George Green's paper

arXiv:0807.0088v1 [physics.hist-ph] 1 Jul 2008An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism.∗ By George Green Fellow of Gonville- and Cains-Colleges at Cambridge. MSC-Class: 31-3 01A50 Preface. After I had composed the following Essay, I naturally felt an xious to be- come acquainted with what had been effected by former writer s on the same subject, and, had it been practicable, I should have bee n glad to have given, in this place, an historical sketch of its progress; m y limited sources of information, however, will by no means permit me to do so; b ut prob- ably I may here be allowed to make one or two observations on th e few works which have fallen in my way, more particularly as an opp ortunity will thus offer itself, of noticing an excellent paper, pres ented to the Royal Society by one of the most illustrious members of that learne d body, which appears to have attracted little attention, but which, on ex amination, will be found not unworthy the man who was able to lay the foundatio ns of pneumatic chymistry, and to discover that water, far from be ing according to the opinions then received, an elementary substance, was a compound of two of the most important gasses in nature. It is almost needless to say the author just alluded to is the c elebrated CAVENDISH , who, having confined himself to such simple methods, as may readily be understood by any one possessed of an elementa ry knowl- edge of geometry and fluxions, has rendered his paper accessi ble to a great number of readers; and although, from subsequent remarks, h e appears dissatisfied with an hypothesis which enabled him to draw some impor- tant conclusions, it will readily be perceived, on an attent ive perusal of his paper, that a trifling alteration will suffice to render the who le perfectly legitimate†. ∗Originally published as book in Nottingham, 1828. Reprinte d in three parts in Jour- nal f ¨ ur die reine und angewandte Mathematik Vol. 39, 1 (1850 ) p. 73–89, Vol. 44, 4 (1852) p. 356–74, and Vol. 47, 3 (1854) p. 161–221. From there transc ribed by Ralf Stephan, eMail: mailto:[email protected] †In order to make this quite clear, let us select one of C AVENDISH ’s propositions, the twentieth for instance, and examine with some attention the method there employed. The object of this proposition is to show, that when two similar c onducting bodies communicate 2 Preface. Little appears to have been effected in the mathematical the ory of elec- tricity, except immediate deductions from known formulae, that first pre- sented themselves in researches on the figure of the earth, of w hich the principal are, — the determination of the law of the electric density on the surfaces of conducting bodies differing little from a spher e, and on those of ellipsoids, from 1771, the date of C AVENDISH ’s paper, until about 1812, when M. P OISSON presented to the French Institute two memoirs of singu- lar elegance, relative to the distribution of electricity o n the surfaces of con- ducting spheres, previously electrified and put in presence o f each other. It would be quite impossible to give any idea of them here: to be d uly appre- ciated they must he read. It will therefore only be remarked, that they are in fact founded upon the consideration of what have, in this E ssay, been termed potential functions, and by means of an equation in va riable differ- ences, which may immediately be obtained from the one given i n our tenth article, serving to express the relation between the two pot ential functions arising from any spherical surface, the author deduces the v alues of these functions belonging to each of the two spheres under conside ration, and thence the general expression of the electric density on the surface of ei- ther, together with their actions on any exterior point. by means of a long slender canal, and are charged with electri city, the respective quantities of redundant fluid contained in them, will be proportional to then−1 power of their cor- responding diameters: supposing the electric repulsion to vary inversely as the npower of the distance. This is proved by considering the canal as cyli ndrical, and filled with incom- pressible fluid of uniform density: then the quantities of el ectricity in the interior of the two bodies are determined by a very simple geometrical construc tion, so that the total action exerted on the whole canal by one of them, shall exactly balan ce that arising from the other; and from some remarks in the 27thproposition, it appears the results thus obtained, agree very well with experiments in which real canals are employed , whether they are straight or crooked, provided, as has since been shown by C OULOMB ,nis equal to two. The author however confesses he is by no means able to demonstrate this, although, as we shall see immediately, it may very easily be deduced from the proposit ions contained in this paper. For this purpose, let us conceive an incompressible fluid of u niform density, whose parti- cles do not act on each other, but which are subject to the same actions from all the electricity in their vicinity, as real electric fluid of like density woul d be; then supposing an infinitely thin canal of this hypothetical fluid, whose perpendicular s ections are all equal and similar, to pass from a point aon the surface of one of the bodies, through a portion of its ma ss, along the interior of the real canal, and through a part of the other body, so as to reach a point Aon its surface, and then proceed from Atoain a right line, forming thus a closed circuit, it is evident from the principles of hydrostatics, and may be proved from our au- thor’s 23dproposition, that the whole of the hypothetical canal will b e in equilibrium, and as every particle of the portion contained within the system is necessarily so, the rectilinear portion aAmust therefore be in equilibrium. This simple consideratio n serves to complete CAVENDISH ’s demonstration, whatever may be the form or thickness of th e real canal, pro- vided the quantity of electricity in it is very small compare d with that contained in the bodies. An analagous application of it will render the demon stration of the 22dproposi- tion complete, when the two coatings of the glass plate commu nicate with their respective conducting bodies, by fine metallic wires of any form. Preface. 3 I am not aware of any material accessions to the theory of elec tricity, strictly so called, except those before noticed; but since t he electric and magnetic fluids are subject to one common law of action, and th eir the- ory, considered in a mathematical point of view, consists me rely in devel- oping the consequences which flow from this law, modified only b y con- siderations arising from the peculiar constitution of natu ral bodies with respect to these two kinds of fluid, it is evident, the mathema tical theory of the latter must be very intimately connected with that of the former; nev- ertheless, because it is here necessary to consider bodies a s formed of an immense number of insulated particles, all acting upon each other mutu- ally, it is easy to conceive that superior difficulties must, o n this account, present themselves, and indeed, until within the last four o r five years, no successful attempt to overcome them had been published. For this farther extension of the domain of analysis, we are again indebted to M. P OISSON , who has already furnished us with three memoirs on magnetism : the two first contain the general equations on which the magnetic stat e of a body depends, whatever may be its form, together with their compl ete solution in case the body under consideration is a hollow spherical sh ell, of uni- form thickness, acted upon by any exterior forces, and also w hen it is a solid ellipsoid subject to the influence of the earth’s actio n. By supposing magnetic changes to require time, although an exceedingly s hort one, to complete them, it had been suggested that M. A RAGO ’s discovery relative to the magnetic effects developed in copper, wood, glass, et c., by rotation, might be explained. On this hypothesis M. P OISSON has founded his third memoir, and thence deduced formulae applicable to magnetis m in a state of motion. Whether the preceding hypothesis will serve to ex plain the sin- gular phenomena observed by M. A RAGO or not, it would ill become me to decide; but it is probably quite adequate to account for th ose produced by the rapid rotation of iron bodies. We have just taken a cursory view of what has hitherto been wri tten, to the best of my knowledge, on subjects connected with the math ematical theory of electricity; and although many of the artifices empl oyed in the works before mentioned are remarkable for their elegance, i t is easy to see they are adapted only to particular objects, and that some ge neral method, capable of being employed in every case, is still wanting. In deed M. P OIS- SON, in the commencement of his first memoir (M´ em. de l’Institut 1 811), has incidentally given a method for determining the distrib ution of elec- tricity on the surface of a spheroid of any form, which would n aturally present itself to a person occupied in these researches, bei ng in fact nothing more than the ordinary one noticed in our introductory obser vations, as requiring the resolution of the equation ( a). Instead however of supposing, as we have done, that the point pmust be upon the surface, in order that the equation may subsist, M. P OISSON availing himself of a general fact, which was then supported by experiment only, has conceived t he equation to hold good wherever this point may be situated, provided it is within 4 Preface. the spheroid, but even with this extension the method is liab le to the same objection as before. Considering how desirable it was that a power of universal ag ency, like electricity, should, as far as possible, be submitted to cal culation, and re- flecting on the advantages that arise in the solution of many d ifficult prob- lems, from dispensing altogether with a particular examina tion of each of the forces which actuate the various bodies in any system, by confining the attention solely to that peculiar function on whose diff erentials they all depend, I was induced to try whether it would be possible t o discover any general relations, existing between this function and t he quantities of electricity in the bodies producing it. The advantages L APLACE had de- rived in the third book of the M´ ecanique Celeste , from the use of a partial differential equation of the second order, there given, wer e too marked to escape the notice of any one engaged with the present subject , and natu- rally served to suggest that this equation might be made subs ervient to the object I had in view. Recollecting, after some attempts to ac complish it, that previous researches on partial differential equation s, had shown me the necessity of attending to what have, in this Essay, been d enominated the singular values of functions, I found, by combining this consideration with the preceding, that the resulting method was capable of being applied with great advantage to the electrical theory, and was thus, in a short time, enabled to demonstrate the general formulae contained in th e preliminary part of the Essay. The remaining part ought to be regarded pri ncipally as furnishing particular examples of the use of these general f ormulae; their number might with great ease have been increased, but those w hich are given, it is hoped, will suffice to point out to mathematicians , the mode of applying the preliminary results to any case they may wish to investigate. The hypotheses on which the received theory of magnetism is f ounded, are by no means so certain as the facts on which the electrical the ory rests; it is however not the less necessary to have the means of submittin g them to cal- culation, for the only way that appears oppen to us in the inve stigation of these subjects, which seem as it were desirous to conceal the mselves from our view, is to form the most probable hypotheses we can, to de duce rig- orously the consequences which flow from them, and to examine whether such consequences agree numerically with accurate experim ents. The applications of analysis to the physical Sciences, have the double ad- vantage of manifesting the extraordinary powers of this won derful instru- ment of thought, and at the same time of serving to increase th em; number- less are the instances of the truth of this assertion. To sele ct one we may re- mark, that M. F OURIER , by his investigations relative to heat, has not only discovered the general equations on which its motion depend s, but has likewise been led to new analytical formulae, by whose aid M. M. CAUCHY and P OISSON have been enabled to give the complete theory of the motion of the waves in an indefinitely extended fluid. The same formula e have also put us in possession of the solutions of many other interesti ng problems, Introductory observations. 5 too numerous to be detailed here. — It must certainly be regar ded as a pleasing prospect to analists, that at a time when astronomy , from the state of perfection to which it has attained, leaves little room fo r farther applica- tions of their art, the rest of the physical sciences should s how themselves daily more and more willing to submit to it; and, amongst othe r things, probably the theory that supposes light to depend on the undu lations of a luminiferous fluid, and to which the celebrated Dr. T. Y OUNG has given such plausibility, may furnish a useful subject of research , by affording new opportunities of applying the general theory of the motion o f fluids. The number of these opportunities can scarcely be too great, as i t must be evi- dent to those who have examined the subject, that, although w e have long been in possession of the general equations on which this kin d of motion depends, we are not yet well acquainted with the various limi tations it will be necessary to introduce, in order to adapt them to the diffe rent physical circumstances which may occur. Should the present Essay tend in any way to facilitate thc app lication of analysis to one of the most interesting of the physical sci ences, the au- thor will deem himself amply repaid for any labour he may have bestowed upon it; and it is hoped the difficulty of the subject will incli ne mathe- maticians to read this work with indulgence, more particula rly when they are informed that it was written by a young man, who has been ob liged to obtain the little knowledge he possesses, at such interva ls and by such means, as other indispensable avocations which offer but fe w opportuni- ties of mental improvement, afforded. Introductory observations. The object of this Essay is to submit to Mathematical Analysis the phe- nomena of the equilibrium of the Electric and Magnetic Fluids , and to lay down some general principles equally applicable to perfect and imperfect conductors; but, before entering upon the calculus, it may n ot be amiss to give a general idea of the method that has enabled us to arrive at results, remarkable for their simplicity and generality, which it wo uld be very dif- ficult if not impossible to demonstrate in the ordinary way. It is well known, that nearly all the attractive and repulsiv e forces exist- ing in nature are such, that if we consider any material point p, the effect, in a given direction, of all the forces acting upon that point, a rising from any system of bodies Sunder consideration, will be expressed by a partial dif- ferential of a certain function of the co-ordinates which se rve to define the point’s position in space. The consideration of this functi on is of great im- portance in many inquiries, and probably there are none in wh ich its utility is more marked than in those about to engage our attention. In the sequel 6 Introductory observations. we shall often have occasion to speak of this function, and wi ll therefore, for abridgment, call it the potential function arising from the system S. If pbe a particle of positive electricity under the influence of f orces arising from any electrified body, the function in question, as is well known, will be obtained by dividing the quantity of electricity in each e lement of the body, by its distance from the particle p, and taking the total sum of these quotients for the whole body, the quantities of electricity in those elements which are negatively electrified, being regarded as negative . It is by considering the relations existing between the dens ity of the elec- tricity in any system, and the potential functions thence ar ising, that we have been enabled to submit many electrical phenomena to cal culation, which had hitherto resisted the attempts of analysts; and th e generality of the consideration here employed, ought necessarily, and do es, in fact, in- troduce a great generality into the results obtained from it . There is one consideration peculiar to the analysis itself, the nature a nd utility of which will be best illustrated by the following sketch: Suppose it were required to determine the law of the distribu tion of the electricity on a closed conducting surface Awithout thickness, when placed under the influence of any electrical forces whatever : these forces, for greater simplicity, being reduced to three, X,Yand Z, in the direction of the rectangular co-ordinates, and tending to increase th em. Then ̺rep- resenting the density of the electricity on an element dσof the surface, and rthe distance between dσand p, any other point of the surface, the equa- tion for determining ̺which would be employed in the ordinary method, when the problem is reduced to its simplest form, is known to b e (a) cons = a=/integraldisplay̺dσ r−/integraldisplay (X dx+Y dy+Z dz); the first integral relative to dσextending over the whole surface A, and the second representing the function whose complete dif ferential is X dx+Y dy+Z dz,x,yand zbeing the co-ordinates of p. This equation is supposed to subsist, whatever may be the pos ition of p, provided it is situate upon A. But we have no general theory of equations of this description, and whenever we are enabled to resolve o ne of them, it is because some consideration peculiar to the problem rende rs, in that par- ticular case, the solution comparatively simple, and must b e looked upon as the effect of chance, rather than of any regular and scient ific procedure. We will now take a cursory view of the method it is proposed to s ubsti- tute in the place of the one just mentioned. Let us make B=/integraltext(X dx+Y dy+Z dz)whatever may be the position of the point p,V=/integraltext̺dσ rwhen pis situate any where within the sur- face A, and V′=/integraltext̺dσ rwhen pis exterior to it; the two quantities Vand V′, although expressed by the same definite integral, are essenti ally distinct functions of x,y, and z, the rectangular co-ordinates of p; these functions, as is well known, having the property of satisfying the parti al differential Introductory observations. 7 equations 0=d2V dx2+d2V dy2+d2V dy2, 0=d2V′ dx2+d2V′ dy2+d2V′ dy2. If now we could obtain the values of Vand V′from these equations, we should have immediately, by differentiation, the required value of ̺, as will be shown in the sequel. In the first place, let us consider the function V, whose value at the sur- face Ais given by the equation ( a), since this may be written a=V−B the horizontal line over a quantity indicating that it belon gs to the sur- face A. But, as the general integral of the partial differential eq uation ought to contain two arbitrary functions, some other condition is requisite for the complete determination of V. Now since V=/integraltext̺dσ r, it is evident that none of its differential co-efficients can become infinite when pis situate any where within the surface A, and it is worthy of remark, that this is pre- cisely the condition required: for, as will be afterwards sh own, when it is satisfied we shall have generally V=−/integraldisplay (̺)dσV; the integral extending over the whole surface, and (̺)being a quantity dependant upon the respective positions of pand dσ. All the difficulty therefore reduces itself to finding a functio nV, which satisfies the partial differential equation, becomes equal t o the known value ofVat the surface, and is moreover such that none of its differen tial coeffi- cients shall be infinite when pis within A. In like manner, in order to find V′, we shall obtain V′, its value at A, by means of the equation ( a), since this evidently becomes a=V′−B, i. e. V′=V. Moreover it is clear, that none of the differential co-efficien ts of V′=/integraltext̺dσ r can be infinite when pis exterior to the surface A, and when pis at an in- finite distance from A,V′is equal to zero. These two conditions combined with the partial differential equation in V′, are sufficient in conjunction with its known value V′at the surface Afor the complete determination of V′, since it will be proved hereafter, that when they are satisfied we shall have V′=−/integraldisplay (̺)dσV′; the integral, as before, extending over the whole surface A, and(̺)being a quantity dependant upon the respective position of pand dσ. 8 Introductory observations. It only remains therefore to find a function V′which satisfies the partial differential equation, becomes equal to V′when pis upon the surface A, vanishes when pis at an infinite distance from A, and is besides such, that none of its differential co-efficients shall be infinite, when t he point pis exterior to A. All those to whom the practice of analysis is familiar, will r eadily per- ceive that the problem just mentioned, is far less difficult th an the direct res- olution of the equation ( a), and therefore the solution of the question origi- nally proposed has been rendered much easier by what has prec eded. The peculiar consideration relative to the differential co-ef ficients of Vand V′, by restricting the generality of the integral of the partial differential equa- tion, so that it can in fact contain only one arbitrary functi on, in the place of two which it ought otherwise to have contained, and, which has thus enabled us to effect the simplification in question, seems wor thy of the at- tention of analysts, and may be of use in other researches whe re equations of this nature are employed. We will now give a brief account of what is contained in the fol lowing Essay. The first seven articles are employed in demonstrating some very general relations existing between the density of the elect ricity on surfaces and in solids, and the corresponding potential functions. T hese serve as a foundation to the more particular applications which follo w them. As it would be difficult to give any idea of this part without employi ng analyt- ical symbols, we shall content ourselves with remarking, th at it contains a number of singular equations of great generality and simpl icity, which seem capable of being applied to many departments of the elec trical theory besides those considered in the following pages. In the eighth article we have determined the general values o f the densi- ties of the electricity on the inner and outer surfaces of an i nsulated electri- cal jar, when, for greater generality, these surfaces are su pposed to be con- nected with separate conductors charged in any way whatever ; and have proved, that for the same jar, they depend solely on the diffe rence existing between the two constant quantities, which express the valu es of the po- tential functions within the respective conductors. After wards, from these general values the following consequences have been deduce d: — When in an insulated electrical jar we consider only the elec tricity ac- cumulated on the two surfaces of the glass itself, the total q uantity on the inner surface is precisely equal to that on the outer surface , and of a con- trary sign, notwithstanding the great accumulation of elec tricity on each of them: so that if a communication were established between th e two sides of the jar, the sum of the quantities of electricity which would manifest them- selves on the two metallic coatings, after the discharge, is exactly equal to that which, before it had taken place, would have been observ ed to have existed on the surfaces of the coatings farthest from the gla ss, the only por- tions then sensible to the electrometer. Introductory observations. 