of a homogeneous fluid mass that revolves upon an axis . 137 
In the expression under the sign of integration, r increases 
from its value at the given attracted point till it becomes in- 
finitely great ; the angles which it makes with the axes of the 
co-ordinates remaining constantly the same. But if we sub- 
stitute the values of a, h, c in the equation, 
a ~ I ^ _l_ C® ____ 
"I® » A 7 ® ■ 1 "® 1 ’ 
we shall get, 
-h = — 4 - 
r® A® * A® -f- e® 
and, by writing x 2 for h 2 , 
1 ( 1 — /n- a ) Sin.® ct , ( 1 — jw,®) Cos.® xj _ 
r® x® "■ x® + e2 x® -f- c'® ’ 
in which expression, (x and rs remaining the same, x will vary 
from h to be infinite, while r increases from its value at the 
given attracted point to be infinite. Wherefore, by substi- 
tution, we get, 
(!— /* a ) Sin.® v s , ( I - r?) Cos.® ■a . 
1 h- + e' s ’ 
V(r) = 2 n.kkk'.J'- 
■ d x 
( [x 2 + e®) 2 (*® + e'®) s 
— c? x 
27 t . kk' k". r 2 p. P 
X® (.r 
+ e®) 2 0® + e'®) 2 
2 7 T . k k k". r 3 ( 1 — Sin. 2 ot. P 1~~ 
U O® -f e®) 2 ( x a ~ 
• 2 tt . k k' k". r 9 (1-— fl) Cos . 3 vs. P 
+ e' 9 ) 
■ d x 
(x 2 -f e®) 5 ( x + e'®)^ 
Let M denote the mass of the ellipsoid, then M = £ & . k k' k" 
wherefore, by substituting a, b, c, for the equivalent quanti- 
ties, we finally get, 
\T f \ 3 M P —C 
V( ’' )= V; 7 ^ r 
■ dx 
a 3M 
a 3 . 
(x* -f e ®) 2 (p -f- e '®) 2 
— d x 
2 x 2 (x 2 + e®) 2 (x 2 + e' 2 )$ 
jm. r =*£ 
2 {x 2 -f- e ®) 2 (x + e'®) 
3 M 
/; 
e®) 2 (** e'®) 2 ' 
T 
MDCCCXXIV. 
