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A Mathematical Theory of Magnetism.   By WILLIAM THOMSON, Esq.,

 F.R.S.E., Fellow of St. Peter's College, Cambridge, [ https://en.wikipedia.org/wiki/Peterhouse,_Cambridge ]
 and Professor of Nat Philosophy in the University of Glasgow. [ https://en.wikipedia.org/wiki/University_of_Glasgow ]
 

    Received June 21,-Read June 21, 1849.

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 Introduction;

THE existence of magnetism is recognized by certain phenomena of force

 are attributed to it as their cause. Other physical effects are found to be pr  by the same agency; as in the operation of magnetism with reference to po  light, recently discovered by Mr. FARADAY; but we must still regard magnetic  as the characteristic of magnetism, and, however interesting such other phe  may be in themselves, however essential a knowledge of them may be for e  us to arrive at any satisfactory ideas regarding the physical nature of mag  and its connection with the general properties of matter, we must still cons  investigation of the laws, according to which the development and the act  magnetic force are regulated, to be the primary object of a Mathematical Th  this branch of Natural Philosophy.


2. Magnetic bodies, when ptt near one another, in general exert very se  mutual forces; but a body which is not magnetic, can experience no force in  of the magnetism of bodies in its neighbourhood. It may indeed be observe  body, M, will exert a force upon another body A; and again, on a third bo  although when A and B are both removed to a considerable distance from  mutual action can be discovered between themselves: but in all such cases A and B  are, when in the neighbourhood of M, temporarily magnetic; and when both a  under the influence of M at the same time, they are found to act upon one anot  with a mutual force. All these phenomena are investigated in the mathematica  theory of magnetism, which therefore comprehends two distinct kinds of magn  action:-the mutual forces exercised between bodies possessing magnetism, and t  magnetization induced in other bodies through the influence of magnets. The Fi

 Part of this paper is confined to the more descriptive and positive details of the sub  ject, with reference to the former class of phenomena. After a sufficient foundatio  has been laid in it, by the mathematical exposition of the distribution of magnetism  in bodies, and by the determination and expression of the general laws of magn  force, a Second Part will be devoted to the theory of magnetization by influence  magnetic induction.

 

 2 2 (PAGE 2)

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CHAPTER I. Preliminary DeJinitions and Explanations.

3. A magnet is a substance which intrinsically possesses magnetic properties. A piece of loadstone, a piece of magnetized steel, a galvanic circuit, are examples of the varieties of natural and artificial magnets at present known; but a piece of soft iron, or a piece of bismuth temporarily magnetized by induction, cannot, in unqualified terms, be called a magnet. A galvanic circuit is frequently, for the sake of distinction, called an " electro-magnet;' but, according to the preceding definition of a magnet, the simple term, without qualification, may be applied to such an arrangement. On the other hand, a piece of apparatus consisting of a galvanic coil, with a soft iron core, although often called simply " an electro-magnet," is in reality a complex arrangement involving an electro-magnet (which is intrinsically magnetic as long as the electric cur- rent is sustained) and a body transiently magnetized by induction.

4. In the following analysis of magnets, the magnetism of every magnetic sub- stance considered, will be regarded as absolutely permanent under all circumstances. This condition is not rigorously fulfilled either for magnetized steel or for loadstone, as the magnetism of any such substance is always liable to modification by induction, and may therefore be affected either by bringing another magnet into its neighbour- hood, or by breaking the mass itself and separating the fragments. When, however, we consider the magnetism of any fragment taken from a steel or loadstone magnet, the hypothesis will be that it retains without any alteration the magnetic state which it actually had in its position in the body. The general theory of the distribu- tion of magnetism founded upon conceptions of this kind, will be independent of the truth or falseness of any such hypothesis which may be made for the sake of con- venience in studying the subject; but of course any actual experiments in illustration of the analysis or synthesis of a magnet would be affected by a want of rigidity in the magnetism of the matter operated on. For such illustrations, electro-magnets are extremely appropriate, as in them, except during the motion by which any alteration in their form or arrangement is effected, no appreciable inductive action can exist.

