438 
ON THE ATTRACTION OF AN ELLIPSOID. 
[7 5 
Part II.—On a Formula for the Transformation of Certain Multiple Integrals. 
Consider the integral 
where the number of variables x, y,... is equal to n, and F (x, y, ...) is a homogeneous 
function of the order fx. 
Suppose that x, y, ... are connected by a homogeneous equation yfr (x, y, ...) = 0 
containing a variable parameter m (so that « is a homogeneous function of the order 
zero in the variables x, y, ...). Then, writing 
F = X" + y° + ... , X — TOL, y = r/3,... 
the quantities a, /3, ... are connected by the equations 
a 2 + /3 2 + ... = 1, yjr(a, 0, ...) = 0, 
and we may therefore consider them as functions of co and of (n — 2) independent 
variables d, &c.; whence 
dxdy ... = r w_1 V dr dcodd ... , 
where 
V = a , ß , ... 
da df3 
dw ’ da> ’ 
da. dß 
dd ’ Td’ 
F{x, y, ...) = r*F(a, ß...), 
Also 
and therefore 
r n+n-1 F 7 (a, /3, ... ) V dr dw dd ... , 
or, integrating with respect to r, 
which, taken between the proper limits, is a function of a, /3,..., equal /(a, /3, ...) 
suppose; this gives 
in which I shall assume that the limits of co are constant. If, in order to get rid 
of the condition a 2 + /3 2 ... = l, we assume 
a 
p 2 =p 2 + q 2 + ...