PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 267 
given. For this purpose, let us consider points P and P', in the two magnets respect- 
ively, and let their coordinates with reference to three fixed rectangular axes be de- 
noted by X, y, % and x', y’, z ' ; let also the intensity of magnetization at P be denoted 
by i, and its direction cosines by and let the corresponding quantities, with 
reference to P', be denoted by {', I’, m', n. Then it may be demonstrated without 
difficulty that 
/ ''‘x 
Q —ffffffdxdydzdx'dy'dz'ii' 
d'^\ 
. / ^ 
1 
4- 
^^^dzdz<y 
where, for brevity, A is taken to denote {{x—3j)'^-\-{y—y’)'^-\-{z—z'Y]^, and the diffe- 
rentiations upon ^ are merely indicated. Now, by any of the ordinary formulae for 
the transformation of coordinates, the values o^x,y^ z, and x',y', z', may be expressed in 
terms of coordinates of the point P with reference to axes fixed in the magnet to which 
it belongs, of the coordinat^es of the point P' with reference to axes fixed in the 
other, and of the coordinates adopted to express the relative position of the two 
magnets : and so the preceding expression for Q may be transformed into an expression 
involving explicitly the relative coordinates, and containing the coordinates of the 
points P and P' in the two bodies only as variables in integrations, the limits of which, 
depending only on the forms and dimensions of the two bodies, are absolutely con- 
stant. Thus Q is obtained as a function of the relative coordinates of the bodies, 
and the solution of the problem is complete. 
There is no difficulty in working out the result by this method, so as actually to 
obtain either the expressions (6) and (7) of § 61, or the expressions indicated in § 62, 
although the process is somewhat long. 
The method just explained for expressing the mutual action between two magnets in 
terms of a function of their relative position, has been added to this chapter rather for 
the sake of completing the mathematical theory of the division of the subject to which 
it is devoted, than for its practical usefulness in actual problems regarding magnetic 
force, for which the most convenient solutions may generally be obtained by some of 
the more synthetical methods explained in the preceding parts of the chapter. There 
is however a far more important application of the principles upon which this last 
method is founded which remains to be made. The mechanical value of a distribu- 
2 M 2 