5. In selecting from the known phenomena of magnetism those elementary facts which are to serve for the foundation of the theory, all complex actions, depending on the irregularities of the bodies made use of, should be excluded. Thus if we were to attemtpt an experimental investigation of the action between two amorphous frag- ments of loadstone, or between two pieces of steel magnetized by ordinary processes, we should probably fail to recognize the simple laws on which the actions, resulting from such complicated circumstances, depend; and we must look for a simpler case of magnetic action before we can make an analysis which may lead to the establish- metit of the fundamental principles of the theory. Much complication will be avoided if we take a case in which the irregularities of one at least of the bodies do not affect the phenomena to be considered. Now the earth, as was first shown by GILBERT, is a magnet; and its dimensions are so great that there is no sensible variation in its action on different parts of any ordinary magnet upon which we can experiment, and consequently, in the circumstances, no complicacy depending on the actual distribution of terrestrial magnetism. We may therefore, with advantage, commence by examining the action which the earth produces upon a magnet of any kind at its surface.

6. At a very early period in the history of magnetic discovery, the remarkable property of " pointing north and south " was observed to be possessed by fiagments of loadstone and magnetized steel needles. To form a clear conception of this phenomenon, we must consider the total action produced by the earth upon a magnet of any kind, and endeavour to distinguish between the effects of gravitation which the earth exerts upon the body in virtue of its weight, and those which result from the magnetic agency

7. In the first place, it is to be remarked that the magnetic agency of the earth gives rise to no resultant force of sensible magnitude, upon any magnet with reference to which we can perform experiments, as is proved by the following observed facts. (1.) A magnet placed in any manner, and allowed to move with perfect freedom in any horizontal direction (by being floated, for example, on the surface of a liquid), experlences no action which tends to set its centre of gravity in motion, and there is therefore no horizontal force upon the body. (2.) The magnetism of a body may be altered in any way, without affecting its weight as indicated by a balance. Hence there can be no vertical force upon it depending on its magnetism.

8. It follows that any magnetic action which the earth can exert upon a magnet must be a couple. To ascertain the manner in which this action takes place, let us conceive a magnet to be supported by its centre of gravity* and left perfectly free to turn round this point, so that without any constraint being exerted which could balance the magnetic action, the body may be in circumstances the same as if it were without weight. The magnetic action of the earth upon the magnet gives rise to the following phenomena:-
(1.) The body does not remain in equilibrium in every position in which it may be brought to rest, as it would do did it experience no action but that of gravitation.
(2.) If the body be placed in a position of equilibrium, there is a certain axis (which, for the pre- sent, we may conceive to be found by trial), such, that if the body be turned round it, through any angle, and be brought to rest, it will remain in equilibrium.
(3.) If the body be turned through 180?, about an axis perpendicular to this, it will again be in a position of equilibrium.

* The ordinary process for finding experimentally the centre of gravity of a body, fails when there is any magnetic action to interfere with the effects of gravitation. It is, however, for our present purpose, sufficient to know that the centre of gravity exists; that is, that there is a point such that the vertical line of the resultant action of gravity passes through it, in whatever position the body be held. If it were of any consequence, a process, somewhat complicated by the magnetic action, for actually determining, by experiment, the centre of gravity of a magnet might be indicated, and thus the experimental treatment of the subject in the text would be completed.

(4.) Any motion of the body whatever, which is not of either of the kinds just described, nor compounded of the two, will bring it into a position in which it will not be in equilibrium.
(5.) The directing couple experienced by the body in any position depends solely on the angle of inclination of the axis described in (1.) to the line along which it lies when the body is in equili- brium; being independent of the position of the plane of this angle, and of the position of the body with reference to that axis.

9. From these observations we draw the conclusion that a magnet always expe- riences a directing couple from the earth, unless a certain axis in the body is placed in a determinate position. This line in the body is called its magnetic axis*.
 