9 If an electrical jar communicates by means of a long slender w ire with a spherical conductor, and is charged in the ordinary way, the density of the electricity at any point of the interior surface of the jar, i s to the density on the conductor itself, as the radius of the spherical conduct or to the thickness of the glass in that point. The total quantity of electricity contained in the interior of any number of equal and similar jars, when one of them communicates with the prime conductor and the others are charged by cascade, is precisel y equal to that, which one only would receive, if placed in communication wit h the same conductor, its exterior surface being connected with the co mmon reservoir. This method of charging batteries, therefore, must not be em ployed when any great accumulation of electricity is required. It has been shown by M.P OISSON , in his first Memoir on Magnetism (M´ em. de l’Acad. de Sciences, 1821 et 1822), that when an ele ctrified body is placed in the interior of a hollow spherical conducting sh ell of uniform thickness, it will not be acted upon in the slightest degree b y any bodies exterior to the shell, however intensely they may be electri fied. In the ninth article of the present Essay this is proved to be generally tr ue, whatever may be the form or thickness of the conducting shell. In the tenth article there will be found some simple equation s, by means of which the density of the electricity induced on a spherica l conducting surface, placed under the influence of any electrical forces whatever, is immediately given; and thence the general value of the poten tial function for any point either within or without this surface is determ ined from the arbitrary value at the surface itself, by the aid of a definite i ntegral. The proportion in which the electricity will divide itself betw een two insulated conducting spheres of different diameters, connected by a v ery fine wire, is afterwards considered; and it is proved, that when the rad ius of one of them is small compared with the distance between their surfa ces, the prod- uct of the mean density of the electricity on either sphere, b y the radius of that sphere, and again by the shortest distance of its surfac e from the centre of the other sphere, will be the same for both. Hence when thei r distance is very great, the densities are in the inverse ratio of the radi i of the spheres. When any hollow conducting shell is charged with electricit y, the whole of the fluid is carried to the exterior surface, without leavi ng any portion on the interior one, as may be immediately shown from the four th and fifth articles. In the experimental verification of this, it is nece ssary to leave a small orifice in the shell: it became therefore a problem of som e interest to determine the modification which this alteration would pro duce. We have, on this account, terminated the present article, by in vestigating the law of the distribution of electricity on a thin spherical co nducting shell, having a small circular orifice, and have found that its densit y is very nearly constant on the exterior surface, except in the immediate vi cinity of the orifice; and the density at any point pof the inner surface, is to the constant density on the outer one, as the product of the diameter of a ci rcle into 10 Introductory observations. the cube of the radius of the orifice, is to the product of three t imes the circumference of that circle into the cube of the distance of pfrom the centre of the orifice; excepting as before those points in its immedia te vicinity. Hence, if the diameter of the sphere were twelve inches, and t hat of the orifice one inch, the density at the point on the inner surface o pposite the centre of the orifice, would be less than the hundred and thirty thousandth part of the constant density on the exterior surface. In the eleventh article some of the effects due to atmospheri cal electricity are considered; the subject is not however insisted upon, as the great vari- ability of the cause which produces them, and the impossibil ity of measur- ing it, gives a degree of vagueness to these determinations. The form of a conducting body being given, it is in general a pr oblem of great difficulty, to determine the law of the distribution of t he electric fluid on its surface: but it is possible to give different forms, of almost every imaginable variety of shape, to conducting bodies; such, th at the values of the density of the electricity on their surfaces may be rigor ously assignable by the most simple calulations: the manner of doing this is ex plained in the twelfth article, and two examples of its use are given. In the last, the re- sulting form of the conducting body is an oblong spheroid, an d the density of the electricity on its surface, here found, agrees with th e one long since deduced from other methods. Thus far perfect conductors only have been considered. In or der to give an example of the application of theory to bodies which are no t so, we have, in the thirteenth article, supposed the matter of which they are formed to be endowed with a constant coercive force equal to β, and analagous to friction in its operation, so that when the resultant of the e lectric forces act- ing upon any one of their elements is less than β, the electrical state of this element shall remain unchanged; but, so soon as it begins to e xceed β, a change shall ensue. Then imagining a solid of revolution to t urn continu- ally about its axis, and to be subject to a constant electrica l force facting in parallel right lines, we determine the permanent electri cal state at which the body will ultimately arrive. The result of the analysis i s, that in conse- quence of the coercive force β, the solid will receive a new polarity, equal to that which would be induced in it if it were a perfect conduc tor and acted upon by the constant force β, directed in lines parallel to one in the body’s equator, making the angle 90◦+γ, with a plane passing through its axis and parallel to the direction of f:fbeing supposed resolved into two forces, one in the direction of the body’s axis, the other bdirected along the intersection of its equator with the plane just mentione d, and γbeing determined by the equation sinγ=β b. In the latter part of the present article the same problem is c onsidered under a more general point of view, and treated by a different analysis: the Introductory observations. 11 body’s progress from the initial, towards that permanent st ate it was the object of the former part to determine is exhibited, and the g reat rapidity of this progress made evident by an example. The phenomena which present themselves during the rotation of iron bodies, subject to the influence of the earth’s magnetism, ha ving lately en- gaged the attention of experimental philosophers, we have b een induced to dwell a little on the solution of the preceeding problem, sin ce it may serve in some measure to illustrate what takes place in these cases . Indeed, if there were any substances in nature whose magnetic powers, l ike those of iron and nickel, admit of considerable developement, and in which more- over the coercive force was, as we have here supposed it, the s ame for all their elements, the results of the preceding theory ought sc arcely to differ from what would be observed in bodies formed of such substanc es, pro- vided no one of their dimensions was very small, compared wit h the oth- ers. The hypothesis of a constant coercive force was adopted in this article, in order to simplify the calculations: probably, however, t his is not exactly the case of nature, for a bar of the hardest steel has been show n (I think by Mr. B ARLOW ) to have a very considerable degree of magnetism induced in it by the earth’s action, which appears to indicate, that alt hough the coer- cive force of some of its particles is very great, there are ot hers in which it is so small as not to be able to resist the feeble action of the e arth. Nev- ertheless, when iron bodies are turned slowly round their ax es, it would seem that our theory ought not to differ greatly from observa tion; and in particular, it is very probable the angle γmight be rendered sensible to experiment, by sufficiently reducing bthe component of the force f. The remaining articles treat of the theory of magnetism. Thi s theory is here founded on an hypothesis relative to the constitutio n of magnetic bodies, first proposed by C OULOMB , and afterwards generally received by philosophers, in which they are considered as formed of an in finite num- ber of conducting elements, separated by intervals absolut ely impervious to the magnetic fluid, and by means of the general results cont ained in the former part of the Essay, we readily obtain the necessary equ ations for de- termining the magnetic state induced in a body of any form, by the action of exterior magnetic forces. These equations accord with th ose M. P OISSON has found by a very different method. (M´ em. de l’Acad. des Sc iences, 1821 et 1822.) If the body in question be a hollow spherical shell of constan t thickness, the analysis used by L APLACE (M´ ec. Cel. Liv. 3) is applicable, and the prob- lem capable of a complete solution, whatever may be the situa tion of the centres of the magnetic forces acting upon it. After having g iven the gen- eral solution, we have supposed the radius of the shell to bec ome infinite, its thickness remaining unchanged, and have thence deduced formula be- longing to an indefinitely extended plate of uniform thicknes s. From these it follows, that when the point p, and the centres of the magnetic forces are situate on opposite sides of a soft iron plate of great ext ent, the total 12 Introductory observations. action on pwill have the same direction as the resultant of all the force s, which would be exerted on the points p,p′,p′′,p′′′etc. in infinitum if no plate were interposed, and will be equal to this resultant mu ltiplied by a very small constant quantity: the points p,p′,p′′,p′′′etc. being all on a right line perpendicular to the flat surfaces of the plate, and rece ding from it so, that the distance between any two consecutive points may be e qual to twice the plate’s thickness. What has just been advanced will be sensibly correct, on the s upposition of the distances between the point pand the magnetic centres not being very great, compared with the plate’s thickness, for, when t hese distances are exceedingly great, the interposition of the plate will m ake no sensible alteration in the force with which pis solicited. When an elongated body, as a steel wire for instance, has, und er the in- fluence of powerful magnets, received a greater degree of mag netism than it can retain alone, and is afterwards left to itself, it is sa id to be magnetized to saturation. Now if in this state we consider any one of its c onducting elements, the force with which a particle pof magnetism situate within the element tends to move, will evidently be precisely equal to its coercive force f, and in equilibrium with it. Supposing therefore this force to be the same for every element, it is clear that the degree of magneti sm retained by the wire in a state of saturation, is, on account of its elon gated form, ex- actly the same as would be induced by the action of a constant f orce, equal tof, directed along lines parallel to its axis, if all the elemen ts were perfect conductors; and consequently, may readily be determined by the general theory. The number and accuracy of C OULOMB ’s experiments on cylindric wires magnetized to saturation, rendered an application of theory to this particular case very desirable, in order to compare it with e xperience. We have therefore effected this in the last article, and the res ult of the compar- ison is of the most satisfactory kind. 1. 13 General preliminary results. 1. The function which represents the sum of all the electric par ticles act- ing on a given point divided by their respective distances fr om this point, has the property of giving, in a very simple form, the forces b y which it is solicited, arising from the whole electrified mass. We shal l, in what fol- lows, endeavour to discover some relations between this fun ction, and the density of the electricity in the mass or masses producing it , and apply the relations thus obtained, to the theory of electricity. Firstly, let us consider a body of any form whatever, through which the electricity is distributed according to any given law, and fix ed there, and let x′,y′,z′, be the rectangular co-ordinates of a particle of this body, ̺′the den- sity of the electricity in this particle, so that dx′dy′dz′being the volume of the particle, ̺′dx′dy′dz′shall be the quantity of electricity it contains: more- over, let r′be the distance between this particle and a point pexterior to the body, and Vrepresent the sum of all the particles of electricity divide d by their respective distances from this point, whose co-ordin ates are supposed to be , x,y,z, then shall we have r′=/radicalBig/parenleftbig(x′−x)2+ (y−y′)2+ (z−z′)2/parenrightbig , and V=/integraldisplay̺′dx′dy′dz′ r′; the integral comprehending every particle in the electrified mass under consideration. LAPLACE has shown, in his M´ ec. Celeste , that the function Vhas the property of satisfying the equation 0=d2V dx2+d2V dy2+d2V dz2 and as this equation will be incessantly recurring in what fo llows, we shall write it in the abridged form 0 =δV; the symbol δbeing used in no other sense throughout the whole of this Essay. In order to prove that 0 =δV, we have only to remark, that by differenti- ation we immediately obtain 0 =δ1 r′, and consequently each element of V substituted for Vin the above equation satisfies it; hence the whole inte- gral (being considered as the sum of all these elements) will also satisfy it. This reasoning ceases to hold good when the point pis within the body, for then, the co-efficients of some of the elements which enter int oVbecoming infinite, it does not therefore necessarily follow that Vsatisfies the equation 0=δV, although each of its elements, considered separately, may d o so. 14 General preliminary results. In order to determine what δVbecomes for any point within the body, conceive an exceedingly small sphere whose radius is ainclosing the point pat the distance bfrom its centre, aand bbeing exceedingly small quanti- ties. Then, the value of Vmay be considered as composed of two parts, one due to the sphere itself, the other due to the whole mass exter ior to it: but the last part evidently becomes equal to zero when substitut ed for VinδV, we have therefore only to determine the value of δVfor the small sphere itself, which value is known to be δ(2πa2̺−2 3πb2̺); ̺being equal to the density within the sphere and consequentl y to the value of ̺′atp. If now x′,y′,z′, be the co-ordinates of the centre of the sphere, we have b2= (x′−x)2+ (y′−y)2+ (z′−z)2, and consequently δ(2πa2̺−2 3πb2̺) =−4π̺. Hence, throughout the interior of the mass 0=δV+4π̺; of which, the equation 0 =δVfor any point exterior to the body is a partic- ular case, seeing that, here ̺=0. Let now qbe any line terminating in the point p, supposed without the body, then −(dV dq) =the force tending to impel a particle of positive electric- ity in the direction of q, and tending to increase it. This is evident, because each of the elements of Vsubstituted for Vin−(dV dq), will give the force arising from this element in the direction tending to increa seq, and conse- quently, −(dV dq)will give the sum of all the forces due to every element of V, or the total force acting on pin the same direction. In order to show that this will still hold good, although the point pbe within the body; conceive the value of Vto be divided into two parts as before, and moreover let p be at the surface of the small sphere or b=a, then the force exerted by this small sphere will be expressed by 4 3πa̺/parenleftBigda dq/parenrightBig ; dabeing the increment of the radius a, corresponding to the increment dq ofq, which force evidently vanishes when a=0 : we need therefore have regard only to the part due to the mass exterior to the sphere, and this is evidently equal to V−4 3πa2̺. But as the first differentials of this quantity are the same as t hose of Vwhen ais made to vanish, it is clear, that whether the point pbe within or without 2. 15 the mass, the force acting upon it in the direction of qincreasing, is always given by −(dV dq). Although in what precedes we have spoken of one body only, the rea- soning there employed is general, and will apply equally to a system of any number of bodies whatever, in those cases even, where the re is a finite quantity of electricity spread over their surfaces, and it i s evident that we shall have for a point pin the interior of any one of these bodies (1.) 0 =δV+4π̺. Moreover, the force tending to increase a line qending in any point pwithin or without the bodies, will be likewise given by −(dV dq); the function Vrep- resenting the sum of all the electric particles in the system divided by their respective distances from p. As this function, which gives in so simple a form the values of the forces by which a particle pof electricity, any how situated, is impelled, will recur very frequently in what fo llows, we have ventured to call it the potential function belonging to the s ystem, and it will evidently be a function of the co-ordinates of the particle punder consider- ation. 2. It has been long known from experience, that whenever the ele ctric fluid is in a state of equilibrium in any system whatever of perfect ly conduct- ing bodies, the whole of the electric fluid will be carried to t he surface of those bodies, without the smallest portion of electricity r emaining in their interior: but I do not know that this has ever been shown to be a necessary consequence of the law of electric repulsion, which is found to take place in nature. This however may be shown to be the case for every im aginable system of conducting bodies, and is an immediate consequenc e of what has preceded. For let x,y,z, be the rectangular co-ordinates of any particle pin the interior of one of the bodies; then will −(dV dx), be the force with which p is impelled in the direction of the co-ordinate x, and tending to increase it. In the same way −dV dyand−dV dzwill be the forces in yand z, and since the fluid is in equilibrium all these forces are equal to zero : hence 0=dV dxdx+dV dydy+dV dzdz=dV, which equation being integrated gives V= const. This value of Vbeing substituted in the equation (1) of the preceding num- ber gives ̺=0, and consequently shows, that the density of the electricity at any point in the interior of any body in the system is equal to zero . 16 General preliminary results. The same equation (1) will give the value of ̺the density of the electricity in the interior of any of the bodies, when there are not perfec t conductors, provided we can ascertain the value of the potential functio nVin their interior. 3. Before proceeding to make known some relations which exist b etween the density of the electric fluid at the surfaces of bodies, an d the corre- sponding values of the potential functions within and witho ut those sur- faces, the electric fluid being confined to them alone, we shall in the first place, lay down a general theorem which will afterwards be ve ry useful to us. This theorem may be thus enunciated: LetUand Vbe two continuous functions of the rectangular co-ordinate s x,y,z, whose differential co-efficients do not become infinite at any point within a solid body of any form whatever; then will /integraldisplay dx dy dz U δV+/integraldisplay dσU/parenleftBigdV dw/parenrightBig =/integraldisplay dx dy dz V δU+/integraldisplay dσV/parenleftBigdU dw/parenrightBig ; the triple integrals extending over the whole interior of th e body, and those relative to dσ, over its surface, of which dσrepresents an element: dwbeing an infinitely small line perpendicular to the surface, and mea sured from this surface towards the interior of the body. To prove this let us consider the triple integral /integraldisplay dx dy dz/braceleftBig/parenleftBigdV dx/parenrightBig/parenleftBigdU dx/parenrightBig +/parenleftBigdV dy/parenrightBig/parenleftBigdU dy/parenrightBig +/parenleftBigdV dz/parenrightBig/parenleftBigdU dz/parenrightBig/bracerightBig . The method of integration by parts, reduces this to /integraldisplay dy dz V′′dU′′ dx−/integraldisplay dy dz V′dU′ dx+/integraldisplay dx dz V′′dU′′ dy−/integraldisplay dx dz V′dU′ dy +/integraldisplay dx dy V′′dU′′ dz−/integraldisplay dx dy V′dU′ dz−/integraldisplay dx dy dz V/braceleftBigd2U dx2+d2U dy2+d2U dz2/bracerightBig ; the accents over the quantities indicating, as usual, the va lues of those quantities at the limits of the integral, which in the presen t case are on the surface of the body, over whose interior the triple integral s are supposed to extend. Let us now consider the part/integraltext dy dz V′′dU′′ dxdue to the greater values of x. It is easy to see since dwis every where perpendicular to the surface of the solid, that if dσ′′be the element of this surface corresponding to dy dz , we shall have dy dz=−dx dwdσ′′ 3. 17 and hence by substitution /integraldisplay dy dz V′′dU′′ dx=−/integraldisplay dσ′′dx dwV′′dU′′ dx. In like manner it is seen, that in the part −/integraltext dy dz V′dU′ dxdue to the smaller values of x, we shall have dy dz= +dx dwdσ′, and consequently −/integraldisplay dy dz V′dU′ dx=/integraldisplay dσdx dwV′dU′ dx. Then, since the sum of the elements represented by dσ′, together with those represented by dσ′′, constitute the whole surface of the body, we have by adding these two parts /integraldisplay dy dz/parenleftBig V′′dU′′ dx−V′dU′ dx/parenrightBig =−/integraldisplay dσdx dwVdU dx; where the integral relative to do is supposed to extend over t he whole sur- face, and dxto be the increment of xcorresponding to the increment dw. In precisely the same way we have /integraldisplay dx dz/parenleftBig V′′dU′′ dy−V′dU′ dy/parenrightBig =−/integraldisplay dσdy dwVdU dy, and /integraldisplay dx dy/parenleftBig V′′dU′′ dz−V′dU′ dz/parenrightBig =−/integraldisplay dσdz dwVdU dz; therefore, the sum of all the double integrals in the express ion before given will be obtained by adding together the three parts just foun d; we shall thus have −/integraldisplay dσV/braceleftBigdU dxdx dw+dU dydy dw+dU dzdz dw/bracerightBig =−/integraldisplay dσVdU dw; where VanddU dwrepresent the values at the surface of the body. Hence, the integral/integraldisplay dx dy dz/braceleftBigdV dxdU dx+dV dydU dy+dV dzdU dz/bracerightBig , by using the characteristic δin order to abridge the expression, becomes −/integraldisplay dσVdU dw−/integraldisplay dx dy dz V δU. Since the value of the integral just given remains unchanged when we substitute Vin the place of Uand reciprocally, it is clear, that it will also be expressed by −/integraldisplay dσUdV dw−/integraldisplay dx dy dz U δV. Hence, if we equate these two expressions of the same quantit y, after hav- ing changed their signs, we shall have (2.)