced from the elementary formula for fluid pressure in hydrokinetics. 739. I have long ago shown [ 646 above] that an elongated diamagnetic body in a uniform magnetic field tends, as tends an elongated ferromagnetic body, to place its length along the lines of force. Hence a long solid, pivoted on a fixed axis through its middle in a uniform stream of liquid, tends to place its length perpendicularly across the direction of motion ; a known result (Thomson and Tait's Natural Philosophy, 335). Again, two globes held in a uniform stream with the line join- ing their centres perpendicular to the stream, require force to prevent them from mutually approaching one another. In the magnetic analogue, two spheres of diamagnetic or ferromagnetic inductive capacity repel one another when held in a line at right angles to the lines of force. A hydrokinetic result similar 576 A Mathematical Theory of Magnetism. [XLI. to this applied to the case of two equal globes, is to be found in Thomson and Tait's Natural Philosophy, 332. ' 740. IV. (Proposition.) If the body considered in III., 738 [be an infinitely small globe*, and] be acted on by force applied so as always to balance the resultant of the fluid pressure, cal- culated for it according to II. and III. for whatever position it may come to at any time, and if it be influenced, besides, by any other system of applied forces, superimposed on the former, it will move just as it would move, under the influence of the latter system of forces alone, were the fluid at rest, except in so far as compelled to move by the body's own motion through it. A particular case of this proposition was first published many years ago, by Professor James Thomson, on account of which he gave the name of "vortex of free mobility" to the cyclic irrotational motion symmetrical round a straight axis. [Additional, Sept. 14, 1872. The same proposition holds for a globe of any dimensions, in a field of fluid motion consisting of circulation or circulations with infinitely fine rigid endless curve or curves for core, and no other rigid body in the liquid. Demonstration to appear in the Proceedings of the Royal Society of Edinburgh for 1871-2. f] Extracts from two Letters to Professor Frederick Guthrie. [From the Philosophical Magazine for June 1871.] GLASGOW, Nov. 14t7i, 1870. I HAVE to-day received the Proceedings of the Royal Society containing your paper " On Approach caused by Vibration," which I have read with great interest. The experiments you describe constitute very beautiful illustrations of the known theorem for fluid pressure in abstract hydrokinetics, with which I have been much occupied in mathematical investigations connected with vortex-motion. 741. According to this theorem, the average pressure at any point of an incompressible frictionless fluid originally at rest, * [The proposition as originally published without limitation is obviously false, although that it is so I have only perceived to-day. Sept. 2, 1872.] t Proceedings of the Royal Society of Edinburgh, March 4, 1872, XLI.] HydroUnetic Analogy for Extreme Diamagnetic. 577 but set in motion and kept in motion by solids moving to and fro, or whirling round in any manner, through a finite space of it, is equal to a constant diminished by the product of the density into half the square of the velocity. This immediately explains the attractions demonstrated in your experiments; for in each case the average square of velocity is greater on the side of the card nearest the tuning-fork than on the remote side. Hence obviously the card must be attracted by the fork as you have found it to be ; but it is not so easy at first sight to perceive that the square of the average velocity must be greater on the surfaces of the tuning-fork next to the card than on the remote portions of the vibrating surface. Your theoretical observation, however, that the attraction must be mutual, is beyond doubt valid, as we may convince ourselves by imagining, the stand which bears the tuning-fork and the card to be perfectly free to move through the fluid. If the card were attracted towards the tuning-fork, and there were not an equal and opposite force on the remainder of the whole surface of the tuning-fork and support, the whole system would commence moving, and continue moving with an accelerated velocity in the direction of the force acting on the card an impossible result. It might, indeed, be argued that this result is not impossible, as it might be said that the kinetic energy of the vibrations could gradually transform itself into kinetic energy of the solid mass moving through the fluid, and of the fluid escaping before and closing up behind the solid. But "common sense" almost suffices to put down such an argu- ment, and elementary mathematical theory, especially the theory of momentum in hydrokinetics explained in my article on " Vortex-motion," * negatives it. 742. The law of the attraction which you observed agrees per- fectly with the law of magnetic attraction in a certain ideal case which may be fully specified by the application of a principle explained in a short article [ 733... 740] communicated to the Royal Society of Edinburgh in February last [1870], as an abstract of an intended continuation of my paper on "Vortex-motion." Thus, if we take as an ideal tuning-fork two globes or disks * Transactions of the Royal Society of Edinburgh, read 29th April, 1867. T. E. 37 578 A Mathematical Theory of Magnetism. [XLI. moving rapidly to-and-fro in the line joining their centres, the corresponding magnet will be a bar with poles of the same name as its two ends and a double opposite pole in its middle. Again, the analogue of your paper disk is an equal and similar dia- magnetic of extreme diamagnetic inductive capacity [ 734]. The mutual force between the magnetic and the diamagnetic will be equal and opposite to the corresponding hydrokinetic force at each instant. To apply the analogy, we must suppose the magnet to gradually vary from maximum magnetization to zero, then through an equal and opposite magnetization back through zero to the primitive magnetization, and so on periodi- cally. The resultant of fluid pressure on the disk is not at each instant equal and opposite to the magnetic force at the corresponding instant, but the average resultant of the fluid pressure is equal to the average resultant of the magnetic force. Inasmuch as the force on the diamagnetic is generally repul- sion from the magnet, however the magnet be held, and is unaltered in amount by the reversal of the magnetization, it follows that the average resultant of the fluid pressure is an attraction on the whole towards the tuning-fork, into whatever position the tuning-fork be turned relatively to it. ... Nov. 23, 1870. 