END OF PROOFING THIS DATE  3-24-2021  ::  see  https://hansandcassady.org/William-Thomson-108396.pdf  

10. The direction towards which the magnetic axis of the body tends in virtue of the earth's action, is called " the line of dip," or " the direction of the total terrestrial magnetic force," at the locality of the observation. 11. No further explanation regarding phenomena which depend on terrestrial magnetism is required in the present chapter; but, as the facts have been stated in part, it may be right to complete the statement, as far as regards the action expe- rienced by a magnet of any kind when held in different positions in a given locality, by mentioning the following conclusions, deduced in a very obvious manner from the general laws of magnetic action stated below, and verified fully by experiment. If a magnet be held with its magnetic axis inclined at any angle to the line of dip, it will experience a couple, the moment of which is proportional to the sine of the angle of inclination, acting in a plane containing the magnetic axis and the line of dip. The position of equilibrium towards which this couple tends to bring the mag- netic axis is stable, and if the direction of the magnetic axis be reversed, the body may be left balanced, but it will be in unstable equilibriu






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THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 251

 of ordinary matter, and it would be wrong to call it either a solid, or the " mag  fluid," or "fluids"; but, without making any hypothesis whatever, we may  "magnetic matter," on the understanding that it possesses only the proper  attracting or repelling magnets, or other portions of " matter " of its own kind  cording to certain determinate laws, which may be stated as follows:-

1. There are two kinds of imaginary magnetic matter, northern and souther represent respectively the northern and southern magnetic polarities of the ear  the similar polarities of any magnet whatever.
2. Like portions of magnetic matter repel and unlike portions attract, m
3. Any two small portions of magnetic matter exert a mutual force whic inversely as the square of the distance between them.
4. Two units of magnetic matter, at a unit of distance from one another, unit of force, mutually.

33. If quantities of magnetic matter be measured nurerically in such uni if the positive or negative sign be prefixed to denote the species of matter,  northern (which, by convention, we may call positive) or southern, all the pr  laws are expressed in the following proposition:-

 If quantities, m and m', of magnetic matter be concentrated respectively at p  a distance, f,from one another, they will repel with aforce algebraically equal to

 mm!  f2'

34. It appears from the explanations given above, that the circu uniformly magnetized needle may be represented if we imagine e  northern and southern magnetic matter to be concentrated at it  numerical measure of these equal quantities being the same as that  of the magnet.

 The mutual action between two needles would thus be reduced to  tion and repulsion between the portions of magnetic matter by which  represented.

35. Any magnetic mass whatever may, as we have seen, be regar of infinitely small bar-magnets put together in such a way as to prod  tion of magnetism which it actually possesses; and hence, by repla  these magnets by imaginary magnetic matter, we obtain a distri  quantities of northern and southern magnetic matter through the  stance, by which its actual magnetic condition may be represented  of this matter becomes very much simplified from the circumstance  general unlike poles of the elementary magnets in contact, by wh  kinds of magnetic matter are partially (or in a class of cases who  through the interior of the body. The determination of the resulting

 * In all cases when the distribution is " solenoidal." See below, Chap. V. ? 68. C

 Royal Society, June 20, 1850.

 2K2

 252 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 magnetic matter, which represents in the simplest possible manner the polarity  any given magnet, is of much interest, and even importance, in the theory of mag  netism, and we may therefore make this an object of investigation, before go  farther.

 36. Let it be required to find the distribution of imaginary magnetic matter to rep  sent the polarity of any number of uniformly magnetized needles, SI N1,, s N,,... Sn  of strengths p1, 2,, ... ^ respectively, when they are placed together, end to end (no  necessarily in the same straight line).

 If A denote the position occupied by 81 when the bars are in their places; if  and S2 are placed in contact at KI; N2 and S3, at K2; and so on until we have t

 last magnet, with its end S, in contact with N,_,, at K._,, and its other end, N1, fr  at a point B; we shall have to imagine

 1i, units of southern magnetic matter to be placed at A;  p1 units of northern, and p2 units of southern matter at K,;  pk2 units of northern, and 3 of southern matter at K2;

 ?-, units of northern, and pi, of southern matter at K,. ,;

 and lastly, p. units of northern matter at B.

 Hence the final distribution of magnetic matter is as follows:-

 --*fk . . ........ at A

 1,--p2 ......... Ki  P2--3 ..... . K2

 -pn--P4 . .... . .. K,n

 and , pn ........... B.

