PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 281 
which is precisely analogous to Poisson’s original investigation (shown in ^ 56. of 
this paper) of the formula of § 51. 
79. Equations (3) and (4) of §§51. and 52, lead to expressions for the components 
of the resultant force at any point in the neighbourhood of a magnet. Taking only 
one of them, (since the three expressions are symmetrical) that for X for instance, 
we have 
Now if the factor of dxdydz in the second member of this equation be differentiated 
with reference to an expression is obtained which does not become infinitely great 
for any values of x, y, z included within the limits of integration, since the point 
(I, ^) is considered to be external in the present investigation. Hence the differen- 
tiation with reference to | may be performed under the integral sign ; and, since 
dx ’ 
we thus obtain 
Now, for all points included within the limits of integration, we have, from Laplace’s 
well-known equation, 
and therefore 
* If the point (0, ij, ?) be either within the magnet, or infinitely near it, the factor of dxdydz in this integral 
is infinitely great for values of {x, y, z) included within the limits of integration ; and it may be demonstrated 
that the value of a part of the integral corresponding to any infinitely small portion of the magnet infinitely 
near the point (^, ij, Q is in general finite, and that it depends on the form of this portion, on its position with 
reference to the line of magnetization through (0, ij, Q, and on the proportions of the distances of its different 
parts from this point. It follows that if the point 0, ij, ? be internal, and if a portion of the magnet round it 
be omitted from the integral, the value of the integral will be affected by the form of the omitted portion, 
however small its dimensions may be, and consequently the complete integral has no determinate value if the 
point (0, ij, ?) be internal. Hence, although as we have seen above (§§ 51, 51.), 
has in all cases a determinate value, which, by the definition (§ 48.), is called the component parallel to OX of 
the resultant force at (^, ?). the expression 
has no meaning when (0, ij, ?) is in the substance of the magnet. 
MDCCCLI. 
2 O 