743. ... There are, no doubt, curiously close analogies between some of the circumstances of motion in contiguous fluids of different densities, and the distribution of magnetic force in a field occupied by substances of different inductive capacities. Thus, if in a great space occupied by frictionless incompressible liquid denser in some portions than in others, a solid be suddenly set in motion, the lines of the fluid motion first generated agree perfectly [compare 751... 763 below] with the permanent lines of magnetic force in a correspond- ingly heterogeneous medium under the influence of a bar- magnet, to be substituted for the moveable solid and placed with its magnetic axis in the line of the solid's motion. As to amounts, the fluid velocity multiplied into the density is simply equal to the resultant magnetic force at each point, if the particular definition [the "electromagnetic definition" (517, Postscript)] of the resultant magnetic force in a medium of XLL] Hydrokinetic Analogy for Extreme Diamagnetic. 579 heterogeneous inductive capacity, given in the foot-note to [ 516 above] 48 of my paper on the "Mathematical Theory of Magnetism,*" be adopted. But here the analogy ends; the rigidity in virtue of which a solid moveable in a fluid medium differing from it in magnetic inductive capacity keeps its form, does not exist [contrast 751 below] in the hydro- kinetic analogue. . . . Report of an Address on the Attractions and Repulsions due to Vibration, observed by Guthrie and Schellbach. [From the North British Daily Mail for Dec. 15, 1870 ; and Proceedings of the Philosophical Society of Glasgow for Dec. 14, 1870.] 744. The speaker began by stating that interesting papers had recently appeared in the Proceedings of the Royal Society and the Philosophical Magazine, by Professor Guthrie, in which some very curious hydrokinetic phenomena were described. From hints and suggestions in his paper, it seems that Prof. Guthrie connected in his own mind these phenomena with possibilities of explaining some of the more recondite actions in nature; and he (the speaker) believed that what gave the great charm to these investigations for Prof. Guthrie himself, and no doubt also for many of those who heard his expositions and saw his experiments, was, that the results belong to a class of phenomena to which we may hopefully look for discover- ing the mechanism of magnetic force, and possibly also the mechanism by which the forces of electricity and of gravity are transmitted. The speaker, however, did not lay any stress at present upon the possibility of applying these results directly to explain magnetism. He believed, on the contrary, that the true kinetic theory of magnetism (and the ultimate theory of magnetism is undoubtedly kinetic) [compare 290 and 546, foot-note above] involves quite a different class of motions from those to which the beautiful phenomena discovered by Prof. Guthrie are due. He rather wished to point out the close con- nexion that existed between the laws of some of these actions and the laws of magnetism, which, while involving some remark- * Philosophical Transactions, June 21, 1849. Published in Part I. for 1851. [ 504523 above.] _Q7_9 580 A Mathematical Theory of Magnetism, [XLI. able coincidences, involves certain contrasts decisive against any hypothesis, such as the ingenious one [ 573 above] of Euler, explaining magnetism \>y fluid motion directly comparable with that which forms the subject of the present communica- tion. - 745. One of the most brilliant steps made in philosophical exposition of which any instance existed in the history of science, was that [ 634 foot-note, and 643 above] in which Faraday stated, in three or four words, intensely full of meaning, the law of the magnetic attraction or repulsion experienced by inductively magnetized bodies. He pointed out that a small globe or cube of soft iron tended in a certain direction when free to move in the magnetic field ; while small detached frag- ments of inductively magnetized substances of the kind which he called diamagnetic, tended in the contrary direction ; and that the precise specification of the direction in which the diamagnetic tended "was from places of stronger to places of weaker force." 746. By means of diagrams, the speaker then showed the action of magnets upon small pieces of soft iron in various posi- tions, in the several cases in which the magnetic force is due to a bar-magnet, a horse-shoe magnet, and two bar- magnets placed side by side with their similar poles in the same direc- tion. A diagrammatic illustration of "the lines of magnetic force," in the case of a bar-magnet, was also given. In the case of the horse-shoe magnet, it was pointed out that the small globe of soft iron would have a position of stable equilibrium in the line joining the poles, if free to move in the horizontal line bisecting that line at right angles ; this stable position being the point of greatest force. The attraction experienced would be towards this point ; so that if the globe were placed inside this point that is to say, nearer the bend of the magnet it would seem to be repelled on the whole by the mass of steel while moving towards the place of strongest force. In the case of two bar-magnets placed side by side [ 645 above] with their similar poles in the same direction, it was pointed out that, for each pair of similar poles, there is a zero, or place of no force, mid-way between the two bars, and nearly in the line joining the ends. A globe of soft iron moveable midway between the XLI.] Hydrokinetic Analogy for Extreme Diamagnetic. 