37. The complex magnet AK,K which is uniformly and longitudi  a simple bar of the same length,  which represents a bar-magnet A  at the ends of the whole bar, an  strength of its magnetization.
    
    
     
38. If the length of each part t constant, be diminished without  creased indefinitely, a straight or  produced, which shall possess a d  tinuously from one end to the ot  of the magnetism at any point P

 ? This expression is equivalent to the produ  by retaining it we are enabled to include ca

 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 253

 the values of p at the points A and B, the investigation of ? 36, with the ele

 principles and notation of the differential calculus, leads at once to the dete  of the ultimate distribution of magnetic matter by which such a bar-magne  represented. Thus if AP be denoted by s; , will be a function of s, which ma  posed to be known, and its differential coefficient will express the continuo  bution of magnetic matter which replaces the group of material points a  so that the entire distribution of polarity in the bar and at its ends will be  in any infinitely small length, a, of the bar, a quantity of matter equal to  ds

 and, besides, terminal accumulations, of quantities

 - [y] at A

 and (~) at B.


 

 It follows that if through an  netism is constant, there wi  portion of the magnet; but if  as it diminishes or increases  there will be a distribution of  polarity which results from t  Corresponding inferences ma  tion of magnetism, when the  Thus COULOMB found that  another as if each had a symm  northern within a limited s  from the other, the intermed  bar) being unoccupied; from w  sensible through the middle

 each side, the intensity of th  the ends*.

 39. The distribution of magnetic matter which represents the polarity of a uni formly magnetized body of any formn, may be immediately determined if we imagine

 * This circumstance was alluded to above, in ? 21. Interesting views on the subject of the distribution  of magnetism in bar-magnets are obtained by taking arbitrary examples to illustrate the investigation of the  text. Thus we may either consider a uniform bar variably magnetized, or a thin bar of varying thickness, cut

 from a uniformly magnetized substance; and according to the arbitrary data assumed, various.remarkable  results may be obtained. We shall see afterwards that any such data, however arbitrary, may be actually pro duced in electro-magnets, and we have therefore the means of illustrating the subject experimentally, in as  complete a manner as can be conceived, although from the practical non-rigidity of the magnetism of magnetized

 substances, ordinary steel or loadstone magnets would not afford such satisfactory illustrations of arbitrary

 cases as might be desired. The distribution of longitudinal magnetism in steel needles actually magnetized in  different ways, and especially " magnetized to saturation," has been the object of interesting experimental and  theoretical investigations by COULOMB, BIOT, GREEN and RIESS.

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THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 255  manner that the density will be uniform over each face, and that the quant  matter on the six faces will be as follows:

 --il. 3y, and il. Py3; on the two faces parallel to YOZ;  -im . ya, and in . ya; on the two faces parallel to ZOX;  -in. a3, and in. ap3; on the two faces parallel to XOY.

 Now if we consider adjacent parallelepipeds of equal dimensions, touching the  faces of the one we have been considering, we should find from each of them a seco  distribution of magnetic matter, to be placed upon that one of those six faces wh

 it touches. Thus if we consider the first face ry, or that of which the distance from

 YOZ is x- - a; we shall have a second distribution upon it derived from a parallelepipe  the coordinates of the centre of which are x-co, y, z; and the quantity of matte  this second distribution will be

 {il+ (Ze (-dx }0y

 This, added to that which was found above, gives

 d(il) a(il)

 r)(- ) . Y, or d- d)

 for the total amount of matter upon this face. Again, the quantity in the sec

 distribution on the other face, 3y, is equal to  {il.d(il) })3,

 and therefore the total amount of matter on this face will be

 d(il)

 By determining in a similar way the final quantities of matter on the other faces of  the parallelepiped, we find that the total amount of matter to be distributed over its

 surface is

 -2 {d(il) d() + d(in )py dx *dy indz\

 Now as the parallelepipeds into which we imagine the whole mass divided  small, we may substitute a continuous distribution of natter through  of the superficial distributions on their faces which have been deter  making this substitution, the quantity of matter which we must suppose  through the interior of any one of them must be half the total quantity  since each of its faces is common to it and another parallelepip  quantity of matter to be distributed through the parallelepiped apy is

 fd(i-x. d(im). d(in dBesidesdy dcontinuous distribution through the interior of the mgnet, thee must