/integraldisplay dσVdU dw+/integraldisplay dx dy dz V δU=/integraldisplay dσUdV dw+/integraldisplay dx dy dz U δV. 18 General preliminary results. Thus the theorem appears to be completely established, what ever may be the form of the functions Uand V. In our enunciation of the theorem, we have supposed the diffe rentials ofUand Vto be finite within the body under consideration, a condition, the necessity of which does not appear explicitly in the demo nstration, but, which is understood in the method of integration by parts the re employed. In order to show more clearly the necessity of this condition , we will now determine the modification which the formula must undergo, wh en one of the functions, Ufor example, becomes infinite within the body; and let us suppose it to do so in one point p′only: moreover, infinitely near this point letUbe sensibly equal to1 r;rbeing the distance between the point p′and the element dx dy dz . Then if we suppose an infinitely small sphere whose radius is ato be described round p′, it is clear that our theorem is applicable to the whole of the body exterior to this sphere, and since, δU=δ1 r=0 within the sphere, it is evident, the triple integrals may st ill be supposed to extend over the whole body, as the greatest error that this su pposition can induce, is a quantity of the order a2. Moreover, the part of/integraltext dσUdV dw, due to the surface of the small sphere is only an infinitely small qu antity of the order a; there only remains therefore to consider, the part of/integraltext dσVdU dw, due to this same surface, which, since we have heredU dw=dU dr=d1 r dr=−1 r2=−1 a2, becomes −4πV′ when the radius ais supposed to vanish. Thus, the equation (2.) becomes (3.)/integraldisplay dx dy dz U δV+/integraldisplay dσUdV dw=/integraldisplay dx dy dz V δU+/integraldisplay dσVdU dw−4πV′; where, as in the former equation, the triple integrals exten d over the whole volume of the body, and those relative to dσ, over its exterior surface: V′ being the value of Vat the point p′. In like manner, if the function Vbe such, that it becomes infinite for any point p′′within the body, and is moreover, sensibly equal to1 r′, infinitely near this point, as Uis infinitely near to the point p′, it is evident from what has preceded that we shall have (3.’)/integraldisplay dx dy dz U δV+/integraldisplay dσUdV dw−4πU′′ =/integraldisplay dx dy dz V δU+/integraldisplay dσVdU dw−4πV′; the integrals being taken as before, and U′′representing the value of U, at the point p′′where Vbecomes infinite. The same process will evidently apply, however great may be the number of similar points belo nging to the functions Uand V. For abridgment, we shall in what follows, call those singula r values of a given function, where its differential co-efficients become infinite, and the 4. 19 condition originally imposed upon Uand Vwill be expressed by saying, that neither of them has any singular values within the solid body under consideration. 4. We will now proceed to determine some relations existing bet ween the density of the electric fluid at the surface of a body, and the p otential func- tions thence arising, within and without this surface. For t his, let ̺dσbe the quantity of electricity on an element dσof the surface, and V, the value of the potential function for any point pwithin it, of which the co-ordinates arex,y,z. Then, if V′be the value of this function for any other point p′ exterior to this surface, we shall have V=/integraldisplay̺dσ/radicalbig (ξ−x)2+ (η−y)2+ (ζ−z)2; ξ,η,ζbeing the co-ordinates of dσ, and V′=/integraldisplay̺dσ/radicalbig (ξ−x′)2+ (η−y′)2+ (ζ−z′)2: the integrals relative to dσextending over the whole surface of the body. It might appear at first view, that to obtain the value of V′from that of V, we should merely have to change x,y,z, into x′,y′,z′: but, this is by no means the case; for, the form of the potential function chang es suddenly, in passing from the space within to that without the surface. Of this, we may give a very simple example, by supposing the surface to be a sp here whose radius is aand centre at the origin of the co-ordinates; then, if the den sity̺ be constant, we shall have V=4π̺aand V′=4πa2̺/radicalbig x′2+y′2+z′2; which are essentially distinct functions. With respect to the functions Vand V′in the general case, it is clear that each of them will satisfy L APLACE ’s equation, and consequently 0=δVand 0 =δ′V′: moreover, neither of them will have singular values; for any point of the spaces to which they respectively belong, and at the surface itself, we shall have V=V′, the horizontal lines over the quantities indicating that th ey belong to the surface. At an infinite distance from this surface, we shall li kewise have V′=0. We will now show, that if any two functions whatever are taken , satisfy- ing these conditions, it will always be in our power to assign one, and only 20 General preliminary results. one value of ̺, which will produce them for corresponding potential func- tions. For this we may remark, that the equation (3) art. 3 bei ng applied to the space within the body, becomes, by making U=1 r, /integraldisplaydσ r/parenleftbiggdV dw/parenrightbigg =/integraldisplay dσV/parenleftbiggd1 r dw/parenrightbigg −4πV; since U=1 r, has but one singular point, viz. p; and, we have also δV=0 and δ1 r=0 :rbeing the distance between the point pto which Vbelongs, and the element dσ. If now, we conceive a surface inclosing the body at an infinite d istance from it, we shall have, by applying the formula (2) of the same article to the space between the surface of the body and this imaginary exte rior surface (seeing that here1 r=Uhas no singular value) /integraldisplaydσ r/parenleftbiggdV′ dw′/parenrightbigg =/integraldisplay dσV′/parenleftbiggd1 r dw′/parenrightbigg : since the part due to the infinite surface may be neglected, bec ause V′is there equal to zero. In this last equation, it is evident that dw′is measured from the surface, into the exterior space, and hence /parenleftbiggd1 r dw/parenrightbigg =−/parenleftbiggd1 r dw′/parenrightbigg i. e. 0 =/parenleftbiggd1 r dw/parenrightbigg +/parenleftbiggd1 r dw′/parenrightbigg which equation reduces the sum of the two just given to /integraldisplaydσ r/braceleftbigg/parenleftbiggdV dw/parenrightbigg +/parenleftbiggdV′ dw′/parenrightbigg/bracerightbigg =−4πV. In exactly the same way, for the point p′exterior to the surface, we shall obtain/integraldisplaydσ r′/braceleftbigg/parenleftbiggdV dw/parenrightbigg +/parenleftbiggdV′ dw′/parenrightbigg/bracerightbigg =−4πV′. Hence it appears, that there exists a value of ̺, viz. ̺=−1 4π{(dV dw) + (dV′ dw′)}, which will give Vand V′, for the two potential functions, within and with- out the surface. Again, −(dV dw) = force with which a particle of positive electricity p, placed within the surface and infinitely near it, is impelled i n the direction dwperpendicular to this surface, and directed inwards; and −(dV dw) expresses the force with which a similar particle p′placed without this surface, on the same normal with p, and also infinitely near it, is impelled outwards in the direction of this normal: but the sum of these two forces is equal to double the force that an infinite plane would exert u pon p, supposing it uniformly covered with electricity of the same density as at 5. 21 the foot of the normal on which pis; and this last force is easily shown to be expressed by 2 π̺, hence by equating (4.) 4 π̺=−/braceleftbiggdV dw+dV′ dw′/bracerightbigg , and consequently there is only one value of ̺, which can produce Vand V′ as corresponding potential functions. Although in what precedes, we have considered the surface of one body only, the same arguments apply, how great soever may be their number; for the potential functions Vand V′would still be given by the formulae V=/integraldisplay̺dσ rand V′=/integraldisplay̺dσ r′; the only difference would be, that the integrations must now extend over the surface of all the bodies, and, that the number of functio ns represented byV, would be equal to the number of the bodies, one for each. In th is case, if there were given a value of Vfor each body, together with V′belonging to the exterior space; and moreover, if these functions sati sfied to the above mentioned conditions, it would always be possible to determ ine the den- sity on the surface of each body, so as to produce these values as potential functions, and there would be but one density, viz. that give n by (4’.) 0 =4π̺+dV dw+dV′ dw′ which could do so: ̺,dV dwanddV′ dw′belonging to a point on the surface of any of these bodies. 5. From what has been before established (art. 3), it is easy to p rove, that when the value of the potential function Vis given on any closed surface, there is but one function which can satisfy at the same time th e equation 0=δV, and the condition, that Vshall have no singular values within this surface. For the equation (3) art. 3, becomes by supposing δU=0, /integraldisplay dσUdV dw=/integraldisplay dσVdU dw−4πV′. In this equation, Uis supposed to have only one singular value within the surface, viz. at the point p′, and, infinitely near to this point, to be sensibly equal to1 r;rbeing the distance from p′. If now we had a value of U, which, besides satisfying the above written conditions, was equal to zero at the surface itself, we should have U=0, and this equation would become (5.) 0 =/integraldisplay dσVdU dw−4πV′. 22 General preliminary results. which shows, that V′the value of Vat the point p′is given, when Vits value at the surface is known. To convince ourselves, that there does exist such a function as we have supposed Uto be; conceive the surface to be a perfect conductor put in communication with the earth, and a unit of positive electri city to be con- centrated in the point p′; then the total potential function arising from p′ and from the electricity it will induce upon the surface, wil l be the required value of U. For, in consequence of the communication established betw een the conducting surface and the earth, the total potential fu nction at this surface must be constant, and equal to that of the earth itsel f, i. e. to zero (seeing that in this state they form but one conducting body) . Taking, there- fore, this total potential function for U, we have evidently 0 =U, 0=δU, and U=1 rfor those parts infinitely near to p′. As moreover, this function has no other singular points within the surface, it evidentl y possesses all the properties assigned to Uin the preceding proof. Again, since we have evidently U′=0, for all the space exterior to the surface, the equation (4) art. 4 gives 0=4π(̺) +dU dw; where (̺)is the density of the electricity induced on the surface, by t he action of a unit of electricity concentrated in the point p′. Thus, the equa- tion (5) of this article becomes (6.) V′=−/integraldisplay dσ(̺)V. This equation is remarkable on account of its simplicity and singularity, seeing that it gives the value of the potential for any point p′, within the surface, when V, its value at the surface itself is known, together with (̺), the density that a unit of electricity concentrated in p′would induce on this surface, if it conducted electricity perfectly, and were pu t in communication with the earth. Having thus proved, that V′the value of the potential function V, at any point p′within the surface is given, provided its value Vis known at this surface, we will now show, that whatever the value of Vmay be, the general value of Vdeduced from it by the formula just given shall satisfy the equation 0=δV. For, the value of Vat any point pwhose co-ordinates are x,y,z, deduced from the assumed value of V, by the above written formula, is 4πV=/integraldisplay dσV/parenleftBigdU dw/parenrightBig , Ubeing the total potential function within the surface, aris ing from a unit of electricity concentrated in the point p, and the electricity induced on the 5. 23 surface itself by its action. Then, since Vis evidently independent of x,y,z, we immediately deduce 4πδV=/integraldisplay dσVδ/parenleftbiggdU dw/parenrightbigg . Now the general value of Uwill depend upon the position of the point pproducing it, and upon that of any other point p′whose co-ordinates are x′,y′,z′, to which it is referred, and will consequently be a function of the six quantities x,y,z,x′,y′,z′. But we may conceive Uto be divided into two parts, one =1 r(rbeing the distance pp′) arising from the electricity in p, the other, due to the electricity induced on the surface by th e action of p, and which we shall call U′. Then since U′has no singular values within the surface, we may deduce its general value from that at the s urface, by a formula similar to the one just given. Thus 4πU′=/integraldisplay dσU′/parenleftbiggdU′ dw/parenrightbigg ; where U′is the total potential function, which would be produced by a unit of electricity in p′, and therefore, (dU′ dw)is independent of the co-ordinates x,y,z, ofp, to which δrefers. Hence 4πδU′=/integraldisplay dσ/parenleftbiggdU′ dw/parenrightbigg δU′. We have before supposed U=1 r+U′, and as δ1 r=0, we immediately obtain δU=δU′. Again, since we have at the surface itself 0 =U=1 r+U′;rbeing the distance between pand the element dσ, we hence deduce 0=δU′; this substituted in the general value of δU′, before given, there arises δU′= 0, and consequently 0 =δU. The result just obtained being general, and applicable to any point p′′within the surface, gives immediately 0=δ/parenleftbiggdU dw/parenrightbigg , and we have by substituting in the equation determining δV, 0=δV. In a preceding part of this article, we have obtained the equa tion 0=4π(̺) +/parenleftbiggdU dw/parenrightbigg ; 24 General preliminary results. which combined with 0 =δ(dU dw), gives 0=δ(̺) and therefore the density (̺)induced on any element dσ, which is evi- dently a function of the co-ordinates x,y,z, ofp, is also such a function as will satisfy the equation 0 =δ(̺): it is moreover evident, that (̺)can never become infinite when pis within the surface. It now remains to prove, that the formula V=1 4π/integraldisplay dσV/parenleftbiggdU dw/parenrightbigg =−/integraldisplay dσ(̺)V. shall always give V=Vfor any point within the surface and infinitely near it, whatever may be the assumed value of V. For this, suppose the point pto approach infinitely near the surface; then it is clear that the value of (̺), the density of the electricity induced by p, will be insensible, except for those parts infinitely near to p, and in these parts it is easy to see, that the value of (̺)will be independent of the form of the surface, and depend only on the distance p,dσ. But, we shall afterwards show (art. 10), that when this surface is a sphere of any radiu s whatever, the value of (̺)is (̺) =−α 2π·f3; αbeing the shortest distance between pand the surface, and frepresent- ing the distance p,dσ. This expression will give an idea of the rapidity with which (̺)decreases, in passing from the infinitely small portion of the surface in the immediate vicinity of p, to any other part situate at a fi- nite distance from it, and when substituted in the above writ ten value of V, gives, by supposing αto vanish, V=V. It is also evident, that the function V, determined by the above written for- mula, will have no singular values within the surface under c onsideration. What was before proved, for the space within any closed surfa ce, may likewise be shown to hold good, for that exterior to a number o f closed surfaces, of any forms whatever, provided we introduce the c ondition, that V′shall be equal to zero at an infinite distance from these surfaces. For, conceive a surface at an infinite distance from those under con sideration; then, what we have before said, may be applied to the whole spa ce within the infinite surface and exterior to the others; consequently (5’.) 4 πV′=/integraldisplay dσV′/parenleftbiggdU dw/parenrightbigg ; where the sign of integration must extend over all the surfac es, (seeing that the part due to the infinite surface is destroyed by the conditi on, that V′is 6. 25 there equal to zero ) and dwmust evidently be measured from the surfaces, into the exterior space to which V′now belongs. The form of the equation (6) remains also unaltered, and (6’.) V′=−/integraldisplay (̺)dσV′; the sign of integration extending over all the surfaces, and (̺)being the density of the electricity which would be induced on each of t he bodies, in presence of each other, supposing they all communicated wit h the earth by means of infinitely thin conducting wires. 6. Let now Abe any closed surface, conducting electricity perfectly, a ndp a point within it, in which a given quantity of electricity Qis concentrated, and suppose this to induce an electrical state in A; then will V, the value of the potential function arising from the surface only, at any other point p′, also within it, be such a function of the co-ordinates pand p′, that we may change the co-ordinates of p, into those of p′, and reciprocally, without al- tering its value. Or, in other words, the value of the potenti al function at p′, due to the surface alone, when the inducing electricity Qis concentrated inp, is equal to that which would have place at p, if the same electricity Q were concentrated in p′. For, in consequence of the equilibrium at the surface, we hav e evidently, in the first case, when the inducing electricity is concentrat ed in p, Q r+V=β; rbeing the distance between pand dσ′an element of the surface A, and βa constant quantity dependant upon the quantity of electric ity originally placed on A. Now the value of Vatp′is V=−/integraldisplay (̺′)dσ′V, by what has been shown (art. 5); (̺)being, as in that article, the density of the electricity which would be induced on the element dσ′by a unit of electricity in p′, if the surface Awere put in communication with the earth. This equation gives δV=−/integraldisplay (̺′)dσ′δV=0; since δV=−δQ r=0 : the symbol δreferring to the co-ordinates x,y,z, ofp. But we know that 0 =δ′V; where δ′refers in a similar way to the co-ordinates x′,y′,z′, ofp′only. Hence we have simultaneously 0=δVand 0 =δ′V; 26 General preliminary results. where it must be remarked, that the function Vhas no singular values, provided the points pand p′are both situate within the surface A. This being the case the first equation evidently gives (art. 5) V=−/integraldisplay (̺)dσV; Vbeing what Vwould become, if the inducing point pwere carried to do,p′remaining fixed. Where Vis a function of x′,y′,z′, and ξ,η,ζ, the co-ordinates of dσ, whereas (̺)is a function of x,y,z,ξ,η,ζ, independent ofx′,y′,z′; hence by the second equation 0=δ′V=−/integraldisplay (̺)dσδ′V, which could not hold generally whatever might be the situati on of p, unless we had 0=δV; where we must be cautious, not to confound the present value o fV, with that employed at the beginning of this article in provin g the equation 0 =δV, which last, having performed its office, will be no longer employed. The equation 0 =δ′V′gives in the same way V=−/integraldisplay (̺′)dσ′V; Vbeing what Vbecomes by bringing the point p′to any other element dσ′ of the surface A. This substituted for Vin the expression before given, there arises V= +/integraldisplay/integraldisplay (̺)(̺′)dσdσ′V: in which double integral, the signs of integration, relativ e to each of the independent elements dσand dσ′, must extend over the whole surface. If now, we represent by V′, the value of the potential function at parising from the surface A, when the electricity Qis concentrated in p′, we shall evidently have V= +/integraldisplay (̺′)(̺)dσ′dσ′ V′; where the order of integrations alone is changed, the limits remaining un- altered:′ V′being what V′would become, by first bringing the electrical point p′to the surface, and afterward the point pto which V′belongs. This being done, it is clear that Vand′ V′represent but one and the same quan- tity, seeing that each of them serves to express the value of t he potential function, at any point of the surface A, arising from the surface itself, when 7. 27 the electricity is induced upon it by the action of an electri fied point, situate in any other point of the same surface, and hence we have evide ntly V=V′, as was asserted at the commencement of this article. It is evident from art. 5, that our preceding arguments will b e equally ap- plicable to the space exterior to the surfaces of any number o f conducting bodies, provided we introduce the condition, that the poten tial function V, belonging to this space, shall be equal to zero, when either porp′shall remove to an infinite distance from these bodies, which condit ion will evi- dently be satisfied, provided all the bodies are originally in a natural state. Supposing this therefore to be the case, we see that the poten tial function belonging to any point p′of the exterior space, arising from the electricity induced on the surfaces of any number of conducting bodies, b y an electri- fied point in p, is equal to that which would have place at p, if the electrified point were removed to p′. What has been just advanced, being perfectly independent of the num- ber and magnitude of the conducting bodies, may be applied to the case of an infinite number of particles, in each of which the fluid may mo ve freely, but which are so constituted that it cannot pass from one to an other. This is what is always supposed to take place in the theory of magneti sm, and the present article will be found of great use to us when in the seq uel we come to treat of that theory. 