PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 263 
and the density of the continuous distribution at any internal point, are expressed re- 
spectively by — j. Hence we infer that the 
action of the complete magnet upon any external point is the same as would be pro- 
duced by a certain distribution of imaginary magnetic matter, determinable by 
means of these expressions, when the actual distribution of magnetism in the magnet 
is given*. The demonstratioH of the same theorem, given above (§ 42), illustrates 
in a very interesting manner the process of integration by parts applied to a triple 
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 from 
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, |, of the north pole, and '/j, ^ 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 (3 denote the “ strength” of the bar-magnet, the components of 
the forces on its two poles will be respectively, 
/3X, (3Y, jSZ, on the point (|, fj, Q, 
and — (3X', — /3Y', — |8Z', on the point (|', n', ^0* 
The resultant action due to this system of forces may be determined by means of the 
elementary principles of statics. Thus if we conceive the forces to be transferred to 
the middle of the bar by the introduction of couples, the system will be reduced to a 
force, on this point, whose components are 
^(X-X'), ^(Y-Y'), )3(Z-Z'), 
and a couple, whose components are 
{/3(Z+Z') . i _/3(Y+V) . i (?-?')}. 
|/3(x+x') .i (s-r)}, 
{/3(Y + V).i(5-f)-|3(X+X').i(,-,')}. 
* This very remarkable theorem is due to Poisson, and the demonstration, as it has been just given in the 
text, is to be found in his first memoir on Magnetism. The demonstration which I have given in § 42 may be 
regarded as exhibiting, by the theory of polarity, the physical principles expressed in the analytical formulae. 