 Besides this continuous distribution through the interior of the magnet, there must

 256 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 be a superficial distribution to represent the neutralized polarity at its surface  denote the density of this distribution at any point; [1], [mn], [n] the directi  cosines, and [i] the intensity of the magnetization of the solid close to it; and .  the direction cosines of a normal to the surface, we shall have, as in the case of  uniformly magnetized solid previously considered,  f= [i] cos =[il] . X+ [im] . + [inJ . v ......... (1).

 If, according to the usual definition of " density," k denote the density of the m  matter at P, in the continuous distribution through the interior, the expressio  above for the quantity of matter in the element a, 3, y, leads to the formula  k=-{ d(+ z 4 d. (2). - f d(il) d(imn) d(in)

 These two equations express re  tinuous distribution through th  sents the polarity of the given  is equal to the quantity of sout  showing from thlese formulke,  quantity of matter is algebraic

 

43. If there be an abrupt chang frorn one part of the magneti  formule given above will be c  little from a given case, but wh  tensity or direction of magneti  of the magnet, has merely ve  elements: we may conceive th  same as the given distribution,  limit towards which the value o  according to the ordinary rule  limiting case, we may still der  this way, that, besides the cont  to all points of the substance fo  tribution of magnetic matter on  this superficial distribution will

 of magnetization into the cosine  the two sides of the surface.

44. This result, obtained by the interpretation of formula (2) in the extreme case, might have been obtained directly from the original investigation, by taking into ac-

 count the abrupt variation of the magnetization at the surface of discontinuity, as we

 did the abrupt termination of the magnetized substance at the boundary of the mnag net, and representing the un-neutralized polarity which results, by a superficial dis-

 tribution of magnetic matter,


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260 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 where A and [A] are respectively the distances of the points x,y, z and [x,y, z] from  the point P, and are given by the equations

 A2= ( )2+ (4 _-y)2+ (t_ X)2  [A]2= [-E])2+(- [y] )2+( - [Z])2.

 The double and triple integrals in the first and second terms of this expression are  be taken respectively over the whole surface bounding the magnet, and through  the entire magnetized substance. Since, as is easily shown, the value of that port  of the triple integral in the second member which corresponds to an infinitely sma

 portion of the solid containing (a, 7, '), when this point is internal, is infinitely sm  it follows that the magnetic force at any internal point, as defined in ? 48, is deriv  from a potential expressed by equation (3).

 52. The expressions for the resultant force at any point, and its direction, may b  immediately obtained when the potential function has been determined, by the rule  of the differential calculus. Thus, if V has been determined in terms of the rec

 gular coordinates, , a, $, of the point P, the three components, X, Y, Z, of the result  force on this point will be given, in virtue of LAPLACE'S fundamental theorem enu

 ciated in ? 50, by the formulae,

 dV dV dV

 X=--, Y=- -' Z=- ....... (4),

 where the negative signs are introduced, because the pote  way that it diminishes in the direction along which a n  take the expression (3) for V, and actually differentiate  under the integral signs, we obtain expressions for X,  the expressions that might have been obtained directly, b  ciples of statics (see ? 46), and thus the theorem is veri  extended so as to be applicable to a body acting according to  tutes virtually the ordinary demonstration of the theorem

53. The formulae of the preceding paragraphs are applicab of the potential, and the resultant force at any point, whet  substance or not, according to the general definition of  the magnetized substance, according to the conventional  cannot present itself in problems with reference to the  actual magnets. This case being therefore excluded, we m  gations indicated in ? 47.
54. In the method which is now to be followed, the ma sidered must be conceived to be divided into an infinite n  parts, and the actual magnetism of each part will be take  determining the potential of the magnet at a given externa  the mutual action between two magnets. In the first pl  potential due to an infinitely small element of a magneti  purpose we may commence by considering an infinitely thi