581 two bars is repelled, as it were, from each of the points of zero force, and finds a position of maximum force, which is one of stable equilibrium, on either side of either of the zeros. Fara- day's law [ 634, foot-note above] showed that the soft iron was attracted from places of weaker to places of stronger force, quite irrespectively of the directions of the lines of force and thus summed up a great variety of very curious and puzzling phenomena in one sentence. 747. This expression is perfectly applicable to small bodies at rest in an irrotationally moving fluid ; with the substitution of ''stream lines," instead of Faraday's "lines of magnetic force," and "greater or smaller fluid velocity," instead of " stronger or weaker magnetic force." 748. Mathematicians were content to investigate the general expression of the resultant force experienced by a globe of soft iron in all such cases ; but Faraday, without mathematics, divined the result of the mathematical investigation [ 638, 639, and 671... 681 above] ; and, what has proved of infinite value to the mathematicians themselves, he has given them an articulate language in which to express their results. Indeed, the whole language of the magnetic field and "lines of force" is Faraday's. It must be said for the mathematicians that they greedily accepted it, and have ever since been most zealous in using it to the best advantage. 749. Suppose a tube sunk in a perfect fluid, and the fluid by some means set to enter the one end and flow out by the other, the particles of it would follow the lines of magnetic force. The magnetic field of force in the neighbourhood of a bar-magnet corresponded exactly with the straight tube taking water in at one end and discharging it at the other. If two such tubes were presented with like ends to each other, they attracted, but with unlike ends, they repelled, thus acting differently from two magnets placed in similar relative positions. But, except in being precisely opposite in direction, the resul- tant action between the supposed tubes and that between two bar-magnets follows rigorously the same law, both as to magni- tude and as to line of action. This conclusion, and some others, containing the explanation of most of the experiments now to be shown to the Society, had been worked out mathe- 582 A Mathematical TJieory of Magnetism. [XLI. matically by the speaker, and communicated by him to the Royal Society of Edinburgh *. 750. It had been found by Faraday that the lines of magnetic force were diverted outwards from itself by a diamagnetic body placed in the field. If a body existed of extreme diamag- netic inductive capacity, the lines of magnetic force would pass altogether round it, and none of them through it. This is pre- cisely the phenomenon, with reference to stream lines, which is met with in the hydrokinetic analogue. The speaker then drew attention to some small egg-shells which were suspended so as to move freely, each in a horizontal circle. By slightly waving the hand in front of the egg-shells they were attracted, and the same phenomenon was produced by holding in their neighbour- hood a vibrating tuning-fork. This corresponded to the beha- viour of a diamagnetic in the magnetic field, only that the direction of the motion was opposite. By means of a very delicate anemometer it was shown that the phenomena were independent of currents of air. The speaker showed that in whatever position, with one exception, the fork was held, the attraction was produced. The magnetic analogue to this fork would be a non-magnetic frame substituted for the tuning- fork, and bearing two small magnets laid across the ends, with similar poles pointing towards each other. In this case there would be a zero point in the middle, between the near poles. The same is true of the fluid velocity in the case of the tuning- fork. It would repel the suspended egg-shells from the zero point; but the experiment was one of too great delicacy for a lecture- room. Some very interesting experiments upon flames had been made by Mr Tatlock, his assistant, which the speaker had much pleasure in showing to the Society. A vibrating fork was supported horizontally, and the flame of a candle brought near the vibrating ends. All that part of the flame on a level with the fork was repelled, and bent down in the oppo- site direction, as if by a current of air. On the vibration being stopped, the flame at once assumed its upright form. A tall flame, obtained from ordinary coal gas, was next brought into * Proceedings, Royal Society, Edinburgh, February 1870 [ 733740, above.] XLL] HydroMnetic Analogy for Extreme Diamagnetic. 583 proximity to the vibrating fork, when the middle part of the flame was drawn out towards the fork, the upper and lower parts being repelled. In concluding, the speaker remarked, that it would be very wrong if he were to say that these experiments on the hydrokinetic analogue contained a direct opening up of the question of the mechanism of magnetic forces. They did not go any way towards explaining magnetic forces ; but it was impossible to look upon them without feel- ing that they suggested the possibility of some very simple dynamical explanation. XLII. General Hydrokinetic Analogy for Induced Magnetism. February 1872. [Compare 743 above.] 