7. These things being established with respect to electrified su rfaces; the general theory of the relations between the density of the el ectric fluid and the corresponding potential functions, when the electrici ty is disseminated through the interior of solid bodies as well as over their sur faces, will very readily flow from what has been proved (art. 1). For this let V′represent the value of the potential function at a point p′, within a solid body of any form, arising from the whole of the e lectric fluid contained in it, and ̺′be the density of the electricity in its interior; ̺′be- ing a function of the three rectangular coordinates x,y,z: then if ̺be the density at the surface of the body, we shall have V′=/integraldisplaydx dy dz ̺′ r′+/integraldisplaydσ̺ r; r′being the distance between the point p′whose co-ordinates are x′,y′,z′, and that whose co-ordinates are x,y,z, to which ̺′belongs, also rthe dis- tance between p′and dσ, an element of the surface of the body: V′being evidently a function of x′,y′,z′. If now Vbe what V′becomes by changing x′,y′,z′, into x,y,z, it is clear from (art. 1), that ̺′will be given by 0=4π̺′+δV. 28 General preliminary results. Substituting for ̺′, the value which results from this equation, in that im- mediately preceding we obtain V′=/integraldisplaydx dy dz δV 4πr′+/integraldisplaydσ̺ r, which, by means of the equation (3. art. 3), becomes /integraldisplaydσ̺ r=1 4π/braceleftbigg/integraldisplay dσV/parenleftbiggd1 r dw/parenrightbigg −/integraldisplaydσ r/parenleftbiggdV dw/parenrightbigg/bracerightbigg ; the horizontal lines over the quantities, indicating that t hey belong to the surface itself. Suppose V′to be the value of the potential function in the space exterio r to the body, which, by (art. 5), will depend on the value of Vat the surface only; and the equation (2. art. 3), applied to this exterior s pace, will give since δV=0 and δ1 r=0, /integraldisplay dσV/parenleftbiggd1 r dw′/parenrightbigg =/integraldisplay dσV′/parenleftbiggd1 r dw/parenrightbigg =/integraldisplaydσ r/parenleftbiggdV′ dw′/parenrightbigg ; where dw′is measured from the surface into the exterior space to which V′ belongs, as dwis, into the interior space. Consequently dw=−dw′, and therefore /integraldisplay dσV/parenleftbiggd1 r dw/parenrightbigg =−/integraldisplay dσV/parenleftbiggd1 r dw′/parenrightbigg =−/integraldisplaydσ r/parenleftbiggdV′ dw′/parenrightbigg . Hence the equation determining ̺becomes, by substituting for/integraltext dσVd1 r dw its value just given, /integraldisplay̺dσ r=−1 4π/integraldisplaydσ r/braceleftbigg/parenleftbiggdV dw/parenrightbigg +/parenleftbiggdV′ dw′/parenrightbigg/bracerightbigg . an equation which could not subsist generally, unless (7.) ̺=−1 4π/braceleftbiggdV dw+dV′ dw′/bracerightbigg . Thus the whole difficulty is reduced to finding the value V′of the potential function exterior to the body. Although we have considered only one body, it is clear that th e same theory is applicable to any number of bodies, and that the val ues of ̺and ̺′will be given by precisely the same formulae, however great t hat number may be; V′being the exterior potential function common to all the bodi es. In case the bodies under consideration are all perfect condu ctors, we have seen (art. 1), that the whole of the electricity will be c arried to their sur- faces, and therefore there is here no place for the applicati on of the theory contained in this article; but as there are probably no perfe ctly conducting bodies in nature, this theory becomes indispensably necess ary, if we would investigate the electrical phenomena in all their generali ty. 8. 29 Having in this, and the preceding articles, laid down the mos t general principles of the electrical theory, we shall in what follow s apply these prin- ciples to more special cases; and the necessity of confining th is Essay within a moderate extent, will compel us to limit ourselves to a brie f examination of the more interesting phenomena. Application of the preceding results to the theory of electricity. 8. The first application we shall make of the foregoing principle s, will be to the theory of the Leyden phial. For this, we will call the inne r surface of the phial A, and suppose it to be of any form whatever, plane or curved, th en, Bbeing its outer surface, and θthe thickness of the glass measured along a normal to A;θwill be a very small quantity, which, for greater general- ity, we will suppose to vary in any way, in passing from one poi nt of the surface Ato another. If now the inner coating of the phial be put in com- munication with a conductor C, charged with any quantity of electricity, and the outer one be also made to communicate with another con ducting body C′, containing any other quantity of electricity, it is eviden t, in con- sequence of the communications here established, that the t otal potential function, arising from the whole system, will be constant th roughout the interior of the inner metallic coating, and of the body C. We shall here represent this constant quantity by β. Moreover, the same potential function within the substance o f the outer coating, and in the interior of the conductor C′, will be equal to another constant quantity β′. Then designating by V, the value of this function, for the whole of the space exterior to the conducting bodies of the system, and consequ ently for that within the substance of the glass itself; we shall have (art. 4) V=βand V=β′. One horizontal line over any quantity, indicating that it be longs to the inner surface A; and two showing that it belongs to the outer one B. At any point of the surface A, suppose a normal to it to be drawn, and let this be the axes of w: then w′,w′′, being two other rectangular∗axes, which are necessarily in the plane tangent to Aat this point; Vmay be considered ∗orthogonal – RS 30 Application to electricity. as a function of w,w′,w′′, and we shall have by T AYLOR s theorem, since w′=0 and w′′=0 at the axis of walong which θis measured, V=V+dV dw·θ 1+d2V dw2·θ2 1·2+etc.; where, on account of the smallness of θ, the series converges very rapidly. By writing in the above, for Vand Vtheir values just given, we obtain β′−β=dV dw·θ 1+d2V dw2·θ2 1·2+etc.; In the same way, if wbe a normal to B, directed towards A, and θ′be the thickness of the glass measured along this normal, we shall h ave β′−β=dV dw·θ′ 1+d2V dw2·θ2′ 1·2+etc.. But, if we neglect quantities of the order θ, compared with those retained, the following equation will evidently hold good, dnV dwn= (−1)ndnV dwn; nbeing any whole positive number, the factor (−1)nbeing introduced be- cause wand ware measured in opposite directions. Now by article 4 −4π̺=dV dwand −4π̺=dV dw; ̺and ̺being the densities of the electric fluid at the surfaces Aand B respectively. Permitting ourselves, in what follows, to ne glect quantities of the order θ2compared with those retained, it is clear that we may write θ forθ′, and hence by substitution β−β′=−4π̺θ+/parenleftbiggd2V dw2/parenrightbiggθ2 1·2 β−β′=−4π̺θ+/parenleftbiggd2V dw2/parenrightbiggθ2 1·2; where Vand ̺are quantities of the order1 θ;β′and βbeing the ordre θ0or unity. The only thing which now remains to be determined, is t he value ofd2V dw2for any point on the surface A. Throughout the substance of the glass, the potential functi onVwill sat- isfy the equation 0 =δV, and therefore at a point on the surface of A, where of necessity, w,w′, and w′′, are each equal to zero, we have 0=d2V dw2+d2V dw′2+d2V dw′′2=δV; 8. 31 the horizontal mark over w,w′and w′′being, for simplicity, omitted. Then since w′=0, d2V dw′2= (V0−2Vdw′+V2dw′):dw′2, and as Vis constant and equal to βat the surface A, there hence arises V0=β;Vdw′=β+dV dwdw′2 2R,V2dw′=β+dV dw4dw′2 2R; Rbeing the radius of curvature of the surface A, in the plane (w,w′). Sub- stituting these values in the expression immediately prece ding, we get d2V dw′2=1 RdV dw=−4π̺ R. In precisely the same way we obtain, by writing R′for the radius of curva- ture in the plane (w,w′′), d2V dw′′2=−4π̺ R′: both rays being accounted positive on the side where w, i. e. wis negative. These values substituted in 0 =δV, there results d2V dw2=4π̺/parenleftBig1 R+1 R′/parenrightBig for the required value ofd2V dw2, and thus the sum of the two equations into which it enters, yields ̺/braceleftBig 1+/parenleftBig1 R+1 R′/parenrightBig θ/bracerightBig =−̺, and the difference of the same equations, gives β−β′=2π(̺−̺)θ, therefore the required values of the densities ̺and ̺are (8.)  ̺=β−β′ 4πθ/braceleftBig 1+1 2θ/parenleftBig1 R+1 R′/parenrightBig/bracerightBig ̺=β′−β 4πθ/braceleftBig 1−1 2θ/parenleftBig1 R+1 R′/parenrightBig/bracerightBig ; which values are correct to quantities of the order θ2̺or, which is the same thing, to quantities of the order θ; these having been neglected in the latter part of the preceding analysis, as unworthy of notice. Suppose dσis an element of the surface A, the corresponding element ofB, cut off by normals to A, will be dσ{1+θ(1 R+1 R′)}, and therefore the quantity of fluid on this last element will be ̺dσ{1+θ(1 R+1 R′)}substitut- ing for ̺its value before found, ̺=−̺{1−θ(1 R+1 R′)}and neglecting θ2̺, we obtain −̺dσ. 32 Application to electricity. the same quantity as on the element dσof the first surface. If therefore, we conceive any portion of the surface A, bounded by a closed curve, and a corresponding portion of the surface B, which would be cut off by a nor- mal to A, passing completely round this curve; the sum of the two quan ti- ties of electric fluid, on these corresponding portions, wil l be equal to zero; and consequently, in an electrical jar any how charged, the t otal quantity of electricity in the jar may be found, by calculating the qua ntity, on the two exterior surfaces of the metallic coatings farthest fro m the glass, as the portions of electricity, on the two surfaces adjacent to the glass, exactly neu- tralise each other. This results will appear singular, when we consider the immense quantity of fluid collected on these last surfaces, a nd moreover, it would not be difficult to verify it by experiment. As a particular example of the use of this general theory: sup pose a spherical conductor whose radius a, to communicate with the inside of an electrical jar, by means of a long slender wire, the outside b eing in com- munication with the common reservoir; and let the whole be ch arged: then Prepresenting the density of the electricity on the surface o f the conduc- tor, which will be very nearly constant, the value of the pote ntial function within the sphere, and, in consequence of the communication established, at the inner coating Aalso, will be 4 πaPvery nearly, since we may, without sensible error, neglect the action of the wire and jar itself in calculating it. Hence β=4πaP and β′=0, and the equations (8), by neglecting quantities of the order θ, give ̺=β 4πθ=a θPand ̺=−β 4πθ=−a θP. We thus obtain, by the most simple calculation, the values of the densities, at any point on either of the surfaces Aand B, next the glass, when that on the spherical conductor is known. The theory of the condenser, electrophorous, etc. depends u pon what has been proved in this article; but these are details into which the limits of this Essay will not permit me to enter; there is, however, one resu lt, relative to charging a number of jars by cascade, that appears worthy of n otice, and which flows so readily from the equations (8), that I cannot re frain from introducing it here. Conceive any number of equal and similar insulated Leyden ph ials, of uniform thickness, so disposed, that the exterior coating o f the first, may communicate with the interior one of the second; the exterio r one of the second, with the interior one of the third; and so on througho ut the whole series, to the exterior surface of the last, which we will sup pose in commu- nication with the earth. Then, if the interior of the first phia l, be made to communicate with the prime conductor of an electrical machi ne, in a state of action, all the phials will receive a certain charge, and t his mode of op- erating is called charging by cascade . Permitting ourselves to neglect the 8. 33 small quantities of free fluid on the exterior surfaces of the metallic coat- ings, and other quantities of the same order, we may readily d etermine the electrical state of each phial in the series: for thus, the eq uations (8) become ̺=β−β′ 4πθ,̺=β′−β 4πθ. Designating now, by an index at the foot of any letter, the num ber of the phial to which it belongs, so that, ̺1may belong to the first, ̺2to the second phial, and so on; we shall have, by supposing their whole numb er to be n, since θis the same for every one, ̺1=β1−β′ 1 4πθ̺1=β′ 1−β1 4πθ ̺2=β2−β′ 2 4πθ̺2=β′ 2−β2 4πθ etc. etc. ̺n=βn−β′ n 4πθ̺n=β′ n−βn 4πθ Now βrepresents the value of the total potential function, withi n the prime conductor and interior coating of the first phial, and in consequence of the communications established in this system, we have in regular suc- cession, beginning with the prime conductor, and ending wit h the exterior surface of the last phial, which communicates with the earth , β=β1;β′ 1=β2;β′ 2=β3; etc. . . . β′ n−1=βn;β′ n=0 0=̺1+̺2; 0=̺2+̺3; etc. . . . 0 =̺n−1+̺n. But the first system of equations gives 0 =̺s+̺s, whatever whole number smay be, and the second line of that just exhibited is expresse d by 0 = ̺s−1+̺s, hence by comparing these two last equations ̺s=̺s−1, which shows that every phial of the system is equally charged . Moreover, if we sum up vertically, each of the columns of the first system, there will arise in virtue of the second ̺1+̺2+̺3· · · · · · +̺n=β 4πθ ̺1+̺2+̺3· · · · · · +̺n=β 4πθ. We therefore see, that the total charge of all the phials is pr ecisely the same, as that which one only would receive, if placed in communicat ion with the same conductor, provided its exterior coating were conn ected with the earth. Hence this mode of charging, although it may save time , will never produce a greater accumulation of fluid, than would take plac e, if one phial only were employed. 34 Application to electricity. 9. Conceive now, a hollow shell of perfectly conducting matter , of any form and thickness whatever, to be acted upon by any electrified bod ies, situate without it; and suppose them to induce an electrical state in the shell; then will this induced state be such, that the total action on an el ectrified particle, placed any where within it, will be absolutely null. For let Vrepresent the value of the total potential function, at any p oint pwithin the shell, then we shall have at its inner surface, whi ch is a closed one, V=β; βbeing the constant quantity, which expresses the value of th e potential function, within the substance of the shell, where the elect ricity is, by the supposition, in equilibrium, in virtue of the actions of the exterior bodies, combined with that arising from the electricity induced in t he shell itself. Moreover, Vevidently satisfies the equation 0 =δV, and has no singular value within the closed surface to which it belongs: it follo ws therefore, from art. 5, that its general value is V=β, and as the forces acting upon p, are given by the differentials of V, these forces are evidently all equal to zero . If, on the contrary, the electrified bodies are all within the s hell, and its exterior surface is put in communication with the earth, it i s equally easy to prove, that there will not be the slightest action on any el ectrified point exterior to it; but, the action of the electricity induced on its inner surface, by the electrified bodies within it, will exactly balance the d irect action of the bodies themselves. Or more generally: Suppose we have a hollow, and perfectly conducting shell, bo unded by any two closed surfaces, and a number of electrical bodies ar e placed, some within and some without it, at will; then, if the inner surfac e and interior bodies be called the interior system; also, the outer surfac e and exterior bodies the exterior system; all the electrical phenomena of the interior sys- tem, relative to attractions, repulsions, and densities, w ill be the same as would take place if there were no exterior system, and the inn er surface were a perfect conductor, put in communication with the eart h; and all those of the exterior system will be the same, as if the interi or one did not exist, and the outer surface were a perfect conductor, conta ining a quan- tity of electricity, equal to the whole of that originally co ntained in the shell itself, and in all the interior bodies. This is so direct a consequence of what has been shown in artic les 4 and 5, that a formal demonstration would be quite superfluous, as it is easy to see, the only difference which could exist, relative to the i nterior system, between the case where there is an exterior system, and where there is not one, would be in the addition of a constant quantity, to the to tal potential 10. 35 function within the exterior surface, which constant quant ity must neces- sarily disappear in the differentials of this function, and consequently, in the values of the attractions, repulsions, and densities, w hich all depend on these differentials alone. ln the exterior system there i s not even this difference, but the total potential function exterior to th e inner surface is precisely the same, whether we suppose the interior system t o exist or not. 10. The consideration of the electrical phenomena, which arise from spheres variously arranged, is rather interesting, on account of th e case with which all the results obtained from theory, may be put to the test of experiment; but, the complete solution of the simple case of two spheres o nly, previ- ously electrified, and put in presence of each other, requires the aid of a profound analysis, and has been most ably treated by M. P OISSON (M´ em. de l’Institut. 1811). Our object, in the present article, is merely to give one or two examples of determinations, relative to the distribu tion of electricity on spheres, which may be expressed by very simple formulae. Suppose a spherical surface whose radius is a, to be covered with electric matter, and let its variable density be represented by ̺; then if, as in the M´ ec. C´ eleste , we expand the potential function V, belonging to a point pwithin the sphere, in the form V=U(0)+U(1)r a+U(2)r2 a2+U(3)r3 a3+etc.; rbeing the distance between pand the centre of the sphere, and U(0),U(1) etc. functions of the two other polar co-ordinates of p, it is clear, by what has been shown in the admirable work just mentioned, that the potential function V′, arising from the same spherical surface, and belonging to a point p′, exterior to this surface, at the distance r′from its centre, and on the radius rproduced, will be V′=U(0)a r′+U(1)a2 r′2+U(2)a3 r′3+etc. If, therefore, we make V=ϕ(r), and V′=ψ(r′), the two functions ϕand ψ will satisfy the equation ψ(r) =a rϕ/parenleftBiga2 r/parenrightBig or ϕ(r) =a rψ/parenleftBiga2 r/parenrightBig . But (art. 4) 4π̺=−dV dw−dV′ dw′= +dV dr−dV′ dr′=ϕ′(a)−ψ′(a), 36 Application to electricity. and the equation between ϕand ψ, in its first form, gives, by differentiation, ψ′(r) =−a r2ϕ/parenleftBiga2 r/parenrightBig −a3 r3ϕ′/parenleftBiga2 r/parenrightBig . Making now r=athere arises ψ′(a) =−ϕ(a) a−ϕ′(a); ϕ′and ψ′being the characteristics of the differential co-efficients ofϕand ψ, according to L AGRANGE ’s notation. In the same way the equation in its second form yields ϕ′(a) =−ψ(a) a−ψ′(a); These substituted successively, in the equation by which ̺is determined, we have the following (9.)  4π̺=2ϕ′(a) +ϕ(a) a=2dV dr+V a 4π̺=−2ψ′(a)−ψ(a) a=−2dV′ dr′−V′ a. If, therefore, the value of the potential function be known, either for the space within the surface, or, for that without it, the value o f the density ̺ will be immediately given, by one or other of these equations . From what has preceded, we may readily determine how the elec tric fluid will distribute itself, in a conducting sphere whose ra dius is a, when acted upon by any bodies situate without it; the electrical s tate of these bodies being given. In this case, we have immediately the val ue of the po- tential function arising from them. Let this value, for any p oint pwithin the sphere, be represented by A;Abeing a function of the radius r, and two other polar co-ordinates. Then the whole of the electric ity will be car- ried to the surface (art. 1), and if Vbe the potential function arising from this electrified surface, for the same point p, we shall have, in virtue of the equilibrium within the sphere, V+A=βor V=β−A βbeing a constant quantity. This value of Vbeing substituted in the first of the equations (9), there results 4π̺=−2dA dr−A a+β a: the horizontal lines indicating, as before, that the quanti ties under them belong to the surface itself. In case the sphere communicates with the earth, βis evidently equal to zero , and ̺is completely determined by the above: but if the sphere is insulated, and contains any quantity Qof electricity, the value of βmay be ascertained as follows: Let V′be the value of the potential function without 10. 37 the surface, corresponding to the value V=β−Awithin it; then, by what precedes V′=β r′−A′; A′being determined from Aby the following equations: A=ϕ′(r),ψ′(r) =a rϕ′/parenleftBiga2 r/parenrightBig ,A′=ψ′(r′), and r′, being the radius corresponding to the point p′, exterior to the sphere, to which A′belongs. When r′is infinite, we have evidently V′=Q r′. There- fore by equating Q r′=β r′−A′or β=Q+r′A′; r′being made infinite. Having thus the value of β, the value of ̺becomes known. To give an example of the application of the second equation i n̺; let us suppose a spherical conducting surface, whose radius is a, in communica- tion with the earth, to be acted upon by any bodies situate wit hin it, and B′to be the value of the potential function arising from them, f or a point p′ exterior to it. The total potential function, arising from t he interior bodies and surface itself, will evidently be equal to zero at this su rface, and conse- quently (art. 5), at any point exterior to it. Hence V′+B′=0;V′being due to the surface. Thus the second of the equations (9) becomes 4π̺=2dB′ dr′+B′ a. We are therefore able, by means of this very simple equation, to determine the density of the electricity induced on the surface in ques tion. Suppose now, all the interior bodies to reduce themselves to a single point P, in which a unit of electricity is concentrated, and fto be the dis- tance Pp′: the potential function arising from Pwill be1 fand hence B′=1 f; r′being, as before, the distance between p′and the centre Oof the shell. Let now brepresent the distance OP, and θthe angle POp′, then will f2= b2−2br′·cosθ+r′2. From which equation we deduce successively, /parenleftBigd f dr′/parenrightBig =r′−bcosθ f, and 2dB′ dr′=−2 f2/parenleftBigd f dr′/parenrightBig =−2r′+2b·cosθ f3 Making r′=ain this, and in the value of B′before given, in order to obtain those which belong to the surface, there results 2dB′ dr′+B′ a=−2a2+2ab·cosθ+f2 f3=b2−a2 a f3. 38 Application to electricity. This substituted in the general equation written above, the re arises ̺=b2−a2 4πa f3. IfPis supposed to approach infinitely near to the surface, so that b=a−α; αbeing an infinitely small quantity, this would become ̺=−α 2πf3. In the same way, by the aid of the equation between Aand p, the density of the electric fluid, induced on the surface of a sphere whose radius is a, when the electrified point Pis exterior to it, is found to be ̺=b2−a2 4πa f3; supposing the sphere to communicate, by means of an infinitely fine wire, with the earth, at so great a distance, that we might neglect t he influence of the electricity induced upon it by the action of P. If the distance of Pfrom the surface, be equal to an infinitely small quantity α, we shall have in this case, as in the foregoing, ̺=−α 2πf3. From what has preceded, we may readily deduce the general val ue of V, belonging to any point P, within the sphere, when Vits value at the surface is known. For (̺), the density induced upon an element dσof the surface, by a unit of electricity concentrated in P, has just been shown to be b2−a2 4πa f3; fbeing the distance P,dσ. This substituted in the general equation (6), art. 5, gives (10.) V=−/integraldisplay dσ(̺)V=a2−b2 4πa/integraldisplaydσ f3V. In the same way we shall have, when the point Pis exterior to the sphere, (11.) V=b2−a2 4πa/integraldisplaydσ f3V. The use of these two equations will appear almost immediatel y, when we come to determine the distribution of the electric fluid, on a thin spherical shell, perforated with a small circular orifice. The results just given, may be readily obtained by means of L APLACE ’s much admired analysis ( M´ ec. C´ el. Liv. 3, Ch. 2), and indeed, our general equations (9), flow very easily from the equation (2) art. 10 o f that chapter. Want of room compels me to omit these confirmations of our analy sis, and 10. 