 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 261

 bar of finite length. If m denote the strength of the bar, and if N and S be its north  and south poles respectively, its potential at any point, P, will be, according to  ?[ 34 and 50,

 m m

 NP SP

 Let A denote the distance of the point of bisection of the bar f  between this line and the direction of the bar measured from its centre towards its  north pole. Then, if a be the length of the bar, the expression for the potential becomes

 m (A2-aA cos+ a2) ( A2+acos + a <f

 By expanding this in ascending powers of a, and neglecting all the terms after the  first, we find for the potential of an infinitely small bar magnet,

 ma cos

 A2

 If now we suppose any number of such bar-magnets to be put together so as to  constitute a mass magnetized in parallel lines, infinitely small in all its dimensions,

 cos B  the values of 0 and A, and consequently the value of 2-, will be infinitely nearly  the same for all of them, and the product of this into the sum of the values of ma  for all the bar-magnets will express the potential of the entire mass. Hence, if the  total magnetic moment be denoted by ,, the potential will be equal to

 , cos 0

 Now if we conceive the bars to have been arranged so as to constitute a uniformly  magnetized mass, occupying a volume p, we should have (? 30.) for the intensity of

 magnetization, i.=. Hence if p denote the volume of an infinitely small element of

 uniformly magnetized matter, and i the intensity of its magnetization, the potential  which it produces at any point P, at a finite distance from it, will be

 ip .cos

 A~2

 where A denotes the distance of P fromn any point, E, within the elem  angle between E P and a line drawn through E, in the direction of  the element, towards the side of it which has northern polarity.

 55. Let us now suppose the element E to be a part of a magnet of  sions, of which it is required to determine the total potential at an ex  Let f, , % be the coordinates of P, referred to a system of rectang  x, y, z be those of E. We shall have

 A2= (t_X)2+(4_y)2+ (_)2

 262 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 and, if 1, m, n denote the direction cosines of the magnetization at E,  cos =l- +m-- a+nan

 Hence the expression for the potential of the element E becomes  il\l(E-x) + mt(-y) +<n(-z) }

 { (- X)2 + (~- y)2+ (g- )2}'

 Now the potential of a whole is equal to the sum of the potentials of all its p

 and hence, if we take p=dx dy dz, we have, by the integral calculus, the expre

 V dixl(-x)+im.(l-y)d+i.(z- 5),

 for the potential at the point due to the entire magnet-z)

 for the potential at the point P, due to the entire magnet*.

 56. This expression is susceptible of a very remarkable modification, by integration  by parts. Thus we may divide the second member into three terms, of which the  following is one:

 irr. (-x)dx

 fJJJ {(f-^2+ -y)2+(S _2} y

 Integrating here by parts, with reference to x, we obtain

 d(il)  F f'/ril.dyd rrr d x d L JJ A1J J -Jd dxdydz,

 where the brackets enclosing the double integral denote that the variables in it must

 belong to some point of the surface. If X, p, v denote the direction cosines of a  normal to the surface at any point [E, , $], and dS an element of the surface, we  may take dy dz-=.dS, and hence the double integral is reduced to

 JJ LA] ; ffLlTx, dS

 and, as we readily see by tracing the limits of the first integral with reference to x,  for all possible values of y and z this double integral must be extended over the

 entire surface of the magnet. By treating in a similar manner the other two terms  of the preceding expression for V, we obtain, finally,

 d(il) d(im) d(in)

 v=,/f il]x+ [imT]+ jvS Asfff ' x dxdydz.

 The second member of this equation is the expression for the potential of a certain  complex distribution of matter, consisting of a superficial distribution, and a conti nuous internal distribution. The superficial-density of the distribution on the surface,

 * From the form of definition given in the second foot-note on ? 48, for the magnetic force at an internal  point, it may be shown that the expression (5), as well as the expression (3), is applicable to the potential at  any point, whether internal or external. The same thing may be shown by proving, as may easily be done,

 that the investigation of ? 56 does not fail or become nugatory when (g, /, ~) is included in the limits of inte gration.