751. Imagine an infinitely fine-grained porous solid per- meated by a frictionless incompressible liquid. The con- stitution of the supposed porous material will, for brevity, be designated as molecular, and although we might suppose it to depend on perforations in all directions, and every- where opening into one another all through a continuous rigid solid, it will generally be more convenient to imagine it as made up of two classes of constituents ; (1) small detached rigid particles or molecules, each somehow held absolutely at rest, unless we find it convenient to apply force to it and move it : (2) closed infinitely fine curves of solid matter. It will be convenient to suppose each molecule to be a ring (that is to say a solid with at least one perforation through it) ; or at all events to suppose a considerable proportion of the molecules through any finite portion of space to be annular. This sup- position gives the foundation ( 573... 583 above) for the hydro- kinetic analogue to a permanent polar magnet. Thus ( 574) cyclic irrotational motion ["Vortex Motion," 59 (/) and 60 (#)*] through an infinitesimal solid ring constitutes a perfect analogy for an infinitely small portion of a permanent polar magnet. Again, when the kinematic analogy for a linear closed current ( 535 above) is desired, we shall suppose an infinitely fine closed curve, which to avoid circumlocution I shall call an ityoid (Proceedings, Royal Society of Edinburgh, Dec. 18, 1871), of solid material to be placed, threading through among the inter- stices of the molecules and everywhere infinitely near the line of the electric current, but not in any case passing through the perforation of an annular molecule. By using a temporary membrane drawn across such an ityoid (" Vortex Motion," 62) * Transactions, Royal Society of Edinburgh, April, 1867 and Dec. 1869. XLII.] Permeability in Hydrokinetic Analogy. 585 to generate cyclic irrotational motion, with no circulation through any other aperture than that of the ityoid itself, a per- fect hydrokinetic analogue to the electro-magnetic effect of a fixed linear current of constant strength is obtained. An infi- nite number of ityoids placed infinitely near one another, no- where in contact, but everywhere leaving sufficient interstices for the liquid to flow among them, gives the foundation for the hydrokinetic analogue to a solid electro-magnet ( 535 above). 752. Let any cylindrical or prismatic portion of the supposed porous solid, terminated by planes perpendicular to the cylin- drical surface or sides, be fixed in a tube of impermeable mate- rial fitting close to it all round, but leaving its ends free. This porous plug will constitute an obstruction, but not an absolute barrier, against the flow of a liquid through the tube. Imagine now two perfectly fitting frictionless pistons to be placed on the tube at any distance on the two sides of the plug, and let the whole space bounded by the pistons, the tube, and the im- permeable constituents of the porous solid, be occupied by frictionless incompressible liquid. Let the liquid be set in motion by force applied to either or both the pistons. The motion will be determinate in every part of the fluid according to the condition [Thomson and Tait's Natural Philosophy, 317, Example (3)] that the kinetic energy is less than that of any other motion of the liquid consistent with the given motion of the pistons. If the lengths of the clear portions of tube between the pistons and the two ends of the obstructing plug be very great in comparison with the diameter of the tube, it is easily seen that however coarse or heterogeneous be the porous mate- rial, the motion of the liquid will be sensibly uniform and in parallel lines through all the distant parts of the tube. But if the porous material be infinitely fine-grained and homogeneous as to the average structure of all equal and similar finite por- tions, the motion of the liquid will be uniform and in parallel lines at all finite distances on each side of the plug. If, as an extreme case, the plug be a continuous solid, with an infinite number of infinitely fine cylindrical perforations parallel to its length, the velocity of the liquid through it would be uniform, and would be to the velocity through the clear portions of the tube, in the inverse ratio of the areas traversed, that is to say, 586 A Mathematical Theory of Magnetism. [XLII. in the ratio of the sectional area of the clear tube to the sum of the sectional areas of the perforations. The mass of the fluid in the perforations at any instant, would be to the mass in an equal length of the clear tube, as the sectional area of the tube to the sum of the sectional areas of the perforations ; and therefore the kinetic energy of the whole motion in the per- forations would be to the kinetic energy in an equal length of the clear tube, in the inverse ratio of the areas, that is to say, in the ratio of the whole sectional area of the tube to the sum of the sectional areas of the perforations. Hence, generally the greater the obstruction offered by a plug consisting of any kind of porous material, the greater will be the ratio of the kinetic energy of the liquid permeating through it, to that of the liquid moving freely in an equal length of clear tube ; and (borrowing the word "permeability" from Le Sage), we may say that the permeability of the plug is inversely as the kinetic energy of the liquid permeating through it, when the velocity of the fluid in the clear parts of the tube is given. 753. If we were only occupied with hydrokinetics it would be natural to call the permeability of the clear parts of the tube unity. This would make unity the measure of perfect permea- bility, and would give always a proper fraction for the measure of the permeability of a porous solid. But in view of the magnetic analogy it is more convenient to call the permeability of some particular porous material unity, and to define the permeability of any other material as the number by which we must multiply the kinetic energy of the fluid permeating through a plug of it, to find the kinetic energy in a plug of equal length of the standard material fixed in the same tube. And further, for the magnetic analogy (compare 732 above) it is convenient to attribute to the supposed liquid such a density that 4-7T times the kinetic energy of liquid permeating a solid of unit permeability, reckoned per unit volume of the whole space occupied by porous solid and liquid shall be equal to half the square of the "flux;" the word flux being borrowed from Fourier's theory of the conduction of heat and adapted to the use we have to make of it by the following definition : 754. The component flux in any direction is the whole volume of the liquid traversing a plane perpendicular to this direction XLII.] Permeability in Hydrokinetic Analogy. 