39 this I do the more freely, as the manner of deducing them must i mmediately occur, to any one who has read this part of the M´ ecanique C´ eleste . Conceive now, two spheres Sand S′, whose radii are aand a′, to com- municate with each other by means of an infinitely fine wire: it is required to determine the ratio of the quantities of electric fluid on t hese spheres, when in a state of equilibrium; supposing the distance of the ir centres to be represented by b. The value of the potential function, arising from the electr icity on the surface of S, at a point p, placed in its centre, is /integraldisplay̺dσ a=1 a/integraldisplay ̺dσ=Q a; dσbeing an element of the surface of the sphere, ̺the density of the fluid on this element, and Qthe total quantity on the sphere. If now, we represent by F′, the value of the potential function for the same point p, arising from S′, we shall have, by adding together both parts, F′+Q a; the value of the total potential function belonging to p, the centre of S. In like manner, the value of this function at p′, the centre of S′, will be F+Q′ a′: Fbeing the part arising from S, and Q′the total quantity of electricity on S′. But in consequence of the equilibrium of the system, the tota l potential function throughout its whole interior is a constant quanti ty. Hence F′+Q a=F+Q′ a′. Although it is difficult to assign the rigorous values of Fand F′; yet, when the distance between the surfaces of the two spheres is c onsiderable, compared with the radius of one of them, it is easy to see, that Fand F′will be very nearly the same, as if the electricity on each of the sp heres produc- ing them, was concentrated in their respective centres, and therefore, we have very nearly F=Q band F′=Q′ b. These substituted in the above, there arises Q b+Q′ a′=Q′ b+Q ai.e. Q/parenleftBig1 a−1 b/parenrightBig =Q′/parenleftBig1 a′−1 b/parenrightBig . Thus the ratio of QtoQ′is given by a very simple equation, whatever may be the form of the connecting wire, provided it be a very fine one . If we wished to put this result of calculation to the test of ex periment, it would be more simple to write Pand P′for the mean densities of the fluid 40 Application to electricity. on the spheres, or those which would be observed when, after b eing con- nected as above, they were separated to such a distance, as no t to influence each other sensibly. Then since Q=4πa2Pand Q′=4πa′2P′, we have by substitution, etc. P P′=a(b−a) a′(b−a′). We therefore see, that when the distance bbetween the centres of the spheres is very great, the mean densities will be inversely a s the radii; and these last remaining unchanged, the density on the smaller s phere will decrease, and that on the larger increase in a very simple way , by making them approach each other. Lastly, let us endeavour to determine the law of the distribu tion of the electric fluid, when in equilibrium on a very thin spherical s hell, in which there is a small circular orifice. Then, if we neglect quantiti es of the order of the thickness of the shell, compared with its radius, we ma y consider it as an infinitely thin spherical surface, of which the greater s egment Sis a perfect conductor, and the smaller one sconstitutes the circular orifice. In virtue of the equilibrium, the value of the potential functi on, on the con- ducting segment, will be equal to a constant quantity, as F, and if there were no orifice, the corresponding value of the density would b e F 4πa; abeing the radius of the spherical surface, Moreover on this su pposition, the value of the potential function for any point P, within the surface, would be F. Let therefore,F 4πa+̺represent the general value of the density, at any point on the surface of either segment of the sphere, and F+V, that of the cor- responding potential function for the point P. The value of the potential function for any point on the surface of the sphere, will be F+V, which equated to F, its value on S, gives for the whole of this segment 0=V. Thus the equation (10) of this article becomes V=a2−b2 4πa/integraldisplaydσ f3V; the integral extending over the surface of the smaller segme ntsonly, which, without sensible error, may be considered as a plane. 10. 41 But, since it is evident, that ̺is the density corresponding to the potential function V, we shall have for any point on the segments, treated as a plan e, ̺=−1 2πdV dw, as it is easy to see, from what has been before shown (art. 4); dwbeing perpendicular to the surface, and directed towards the cent re of the sphere; the horizontal line always serving to indicate quantities b elonging to the surface. When the point Pis very near the plane s, and zis a perpendicular from Pupon s,zwill be a very small quantity, of which the square and higher powers may be neglected. Thus b=a−z, and by substitution V=z 4π/integraldisplaydσ f3V; the integral extending over the surface of the small plane s, and fbeing, as before, the distance P,dσ. NowdV dw=dV dzat the surface of s, andz f3=−d dz1 f; hence ̺=−1 2πdV dw=−1 2πdV dz=−1 4π2d dz/integraldisplayz dσ f3V=1 4π2d2 dz2/integraldisplaydσ fV; provided we suppose z=0 at the end of the calculus. Now the density F 4πa+̺, upon the surface of the orifice s, is equal to zero, and therefore, we have for the whole of this surface ̺=−F 4πa. Hence by substitution (12.)−Fπ a=d2 dz2/integraldisplaydσ fV; the integral extending over the whole of the plane s, of which dσis an element, and zbeing supposed equal to zero, after all the operations have been effected. It now only remains to determine the value of Vfrom this equation. For this, let βnow represent the linear radius of s, and y, the distance between its centre Cand the foot of the perpendicular z: then if we conceive an infinitely thin oblate spheroid, of uniform density, of which the circular plane sconstitutes the equator, the value of the potential functio n at the point P, arising from this spheroid, will be ϕ=k/integraldisplaydσ f/radicalBig β2−η2; ηbeing the distance dσ,C, and ka constant quantity. The attraction exerted by this spheroid, in the direction of the perpendicular z, will be −dϕ dz, and by the known formulae relative to the attractions of homogeneo us spheroids, we have −dϕ dz=3Mz β3(tanθ−θ); 42 Application to electricity. Mrepresenting the mass of the spheroid, and θbeing determined by the equations α2=1 2(z2+y2−β2) +1 2/radicalBig (z2+y2−β2)2+4β2z2 tanθ=β α. Supposing now zvery small, since it is to vanish at the end of the calculus, and y<β, in order that the point Pmay fall within the limits of s, we shall have by neglecting quantities of the order z2compared with those retained θ=1 2π−z/radicalbig β2−y2; and consequently −dϕ dz=−d dzk/integraldisplaydσ f/radicalBig β2−η2=3M/radicalbig β2−y2 β3−3Mπ 2β3z. This expression, being differentiated again relative to z, gives d2 dz2k/integraldisplaydσ f/radicalBig β2−η2=3Mπ 2β3. But the mass Mis given by M=k/integraldisplay dσ/radicalBig β2−η2=2πk/integraldisplay ηdη/radicalBig β2−η2=2πkβ3 3. Hence by substitution d2 dz2k/integraldisplaydσ f/radicalBig β2−η2=π2k: which expression is rigorously exact when z=0. Comparing this result with the equation (12) of the present article, we see that if V=k/radicalbig β2−η2, the constant quantity kmay be always determined, so as to satisfy (12). In fact, we have only to make π2k=−Fπ ai. e. k=−F aπ. Having thus the value of V, the general value of Vis known, since V=a2−b2 4πa/integraldisplaydσ f3V=−a2−b2 4πazd dz/integraldisplaydσ f/braceleftBig V=k/radicalBig β2−η2/bracerightBig =a2−b2 4πaz× −dϕ dz=a2−b2 4πaz×3Mz β3(tanθ−θ) =−a2−b2 2πa2F(tanθ−θ). The value of the potential function, for any point Pwithin the shell, be- ingF+V, and that in the interior of the conducting matter of the shel l being constant, in virtue of the equilibrium, the value ̺′of the density, at 11. 43 any point on the inner surface of the shell, will be given imme diately by the general formula (4) art. 4. Thus ̺′=−1 4πdV dw=1 4πdV db=+F 4π2a(tanθ−θ): in which equation, the point Pis supposed to be upon the element dσ′of the interior surface, to which ̺′belongs. If now, Rbe the distance between C, the centre of the orifice, and dσ′, we shall have R2=y2+z2, and by ne- glecting quantities of the orderβ2 R2compared with those retained, we have successively α=R,θ=β Rand tan θ−θ=1 3θ3=β3 3R3. Thus the value of ̺′becomes ̺′=F 12π2aβ3 R3. In the same way, it is easy to show from the equation (11) of thi s article, that ̺′′, the value of the density on an element dσ′′of the exterior surface of the shell, corresponding to the element dσ′of the interior surface, will be ̺′′=F 4πa+̺′, which, on account of the smallness of ̺′for every part of the surface, except very near the orifice s, is sensibly constant and equal toF 4πa, therefore ̺′ ̺′′=β3 3π·R3: which equation shows, how very small the density within the s hell is, even when the orifice is considerable. 11. The determination of the electrical phenomena, which resul t from long metallic wires, insulated and suspended in the atmosphere, depends upon the most simple calculations. As an example, let us conceive two spheres Aand B, connected by a long slender conducting wire; then ̺dx dy dz rep- resenting the quantity of electricity in an element dx dy dz of the exterior space, (whether it results from the ground in the vicinity of the wire hav- ing become slightly electrical, or from a mist, or even a pass ing cloud,) and rbeing the distance of this element from A’s centre; also r′its distance from B’s, the value of the potential function at A’s centre, arising from the whole exterior space, will be /integraldisplay̺dx dy dz r, 44 Application to electricity. and the value of the same function at B’s centre, will be /integraldisplay̺dx dy dz r′, the integrals extending over all the space exterior to the co nducting system under consideration. If now, Qbe the total quantity of electricity on A’s surface, and Q′that on B’s, their radii being aand a′; it is clear, the value of the potential function atA’s centre, arising from the system itself, will be Q a; seeing that, we may neglect the part due to the wire, on accoun t of its fine- ness, and that due to the other sphere, on account of its dista nce. In a similar way, the value of the same function at B’s centre, will be found to be Q′ a′. But (art. 1), the value of the total potential function must b e constant troughout the whole interior of the conducting system, and t herefore, its value at the two centres must be equal; hence Q a+/integraldisplay̺dx dy dz r=Q′ a′+/integraldisplay̺dx dy dz r′. Although ̺, in the present case, is exceedingly small, the integrals co n- tained in this equation, may not only be considerable, but ve ry great, since they are of the second dimension relative to space. The spher es, when at a great distance from each other, may therefore become highly electrical, ac- cording to the observations of experimental philosophers, and the charge they will receive in any proposed case may readily be calcula ted; the value of̺being supposed given. When one of the spheres, Bfor instance, is con- nected with the ground, Q′will be equal to zero , and consequently Qim- mediately given. If, on the contrary, the whole system were i nsulated and retained its natural quantity of electricity, we should hav e, neglecting that on the wire, 0=Q+Q′, and hence Qand Q′would be known. If it were required, to determine the electrical state of the sphere A, when in communication with a wire, of which one extremity is eleva ted into the atmosphere, and terminates in a fine point p, we should only have to make the radius of B, and consequently, Q′, vanish in the expression before given. Hence in this case Q a=/integraldisplay̺dx dy dz r′−/integraldisplay̺dx dy dz r; r′being the distance between pand the element dx dy dz . Since the object of the present article, is merely to indicate the cause of som e phenomena of 12. 45 atmospherical electricity, it is useless to extend it to a gr eater length, more particularly, as the extreme difficulty of determining corre ctly the electri- cal state of the atmosphere at any given time, precludes the p ossibility of putting this part of the theory to the test of accurate experi ment. 12. Supposing the form of a conducting body to be given, it is in ge neral im- possible to assign, rigorously, the law of the density of the electric fluid on its surface in a state of equilibrium, when not acted upon by a ny exterior bodies, and, at present, there has not even been found any con venient mode of approximation applicable to this problem. It is, however , extremely easy to give such forms to conducting bodies, that this law shall b e rigorously assignable by the most simple means. The following method, d epending upon art. 4 and 5, seems to give to these forms the greatest deg ree of gen- erality of which they are susceptible, as, by a tentative pro cess, any form whatever might be approximated indefinitely. Take any continuous function V′, of the rectangular co-ordinates x′,y′,z′, of a point p′, which satisfies the partial differential equation 0 =δV′, and vanishes when p′is removed to an infinite distance from the origin of the co-ordinates. Choose a constant quantity b, such that V′=bmay be the equation of a closed surface A, and that V′may have no singular values, so long as p′ is exterior to this surface: then if we form a conducting body , whose outer surface is A, the density of the electric fluid in equilibrium upon it, wil l be represented by ̺=−h 4πdV′ dw′, and the potential function due to this fluid, for any point p′, exterior to the body, will be hV′; hbeing a constant quantity dependant upon the total quantity of electric- ityQ, communicated to the body. This is evident from what has been proved in the articles cited. LetRrepresent the distance between p′, and any point within A; then the potential function arising from the electricity upon it , will be expressed byQ R, when Ris infinite. Hence the condition Q R=hV′(Rbeing infinite ) which will serve to determine h, when Qis given. In the application of this general method, we may assume for V′, ei- ther some analytical expression containing the co-ordinat es of p′, which is known to satisfy the equation 0 =δV′, and to vanish when p′is removed to an infinite distance from the origin of the co-ordinates; as , for instance, 46 Application to electricity. some of those given by L APLACE (M´ ec. C´ eleste , Liv. 3, Ch. 2), or, the value [of] a potential function, which would arise from a quantity of electricity any how distributed within a finite space, at a point p′without that space; since this last will always satisfy the conditions to which V′is subject. It may be proper to give an example of each of these cases. In th e first place, let us take the general expression given by L APLACE , V=U(0) r+U(1) r2+U(2) r3+etc., then, by confining ourselves to the two first terms, the assumed v alue of V′ will be V′=U(0) r+U(1) r2; rbeing the distance of p′from the origin of the co-ordinates, and U(0),U(1), etc. functions of the two other polar co-ordinates θand ̟. This expression by changing the direction of the axes, may always be reduced t o the form V′=2a r+h2cosθ r2; aand kbeing two constant quantities, which we will suppose positi ve. Then if bbe a very small positive quantity, the form of the surface giv en by the equation V′=b, will differ but little from a sphere, whose radius is2a b: by gradually increasing b, the difference becomes greater, until b=a2 k2; and afterwards, the form assigned by V′=b, becomes improper for our pur- pose. Making therefore b=a2 k2, in order to have a surface differing as much from a sphere, as the assumed value of V′admits, the equation of the sur- face Abecomes V′=2a r+h2cosθ r2=a2 k2. From which we obtain r=k2 a(1+√ 2 cos1 2θ). If now ϕrepresents the angle formed by drand dw′, we have −dr r dθ=√ 2 sin1 2θ 2+2√ 2 cos1 2θ=tanϕ, and as the electricity is in equilibrium upon A, the force with which a par- ticle p, infinitely near to it, would be repelled, must be directed alo ngdw′: but the value of this force is −dV′ dw′, and consequently its effect in the direc- tion of the radius r, and tending to increase it, will be −dV′ dw′cosϕ. This last quantity is equally represented by −dV′ dr, and therefore −dV′ dr=−dV′ dw′cosϕ; 12. 47 the horizontal lines over quantities, indicating, as befor e, that they belong to the surface itself. The value of −(dV′ dw′), deduced from this equation, is dV′ dw′=1 cosϕdV′ dr=1 cosϕ/braceleftbigg2a r2+2k2cosθ r3/bracerightbigg =2a√ 2 cos1 2θ r2cosϕ, this substituted in the general value of ϕ, before given, there arises ̺=−h 4πdV′ dw′=ha√ 2 cos1 2θ 2πr2cosϕ. Supposing Qis the quantity of electricity communicated to the surface, the condition Q R=hV (where Ris infinite) before given, becomes, since rmay here be substituted for R, seeing that it is measured from a point within the surface, Q r=2ah ri. e. h=Q 2a. We have thus the rigorous value of ̺for the surface Awhose equation is r=k2 a(1+√ 2 cos1 2θ)when the quantity Qof electricity upon it is known, and by substituting for rand htheir values just given, there results ̺=Qa2√ 2 cos1 2θ 4πk4cosϕ(1+√ 2 cos1 2θ)2. Moreover the value of the potential function for the point p′whose polar co-ordinates are r,θ, and ̟, is hV′=Q r+Qk2cosθ 2ar2. From which we may immediately deduce the forces acting on any point p′ exterior to A. In tracing the surface A,θis supposed to extend from θ=0 to θ=π, and ̟, from ̟=0 to ̟=2π: it is therefore evident, by constructing the curve whose equation is r=k2 a(1+√ 2 cos1 2θ), that the parts about P, where θ=π, approximate continually in form to- wards a cone whose apex is P, and as the density of the electricity at Pis null, in the example before us, we may make this general infer ence: when any body whatever, has a part of its surface in the form of a con e, directed inwards; the density of the electricity in equilibrium upon it, will be null at 48 Application to electricity. its apex, precisely the reverse of what would take place, if i t were directed outwards, for then, the density at the apex would become infini te.∗ As a second example, we will assume for V′, the value of the potential function arising from the action of a line uniformly covered with electricity. Let 2 abe the length of the line, ythe perpendicular falling from any point p′ upon it, xthe distance of the foot of this perpendicular from the middl e of the line, and x′that of the element dx′from the same point: then taking the element dx′, as the measure of the quantity of electricity it contains, t he assumed value of V′will be V′=/integraldisplaydx′ /radicalbig y2+ (x−x′)2=loga−x+/radicalbig y2+ (a−x)2 −a−x+/radicalbig y2+ (a+x)2; the integral being taken from x′=−atox′= + a. Making this equal to a constant quantity log b, we shall have, for the equation of the surface A, a−x+/radicalbig y2+ (a−x)2 −a−x+/radicalbig y2+ (a+x)2=b, which by reduction becomes 0=y2(1−b2)2+x2·4b(1−b)2−a2·4b(1+b)2. We thus see that this surface is a spheroid produced by the rev olution of an ellipsis about its greatest diameter; the semi-transverse axis being a1+b 1−b=β and semi-conjugate a2√ b 1−b=γ. By differentiating the general value of V′, just given, and substituting for yits value at the surface A, we obtain dV′ dx=−2x1−b 1+b (1+b 1−b)2a2−(1−b 1+b)2x2=−2aβx β4−a2x2. ∗Since this was written, I have obtained formulae serving to e xpress, generally, the law of the distribution of the electric fluid near the apex Oof a cone, which forms part of a conducting surface of revolution having the same axis. Fro m these formulae it results that, when the apex of the cone is directed inwards, the densi ty of the electric fluid at any point p, near to it, is proportional to rn−1;rbeing the distance Op, and the exponent nvery nearly such as would satisfy the simple equation (4n+2)β=3π: where 2 βis the angle at the summit of the cone. If 2 βexceeds π, this summit is directed outwards, and when the excess is not very considerable, nwill be given as above: but 2 βstill increasing, until it becomes 2 π−2γ; the angle 2 γat the summit of the cone, which is now directed outwards, being very small, nwill be given by 2 nlog2 γ=1, and in case the conducting body is a sphere whose radius is b, on which Prepresents the mean density of the electric fluid, ̺, the value of the density near the apex O, will be determined by the formula ̺=2Pbn (a+b)γ/parenleftbiggr a/parenrightbiggn−1 ; abeing the length of the cone. 13. 49 Now writing ϕfor the angle formed by dwand dw′, we have 1 cosϕ=ds −dy=1−b 2x√ b/radicalBigg/parenleftbigg1+b 1−b/parenrightbigg4 a2−x2=/radicalbig β4−a2x2 γx; dsbeing an element of the generating ellipsis. Hence, as in the preceding example, we shall have, dV′ dw′=1 cosϕ·dV′ dx=−2aβ γ/radicalbig β4−a2x2. On the surface Atherefore, in this example, the general value of ̺is ̺=−h 4πdV′ dw′=ahβ 2πγ/radicalbig β4−a2x2, and the potential function for any point p′, exterior to A, is hV′=hloga−x+/radicalbig y2+ (a−x)2 −a−x+/radicalbig y2+ (a+x)2. Making now xand yboth infinite, in order that p′may be at an infinite distance, there results hV′=2ah/radicalbig x2+y2, and thus the condition determining h, inQ, the quantity of electricity upon the surface, is, since Rmay be supposed equal to/radicalbig x2+y2, Q R=hV′=2ah/radicalbig x2+y2i. e. h=Q 2a. These results of our analysis, agree with what has been long k nown con- cerning the law of the distribution of electric fluid on the su rface of a spher- oid, when in a state of equilibrium. 13. In what has preceded, we have confined ourselves to the conside ration of perfect conductors. We will now give an example of the appl ication of our general method, to a body that is supposed to conduct elec tricity im- perfectly, and which will, moreover, be interesting, as it s erves to illustrate the magnetic phenomena, produced by the rotation of bodies u nder the influence of the earth’s magnetism. If any solid body whatever of revolution, turn about its axis , it is required to determine what will take place, when the matter of this sol id is not per- fectly conducting, supposing it under the influence of a cons tant electrical force, acting parallel to any given right line fixed in space, t he body being originally in a natural state. 50 Application to electricity. Letβdesignate the coercive force of the body, which we will suppo se analogous to friction in its operation, so that as long as the total force acting upon any particle within the body is less than β, its electrical state shall remain unchanged, but when it begins to exceed β, a change shall ensue. In the first place, suppose the constant electrical force, whi ch we will designate by b, to act in a direction parallel to a line passing through the centre of the body, and perpendicular to its axis of revoluti on; and let us consider this line as the axis of x, that of revolution being the axis of z, and ythe other rectangular co-ordinate of a point p, within the body and fixed in space. Thus, if Vbe the value of the total potential function for the same point p, at any instant of time, arising from the electricity of the b ody and the exterior force, bx+V will be the part due to the body itself at the same instant; sin ce−bxis that due to the constant force b, acting in the direction of x, and tending to increase it. If now we make z=rcosθ,x=rsinθcos̟,y=rsinθsin̟; the angle ̟being supposed to increase in the direction of the body’s rev o- lution, the part due to the body itself becomes brsinθcos̟+V. Were we to suppose the value of the potential function Vgiven at any in- stant, we might find its value at the next instant, by conceivin g, that whilst the body moves forward through the infinitely small angle dω, the electric- ity within it shall remain fixed, and then be permitted to move, until it is in equilibrium with the coercive force. Now the value of the potential function at p, arising from the body itself, after having moved through the angle dω(the electricity being fixed), will evidently be obtained by changing ̟into ̟−dωin the expression just given, and is therefore brsinθcos̟+V+brsinθsin̟dω−dV d̟dω, adding now the part −bx=−brsinθcos̟due to the exterior bodies, and restoring x,y, etc. we have sincedV d̟=−ydV dx+xdV dy, V+dω/braceleftbigg by+ydV dx−xdV dy/bracerightbigg for the value of the total potential function at the end of the next instant, the electricity being still supposed fixed. We have now only to determine what this will become, by allowing the electricity to move fo rward until the total forces acting on points within the body, which may n ow exceed the coercive force by an infinitely small quantity, are again r educed to an equilibrium with it. If this were done, we should, when the in itial state of 13. 