284 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
where denotes an absolutely arbitrary function ; and the indicated integrations are 
indefinite, with the arbitraries which they introduce subject to the equations (XIV,). 
The demonstration of these equations follows immediately from the results obtained 
by differentiating the three equations (XIV.) with reference to x, y and z respectively. 
The simplest final forms for F, G and H are the following, which are deduced from 
the preceding by integration : — 
¥=f{^dz-ydy)^% ■ 
^=J{ydx—ad%)-\-^~ > 
Yl=^{ady—^dx) +2 
(XVI.) 
Making substitutions according to the formulae (XIV.) for a, /3, 7 in the general ex- 
pression for the potential, we have 
1*4 
<*4 
V=fff )^ + (f _f 
Dividing the second member into six terms, and integrating each by parts, com- 
dW. 
mencing upon the factors such as dy, we obtain an expression, with a triple inte- 
gral involving six terms which destroy one another two and two because of properties 
such as 
7I A 
dy dx dx dy ’ 
and besides, a double integral, which may be reduced in the usual manner to a form 
involving dS, an element of the surface. We thus obtain, finally. 
d\ 
. . . (XVII.) 
83. The second member of this equation expresses the potential of a certain distri- 
bution of magnetism in an infinitely thin sheet coinciding with the surface of the 
body ; the total magnetic moment of the magnetism in the area d^ being 
{ (mH — 72G)^-1- (wF — /H)^-l- (^G — 'niFy}^dS, 
and its direction cosines proportional to 
wF — ZH, ZG~wF. 
Now we have identically, 
Z(mH— /zG)-l-m(72F— ZH)-l-w(ZG — mF)=0 ; 
and hence the direction of this imaginary magnetization at every point of the surface 
is perpendicular to the normal. It follows that we have found a distribution of tan- 
gential magnetism in an infinitely thin sheet coinciding with the bounding surface 
