MAGNETISM -.ses of For an ellipsoid this gives at once for the components of lipsoid, moment An interesting particular case is that of an infinite cylinder. If P be the component of the moment parallel to its axis, P= - xffdydz (--). If the inducing system be magnetic bodies at a finite distance, then Vc = V-( = 0, and P = 0. If the cylinder be magnet ized by a spiral current i, of n windings, of whatever form, then, r being the radius of the cylinder, P = 4:Tr 2 Kr 2 ni. 1 Generalization of the Theory for Isotropic Media in which K is not constant. In the above theory we have supposed the magnetic susceptibility to be constant. This irch-. is by no means the case in nature, however. It is of im- lff>s . portance therefore to consider how the theory must be tn modified when we assume K to be a function of the >isson s magnetization already induced. Subject to the restriction eory. (obviously necessary for isotropic media) that the resultant magnetization shall coincide in direction with the total resultant magnetic force (|p), the most general assumption that can be made is A=/(I)a, B=/(I)/3, C=/(I) 7 . . . . (100), where/ is a function depending on the nature of the sub stance. From these equations, by squaring, adding, and extracting the square root, we get /(!)/! = 1/p, in other words, I, and therefore /(I), are functions of |:j. Hence we may write the equations (95). Aj-FdSK B^FUB)^, C 1 = F(i) y . . . (101). It is easy by means of these to introduce the requisite modifications into the general equations of magnetic equili brium. For the details we refer our readers to Kirchhoff s memoir in Crelle s Journal, 12 where the matter was first fully worked out. It will bs seen at once that the induced magnetization is in general neither solenoidal nor lamellar. There is one important class of cases in which the con- vhich elusions arrived at on the assumption that K is constant he mag- g |.-}} ho]^ y z -} those in which the induced magnetization ion j s ~ is uniform. In such cases I has the same value throughout miform. the body, and K is therefore constant throughout the body in any one case, although it differs from one case to another. For example, in the case of an ellipsoid the equations (85) above given for the components of magnetization still hold good, provided we understand K to be defined by the equation f. /CL I 1 ] It is clear, therefore, that by experiments on an ellipsoid placed in a uniform field we could determine the function F(||), and also test the truth of the mathematical theory. For, ApBpCj being obtained by observation, one of the equations (85) will enable us to determine K, and the argument P can be calculated from a , /? , y and A^B^Cj ; the test of the truth of the theory would be the agreement of the three values of K obtained from the three equations (85). Historical Remarks on the History of the Mathematical Theory. Although the Tentamen of JEpinus, published in 1759, and the discoveries of Mayer and Lambert did much to make clear the exact nature of the problems in volved in the modern mathematical theory of magnetism, yet the origin of that theory is usually, and with justice, Coulomb, dated from Coulomb. 3 Not only did the results of his careful and judicious experiments afford the means of bring ing a mathematical theory to the test, but the marvellous sagacity he displayed in analysing the phenomena enabled him actually to lay the foundations upon which such a 1 Kirchhoff, I. c. 2 xlviii. 370, 1854. , 3 Mem. de I Acad. de Paris, 1780, 1785, &c. theory could be constructed. After him, Biot 4 and Han- Biot. steen, 5 of whose services we have already spoken, are to be Hansteen reckoned as pioneers. The theory as it now stands was virtually created by Poisson in four of the most admirable Poisson. memoirs 6 to be found in the whole literature of mathemati cal physics. In the first two he investigates expressions for the force due to bodies magnetized in any manner; he then applies his formulae to the case of bodies inductively mag netized but having no coercive force. Although he con fines his investigations to the case of isotropic bodies, he is quite aware of the general nature of the consequences of 93olotropy, and in fact distinctly predicts as possible the mag- necrystallic phenomena afterwards discovered by Pliicker and Faraday. The formulae he gives are practically identi cal with those given above (p. 248). He works out in detail the solution for the case of a hollow or solid sphere exposed to any system of inducing forces having a poten tial, 7 and in particular compares the results, when the inducing field is uniform, with the experiments of Barlow. In the second memoir he works out the solution of his equations for an ellipsoid in a uniform field, examining specially the case of an ellipsoid of revolution "and its extreme cases (see above, p. 245). At the end of this memoir he discussers the disturbing forces on a compass, arising from the earth s induction on any distribution of soft iron, and shows that the given components of the disturbing force are expressed by linear functions of the components of the earth s force, involving nine constants which depend on the quantity and distribution of the iron. The third memoir, on magnetism in motion, is an attempt to explain the phenomena of the deviation of the magnetic needle caused by rotating metal spheres or disks. Although the physical interest of this memoir was in a great measure destroyed by the discoveries of Faraday as to the true nature of this action, yet, as a piece of profound mathematical investigation, this work of Poisson s is still worthy of study ; nor is it perfectly certain that his theory will not after all be required to explain certain residual phenomena. The fourth memoir develops the mathematical theory of the deviation of the compass caused by the iron of ships. After Poisson the most im portant investigators are Green and Gauss. Green s ser- Green, vices have already been alluded to in the article ELEC- Gauss. TRICITY ; we need only mention here his approximate solution of the problem of the magnetic distribution on cylindrical bars, which gives a formula agreeing with that of Biot. The all-important work of Gauss has already been detailed. In Crelle s Journal for 1848 J. Neumann worked out Neumann, the solution of the induction problem for an ellipsoid of- revolution under the action of any conservative system ; and six years later, in the same journal, Kirchhoff worked Kirdihoff. out the case of a circular cylinder of infinite length. We are not aware that the solution of Poisson s equations in particular cases has been carried any farther, unless we include as new the case of a hollow ellipsoid treated by A. Greenhill. G. Greenhill in the Journal de Physique for 1881. The most important contributions to the general theory of magnetism since Poisson are to be found in a series of memoirs 8 by Sir William Thomson, the first of which Thomson, appeared in the Philosophical Transactions for 1851. He divests the theory of Poisson of all particular assumptions connected with the two-fluid theory, and bases it on a 4 Traite de Physique, 1816. 5 Magnetismus der Erde, 1819. 6 Mhn. de I lnst., v., 1821 (two memoirs); vi., 1823; and xvi., 1838. 7 He did not use the word "potential," although he uses the corre sponding function. 8 Reprinted in 1872 under the title of Papers on Electrostatics and Magnetism.
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