51 the body was given, be able to determine, successively, its s tate for every one of the following instants. But since it is evident from th e nature of the problem, that the body, by revolving, will quickly arrive at a permanent state, in which the value of Vwill afterwards remain unchanged and be independent of its initial value, we will here confine ourselv es to the de- termination of this permanent state. It is easy to see, by con sidering the forces arising from the new total potential function, whose value has just been given, that in this case the electricity will be in motio n over the whole interior of the body, and consequently β2=/parenleftbiggdV dx/parenrightbigg2 +/parenleftbiggdV dy/parenrightbigg2 +/parenleftbiggdV dz/parenrightbigg2 , which equation expresses that the total force to move any par ticle p, within the body, is just equal to β, the coercive force. Now if we can assume any value for V, satisfying the above, and such, that it shall reproduce its elf after the electricity belonging to the new total potential f unction (art. 7), is allowed to find its equilibrium with the coercive force, it is e vident this will be the required value, since the rest of the electricity is ex actly in equilib- rium with the exterior force b, and may therefore be here neglected. To be able to do this the more easily, conceive two new axes X′,Y′, in advance of the old ones X,Y, and making the angle γwith them; then the value of the new potential function, before given, becomes V+dω/braceleftbigg by′cosγ+bx′sinγ+y′dV dx′−x′dV dy′/bracerightbigg , which, by assuming V=βγ′, and determining γby the equation 0=bsinγ−β reduces itself to y′(β+bcosγdω). Considering now the symmetrical distribution of the electr icity belonging to this potential function, with regard to the plane whose eq uation is 0 =y′, it will [be] evident that, after the electricity has found it s equilibrium, the value of Vat this plane must be equal to zero : a condition which, combined with the partial differential equation before given, will s erve to determine, completely, the value of Vat the next instant, and this value of Vwill be V=βy′. We thus see that the assumed value of Vreproduces itself at the end of the following instant, and is therefore the one required bel onging to the permanent state. If the body had been a perfect conductor, the value of Vwould evidently have been equal to zero , seeing that it was supposed originally in a natural state: that just found is therefore due to the rotation combi ned with the coercive force, and we thus see that their effect is to polari se the body in 52 Application to electricity. the direction of y′positive, making the angle1 2π+γwith the direction of the constant force b; and the degree of polarity will be the same as would be produced by a force equal to β, acting in this direction on a perfectly conducting body of the same dimensions. We have hitherto supposed the constant force to act in a direc tion paral- lel to the equatorial plane of the body, but whatever may be it s direction, we may conceive it decomposed into two; one equal to bas before, and par- allel to this plane, the other perpendicular to it, which las t will evidently produce no effect on the value of V, as this is due to the coercive force, and would still be equal to zero under the influence of the new forc e, if the body conducted electricity perfectly. Knowing the value of the potential function at the surface of t he body, due to the rotation, its value for all the exterior space may b e considered as determined (art. 5), and if the body be a solid sphere, may e asily be expressed analytically; for it is evident (art. 7), from the value of Vjust given, that even in the present case all the electricity will be confined to the surface of the solid; and it has been shown (art. 10), that whe n the value of the potential function for the point pwithin a spherical surface, whose radius is a, is represented by ϕ(r), the value of the same function for a point p′, situate without this sphere, on the prolongation of r, and at the distance r′from its centre, will be a r′ϕ/parenleftbigga2 r′/parenrightbigg . But we have seen that the value of Vdue to the rotation, for the point p, is V=βγ′=βrcosθ′; θ′being the angle formed by the ray rand the axis of y′; the corresponding value for the point p′will therefore be V′=βa3cosθ′ r′2. And hence, by differentiation, we immediately obtain the va lue of the forces acting on any particle situate without the sphere, wh ich arise from its rotation; but, if we would determine the total forces ari sing from the sphere, we must, to the value of the potential function just f ound, add that part which would be produced by the action of the constant for ce upon this sphere, when it is supposed to conduct electricity perf ectly, which will be given in precisely the same way as the former. In fact, fdesignating the constant force, and θ′′the angle formed by rand a line parallel to the direction of f, the potential function arising from it, for the point p, will be −r fcosθ′′, 13. 53 and consequently the part arising from the electricity, ind uced by its action, must be +f rcosθ′′, seeing that their sum ought to be equal to zero. The correspon ding value for the point p′, exterior to the sphere, is therefore f a3cosθ′′ r′2, this added to the value of V′, before found, will give the value of the total potential function for the point p′, arising from the sphere itself. It will be seen when we come to treat of the theory of magnetism , that the results of this theory, in general, agree very nearly with th ose which would arise from supposing the magnetic fluid at liberty to move fro m one part of a magnetized body to another; at least, for bodies whose ma gnetic pow- ers admit of considerable developement, as iron and nickel f or example; the errors of the latter supposition being of the order 1 −gonly; gbeing a constant quantity dependant on the nature of the body, which in those just mentioned, differs very little from unity. It is therefore e vident that when a solid of revolution, formed of iron, is caused to revolve sl owly round its axis, and placed under the influence of the earth’s magnetic f orce f, the act of revolving, combined with the coercive force βof the body, will produce a new polarity, whose direction and quantity will be very nea rly the same as those before determined. Now fhaving been supposed resolved into two forces, one equal to bin the plane of the body’s equator, and another perpendicular to this plane; if βbe very small compared with b, the angle γ will be very small, and the direction of the new polarity will be very nearly at right angles to the direction of b, a result which has been confirmed by many experiments: but by our analysis we moreover see that wh enbis suf- ficiently reduced, the angle γmay be rendered sensible, and the direction of the new polarity will then form with that of bthe angle1 2π+γ;γbeing determined by the equation sinγ=β b. This would be very easily put to the test of experiment by empl oying a solid sphere of iron. The values of the forces induced by the rotation of the body, w hich would be observed in the space exterior to it, may be obtained by differentiating that of V′before given, and will be found to agree with the observations of Mr. B ARLOW (Phil. Tran. 1825), on the supposition of β being very small. As the experimental investigation of the magnetic phenomen a devel- oped by the rotation of bodies, has lately engaged the attent ion of several distinguished philosophers, it may not be amiss to consider the subject in a more general way, as we shall thus not only confirm the precedin g analysis, 54 Application to electricity. but be able to show with what rapidity the body approaches tha t perma- nent state, which it has been the object of the preceding part of this article to determine. Let us now, therefore, consider a body Afixed in space, under the in- fluence of electric forces which vary according to any given l aw; then we might propose to determine the electrical state of the body, after a certain interval of time, from the knowledge of its initial state; su pposing a con- stant coercive force to exist within it. To resolve this in it s most general form, it would be necessary to distinguish between those par ts of the body where the fluid was at rest, from the forces acting there being less than the coercive force, and those where it would be in motion; moreov er these parts would vary at every instant, and the problem therefore becom e very intri- cate: were we however to suppose the initial state so chosen, that the total force to move any particle pwithin A, arising from its electric state and exterior actions, was then just equal to the coercive force β; also, that the al- teration in the exterior forces should always be such, that i f the electric fluid remained at rest during the next instant, this total force sh ould no where be less than β; the problem would become more easy, and still possess a great degree of generality. For in this case, when the fluid is moveable, the whole force tending to move any particle pwithin A, will, at every instant, be exactly equal to the coercive force. If therefore x,y,z, represent the co- ordinates of p, and Vthe value of the total potential function at any instant of time t, arising from the electric state of the body and exterior for ces, we shall have the equation (a.) β2=/parenleftBigdV dx/parenrightBig2 +/parenleftBigdV dy/parenrightBig2 +/parenleftBigdV dz/parenrightBig2 , whose general integral may be thus constructed: Take the value of Varbitrarily over any surface whatever S, plane or curved, and suppose three rectangular co-ordinates w,w′,w′′, whose origin is at a point PonS: the axis of wbeing a normal to S, and those of w′,w′′, in its plane tangent. Then the values ofdV dw′anddV dw′′are known at the point P, and the value ofdV dwwill be determined by the equation /parenleftBigdV dw/parenrightBig2 +/parenleftBigdV dw′/parenrightBig2 +/parenleftBigdV dw′′/parenrightBig2 =β2, which is merely a transformation of the above. Take now another point P′, whose co-ordinates referred to these axes are dV dw,dV dw′anddV dw′′, and draw a right line Lthrough the points P,P′, then will the value of Vat any point p, on L, be expressed by V0+βλ; λbeing the distance Pp, measured along the line L, considered as increas- ing in the direction PP′, and V0, the given value of VatP. For it is very easy to see that the value of Vfurnished by this construction, satisfies the partial 13. 55 differential equation ( a), and is its general integral, moreover the system of lines L,L′,L′′, etc. belonging to the points P,P′,P′′, etc. on S, are evidently those along which the electric fluid tends to move, and will mo ve during the following instant. Let now V+DVrepresent what Vbecomes at the end of the time t+dt; substituting this for Vin (a) we obtain (b.) 0 =dV dx·dDV dx+dV dy·dDV dy+dV dz·dDV dz. Then, if we designate by D′V, the augmentation of the potential function, arising from the change which takes place in the exterior for ces during the element of time dt, DV−D′V will be the increment of the potential function, due to the co rresponding al- terations D̺and D̺′in the densities of the electric fluid at the surface of A and within it, which may be determined from DV−D′Vby art. 7. But, by the known theory of partial differential equations, the mos t general value ofDVsatisfying ( b), will be constant along every one of the lines L,L′,L′′, etc., and may vary arbitrarily in passing from one of them to a nother: as it is also along these lines the electric fluid moves during th e instant dt, it is clear the total quantity of fluid in any infinitely thin nee dle, formed by them, and terminating in the opposite surfaces of A, will undergo no alteration during this instant. Hence therefore (c.) 0 =/integraldisplay D̺′dv+D̺dσ+D̺′dσ′; dvbeing an element of the volume of the needle, and dσ,dσ′, the two ele- ments of A’s surface by which it is terminated. This condition, combin ed with the equation ( b), will completely determine the value of DV, and we shall thus have the value of the potential function V+DV, at the instant of time t+dt, when its value V, at the time t, is known. As an application of this general solution; suppose the body Ais a solid of revolution, whose axis is that of the co-ordinate z, and let the two other axes X,Ysituate in its equator, be fixed in space. If now the exterior el ectric forces are such that they may be reduced to two, one equal to c, acting parallel to z, the other equal to b, directed parallel to a line in the plane (xy), making the variable angle ϕwith X; the value of the potential function arising from the exterior forces, will be −zc−xbcosϕ−ybsinϕ; where band care constant quantities, and ϕvaries with the time so as to be constantly increasing. When the time is equal to t, suppose the value of V to be V=β(xcos̟+ysin̟): 56 Application to electricity. then the system of lines L,L′,L′′, will make the angle ̟with the plane (xz), and be perpendicular to another plane whose equation is 0=xcos̟+ysin̟. If during the instant of time dt,ϕbecomes ϕ+Dϕ, the augmentation of the potential function due to the elementary change in the exter ior forces, will be D′V= (xsinϕ−ycosϕ)b Dϕ; moreover the equation ( b) becomes (b′.) 0 =cos̟·dDV dx+sin̟·dDV dy, and therefore the general value of DVis DV=DF{ycos̟−xsin̟;z}; DFbeing the characteristic of an infinitely small arbitrary fun ction. But, it has been before remarked that the value of DV will be completely de- termined, by satisfying the equation ( b) and the condition ( c). Let us then assume DF{ycos̟−xsin̟;z}=h Dϕ(ycos̟−xsin̟); hbeing a quantity independent of x,y,z, and see if it be possible to deter- mine hso as to satisfy the condition ( c). Now on this supposition DV−D′V=h Dϕ(ycos̟−xsin̟)−(xsinϕ−ycosϕ)b dϕ =Dϕ/bracketleftbig y(hcos̟+bcosϕ)−x(hsin̟+bcosϕ)/bracketrightbig . The value of D̺′corresponding to this potential function is (art. 7) D̺′=0, and on account of the parallelism of the lines L,L′etc. to each other, and to A’s equator dσ=dσ′. The condition ( c) thus becomes (c′.) 0 =D̺+D̺1: D̺and D̺1, being the elementary densities on A’s surface at opposite ends of any of the lines L,L′, etc. corresponding to the potential func- tion DV−D′V. But it is easy to see from the form of this function, that these elementary densities at opposite ends of any line perp endicular to a plane whose equation is 0=y(hcos̟+bcosϕ)−x(hsin̟+bcosϕ), are equal and of contrary signs, and therefore the condition (c) will be sat- isfied by making this plane coincide with that perpendicular t oL,L′, etc., whose equation, as before remarked, is 0=xcos̟+ysin̟; 13. 57 that is the condition ( c) will be satisfied, if hbe determined by the equation hcos̟+bcosϕ sin̟=−hsin̟+bsinϕ cos̟, which by reduction becomes 0=h+bcos(ϕ−̟), and consequently V+DV=β(xcos̟+ysin̟) +h Dϕ(ycos̟−xsin̟) =βx/parenleftbig cos̟+b βsin̟cos(ϕ−̟)Dϕ/parenrightbig+βy/parenleftbig sin̟−b βcos̟cos(ϕ−̟)Dϕ/parenrightbig =βxcos/parenleftbig ̟+b βcos(ϕ−̟)Dϕ/parenrightbig+βysin/parenleftbig ̟−b βcos(ϕ−̟)Dϕ/parenrightbig When therefore ϕis augmented by the infinitely small angle Dϕ,̟receives the corresponding increment −b βcos(ϕ−̟)Dϕ, and the form of Vremains unaltered; the preceding reasoning is consequently applic able to every in- stant, and the general relation between ϕand ̟[is] expressed by 0=D̟+b βcos(ϕ−̟)Dϕ: a common differential equation, which by integration gives H·eϕcotan γ=sin(3 4π−1 2γ+1 2̟−1 2ϕ) sin(1 4π+1 2γ+1 2̟−1 2ϕ); Hbeing an arbitrary constant, and γ, as in the former part of this article, the smallest root of 0=bsinγ−β. Let̟0and ϕ0, be the initial values of ̟and ϕ; then the total potential function at the next instant, if the electric fluid remained fix ed, would be V′=β(xcos̟0+ysin̟0) + ( xsinϕ0−ycosϕ0)b dϕ, and the whole force to move a particle p, whose co-ordinates are , x,y,z, /radicalBigg/parenleftBigdV′ dx/parenrightBig2 +/parenleftBigdV′ dy/parenrightBig2 +/parenleftBigdV′ dz/parenrightBig2 =β+dϕ·bsin(ϕ0−̟0), which, in order that our solution may be applicable, must not be less than β, and consequently the angle ϕ0−̟0must be between 0 and π: when this is the case, ̟is immediately determined from ϕby what has preceded. In fact, by finding the value of Hfrom the initial values ̟0and ϕ0, and making ζ=1 4π+1 2γ+1 2̟−1 2ϕ, we obtain tanζ=tanζ0 e(ϕ−ϕ0)cotγ+tanγtanζ0(e(ϕ−ϕ0)cotγ−1); ζ0being the initial value of ζ. 58 Application to electricity. We have, in the latter part of this article, considered the bo dyAat rest, and the line X′, parallel to the direction of b, as revolving round it: but if, as in the former, we now suppose this line immovable and th e body to turn the contrary way, so that the relative motion of X′toXmay remain unaltered, the electric state of the body referred to the axe sX,Y,Z, evi- dently depending on this relative motion only, will consequ ently remain the same as before. In order to determine it on the suppositio n just made, letX′be the axis of x′, one of the co-ordinates of p, referred to the rectan- gular axes X′,Y′,Z′, also y′,z, the other two; the direction X′Y′, being that in which Arevolves. Then, if ̟′be the angle the system of lines L,L′, etc. forms with the plane (x′,z), we shall have ̟+̟′=ϕ; ϕ, as before stated, being the angle included by the axes X,X′. Moreover the general values of Vand Qwill be V=β(x′cos̟′+y′sin̟′)and ζ=1 4π+1 2γ−1 2̟′, and the initial condition, in order that our solution may be a pplicable, will evidently become ϕ0−̟0=̟′ 0=a quantity betwixt 0 and π. As an example, let tan γ=1 10, since we know by experiment that γis generally very small; then taking the most unfavorable case , viz. where ̟′ 0=0, and supposing the body to make one revolution only, the val ue ofζ, determined from its initial one, ζ0=1 4π+1 2γ−1 2̟′, will be found extremely small and only equal to a unit in the 27th decimal pl ace. We thus see with what rapidity ζdecreases, and consequently, the body approaches to a permanent state, defined by the equation 0=ζ=1 4π+1 2γ−1 2̟′. Hence, the polarity induced by the rotation is ultimately di rected along a line, making an angle equal to1 2π+γwith the axis X′, which agrees with what was shown in the former part of this article. The value of Vat the body’s surface being thus known at any instant whatever, that of the potential function at a point p′exterior to the body, together with the forces acting there, will be immediately d etermined as before. 14. 59 Application of the preceding results to the theory of magnetism. 14. The electric fluid appears to pass freely from one part of a con tinuous conductor to another, but this is by no means the case with the magnetic fluid, even with respect to those bodies which, from their ins tantly return- ing to a natural state the moment the forces inducing a magnet ic one are removed, must be considered, in a certain sense, as perfect c onductors of magnetism. C OULOMB , I believe, was the first who proposed to consider these as formed of an infinite number of particles, each of whic h conducts the magnetic fluid in its interior with perfect freedom, but w hich are so con- stituted that it is impossible there shall be any communicat ion of it from one particle to the next. This hypotesis is now generally adopte d by philoso- phers, and its consequences, as far as they have hitherto bee n developed, are found to agree with observation; we will therefore admit it in what follows, and endeavour thence to deduce, mathematically, t he laws of the distribution of magnetism in bodies of any shape whatever. Firstly, let us endeavour to determine the value of the poten tial function, arising from the magnetic state induced in a very small body A, by the action of constant forces directed parallel to a given right line; the body being composed of an infinite number of particles, all perfect conductors of magnetism and originally in a natural state. In order to de duce this more immediately from art. 6, we will conceive these forces t o arise from an infinite quantity Qof magnetic fluid, concentrated in a point pon this line, at an infinite distance from A. Then the origin Oof the rectangular co-ordinates being any where within A, ifx,y,z, be those of the point p, and x′,y′,z′, those of any other exterior point p′, to which the potential function Varising from Abelongs, we shall have (vide M´ ec. C´ el. Liv. 3) V=U(0) r′+U(1) r′2+U(2) r′3+etc.; r′=/radicalbig x′2+y′2+z′2being the distance Op′. Moreover, since the total quantity of magnetic fluid in Ais equal to zero , U(0)=0. Supposing now r′very great compared with the dimensions of the body, all the terms afterU(0) r′in the expression just given will be exceed- ingly small compared with this, by neglecting them, therefo re, and substi- tuting for U(1)its most general value, we obtain V=U(1) r′2=Ax′+By′+Cz′ r′3; A,B,C, being quantities independent of x′,y′,z′, but which may contain x,y,z. 60 Application to magnetism. Now (art. 6) the value of Vwill remain unaltered, when we change x,y,z, into x′,y′,z′, and reciprocally. Therefore, V=Ax′+By′+Cz′ r′3=A′x+B′y+C′z r3; A′,B′,C′, being the same functions of x′,y′,z′, as A,B,C, are of x,y,z. Hence it is easy to see that Vmust be of the form V=a′′xx′+b′′yy′+c′′zz′+e′′(xy′+x′y) +f′′(xz′+x′z) +g′′(yz′+y′z) r3r′3; a′′,b′′,c′′,e′′,f′′,g′′, being constant quantities. IfX,Y,Z, represent the forces arising from the magnetism concentra ted inp, in the directions of x,y,z, positive, we shall have X=−Qx r3;Y=−Qy r3;Z=−Qz r3; and therefore Vis of the form V=a′Xx′+b′Yy′+c′Zz′+e′(Xy′+Yx′) +f′(Xz′+Zx′) +g′(Yz′+Zy′) r′3 a′,b′, etc. being other constant quantities. But it will always be possible to determine the situation of three rectangular axes, so that e,f, and gmay each be equal to zero, and consequently Vbe reduced to the following sim- ple form (a.) V=aXx′+bYy′+cZz′ r′3; a,b, and cbeing three constant quantities. When Ais a sphere, and its magnetic particles are either spherical , or, like the integrant particles of non-crystalized bodies , arranged in a confused manner; it is evident the constant quantities a′,b′,c′, etc. in the general value of V, must be the same for every system of rectangular co-ordinates, and consequently we must have a′=b′=c′,e′=0,f′=0, and g′=0, therefore in this case (b.) V=a′(Xx′+Yy′+Zz′) r′3; a′being a constant quantity dependant on the magnitude and nat ure of A. The formula ( a) will give the value of the forces acting on any point p′, arising from a mass Aof soft iron or other similar matter, whose magnetic state is induced by the influence of the earth’s action; suppo sing the dis- tance Ap′to be great compared with the dimensions of A, and if it be a solid of revolution, one of the rectangular axes, say X, must coincide with the axis of revolution, and the value of Vreduce itself to V=a′Xx′+b′(Yy′+Zz′) r′3; 14. 