282 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM, 
Dividing the second member into four terms, and applying an obvious process of 
integration by parts, we deduce 
X= 
f 
4 
4 
1 
J'fi^-a-^dxdz-a-j-Jxdy+^-j^dydz+Y-^dydz^ 
\ 1 j , d\ 
PPf* Ida. I\ da. ^ £?3 A dj 
y \dy dy~^dz dz dx dy dz 
dy dy ' dz dz dx dy 
Modifying the double integral by assuming, in its different terms 
dydz = ldS; dzd.v=mdS; dxd^=ndS, 
and altering the order of all the terms, v/e obtain 
dz 
]■ 
f4 
^4- 
A 
This expression, when the indicated diffeientiations are actually performed upon-^? 
becomes identical with the expression for X at the end of 78, and the formulee 
which it was required to prove are therefore established. 
80, The triple integrals in these expressions vanish in the case of a lamellar distri- 
bution, in virtue of the equations (III.) of §§ 75 ; and we have simply 
X 
4 
Y=-[^[-A(„/3-my)-A(^„_ //3)|ds] 
4 
z=- h- 
(X.). 
dy 
To interpret these expressions, let us assume, for brevity, 
U=w/3 — V = /y — wa ; W = fnci — l(3 (XL)- 
From these we deduce 
mW — nV =a — I (^lDi-\-m(i-\-n'y)=ciA 
n\] —IW=[^ — m(/a+m(3+wy)=/3^ !> (XII.): 
where a^, (3^, denote the rectangular components of the tangential component of the 
magnetization at a point infinitely near the surface. Conversely, from these equations 
we deduce 
U=w/3, — my^-, \ = ly—noip, W (XIII.). 
Now the direct data required for obtaining the values of X, Y, and Z, by means of 
formulee (X.), are simply the values of U, V, W at all points of its surface. Eqiia- 
