of a homogeneous fluid mass which revolves upon an axis. 125 
Now, in the case of a homogeneous mass of fluid in equi- 
librio, these forces must be evanescent; and that too, for 
every point within the stratum, which requires that 
V ( r ) — V/( r )> 
shall be a constant quantity independent of a , h, c. And this 
is the remaining condition necessary to the equilibrium. 
v 
Now, if we put. 
c\ r _ d l 
^ J r if 2 r R 7 7 + R' 2 
Or= r 'J~T-ff- v 
R ', 3 d^rUz' 
V r 2 — 2 r R‘,y + R' 2 5 
we shall get, 
V(r) = Q + Kr a 
v,('-)=Q, + K/ 2 :* 
consequently, 
V(r) - V, (r) = Q _ Q, + r- (K- K,). 
A very little attention will show that Q and Q,, contain no 
terms multiplied by r \ For if we expand O into a series of 
the ascending powers of r, the term containing r 3 , will be 
r 2 x flfC^ ) d f d'ay' , 
which, by the nature of the function C (2 \ is equal to zero. 
Wherefore the foregoing expression cannot be independent 
of a, b, c, unless 
K — K 7 = o ; and K = K ; . 
But it has already been shown that the equality of K to K, 
requires that the radii R' and R' /5 which make the same angles 
with the axes of the co-ordinates, and consequently are in the 
same straight line, be constantly in the same proportion to 
one another :* and hence we obtain this property of a homo_ 
* Section 3. 