61 a′and b’ being two constant quantities dependant on the form and nat ure of the body. Moreover the forces acting in the directions of x′,y′,z′, positive, are expressed by −/parenleftBigdV dx′/parenrightBig ,−/parenleftBigdV dy′/parenrightBig ,−/parenleftBigdV dz′/parenrightBig . We have thus the means of comparing theory with experiment, b ut these are details into which our limits will not permit us to enter. The formula ( b), which is strictly correct for an infinitely small sphere, on the supposition of its magnetic particles being arranged in a confused manner, will, in fact, form the basis of our theory, and altho ugh the preced- ing analysis seems sufficiently general and rigorous, it may n ot be amiss to give a simpler proof of this particular case. Let, therefore , the origin Oof the rectangular co-ordinates be placed at the centre of the i nfinitely small sphere A, and OBbe the direction of the parallel forces acting upon it; then, since the total quantity of magnetic fluid in Ais equal to zero , the value of the potential function V, at the point p′, arising from A, must evidently be of the form V=kcosθ r′2; r′representing as before the distance Op′, and θthe angle formed between the line Op′, and another line OD fixed in A. If now fbe the magni- tude of the force directed along OB, the constant kwill evidently be of the form k=a′f;a′being a constant quantity. The value of V, just given, holds good for any arrangement, regular or irregular, of the magne tic particles composing A, but on the latter supposition, the value of Vwould evidently remain unchanged, provided the sphere, and consequently th e line OD, re- volved round OBas an axis, which could not be the case unless OBand OD coincided. Hence θ=angle BOp′and V=a′fcosθ r′2. Let now α,β,γ, be the angles that the line Op′=r′makes with the axes ofx,y,z, and α′,β′,γ′, those which OBmakes with the same axes; then substituting for cos θits value cos αcosα′+cosβcosβ′+cosγcosγ′, we have, since fcosα=X,fcosβ=Y,fcosγ=Z, (b.) V=a′(Xcosα+Ycosβ+Zcosγ) r′2. Which agrees with the equation (b), seeing that cos α=x′ r′, cos β=y′ r′, cosγ=z′ r′. 62 Application to magnetism. 15. Conceive now, a body A, of any form, to have a magnetic state induced in its particles by the influence of exterior forces, it is cle ar that if dvbe an element of its volume, the value of the potential function ar ising from this element, at any point p′whose co-ordinates are x′,y′,z′, must, since the total quantity of magnetic fluid in dvis equal to zero , be of the form (a.)dv/bracketleftbig X(x′−x) +Y(y′−y) +Z(z′−z)/bracketrightbig r3; x,y,z, being the co-ordinates of dv,rthe distance p′,dvand X,Y,Z, three quantities dependant on the magnetic state induced in dv, and serving to define this state. If therefore dv′be an infinitely small volume within the body Aand inclosing the point p′, the potential function arising from the whole Aexterior to dv′, will be expressed by /integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3; the integral extending over the whole volume of Aexterior to dv′. It is easy to show from this expression that, in general, alth ough dv′be infinitely small, the forces acting in its interior vary in mag nitude and di- rection by passing from one part of it to another; but, when dv′is spherical, these forces are sensibly constant in magnitude and directi on, and conse- quently, in this case, the value of the potential function in duced in dv′by their action, may be immediately deduced from the preceding article. Letψ′represent the value of the integral just given, when dv′is an in- finitely small sphere. The force acting on p′arising from the mass exterior todv′, tending to increase x′, will be −/parenleftbiggdψ′ dx′/parenrightbigg ; the line above the differential co-efficient indicating that it is to be obtained by supposing the radius of dv′to vanish after differentiation, and this may differ from the one obtained by first making the radius vanish, and after- wards differentiating the resulting function of x′,y′,z′, which last being rep- resented as usual bydψ′ dx′, we have dψ′ dx′=d dx′/integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3 dψ′ dx′=d dx′/integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3; the first integral being taken over the whole volume of Aexterior to dv′, and the second over the whole of Aincluding dv′. Hence dψ′ dx′−dψ′ dx′==d dx′/integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3; 15. 63 the last integral comprehending the volume of the spherical particle dv′ only, whose radius ais supposed to vanish after differentiation. In order to effect the integration here indicated, we may remark that X,Yand Zare sensibly constant within dv′, and may therefore be replaced by X′,Y′and Z′, their values at the centre of the sphere dv′, whose co-ordinates are x′,y′,z′; the required integral will thus become /integraldisplay dx dy dzX′(x′−x) +Y′(y′−y) +Z′(z′−z) r3. Making for a moment E=X′x+Y′y+Z′y, we shall have X′=dE dx,Y′= dE dy,Z′=dE dz, and as alsox′−x r3=d1 r dx;y′−y r3=d1 r dy;z′−z r3=d1 r dzthis integral may be written/integraldisplay dx dy dz/braceleftbiggdE dx·d1 r dx+dE dy·d1 r dy+dE dz·d1 r dz/bracerightbigg , which since δE=0, and δ1 r=0, reduces itself by what is proved in art. 3, to −/integraldisplaydσ r/parenleftBigdE dw/parenrightBig =(because dw=−da)/integraldisplaydσ rdE da; the integral extending over the whole surface of the sphere dv′, of which dσis an element; rbeing the distance p′,dσ, and dwmeasured from the surface towards the interior of dv′. Now/integraltextdσ rdE daexpresses the value of the potential function for a point p′, within the sphere, supposing its surface every where covered with electricity whose density isdE da, and may very easily be obtained by No. 13, Liv. 3, M´ ec. C´ eleste . In fact, using for a moment the notation there employed, supposing the origin of the pol ar co-ordinates at the centre of the sphere, we have E=E′+a[X′cosθ+Y′sinθcos̟+Z′sinθsin̟]; E′being the value of Eat the centre of the sphere. Hence dE da=X′cosθ+Y′sinθcos̟+Z′sinθsin̟, and as this is of the form U(1)(Vide M´ ec. C´ eleste Liv. 3.), we immediately obtain/integraldisplaydσ rdE da=4 3πr′{X′cosθ′+Y′sinθ′cos̟′+Z′sinθ′sin̟′}, where r′,θ′,̟′, are the polar co-ordinates of p′. Or by restoring x′,y′, and z′ /integraldisplaydσ rdE da=4 3π/braceleftbig X′(x′−x′) +Y′(y′−y′) +Z′(z′−z′)/bracerightbig . 64 Application to magnetism. Hence we deduce successively dψ′ dx′−dψ′ dx′=d dx′/integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3 =d dx′/integraldisplaydσ rdE da=4 3π/braceleftbig X′(x′−x′) +Y′(y′−y′) +Z′(z′−z′)/bracerightbig=4 3πX′. If now we make the radius avanish, X′must become equal to X′, the value ofXat the point p′, and there will result dψ′ dx′−dψ′ dx′=4 3πX′i. e.dψ′ dx′=dψ′ dx′−4 3πX′. Butdψ′ dx′expresses the value of the force acting in the direction of xpos- itive, on a point p′within the infinitely small sphere dv′, arising from the whole of Aexterior to dv′; substituting now fordψ′ dx′its value just found, the expression of this force becomes 4 3πX′−dψ′ dx′. Supposing V′to represent the value of the potential function at p′, arising from the exterior bodies which induce the magnetic state of A, the force due to them acting in the same direction, is −dV′ dx′, and therefore the total force in the direction of x′positive, tending to induce a magnetic state in the spherical element dv′, is 4 3πX′−dψ′ dx′−dV′ dx′=X. In the same way, the total forces in the directions of y′and z′positive, acting upon dv′, are shown to be 4 3πY′−dψ′ dy′−dV′ dy′=Y, and,4 3πZ′−dψ′ dz′−dV′ dz′=Z. By the equation ( b′) of the preceding article, we see that when dv′is a per- fect conductor of magnetism, and its particles are not regul arly arranged, the value of the potential function at any point p′′, arising from the mag- netic state induced in dv′by the action of the forces X,Y,Z, is of the form a′(Xcosα+Ycosβ+Zcosγ) r′2; r′being the distance p′′,dv′, and α,β,γ, the angles which r′forms with the axes of the rectangular co-ordinates. If then x′′,y′′,z′′, be the co-ordinates ofp′′, this becomes, by observing that here a′=kdv′, kdv′/bracketleftbig X(x′′−x′) +Y(y′′−y′) +Z(z′′−z′)/bracketrightbig r′3, 15. 65 kbeing a constant quantity dependant on the nature of the body . The same potential function will evidently be obtained from the expr ession ( a) of this article, by changing dv,p′, and their co-ordinates, into dv′,p′′, and their co- ordinates; thus we have dv′/bracketleftbig X′(x′′−x′) +Y′(y′′−y′) +Z′(z′′−z′)/bracketrightbig r′3. Equating these two forms of the same quantity, there results the three fol- lowing equations: X′=kX=4 3πk X′−kdψ′ dx′−kdV′ dx′ Y′=kY=4 3πk Y′−kdψ′ dy′−kdV′ dy′ Z′=kZ=4 3πk Z′−kdψ′ dz′−kdV′ dz′, since the quantities x′′,y′′,z′′, are perfectly arbitrary. Multiplying the first of these equations by dx′, the second by dy′, the third by dz′, and taking their sum, we obtain 0= (1−4 3πk)(X′dx′+Y′dy′+Z′dz′) +k dψ′+k dV′. Butdψ′and dV′being perfect differentials, X′dx′+Y′dy′+Z′dz′must be so likewise, making therefore dϕ′=X′dx′+Y′dy′+Z′dz′ the above, by integration, becomes const = (1−4 3πk)ϕ′+kψ′+kV′. Although the value of kdepends wholly on the nature of the body under consideration, and is to be determined for each by experimen t, we may yet assign the limits between which it must fall. For we have, in t his theory, supposed the body composed of conducting particles, separa ted by inter- vals absolutely impervious to the magnetic fluid; it is there fore clear the magnetic state induced in the infinitely small sphere dv′, cannot be greater than that which would be induced, supposing it one continuou s conduct- ing mass, but may be made less in any proportion, at will, by au gmenting the non-conducting intervals. When dv′is a continuous conductor, it is easy to see the value of the potential function at the point p′′, arising from the magnetic state induced in it by the action of the forces X,Y,Zwill be 3dv 4π·X(x′′−x′) +Y(y′′−y′) +Z(z′′−z′) r′3, seeing that3dv 4π=a3;arepresenting, as before, the radius of the sphere dv′. By comparing this expression with that before found, when dv′was not a continuous conductor, it is evident kmust be between the limits 0 and3 4π, 66 Application to magnetism. or, which is the same thing, k=3g 4π;gbeing any positive quantity less than 1. The value of k, just found, being substituted in the equation serving to determine ϕ′, there arises const = (1−g)ϕ′+3g 4π(ψ′+V′). Moreover ψ′=/integraldisplay dx dy dzX(x′−x) +Y(y′−y) +Z(z′−z) r3 =/integraldisplay dx dy dz/braceleftbiggdϕ dx·d1 r dx+dϕ dy·d1 r dy+dϕ dz·d1 r dz/bracerightbigg =4πϕ′−/integraldisplay dσϕ/parenleftbiggd1 r dw/parenrightbigg (art. 3); the triple integrals extending over the whole volume of A, and that rela- tive to dσover its surface, of which dσis an element; the quantities ϕand d1 r dwbelonging to this element. We have, therefore, by substitut ion const = (1+2g)ϕ′+3g 4π/parenleftBigg V′−/integraldisplay dσϕ/parenleftbiggd1 r dw/parenrightbigg/parenrightBigg . Now δ′V′=0, and δ′/integraltext dσϕ/parenleftbigd1 r dw/parenrightbig and consequently δ′ϕ′=0; the symbol δ′ referring to x′,y′,z′, the co-ordinates of p′; or, since x’,y′and z′are arbitrary, by making them equal to x,y, and z, respectively, there results 0=δϕ, in virtue of which, the value of ψ′, by article 3, becomes (b.) ψ′=−/integraldisplaydσ r/parenleftbiggdϕ dw/parenrightbigg ; rbeing the distance p′,dσ, and(dϕ dw)belonging to dσ. The former equation serving to determine ϕ′gives, by changing x′,y′,z′, into x,y,z, (c.) const = (1−g)ϕ+3g 4π(ψ+V); ϕ,ψand Vbelonging to a point p, within the body, whose co-ordinates arex,y,z. It is moreover evident from what precedes that, the functio nsϕ, ψand Vsatisfy the equations 0 =δϕ, 0=δψand 0 =δVand have no singular values in the interior of A. The equations ( b) and ( c) serve to determine ϕand ψ, completely, when the value of Varising from the exterior bodies is known, and therefore they enable us to assign the magnetic state of every part of th e body A, seeing that it depends on X,Y,Z, the differential co-efficients of ϕ. It is also 15. 67 evident that ψ′, when calculated for any point p′, not contained within the body A, is the value of the potential function at this point arising from the magnetic state induced in A, and therefore this function is always given by the equation ( b). The constant quantity g, which enters into our formulae, depends on the nature of the body solely, and, in a subsequent article, its v alue is deter- mined for a cylindric wire used by C OULOMB . This value differs very little from unity: supposing therefore g=1, the equations ( b) and ( c) become (b′.) ψ′=−/integraldisplaydσ r/parenleftbiggdϕ dw/parenrightbigg ; (c′.) const =ψ+V, evidently the same, in effect, as would be obtained by consid ering the mag- netic fluid at liberty to move from one part of the conducting b ody to an- other; the density ̺being here replaced by/parenleftbigdϕ dw/parenrightbig , and since the value of the potential function for any point exterior to the body is, on e ither suppo- sition, given by the formula ( b), the exterior actions will be precisely the same in both cases. Hence, when we employ iron, nickel, or sim ilar bodies, in which the value of gis nearly equal to 1, the observed phenomena will differ little from those produced on the latter hypothesis, except when one of their dimensions is very small compared with the others, i n which case the results of the two hypotheses differ widely, as will be se en in some of the applications which follow. If the magnetic particles composing the body were not perfec t conduc- tors, but indued with a coercive force, it is clear there migh t always be equi- librium, provided the magnetic state of the element dv′was such as would be induced by the forcesdψ dx′+dV′ dx′+A′,dψ dy′+dV′ dy′+B′anddψ dz′+dV′ dz′+C′, instead ofdψ dx′+dV′ dx′,dψ dy′+dV′ dy′anddψ dz′+dV′ dz′; supposing the resultant of the forces A′,B′,C′, no where exceeds a quantity β, serving to measure the co- ercive force. This is expressed by the condition A′2+B′2+C′2<β2 The equation ( c) would then be replaced by 0= (1−g)dϕ+3g 4π(dψ+dV+A dx+B dy+C dz); A,B,C, being any functions of x,y,z, as A′,B′,C′, are of x′,y′,z′, subject only to the condition just given. It would be extremely easy so to modify the preceding theory, as to adapt it to a body whose magnetic particles are regularly arranged , by using the equation ( a) in the place of the equation ( b) of the preceding article; but, as observation has not yet offered any thing which would indica te a regular arrangement of magnetic particles, in any body hitherto exa mined, it seems 68 Application to magnetism. superfluous to introduce this degree of generality, more par ticularly as the omission may be so easily supplied. 16. As an application of the general theory contained in the prec eding arti- cle, suppose the body Ato be a hollow spherical shell of uniform thickness, the radius of whose inner surface is a, and that of its outer one a′; and let the forces inducing a magnetic state in A, arise from any bodies whatever, situate at will, within or without the shell. Then since in th e interior of A’s mass 0 =δϕ, and 0 =δV, we shall have ( M´ ec. C´ el. Liv. 3) ϕ=∑ϕ(i)ri+∑ϕ(i) ′r−i−1V=∑U(i)ri+∑U(i) ′r−i−1; rbeing the distance of the point p, to which ϕand Vbelong, from the shell’s centre, ϕ(0),ϕ(1), etc. — U(0),U(1), etc. functions of θand ̟, the two other polar co-ordinates of p, whose nature has been fully explained by L APLACE in the work just cited; the finite integrals extending from i=0 toi=∞. If now, to prevent ambiguity, we enclose the rof equation ( b) art. 15 in a parenthesis, it will become ψ=/integraldisplaydσ (r)/parenleftbiggdϕ dw/parenrightbigg ; (r)representing the distance p,dσ, and the integral extending over both surfaces of the shell. At the inner surface we havedϕ dw=dϕ drand r=a: hence the part of ψdue to this surface is −/integraldisplaydσ (r)dϕ dr=/integraldisplaydσ (r)∑iϕ(i)ai−1+/integraldisplaydσ (r)∑(i+1)ϕ(i) ′a−i−2 the integrals extending over the whole of the inner surface, and dσbeing one of its elements. Effecting the integrations by the formu lae of L APLACE (M´ ec. C´ eleste , Liv. 3), we immediately obtain the part of ψ, due to the inner surface, viz. 4πa2 r∑ai (2i+1)ri/parenleftbig−iai−1ϕ(i)+ (i+1)ϕ(i) ′a−i−2/parenrightbig . In the same way the part of ψdue to the outer surface, by observing that for itdϕ dw=−dϕ drand r=a′, is found to be 4πa′∑ri (2i+1)ai′/parenleftbig iai−1 ′ϕ(i)−(i+1)ϕ(i) ′a−i−2 ′/parenrightbig . The sum of these two expressions is the complete value of ψ, which, to- gether with the values of ϕand Vbefore given, being substituted in the 16. 69 equation ( c) art. 15, we obtain const =(1−g)∑ϕ(i) ′r−i−1+ (1−g)∑ϕ(i)ri +3g 4π∑U(i) ′r−i−1+3g 4π∑U(i)ri +3ga2 r∑ai (2i+1)ri/parenleftbig−iai−1ϕ(i)+ (i+1)ϕ(i) ′a−i−2/parenrightbig +3ga′∑ri (2i+1)ai′/parenleftbig iai−1 ′ϕ(i)−(i+1)ϕ(i) ′a−i−2 ′/parenrightbig . Equating the co-efficients of like powers of the variable r, we have gener- ally, whatever imay be, 0= (1−g)ϕ(i) ′+3gai+2 2i+1/parenleftbig−iai−1ϕ(i)+ (i+1)ϕ(i) ′a−i−2/parenrightbig+3g 4πU(i) ′ 0= (1−g)ϕ(i)+3g (2i+1)ai−1 ′/parenleftbig iai−1 ′ϕ(i)−(i+1)ϕ(i) ′a−i−2 ′/parenrightbig+3g 4πU(i); neglecting the constant on the right side of the equation in ras superfluous, since it may always be made to enter into ϕ(0). If now, for abridgment, we make D= (2i+1)2(1+g) + ( i−1)(i+2)g2−9g2i(i+1)/parenleftBiga a′/parenrightBig2i+1 , we shall obtain by elimination ϕ(i)=−3g 4πU(i)(2i+1)/parenleftbig 2i+1+ (i+2)g/parenrightbig D−3g 4πU(i) ′3g(i+1)(2i+1)a−2i−1 ′ D ϕ(i) ′=−3g 4πU(i)3gi(2i+1)a2i+1 D−3g 4πU(i) ′(2i+1)/parenleftbig 2i+1+ (i−1)g/parenrightbig D. These values substituted in the expression ϕ=∑ϕ(i)ri+∑ϕ(i) ′r−i−1, give the general value of ϕin a series of the powers of r, when the potential function due to the bodies inducing a magnetic state in the sh ell is known, and thence we may determine the value of the potential functi on or arising from the shell itself, for any point whatever, either within or without it. When all the bodies are situate in the space exterior to the sh ell, we may obtain the total actions exerted on a magnetic particle in it s interior, by the following simple method, applicable to hollow shells of any shape and thickness. The equation ( c) art. 15 becomes, by neglecting the superfluous constant, 0= (1−g)ϕ+3g 4π(ψ+V). If now (ϕ)represent the value of the potential function, correspondi ng to ϕ the value of ϕat the inner surface of the shell, each of the functions (ϕ), 70 Application to magnetism. ψand V, will satisfy the equations 0 =δ(ϕ), 0=δψand 0 =δV, and moreover, have no singular values in the space within the she ll; the same may therefore be said of the function (1−g)(ϕ) +3g 4π(ψ+V), and as this function is equal to zero at the inner surface, it follows (art. 5) that it is so for any point pof the interior space. Hence 0= (1−g)(ϕ) +3g 4π(ψ+V). But ψ+Vis the value of the total potential function at the point p, aris- ing from the exterior bodies and shell itself: this function will therefore be expressed by −4π(1−g) 3g(ϕ). In precisely the same way, the value of the total potential fu nction at any point p′, exterior to the shell, when the inducing bodies are all with in it, is shown to be −4π(1−g) 3g(ϕ′); (ϕ′)being the potential function corresponding to the value of ϕat the exterior surface of the shell. Having thus the total potenti al functions, the total action exerted on a magnetic particle in any direction , is immediately given by differentiation. To apply this general solution to our spherical shell, the in ducing bodies being all exterior to it, we must first determine ϕ, the value of ϕat its inner surface, making 0 =∑U(i) ′r−i−1since there are no interior bodies, and thence deduce the value of (ϕ). Substituting for ϕ(i)and ϕ(i) ′their values before given, making U(i) ′=0 and r=a, we obtain ϕ=−3g 4π(1+2g)∑U(i)(2i+1)2ai D, and the corresponding value of (ϕ)is (M´ ec. C´ el. Liv. 3) (ϕ) =−3g 4π(1+2g)∑U(i)(2i+1)2ri D. The value of the total potential function at any point pwithin the shell, whose polar coordinates are r,θ,̟is −3g 4π(1−g)(ϕ) = ( 1−g)(1+2g)∑U(i)(2i+1)2ri D. In a similar way, the value of the same function at a point p′exterior to the shell, all the inducing bodies being within it, is found to be (1−g)(1−2g)∑U(i) ′(2i+1)2 D·ri+1; 16. 71 r,θand ̟in this expression representing the polar co-ordinates of p′. To give a very simple example of the use of the first of these form ulae, suppose it were required to determine the total action exert ed in the inte- rior of a hollow spherical shell, by the magnetic influence of the earth; then making the axis of xto coincide with the direction of the dipping needle, and designating by f, the constant force tending to impel a particle of pos- itive fluid in the direction of xpositive, the potential function V, due to the exterior bodies, will here become V=−f·x=−fcosθ·r=U(1)·r. The finite integrals expressing the value of Vreduce themselves therefore, in this case, to a single term, in which i=1, and the corresponding value ofDbeing 9 (1+g−2g2a3 a3′), the total potential function within the shell is −(1−g)(1+2g)U(1) r 1+g−2g2a3 a3′=1+g−2g2 1+g−2g2a3 a3′f·x. We therefore see that the effect produced by the intervening shell, is to reduce the directive force which would act on a very small mag netic needle, from f, to1+g−2g2 1+g−2g2a3 a3′f. In iron and other similar bodies, gis very nearly equal to 1, and therefore the directive force in the interior of a hollow spherical she ll is greatly dimin- ished, except when its thickness is very small compared with its radius, in which case, as is evident from the formula, it approaches tow ards the orig- inal value f, and becomes equal to it when this thickness is infinitely smal l. To give an example of the use of the second formula, let it be pr oposed to determine the total action upon a point p, situate on one side of an in- finitely extended plate of uniform thickness, when another po intP, con- taining a unit of positive fluid, is placed on the other side of the same plate considering it as a perfect conductor of magnetism. For this , let fall the per- pendicular PQupon the side of the plate next P, on PQprolonged, demit the perpendicular pq, and make PQ=b,Pq=u,pq=v, and t=the thickness of the plate; then, since its action is evidently e qual to that of an infinite sphere of the same thickness, whose centre is upon the line QPat an infinite distance from P, we shall have the required value of the total potential function at pby supposing a′=a+t,ainfinite, and the line PQ prolonged to be the axis from which the angle θis measured. Now in the present case V=1 Pp=1/radicalbig r2−2r(a−b)cosθ+ (a−b)2=∑U(i) ′r−i−1, 72 Application to magnetism. and the value of the potential function, as before determine d, is (1−g)(1−2g)∑(2i+1)2 DU(i) ′r−i−1. From the first expression we see that the general term U(i) ′r−i−1is a quan- tity of the order (a−b)ir−i−1. Moreover, by substituting for rits value in u (a−b)ir−i−1= (a−b)i(a−b+u)−i−1=1 ae−iu a; neglecting such quantities as are of the order1 acompared with those re- tained. The general term U(i) ′r−i−1, and consequently U(i) ′, ought therefore to be considered as functions ofi a=γ. In the finite integrals just given, the increment of iis 1, and the corresponding increment of γis1 a=dγ (because ais infinite), the finite integrals thus change themselves into o rdi- nary integrals or fluents. In fact ( M´ ec. C´ el. Liv. 3), U(i) ′always satisfies the equation d2U(i) ′ dθ2+cosθ sinθdU(i) ′ dθ+i(i+1)U(i) ′=0, and as θis infinitely small whenever Vhas a sensible value, we may elim- inate it from the above by means of the equation aθ=v, and we obtain by neglecting infinitesimals of higher orders than those retain ed, sincei a=γ, 0=d2U(i) ′ dv2+dU(i) ′ v dv+γ2U(i) ′. Hence the value U(i) ′is of the form U(i) ′=A/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv); seeing that the remaining part of the general integral becom es infinite when vvanishes, and ought therefore to be rejected. It now only rem ains to de- termine the value of the arbitrary constant A. Making, for this purpose, θ=0, i. e. v=0, we have U(i) ′= (a−b)iand/integraldisplay1 0dβ/radicalbig 1−β2=1 2π: hence (a−b)i=1 2(Aπ) i. e. A=2 π(a−b)i. By substituting for Aand rtheir values, there results U(i) ′r−i−1=2 π(a−b)i(a−b+u)−i−1/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv) =2dγ πe−γu/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv); 16. 