262 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
and, if Z, m, n denote the direction cosines of the magnetization at E, 
cos 6=1 
I—X fi—y K — 2 
■m 
\-n- 
A ' A A 
Hence the expression for the potential of the element E becomes 
Now the potential of a whole is equal to the sum of the potentials of all its parts, 
and hence, if we take <p=dx dy dz, we have, by the integral calculus, the expression, 
JJJ + d y 
( 5 ), 
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 : 
rrr d.{i-x)dx , , 
JJJ y - 
Integrating here by parts, with reference to x, we obtain 
d{iT) 
^dxdydz, 
where the brackets enclosing the double integral denote that the variables in it must 
belong to some point of the surface. If X, (jij, v denote the direction cosines of a 
normal to the surface at any point [|, ??, ^], and c?S an element of the surface, v/e 
may take dy dz='k.d^, and hence the double integral is reduced to 
JJ [A] ’ 
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{iT) d{im) d{in) 
dx 
dy 
dz 
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 (0, ij, ?) is included in the limits of inte- 
gration. 