 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 263

 and the density of the continuous distribution at any internal point, are expressed  spectively by [il]x+ [imn]p+ [in]v, and-- d) + )+ Hence weinfer that

 action of the complete magnet upon any external point is the same as would be  duced by a certain distribution of imaginary magnetic matter, determinable  means of these expressions, when the actual distribution of magnetism in the magn  is given*. The demonstration of the same theorem, given above (? 42), illustra  in a very interesting manner the process of integration by parts applied to a tr  integral.

57. The mutual action of any two magnets, considered as the resultant of the mutual actions between the infinitely small elements into which we may conceive  them to be divided, consists of a force and a couple of which the components will  be expressed by means of six triple integrals. Simpler expressions for the same  results may be obtained by employing a notation for subsidiary results derived fromn  triple integration with reference to one of the bodies, in the following manner.
58. Let us in the first place determine the action exerted by a given magnet, upon an infinitely thin, uniformly and longitudinally magnetized bar, placed in a given  position in its neighbourhood.

 We may suppose the rectangular coordinates, g, ?, 9, of the north pole, and I', r', ;' of  the south pole of the bar to be given, and hence the components, X, Y, Z and X', Y', Z',  of the resultant forces, at those points, due to the other given magnet, may be regarded  as known. Then, if i denote the " strength" of the bar-magnet, the components of  the forces on its two poles will be respectively,

 P3X, pY, 3Z, on the point (d, , ),

 and --X', -3Y', -ZZ', on the point (~', ,', 1).

 The resultant action due to this system of forces may be determined by means of  elementary principles of statics. Thus if we conceive the forces to be transferr  the middle of the bar by the introduction of couples, the system will be reduced  force, on this point, whose components are

 0(X-X'), (Y--Y'), p(Z-Z'),

 and a couple, whose components are

 /P(Z+Z ') ( - ' )-/P(Y+Y') .^ (t-- ;),

 {((X+Xeg .(4 )-^(Z+Z) J. }

 {/3(Y+Y') .* (S-- )-3(X+X') -. (-'.)}

 * This very remarkable theorem is due to PoissoN, and the demonstration, as it has been  text, is to be found in his first memoir on Magnetism. The demonstration which I have giv  regarded as exhibiting, by the theory of polarity, the physical principles expressed in the anal

 264 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 59. If 1, m, n denote the direction cosines of a line drawn along the bar, from  middle towards its north pole, and if a be the length of the bar, we shall have  -'--=al, j---'=am, g--'=an.

 Hence, if the bar be infinitely short, and if x, y, z denote the coordinates of  middle point, we have

 dX dX dX  dY dY dY

 -- ' a d l+Z. am+ - -

 ;ancdj Z-Z'= d. al+- dy . am+ z. an. and z-z=' adZ dZ dZ

 Multiplying each member of these equations by 3, we obtain the expressions f  components of the force in this case; and the expressions for the components of  couples are found in their simpler forms, by substituting for a-_', &c. their  given above; and, on account of the infinitely small factor which each term cont  taking 2X, 2Y, and 2Z, in place of X+X', Y+Y', and Z+Z'.


 

 60. Let us now suppose an'infinite number of such infinitely small bar-ma  to be put together so as to constitute a mass, infinitely small in all its dimen  uniformly magnetized in the direction (1, m, n) to such an intensity that its ma  moment is ^. We infer, from the preceding investigation, that the total act  this body, when placed at the point x, y, z, will be composed of a force whos  ponents are

 /dX dX dX

 - + m+dzny

 /dY dYm dY \  dx+ dy m dzn

 /dZ dZ dZ X

 dxl m+ zn)

 acting at the centre of gravity of the solid supposed homogeneou  which the components are

 p(Zm-Yn),

 (^(Xn- Z),  d4(YI-Xm).

