PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 265 
the action which one of the magnets exerts upon an element dxdydz of the other. 
To determine the total resultant action, we may transfer all the forces to the origin 
of coordinates, by introducing additional couples ; and, by the usual process, we find, 
for the mutual action between the two magnets, a force in a line through this point, 
and a couple, of which the components, F, G, H, and L, M, N, are given by the 
equations 

62. If, in the second members of these equations, we employ for X, Y, Z respectively 
their values obtained, as indicated in equations (4) of ^ 52, by the differentiation of the 
expression (5) for V in § 55, we obtain expressions for F, G, H, L, M, N, which may 
readily be put under symmetrical forms with reference to the two magnets, exhibiting 
the parts of those quantities depending on the mutual action between an element of 
one of the magnets, and an element of the other. Again, expressions exhibiting the 
mutual action between any element of the imaginary magnetic matter of one magnet, 
and any element of the imaginary magnetic matter of the other, may be found by 
first modifying by integration by parts, as in § 56, from the expressions which we 
have actually obtained for F, G, H, L, M, N ; and then substituting for X, Y, and Z 
their values obtained by the differentiation of the expression (3) for V. 
It is unnecessary here to do more than indicate how such other formulse may be 
derived from those given above ; for whenever it may be required, there can be no 
difficulty in applying the principles which have been established in this paper to 
obtain any desired form of expression for the mutual action between two given 
magnets. 
2 M 
MDCCCLI. 
