278 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
iiitegration of the general expression for the potential in the case of a lamellar dis- 
tribution, in the following manner : — 
In equation (5) of § 55, which, as was remarked in the foot-note, expresses the 
potential for any point, whether internal or external, let and ^ be substituted 
in place of z7, /m, and in respectively ; and, for the sake of brevity, let 
l — X 
4 
be denoted by A : then observing that and so for the similar terms ; we have 
^=fff 
4 ■'a 4 “'a .4 j ^ 
li+Ty 4+4 (“) 
Dividing the second member into three terms, integrating the first by parts com- 
mencing with the factor ^ dx, and so for the other terms ; we obtain 
dl 
dx^ 
dz^ 
dxdydz, 
(b) 
where the brackets which inclose the double integral denote that it has reference to 
the surface of the body. Now, for any set of values of x, y, z, for which is finite, 
we have, as is well known, 
A . A . A 
dx^ 
dx^ ^ dy^ 
■ 0 ; 
(c) 
and consequently, if the point is not in the space included by the triple integral 
in the expression for V, each element of this integral, and therefore also the whole, 
vanishes. In the contrary case, the simultaneous values x=^, y = n, and will 
be included in the limits of integration, and, as these values make infinitely great, 
the equation (c) will fail for one element of the integral, although it still holds for 
all elements corresponding to points at a finite distance from (|, ri, Q. Hence, if {<p) 
denote the value assumed by the function (p at this point, we have 
dy"^ ~ dz'^ 
jdxdydz = {(p)jyy 
d^^\ 
. 
L- ^ 
y dx"^ 
^ dy‘^ 
' dx'^j' 
j dxdydz. 
where the limits of integration may correspond to any surface whatever which com- 
pletely surrounds the point (|, yi, Q. Now it is easily proved (as is well known) that 
the value of 
Mr 
jdxdydi 
z 