587 per unit of area per unit of time. In the complicated motion of the liquid through the interstices of the porous solid, the com- ponent velocity perpendicular to any plane may be in contrary directions at different points of the plane ; but in reckoning the flux we must take the excess (positive or negative) of the quantity crossing in the direction called positive above that which crosses in the direction called negative. By considering a tetrahedral portion of space (whether clear or occupied by porous solid) bounded by three mutually rectangular planes and a fourth plane cutting them all, we see immediately that the composition of fluxes follows the ordinary law of the com- position of velocities or the composition of forces ; an elemen- tary proposition due to Fourier. 755. Let X, Y, Z denote, for any possible motion of the liquid, the components of flux at any point (x, y, z) referred to rectangular co-ordinates. X t Y, Z must ( 540 above) fulfil the equation dX dY dZ _ n .... fa + ~fy + ~fa~ () ............ Wi called the "equation of continuity." 756. In general the permeability of a porous solid may be supposed to be different in different directions. When it is so the structure is of course to be called aeolotropic (Thomson and Tait's Natural Philosophy, 676; quoted above, 604, foot- note). Still denoting by X, Y, Z the components of flux in three directions at right angles to one another, denote by Q the kinetic energy per unit of volume, which must be a quadratic function of X, Y, Z. Hence, by the ordinary analysis of quad- ratic functions, we see that there are three determinate direc- tions (I, m, ri), (l r , m' t ri} y (I", ra", n"), at right angles to one another, to be called (according to analogy of ordinary usage) the principal axes of permeability, and three determinate con- stants OT, -a/, &" to be called the principal permeabilities, in terms of which we have the following expression for Q : 1 t(lX+mY+nZ)* (VX+m'Y+riZ)* (l"X + m"Y+ri'Z)* Q= 8-*\ -- JT w~ ~w -- 757. Now let us suppose the whole of space to be occupied by a rigid porous solid of infinitely fine-grained texture with different degrees of permeability and seolo tropic quality in 588 A Mathematical Theory of Magnetism. [XLII. different parts; and let a frictionless incompressible liquid initially at rest fill all the interstices. In a portion M of the porous solid (to represent the " inducing magnet " in the mag- netic analogue), let some of the constituent molecules be an- nular, and let the apertures of some of the rings be temporarily closed by infinitely thin flexible and extensible membranes. (It is a matter of indifference whether there be other rings or not either in M or elsewhere.) Let impulsive pressure be applied to these membranes, uniform on each, but not neces- sarily of equal values for the different membranes ; and in- stantly let all the membranes be dissolved. The motion of the fluid will be everywhere irrotational and determinate [" Vortex Motion," 62 and 62 (c)*], and will be of the class called polycyclic [" Yortex Motion," 60 (x) *]. The kinetic energy of the whole fluid motion produced will [Thomson and Tait's Natural Philosophy^ 3\7 Example ($)\ be less than that of any other motion consistent with the incompressibility of the fluid, having the same normal component velocity at each point of the supposed membrane surfaces. A partial application of the same theorem shows that if we leave out of account the fluid motion within any surface 8, completely enclosing M, and consider the normal component velocity as given at each point of this sur- face, the kinetic energy of the fluid motion through the rest of space will be less than that of any other motion with the same normal component velocity at each point of S. 758. To find the analytical expression of this condition let fffdxdydz denote integration through all space except that enclosed by 8. Then X, Y, ^must, subject to equation (1), be such functions of (a?, T/, z) as to make fffQdzdyd* a minimum. Hence, X denoting an indeterminate multiplier, we have + -<> ...... (3). Applying the usual process of integration by parts to the terms involving X, we find SfffdxdydzX ~ + ~+= * Transactions, Royal Society of Edinburgh, April 1867 and Dec. 1869. XLII.] Kinetic Energy a Minimum. 589 where fJdS denotes integration over the whole bounding surface of the space included in the triple integral, and I, m, n are the direction-cosines of the normal. For the infinitely distant parts of the boundary the double integral vanishes, as by hypo- thesis there is no motion there ; and for the boundary of M (which is the remainder of the boundary of the space included in the triple integral) the double integral vanishes, because the condition that the normal component velocity is given over the boundary of M, requires that Hence as Q involves only X, F, Z, and not their differential coefficients, the variation al equation (3) gives dQ_d\ dQ L _d\ dQ_d\ dX~dx> dY~dy> dZ~d~z" These equations, with (1) and (2), 755, 756, and lX + mY+nZ=N .. ................... (5), for every point of the boundary of M, where N denotes the given normal component velocity, suffice to determine X, F, Z for every point of space external to M. Comparing them with equations (43), (42), and (40) of 713 above, we see that they are simply the equations of the magnetic induction through space external to M, due to any distribution . of magnetization or of electric currents within M\ if zr, -zxr', is" be the three principal magnetic permeabilities, and (I, m } n), (I, m', ri} } (I", m", n") the principal axes at any point (x, y, z}\ X, F, Z the components of the resultant force at the same point according to the electro- magnetic definition ; and N its normal component at any point of a surface M t which completely encloses the inducing magnet. 