73 becausei a=γand1 a=dγ. Writing now in the place of iits value aγ, and neglecting infinitesimal quantities, we have (2i+1)2 D=4 4+4g+g2−9g2e−2γt. Hence the value of the total potential function becomes 8 π(1−g)(1+2g)/integraldisplay∞ 0dγ·e−γu 4+4g+g2−9g2e−2γt/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv); where the integral relative to γis taken from γ=0 toγ=∞, to correspond with the limits 0 and ∞ofi, seeing that i=aγ. The preceding solution is immediately applicable to the ima ginary case only, in which the inducing bodies reduce themselves to a sin gle point P, but by the following simple artifice we may give it a much greate r degree of generality: Conceive another point P′, on the line PQ, at an arbitrary distance c from P, and suppose the unit of positive fluid concentrated in P′instead ofP; then if we make r′=Pp, and θ=∠pPQ , we shall have u=r′cosθ′, v=r′sinθ′, and the value of the potential function arising from P′will be 1 P′p=1√ r′2−2r′ccosθ′+c2=Q(0)1 r′+Q(1)c r′2+Q(2)c2 r′3+etc. Moreover, the value of the total potential function at pdue to this, arising from P′and the plate itself, will evidently be obtained by changing uinto u−cin that before given, and is therefore 8 π(1−g)(1+2g)/integraldisplay∞ 0eγcdγe−γu (2+g)2−9g2e−2γt/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv). Expanding this function in an ascending series of the powers ofc, the term multiplied by ciis 8 π(1−g)(1+2g)/integraldisplay∞ 0γiei 1·2·3···ndγe−γu (2+g)2−9g2e−2γt/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv), which, as cis perfectly arbitrary, must be the part due to the term Q(i)ci ri+1in the potential function arising from the inducing bodies. If then this function had been Q(0)k0 r′+Q(1)k1 r′2+Q(2)k2 r′3+Q(3)k3 r′4+etc.; where the successive powers c0,c1,c2etc. of care replaced by the arbitrary constant quantities k0,k1,k2, etc., the corresponding value of the total po- tential function will be given by making a like change in that due to P′. Hence if, for abridgment, we make ϕ(γ) =k0+k1 1γ+k2 1·2γ2+k3 1·2·3γ3+etc., 74 Application to magnetism. the value of this function at the point pwill be 8 π(1−g)(1+2g)/integraldisplay∞ 0ϕ(γ)dγe−γu (2+g)2−9g2e−2γt/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv). Now, if the original one due to the point Pbe called F, it is clear the expres- sion just given may be written ϕ/parenleftBig−d du/parenrightBig ·F; where the symbols of operation are separated from those of qu antity, ac- cording to A RBOGAST ’s method; thus all the difficulty is reduced to the determination of F. Resuming therefore the original supposition of the plate’s magnetic state being induced by a particle of positive fluid concentrated in P, the value of the total potential function at pwill be F=8 π(1−g)(1+2g)/integraldisplay∞ 0dγe−γu (2+g)2−9g2e−2γt/integraldisplay1 0dβ/radicalbig 1−β2cos(βγv), as was before shown. Let3g 2+g=m: we shall have F=2 π(1−m2)/integraldisplay1 0dβ (1−β2)1 2/integraldisplay∞ 0dγe−uγ 1−m2e−2γtcos(βγv) =2 π(1−m2)/integraldisplay1 0dβ (1−β2)1 2/integraldisplay∞ 0dγe−uγ(1+m2e−2γt+m4e−4γt+etc.)cos(βγv) =2 π(1−m2)/integraldisplay1 0dβ (1−β2)1 2/braceleftBigu u2+β2v2+m2(u+2t) (u+2t)2+β2v2+m4(u+4t) (u+4t)2+β2v2+etc./bracerightBig =2 π(1−m2)∑/integraldisplay1 2π 0m2iuidθ u2 i+v2sin2θ, where ui=u+2it = (1−m2)∑m2i u2 i+v2)1 2. Writing now e−vβγ√−1in the place of cos (βγv), we obtain F=8 π(1−g)(1+2g)/integraldisplay1 0dβ/radicalbig 1−β2/integraldisplay∞ 0dγe−γ(u+βv√ −1) (2+g)2−9g2e−2γt, provided we reject the imaginary quantities which may arise . In order to transform this double integral let z=3g 2+ge−γt, and we shall have F=8(1−g)(1+2g) 9πg2t/parenleftBig2+g 3g/parenrightBigu t−2/integraldisplay1 0dβ/radicalbig 1−β2/parenleftBig2+g 3g/parenrightBigβv√−1 t/integraldisplaydz·zu t−1+βv√−1 t 1−z2; the integral relative to zbeing taken from z=0 to z=3g 2+g. 16. 75 The value of 1 −g, for iron and other similar bodies, is very small, ne- glecting therefore quantities which are of the order (1−g)compared with those retained, there results (a.) F=8(1−g) 3πt/integraldisplay1 0dβ/radicalbig 1−β2/integraldisplay1 0dz 1−z2zu t−1+βv t√−1; where uand vmay have any values whatever provided they are not very great and of the ordert 1−g. IfF1represents what Fbecomes by changing u into u+2t, we have F1=8(1−g) 3πt/integraldisplay1 0dβ/radicalbig 1−β2/integraldisplay1 0z2dz 1−z2zu t−1+βv t√ −1; and consequently F−F1=8(1−g) 3πt/integraldisplay1 0dβ/radicalbig 1−β2/integraldisplay1 0dz·zu t−1+βv t√ −1 which, by effecting the integrations and rejecting the imag inary quantities, becomes F−F1=4(1−g) 3√ u2+v2=4(1−g) 3r′. Suppose now pOis a perpendicular falling from the point pupon the sur- face of the plate, and on this line, indefinitely extended in th e direction Op, take the points p1,p2,p3, etc., at the distances 2 t, 4t, 6t, etc. from p; then F1,F2,F3, etc. being the values of F, calculated for the points p1,p2,p3, etc. by the formula ( a) of this article, and r′ 1,r′ 2,r′ 3, etc. the corresponding values ofr′, we shall equally have F1−F2=4(1−g) 3r′ 1,F2−F3=4(1−g) 3r′ 2, etc.; and consequently F=4 3(1−g)/braceleftbigg1 r′+1 r′ 1+1 r′ 2+etc. in infinitum/bracerightbigg ; seeing that F∞=0. From this value of F, it is evident the total action exerted upon the point p, in any given direction pn, is equal to the sum of the actions which would be exerted without the interposition of the plat e, on each of the points p,p1,p2, etc. in infinitum, in the directions pn,p1n1,p2n2, etc. multiplied by the constant factor4 3(1−g): the lines pn,p1n1,p2n2, etc. being all parallel. Moreover, as this is the case whereve r the inducing point Pmay be situate, the same will hold good when, instead of P, we substitute a body of any figure whatever magnetized at will. Th e only condition to be observed, is, that the distance between pand every part of the inducing body be not a very great quantity of the order1 1−g. On the contrary, when the distance between pand the inducing body is great enough to render(1−g)r′ ta very considerable quantity, it will be easy 76 Application to magnetism. to show, by expanding Fin a descending series of the powers of r′, that the actions exerted upon pare very nearly the same as if no plate were interposed. We have before remarked (art. 15), that when the dimensions o f a body are all quantities of the same order, the results of the true t heory differ little from those, which would be obtained by supposing the m agnetic like the electric fluid, at liberty to move from one part of a conduc ting body to another; but when, as in the present example, one of the dim ensions is very small compared with the others, the case is widely dif ferent; for if we make grigorously equal to 1 in the preceding formulae, they will belong to the latter supposition (art. 15), and as Fwill then vanish, the interposing plate will exactly neutralize the action of any magnetic bodies however they may be situate, provided they are on the side opp osite the attracted point. This differs completely from what has been deduced above by employing the correct theory. A like difference between t he results of the two suppositions takes place, when we consider the action ex erted by the earth on a magnetic particle, placed in the interior of a holl ow spherical shell, provided its thickness is very small compared with it s radius, as will be evident by making g=1 in the formulae belonging to this case, which are given in a preceding part of the present article. 17. Since C OULOMB ’s experiments on cylindric wires magnetized to satura- tion are numerous and very accurate, it was thought this litt le work could not be better terminated, than by directly deducing from the ory such con- sequences as would admit of an immediate comparison with the m, and in order to effect this, we will, in the first place, suppose a cy lindric wire whose radius is aand length 2 λ, is exposed to the action of a constant force, equal to f, and directed parallel to the axis of the wire, and then endea vour to determine the magnetic state which will thus be induced in it. For this, letrbe a perpendicular falling from a point pwithin the wire upon its axis, and x, the distance of the foot of this perpendicular from the midd le of the axis; then fbeing directed along or positive, we shall have for the value of the potential function due to the exterior forces V=−f x, and the equations ( b), (c) (art. 15) become, by omitting the superfluous con- stant, (b) ψ′=−/integraldisplaydσ (r)/parenleftbiggdϕ dw/parenrightbigg , (c) 0 = (1−g)ϕ+3g 4πψ−3g f 4πx: 17. 77 (r), the distance p′,dσbeing inclosed in a parenthesis to prevent ambiguity, and p′being the point to which ψ′belongs. By the same article we have 0=δϕand 0 =δψ, and as ϕand ψevidently depend on xand ronly, these equations being written at length are 0=r2d2ϕ dx2+rd dr/parenleftBigr dϕ dr/parenrightBig 0=r2d2ψ dx2+rd dr/parenleftBigr dψ dr/parenrightBig . Since ris always very small compared with the length of the wire, we m ay expand ϕin an ascending series of the powers of r, and thus ϕ=X+X1r+X2r2+etc.; X,X1,X2etc. being functions of xonly. By substituting this value in the equation just given, and comparing the co-efficients of like p owers of r, we obtain ϕ=X−d2X dx2r2 22+d4X dx4r4 22·42+etc. In precisely the same way the value of ψis found to be ψ=Y−d2Y dx2r2 22+d4Y dx4r4 22·42−etc. It now only remains to find the values of Xand Yin functions of x. By supposing p′placed on the axis of the wire, the equation ( c) becomes Y=−/integraldisplaydσ (r)/parenleftbiggdϕ dw/parenrightbigg ; the integral being extended over the whole surface ofthe wir e:Y′belonging to the point p′, whose co-ordinates will be marked with an accent. The part of Y′due to the circular plane at the end of the cylinder, where x=λ, is −dX′′ dx/integraldisplaya 02πr dr (r)=−2πdX′′ dx/braceleftbigg/radicalBig (λ+x′)2+a2−λ−x′/bracerightbigg , since here dσ=2πr dr anddϕ dw=dX′′ dx, by neglecting quantities of the order a2on account of their smallness; X′′representing the value of X when x=−λ. At the other end where x= + λwe have dσ=2πr dr,dϕ dw=dX′′′ dxand consequently the part due to it is dX′′′ dx/integraldisplaya 02πr dr (r)=2πdX′′′ dx/braceleftbigg/radicalBig (λ−x′)2+a2−λ+x′/bracerightbigg , X′′′designating the value of Xwhen x= + λ. 78 Application to magnetism. At the curve surface of the cylinder dσ=2πa dx anddϕ dw=−dϕ dr=1 2ad2X dx2 provided we omit quantities of the order a2compared with those retained. Hence the remaining part due to this surface is −πa2/integraldisplaydx (r)d2X dx2; the integral being taken from x=−λtox= + λ. The total value of Y′is therefore Y′=2πdX′′′ dx/braceleftbigg/radicalBig (λ−x′)2+a2−λ+x′/bracerightbigg −2πdX′′ dx/braceleftbigg/radicalBig (λ+x′)2+a2−λ−x′/bracerightbigg −πa2/integraldisplaydx (r)d2X dx2; the limits of the integral being the same as before. If now we s ubstitute for (r)its value/radicalbig (x−x′)2+a2we shall have −πa2/integraldisplaydx (r)d2X dx2=−πa2/integraldisplaydx/radicalbig (x−x′)2+a2d2X dx2; both integrals extending from x=−λtox= + λ. On account of the smallness of a, the elements of the last integral where x is nearly equal to x′are very great compared with the others, and therefore the approximate value of the expression just given, will be −πa2Ad2X′ dx′2where A=/integraldisplaydx/radicalbig (x−x′)2+a2=2 log2µ avery nearly; the two limits of the integral being −µand+µand µso chosen that when p′is situate any where on the wire’s axis, except in the immedia te vicinity of either end, the approximate shall differ very little from the true value, which may in every case be done without difficulty. Having thus , by sub- stitution, a value of Y′free from the sign of integration, the value of Yis given by merely changing x′into xand X′into X; in this way Y=2πdX′′′ dx/braceleftbigg/radicalBig (λ−x)2+a2−λ+x/bracerightbigg −2πdX′′ dx/braceleftbigg/radicalBig (λ+x)2+a2−λ−x/bracerightbigg −πa2Ad2X dx2; The equation ( c), by making r=0, becomes 0= (1−g)X+3g 4πY−3g f 4πx, 17. 79 or by substituting for Y 0= (1−g)X−3 4(ga2A)d2X dx2−3g f 4πx +3 2gdX′′′ dx/braceleftbigg/radicalBig (λ−x)2+a2−λ+x/bracerightbigg −3 2gdX′′ dx/braceleftbigg/radicalBig (λ+x)2+a2−λ−x/bracerightbigg ; an equation which ought to hold good, for every value of x, from x=−λ tox= + λ. In those cases to which our theory will be applied, 1 −gis a small quan- tity of the same order as a2A, and thus the three terms of the first line of our equation will be of the order a2AX; making now x= + λ,3 2gdX′′′ dxais shown to be of the order a2AX′′′, and thereforedX′′′ dx÷X′′′is a small quan- tity of the order aA; but for any other value of xthe function multiplying dX′′′ dxbecomes of the order a2, and therefore we may without sensible error neglect the term containing it, and likewise suppose dX′′′ dx÷X′′′=0. In the same way by making x=−λ, it may be shown that the term con- tainingdX′′ dxis negligible, and dX′′ dx÷X′′=0. Thus our equation reduces itself to 0= (1−g)X−3 4(ga2A)d2X dx2−3g f 4πx, of which the general integral is X=3g f x 4π(1−g)+Be−βx+Ce+βx; where β2=4(1−g) 3ga2A:Band Cbeing two arbitrary constants. Determining these by the conditions 0 =dX′′′ dx÷X′′′and 0 =dX′′ dx÷X′′, we ultimately obtain X=3g f x 4π(1−g)/braceleftbigg x−eβx−e−βx β(eβλ+e−βλ)/bracerightbigg . But the density of the fluid at the surface of the wire, which wo uld produce the same effect as the magnetized wire itself, is −dϕ dw=dϕ dr=−1 2ad2X dx2very nearly, 80 Application to magnetism. and therefore the total quantity in an infinitely thin section whose breadth isdx, will be −πa2d2X dx2dx=3g fβa2 4(1−g)·eβx−e−βx eβλ+e−βλdx. As the constant quantity fmay represent the coercive force of steel or other similar matter, provided we are allowed to suppose thi s force the same for every particle of the mass, it is clear that when a wir e is magne- tized to saturation, the effort it makes to return to a natura l state must, in every part, be just equal to f, and therefore, on account of its elongated form, the degree of magnetism retained by it will be equal to t hat which would be induced in a conducting wire of the same form by the fo rcef, directed along lines parallel to its axis. Hence the precedi ng formulae are applicable to magnetized steel wires. But it has been shown b y M. B IOT (Trait´ e de Phy. Tome 3, Chap. 6), from C OULOMB ’s experiments, that the apparent quantity of free fluid in any infinitely thin section i s represented by A′(µ′−x−µ′+x)dx. This expression agrees precisely with the one before deduce d from theory, and gives, for the determination of the constants A′and µ′, the equations β=−logµ′;A′=3g fβa2 4(1−g)(eβλ+e−βλ). The chapter in which these experiments are related, contain s also a num- ber of results, relative to the forces with which magnetized wires tend to turn towards the meridian, when retained at a given angle fro m it, and it is easy to prove that this force for a fine wire, whose variable sec tion is s, will be proportional to the quantity /integraldisplay s dxdϕ dx where the wire is magnetized in any way either to saturation o r otherwise, the integral extending over its whole length. But in a cylind ric wire mag- netized to saturation, we have, by neglecting quantities of the order a2, dϕ dx=dX dx=3g f 4π(1−g)/braceleftbigg 1−eβx−e−βx eβλ+e−βλ/bracerightbigg and s=πa2, and therefore for this wire the force in question is proporti onal to 3g f a2 4(1−g)/braceleftbigg 2λ−2(eβλ−e−βλ) β(eβλ+e−βλ)/bracerightbigg . The value of g, dependant on the nature of the substance of which the nee- dles are formed, being supposed given as it ought to he, we hav e only to determine βin order to compare this result with observation. But βde- pends upon A=2 logµ a, and on account of the smallness of a,Aundergoes but little alteration for very considerable variations in µ, so that we shall be 17. 81 able in every case to judge with sufficient accuracy what value ofµought to be employed: nevertheless, as it is always desirable to av oid every thing at all vague, it will be better to determine Aby the condition, that the sum of the squares of the errors committed by employing, as we hav e done, Ad2X′ dx′2for the approximate value of/integraltext+λ −λdx√ (x−x′)2+a2shall be a minimum for the whole length of the wire. In this way I find when λis so great that quantities of the order1 βλmay be neglected: A=, 231863 −2 log aβ+2aβ; where , 231863 etc. =2 log 2 −2(A);(A)being the quantity represented byAin L ACROIX :Trait´ e du Cal. Diff. Tome 3, p. 521. Substituting the value ofAjust found in the equation β2=4(1−g) 3ga2Abefore given, we obtain 4(1−g) 3g·a2β2=, 231863 −2 log aβ+2aβ. We hence see that when the nature of the substance of which the wires are formed remains unchanged, the quantity aβis constant, and therefore β varies in the inverse ratio of a. This agrees with what M. B IOThas found by experiment in the chapter before cited, as will be evident by recollecting that β=−logµ′. From an experiment made with extreme care by C OULOMB , on a magne- tized wire whose radius was1 12inch, M. B IOThas found the value of µ′to be , 517948 ( Trait´ e de Phy. Tome 3, p. 78). Hence we have in this case aβ=−1 12logµ′=, 054823, which, according to a remark just made, ought to serve for all steel wires. Substituting this value in the equation ( a) of the present article, we obtain g=, 986636. With this value of gwe may calculate the forces with which different lengths of a steel wire whose radius is1 12inch, tend to turn towards the meridian, in order to compare the results with the table of C OULOMB ’s observations, given by M. B IOT(Trait´ e de Phy. Tome 3, p. 84). Now we have before proved that this force for any wire may be represented by K/parenleftbigg βλ−eβλ−e−βλ eβλ+e−βλ/parenrightbigg =K/parenleftbigg βλ−1−e−2βλ 1+e−2βλ/parenrightbigg ; where, for abridgment, we have supposed K=3g f a2 2β(1−g). It has also been shown that for any steel wire: aβ=, 0548235, 82 Application to magnetism. the French inch being the unit of space, and as in the present c asea=1 12, there results β=, 657882. It only remains therefore to determine Kfrom one observation, the first for example, from which we obtain K=58◦, 5 very nearly; the forces being measured by their equivalent t orsions. With this value of Kwe have calculated the last column of the following table: Length 2λ. Observed Calculated (inch) Torsion (◦). Torsion (◦). 18 288 287,9 12 172 172,1 9 115 115,3 6 59 59,3 4,5 34 33,9 3 13 13,5 The three last observations have been purposely omitted, be cause the ap- proximate equation ( a) does not bold good for very short wires. The very small difference existing between the observed and calculated results will appear the more remarkable, if we reflect that th e value of β was determined from an experiment of quite a different kind t o any of the present series, and that only one of these has been employed f or the deter- mination of the constant quantity K, which depends on f, the measure of the coercive force. The table page 87 of the volume just cited, contains another s et of ob- served torsions, for different lengths of a much finer wire who se radius a=1 12/radicalBig 38 865: hence we find the corresponding value of β=3, 13880, and the first observation in the table gives K=◦, 6448. With these values the last column of the following table has been calculated as bef ore: Length 2λ. Observed Calculated (inch) Torsion (◦). Torsion (◦). 12 11,50 11,50 9 8,50 8,46 6 5,30 5,43 3 2,30 2,39 2 1,30 1,38 1 ,35 ,42 ,5 ,07 ,084 ,25 ,02 ,012 Here also the differences between the observed and calculat ed values are extremely small, and as the wire is a very fine one, our formula i s applicable to much shorter pieces than in the former case. In general, wh en the length 17. 83 of the wire exceeds 10 or 15 times its diameter, we may employ i t without hesitation. References [1] B ARLOW , Peter: On the Temporary Magnetic Effect Induced in Iron Bodies by Ro tation. Philo- sophical Transactions 115 (1825) 317–27. doi:10.1098/rstl.1825.0016 [2] B IOT, Jean-Baptiste: Trait´ e de physique exp´ erimentale et math´ ematique 3 vols. Paris, Deter- ville 1816. [3] C AVENDISH , Henry: An Attempt to Explain Some of the Principal Phaenomena of Ele c- tricity, by means of an Elastic Fluid. Philosophical Transactions 61 (1771), 564–677. doi:10.1098/rstl.1771.0056 . [4] F OURIER , Jean-Baptiste Joseph: Th´ eorie analytique de la chaleur. Paris, F. Didot p` ere et fils, 1822. [5] F OURIER , Jean-Baptiste Joseph: Note relative aux vibrations des surfaces ´ elastiques . . . Bul- letin des sciences, par la Soci´ et´ e de Paris, 129–36 [Also i nŒuvres , vol. 2, 255–65]. [6] L ACROIX , Sylvestre Franc ¸ois: Trait´ e du calcul diff´ erentiel et du calcul int´ egral. 3 vols. Paris, J.B.M. Duprat. 1797-1800. [7] L APLACE , Pierre Simon: Trait´ e de m´ ecanique c´ eleste, tome premier-[cinqui` eme ].Paris, De l’Imprimerie de Crapelet : Chez J.B.M. Duprat, [1798]–1827 . [8] P OISSON , Sim´ eon-Denis: M´ emoire sur la distribution de l’´ electricit´ e ` a la surfa ce des corps conducteurs. [Lu devant la classe des sciences math´ ematiq ues et physiques les 9 mai et 3 aoˆ ut 1812] M´ emoires de la classe des sciences math´ ematiques et physi ques de l’Institut de France. 1811, t. XII, no1, p. 1–92. [9] P OISSON , Sim´ eon-Denis: Second m´ emoire sur la distribution de l’´ electricit´ e ` a l a surface des corps conducteurs. [Lu le 6 septembre 1813] M´ emoires de la classe des sciences math´ ematiques et physiques de l’Institut de France. 1811, t. XII, no2, p. 163–274. [10] P OISSON , Sim´ eon-Denis: M´ emoire sur la th´ eorie du magn´ etisme. M´ emoires de l’Acad´ emie des sciences de l’Institut de France. Vol. V (1821), p. 247–3 38. [11] P OISSON , Sim´ eon-Denis: Second m´ emoire sur la th´ eorie du magn´ etisme. M´ emoires de l’Acad´ emie des sciences de l’Institut de France. Vol. V (18 21), p. 488–533. [12] P OISSON , Sim´ eon-Denis: M´ emoire sur la th´ eorie du magn´ etisme en mouvement. M´ emoires de l’Acad´ emie des sciences de l’Institut de France. Vol. VI (1823), p. 441–570.