 61. The preceding investigation enables us, by means of the integral calculus, to  determine the total mutual action between any two given magnets. For, if we talke  X, Y, Z to denote the components of the resultant force due to one of the magnets,  at any point (x, y, z) of the other, and if i denote the intensity and (4, m, n) the direc tion of magnetization of the substance of the second magnet at this point, we may  take p;=i. dxdydz in the expressions whicli were obtained, and they will then express

 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 265

 the action which one of the magnets exerts upon an element dxdydz of th  To determine the total resultant action, we may transfer all the forces to th  of coordinates, by introducing additional couples; and, by the usual process, w  for the mutual action between the two magnets, a force in a line through this p  and a couple, of which the components, F, G, H, and L, M, N, are given  equations

 F=fff(i-X * dX d dX d  rrry-. dY . dY . dY

 G ( +m+il +- J im in............ (6)

 G=jjm +ndz d dL(6)

 TorP. .I . *mdZ . dZ . Z .Y . dY . Y 1

 L V =J>> > zmz- inY ++y k nt+^ -Z l d--+im dy+in

 M/'fff inX - i/Z+(, m +in- x . + z -. , -d. (7) M / m dxdyx Xi+y^+1d (7d)y =il imdX i? dX) (ildZ dZ ZindZ)}dxdydz

 N=f ^ilY--imX-{-x i1x+ zmT+n - N =J ay l Z lY- m -x n d d + +n dx y Jd N=Sf { ilY-i i. i Y iY. *dY dY . dX .dX d

 62. If, in the second members of these equations, we employ for X  their values obtained, as indicated in equations (4) of ? 52, by the di  expression (5) for V in ? 55, we obtain expressions for F, G, H, L  readily be put under symmetrical forms with reference to the two  the parts of those quantities depending on the mutual action be  one of the magnets, and an element of the other. Again, expres  mutual action between any element of the imaginary magnetic mat  and any element of the imaginary magnetic mlatter of the othe  first modifying by integration by parts, as in ? 56, from the ex  have actually obtained for F, G, H, L, M, N; and then substitut  their values obtained by the differentiation of the expression (3) fo

 It is unnecessary here to do more than indicate how such othe  derived from those given above; for whenever it may be requir  difficulty in applying the principles which have been establish  obtain any desired form of expression for the mutual action b  magnets.

 MDCCCLI. ,                                                                                                                       2 M

 266 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM.

 ?Q 63 and 64 . On the Expression of lMutual Action between two Magnets by means of  the Differential Coeficients of a Function of their relative Position.

 63. By a simple application of the theory of the potential, it may be shown that  the amount of mechanical work spent or gained in any motion of a permanent mag net, effected under the action of another permanent magnet in a fixed position, depends  solely on the initial and final positions, and not at all upon the positions successively  occupied by the magnet in passing from one to the other. Hence the amount of work  requisite to bring a given magnet fromn being infinitely distant from all magnetic

 bodies, into a certain position in the neighbourhood of a given fixed magnet, depends  solely upon the distributions of magnetism in the two, and on the relative position  which they have acquired. Denoting this amount by Q, we may consider Q as a

 function of coordinates which fix the relative position of the two magnets; and the  variation which Q experiences when this is altered in any way will be the amount of  work spent or lost, as the case may be, in effecting the alteration. This enables us  to express completely the mutual action between the- two magnets, by means of dif ferential coefficients of Q, in the following manner:-

 If we suppose one of the magnets to remain fixed during the alterations of relative  position conceived to take place, the quantity Q will be a function of the linear and an-

 gular coordinates by which the variable position of the other is expressed. Without  specifying any particular systern of coordinates to be adopted, we may denote by dgQ

 the augmentation of Q when the moveable magnet is pushed through an infinitely  small space dE in any given direction, and by dpQ the augmentation of Q when it is  turned round any given axis, through an infinitely small angle d@. Then, if F denote  the force upon the magnet in the direction of dA, and L the moment round the fixed  axis of all the forces acting upon it (or the component, round the fixed axis, of the  resultant couple obtained when all the forces on the different parts of the magnet are  transferred to any point on this axis), we shall have

 -Fd=- dQ, and -Ldp-=dQ,

 since a force equal to -F is overcome through the space d~ in the first case, and a  couple, of which the moment is equal to -L, is overcome through an angle dp in the  second case of motion. Hence we have

 dQ

 F----

 CdJpQ

 64. It only remains to show how the function Q may be determined when the distri butions of magnetism in the two magnets and the relative positions of the bodies are

 * Communicated June 20, 1850.