759. Considering next the fluid motion within the space M, and its electro-magnetic analogue, we see from equations (42) of 713 above, that d_dQ__d_dQ^ d^dQ__d^dQ d dQ d dQ TyTZ fa~dY> dzdX dxTZ* fadY~fydX (b) > where they are not zero are equal to the component intensities of the electric flow ( 539 above), at (a?, y, z\ in a determinate distribution of electric currents, which, with the magnetism induced by it throughout space, produces resultant electro- 590 A Mathematical Theory of Magnetism. [XLII. magnetic force (X, Y, Z) at any point (x, y, z). Suppose now any motion to be given ( 751 above) to solid material in space external to M, or any cyclic irrotational motion of the liquid to be generated by the aid of membranes temporarily stopping aper- tures of solids in the space external to M; this will alter the motion already existing by compounding with it the motion which the supposed actions external to M would produce of themselves in the liquid if given motionless. Now from (4) it follows that throughout M, the values of the functions (6) are zero for the second supposed component of the motion. Hence, throughout M the functions (6) being linear functions of the flux components, remain unchanged in the altered motion of the liquid. It follows that their values through any portion of space, throughout which the molecular constitution of the solid matter is completely given, are determinable from the cyclic constants of the fluid motion through all the rings in this part of space, independently of the molecular constitution, or of circulations through apertures in other parts of space. From this, lastly, we see that if M be moved in any manner, transla- tionally or rotationally, with all its parts kept rigidly connected, and the axes of co-ordinates moving along with it, and if it be brought to rest in an altered position, the values of the functions (6) will be the same as they were before the motion. This motion of M as a rigid body implies, of course, motions and changes of molecular arrangement in the solid matter of sur- rounding space which are altogether arbitrary, subject only to the condition of making way for M. 760. The analogy may be further extended to include the re- sultant force experienced by the inducing magnet, or by any moveable solid portion of matter experiencing its inductive in- fluence. To do this, consider the effect of any variation of the solid matter concerned in the hydrokinetic analogue. First, it must be remarked that the effect of the change in the molecular distribution of the solid matter in the space M upon the motion of the fluid, cannot be determined from mere knowledge of the change which it produces in that average quality of the material which I have defined above ( 752) as its permeability. For without changing the permeability we may so alter the molecu- lar arrangement within M as to change to any degree we please XLIL] Analogy of Force. 591 the flux of the fluid in this space, and therefore also the fluid motion through space external to M. Conceive, for instance, an infinitesimal molecular change to be produced which, without altering the "permeability" of the group, shall very much contract infinitesimal apertures through which there is circulation. This may be done either by altering the shapes of infinitesimal molecular rings, or by bringing other molecules towards the apertures of rings so as to obstruct passage through them. The circulation through each aperture remains ("Yortex Motion," 59*) constant, but it is clear that the whole kinetic energy may be diminished as much as we please by the sup- posed process. 761. Let now A denote the solid matter in any portion of space which may be either the whole of M or altogether external to M. Let the permeability outside of A be uniform through some finite space all round it. Keeping A rigid throughout, alter its position infinitesimally ; keep the permeability un- changed in the space immediately contiguous with it, by forces applied to surrounding molecules obliged to give way to it during its motion; and keep all other portions of solid matter in external space rigidly connected with one another. The work done by forces applied to A and the surrounding mole- cules to produce their supposed motions must be equal to the CO /- CO CO augmentation experienced by the integral I I I Qdxdydz. J 00 J - CO J CO This is the same as the amount of work required to give the corresponding motion to the portion of matter corresponding to A in the magnetic analogue ; a consequence of 731 above, with the consideration that both in the hydrokinetic system , ,, x , f d dQ d dQ , and the magnetic analogue, the values ot -y- ~^y T- -TT^ ? etc., are ( 759 above) not altered by the supposed change of A's position. 762. The necessarily complicated character of the dynamical action required to produce the supposed motion of A and re- arrangement of the surrounding molecules disappears altogether in the case in which a finite shell of space contiguous with A all * Transactions, Royal Society of Edinburgh, April 1867 and Dec. 1869. 