75] ON THE ATTRACTION OF AN ELLIPSOID, 
439 
the preceding expression for V becomes 
in which 
Assume 
q \ 
p’ 
F(P. 
"•) pix+n 
D = 
P . 
q, ... 
dp 
dq 
da) ’ 
da) ’ 
dp 
dq 
dd ’ 
dd ’ 
D da> dd ... , 
p = PÇ + P' V + P // Ç... 
q — Q£ + Q'y + Q'Ç ■■■ 
where the number of variables 77, ^... (functions in general of w, 6, &c.) is n, and 
where the coefficients P, Q, &c. a,re supposed to be functions of co only. We have 
dp pdf 
dd dd 
dq _ r)d£ 
dd~^dd 
I p/ dv i P" ^ I 
+ dd + dd 
V dv 
dd 
+ Q' ¿a + Q'' + 
d£ 
dd 
and, substituting these values as well as those of p, q, &c., but retaining the terms 
dco ’ Icd ’ ^ C ‘ * n ^eir origin form, the determinant D resolves itself into the sum 
of a series of products, 
dp dq 
l 
dw ’ da) ’ 
• V , ç > 
P' , Q' , 
drj dÇ 
P", or, 
• Id’ dd’ 
Let 'P be the function to which yjr (p, q, ...) is changed by the substitution of the 
above values of p, q, ... so that is a homogeneous function of 7], £, ... and we have 
the relation ' V P = 0. (It will be convenient to consider |, rj, as functions of 
a), d, &c., such as to satisfy identically this last equation.) We deduce 
1 
ri 
1 
1, . ... 
= &c. = S suppose, 
X 
y 
■ V > ç , 
~ Y 
f, • ?. 
drj dÇ 
dÇ 
' dd’ Id’ 
dd’ • dd’