592 A Mathematical Theory of Magnetism. [XLII. round is free from solid molecules. In this case the (general- ized) component forces required to give any infinitesimal motion whatever to A (compare 502 above), will be simply the differen- tial co-efficients of Q with reference to the corresponding co-ordinates ; and the forces required to balance A in any posi- tion, will be equal and opposite to these forces. Hence the force required to balance A in this case of the hydrokinetic system will be equal and opposite to the force required to bal- ance a rigid body corresponding to A in the magnetic analogue. In the latter, the analogue to the space round A, clear of solids, but traversed by liquid, may [notwithstanding the different convention ( 753 above) more generally adopted] be air. This particular convention being adopted for an instant, the magnetic analogue for all portions of space occupied by the "porous solid," described in 751 above, or by continuous finite solid substance, will be diamagnetic material of any permeability from unity (that of air) to zero (that of ideal substance of ex- treme diamagnetic quality). The analogue of M may be either a real ordinary electro-magnet consisting of an electric current, or distribution of currents through solid conductors of diamag- netic material ; or an ideal polar-magnet ( 697 above) of dia- magnetic inductive quality. But it is to be remarked that by choosing air for the magnetic analogue of space unobstructed by solids in the hydrokinetic system, we exclude all ferro-magnetic induction from the analogy. 763. Using now the general proposition of 761, and making the proper particular suppositions regarding the moveable body A, we not only prove Propositions II. and III. of 737, 738 above, but extend their application to real bodies of any degrees of diamagnetic inductive capacity instead of the ideal bodies of "extreme" diamagnetic quality (zero magnetic permeability) imagined in those propositions. PI ate 3. INDEX. ACCUMULATOR, uniform current, 408- 411 Action of a small plane closed circuit on an element of another complete electro-magnet or magnet, 546 JSolotropic, 604, foot-note Analogy, Hydrokinetic, 573-583, 733-763 Atmospheric Electricity, early observers of, 267 method of observing, 262- 266 new apparatus for observing, 391 Notes on, 392-399 Observations on, 296-300 on the necessity for inces- sant recording, and for simultaneous observations in different localities to investigate, 295 Atoms, size of, 400 Attractions and repulsions due to vi- bration observed by Guthrie and Schellbach, Beport of an address on the, 744 Attraction of a uniform spherical sur- face on an external point, 87 propositions in the theory of, 187-205 CAPACITY of conductors, 51-56 Cavendish, 34, foot-note ratio of the capacity of a disc to that of a sphere of the same diame- ter, 235, foot-note Certain partial differential equations, theorems with reference to the solu- tion of, 206 Coercive force, 609, 630 Collector, water dropping, 262, 266, 287 burning match, 261, 286 Condenser, sound produced by the dis- charge of a, 302 Conducting and non-conducting elec- trified bodies, on the attractions of, 144, 148 sphere, determination of distri- bution on a, 77 T. E. Conducting surfaces external and in- ternal, 97 Conductors, insulated, 71 of electricity, 68 Conditions to which the distributions of galvanism in solid and superficial electromagnets is subject, investiga- tion of, 539-546 Cone, area of segment cut from a spherical surface by a small, 86 orthogonal and oblique sections of a small, 85 the solid angle of a, 81 Cones, definitions regarding, 80 Contact electricity, new proof of, 400 Coulomb's experiments, 25 Crystalline and non-crystalline bodies, theory of magnetic induction in, 604-624 Cyclic irrotational motion, 733 DENSITY, electric, 330 Diamagnetics, repulsion of, 643- 646 Diamagnetic particles, reciprocal action of, 695, 696 Dielectric, 36, 447 Dip, line of, 441 Distribution of electricity on a circular segment of a sphere, 231-248 Distribution of electricity, mechanical value of, 695, 696 " of magnetic matter necessary to represent the polarity of a given magnet, 473, 474 Distributions of magnetism, solenoidal and lamellar, 504-523 of matter, mechanical value of, 561-563 ELECTRICITY, atmospheric, 249-301 on the elementary laws of statical, 25-50 conductors of, 68 non-conductors of, 67 of a charged conductor rests en- tirely on its surface, 68 two kinds of, 58 594 Index. Electric current, strength of, 532 accumulator, on a uniform, 408-411 equilibrium, 66 machines founded on induction and convection, 416-425 Electrical density at any point of a charged surface, 69, 93, 138 forces, superposition of, 63 influence on an internal spherical conducting surface, 102-105 on a plane conducting sur- face of infinite extent, 106-112 quantity, 61 Electrification of the atmosphere, what is known regarding the, 253, 296- 301 how experiments may be made for ascertaining the, 254- 262 Electrified bodies, law of force between, 64 surface, repulsion on an element of an, 88 spherical conductors, mutual at- traction or repulsion between two, 128-142 Electrometers and electrostatical mea- surements, Keport on, 341-390 classification of, 343-385 Electrometer, definition of, 341 absolute, 307-309, 339, 358, 363 new absolute, 364-367 divided ring, 263-270, 345-357 electro scopic, 305, foot-note long range, 383, 384 standard, 379-382 portable, 263, 277, 368-378 Electromagnet, definition of, 434 Electromagnets, 524-554 linear, 536 superficial, 537 solid, 538 Electromotive force required to pro- duce a spark in air between parallel metal plates at different distances, measurement of, 320-340 Electroscope, Bennet's gold-leaf, 387 Bohnenberger's modification of, 388 Electrostatic force and variations of electric potential, relations between 337 produced by a Daniell's bat- tery, measurement of the, 305- 319 Electrophorus, reciprocal, 427 Elements, division of surfaces into, 79 Equilibrium, electric, 66 Ellipsoid, attraction of a homogeneous, on a point within or without it, 21-24 Ellipsoid, uniform motion of heat in an, 11-20 FAEADAY'S researches, 27 on electrostatic induction, 36 etc. on specific inductive capacity, 46, etc. Law, experimental illustrations of, 654-664 deduced, from the law of energy, 674-687, 745-750 Ferromagnetic and diamagnetic mag- netization, relations of, to the mag- netizing force, 664-668 Ferromagnetics, attraction of, 634- 642 Field of magnetic force, or field of force, 605 Force at a point due to a magnet, 605 analogy of, 760-763 ^ Forces experienced by inductively mag- netized ferromagnetic or diamagnetic non-crystalline substances, remarks on, 647-653 by matter under magnetic influence, 723-732 by solids immersed in a moving liquid, 733, etc. "Frequency" electric, 294 GALVANOMETER, 341 mirror, 350 Gauss, 187-481 Geometrical slide, 346 Green, essay on the application of mathematical analysis to the theo- ries of electricity and magnetism, 25, 156, 163, 167, 481 potential at a point, 37, foot-note quotation from, on some experi- ments by Coulomb, 234 Guthrie, Professor, extracts from let- ters to, 741-743 HARRIS on the law of electric force, examination of, 26 Heat, uniform motion of, 1-24 Heterostatic electrometers, 385 Holtz's electrical machine, 429 IDIOSTATIC electrometers, 385 Images, electric, 127, 208-230 Imaginary electrical points, 116 magnetic matter, 463-475 Induced magnetism in a plate, 156- 162 Induction, magnetic, 604, 624 plate, 357 Index. 595 Inductive action, curved lines of, 39 capacity of a substance, principal, 611 Inductively magnetized bodies in posi- tions of equilibrium, on the stability of, 665 ferromagnetic or diamag- netic non-crystalline substances, re- marks on the forces experienced by, 647-653 Insulated sphere subjected to the in- fluence of an electrical point, 89- 95 Inverse problems of magnetism, 584- 601 Intensity of magnetization, 461, 462 Isothermal surface, 1 Isotropic, 604, foot-note LAPLACE, 481 Lamellar distribution of magnetism, characteristic of, 514 Laws of statical electricity, on the ele- mentary, 25 of magnetic forces, 452-453 Lame's Memoir on Isothermal Sur- faces, 20 Lettres de M. William Thomson, A.M., Liouville, extraits de, 208-220 Lines of electric force, 39, 251, 256 of magnetic force, 605 of force, diagrams of, 632, 633 Liouville, sur un proprie'te' de la couche electrique en equilibre a la surface d'un corp conducteur, 163 note on the subject of electric images, 221, 230 Lightning, on some remarkable effects of, observed in a farmhouse near Monimail, 301 Ley den phial, capacity of a, 51, etc. MAGNET, definition of a, 434 Magnetic agency of the earth on a magnet, 438 axis, 440, 494 centre, 494 field, 605 force at any point, total, 605 the characteristic of mag- netism, 432, 433 axioms of, 606 induction, determination of the conditions of, 610 general problem of, 700- 732 laws of, 607 Magnetic induction, a principal axis of, 611 inductions, superposition of, 607 Magnetic moment, 458-460 polarity, 443-447 shell, 506-512 solenoid, 505, 507, 509, 511 strength, 454-456 susceptibility, 610 permeability, 628 analogues of, 625-631, 751-756 Magnetism, mathematical theory of, 430, etc. Magnetization, direction of, 462 intensity of, 461 intrinsic, 698 Magnetized matter, mutual actions be- tween any given portions of, 476- 501 Mathematical theory of electricity, actual progress in the, 74 of electricity, objects of the, 73 Measurement by electrometer, inter- pretation of, 336 Mechanical theory of electricity, de- monstration of a fundamental pro- position in the, 149-155 value of a distribution of electri- city on a group of insulated con- ductors, 138 Mouse-mill replenisher, 426 Mutual action between two magnets consists of a force and a couple, 496-501 between two magnets ex- pressed in terms of a function of their relative position, 502-505 NICHOLSON'S revolving doubler, 429 OERSTED, 524 PLANE conducting surface, electrical in- fluence on a, 106-112 Pliicker's hypothesis, 666 Polar magnet, 549 inductive suceptibility of a, 697-699 mechanical values of, 564-572 Polarity, 443 Poles of a magnet, 443, 549 Poisson, Memoirs of, on the mathema- tical theory of electricity, 25 theory of magnetic induction, 604 quotations from, regarding mag- necrystallic action, with explana- tions, 620, 621 Potential at a point, 37, foot-note at any point in the neighbour- hood of or within an electrified body, 129, foot-note 596 Index. Potential, electric, 335 of a magnetic shell at any point, 512 of a closed galvanic circuit of any form, 555-560 Potentials, equality and difference of, 249, foot-note Potential-Equalizer, 422-426 Proof plane, 25, foot-note, 35, 330 QUANTITIES of electricity, measurement of, 328 REPLENISHED 352,418-421,427-429 Resultant electric force at a point, de- finition of, 65 due to a uniform sphe- rical shell, vanishes for any interior point, 78 at any point in an in- sulating fluid, 331 magnetic force at any point, 479-515 SIZE of Atoms, 400 Specific inductive capacity, 45, etc. Spherical conductors, geometrical in- vestigations with reference to the distribution of electricity on, 75, etc. geometrical investigations regarding, 113-127 conducting surface, electrical in- fluence on an internal point of a, 102 surfaces of which the density va- ries inversely as the cube of the distance from a given point, attrac- tion of, 90 Solenoidal distribution of magnetism, characteristic of, 513 Statement of the principles on which the mathematical theory of elec- tricity is founded, 57, etc. Stratum of air between two parallel or nearly parallel plane or curved me- tallic surfaces maintained at differ- ent potentials, 338 Strength of electric current, 532 Superficial density of magnetic matter, 471, 472 "Surface of the earth," definition of, 250; generally negatively electrified, 252 TELEGEAPH wire insulated in the axis of a cylindrical conducting sheath, electrostatic capacity of, 54, etc. Terrestrial electrification, extremely rapid variations of, 259 magnetism, on the electric cur- rents by which the phenomena of, may be produced, 602, 603 Thalln, magnetic susceptibility of iron, 630 The earth, a great magnet, 436 The earth's action on a magnet, sen- sibly a couple, 439, 442 Theory of electricity, on certain defi- nite integrals suggested by problems in the, 166-185 of magnetic force, elementary demonstrations of propositions in the, 669 Tyndall, Professor, correspondence with, 694-696 UNIT strength, 647, foot-note VARLET'S instrument for generating electricity, 428 Volta connection by flame, 412-415 CAMBRIDGE: PRINTED BY c. j. 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