of a homogeneous fluid mass that revolves upon an axis. 1 29 
it follows that the expansion of—! — will produce no quanti- 
1 — 2 
R' 
ties in the integral except such as are evanescent by the 
theorem. 
The most general value of T_, consistent with one of the 
conditions of the equilibrium, has now been found. If we 
write Y" W’ for A ’ B > C > we sha11 S et » 
j 1 
R/* k* I 
( i — Y 1 ) Sin . 1 Y . (1 - Y z ) Cos . 1 Y 
+ 
A ri 
an equation which belongs to an ellipsoid of which k , k', k 1 ', 
are the three semi-axes. It is therefore proved that a homo- 
geneous fluid mass cannot be in equilibrio by the attraction 
of its particles, and a centrifugal force of rotation, unless its 
figure be included in the ellipsoids. But it is still to be shown 
that the same figure is consistent with the other condition of 
the equilibrium. For the sake of abridging, put 
S = ps ( 1 — ^. 2 ) Sin. 2 s/-{- pi ( 1 ) Cos. 2 3/; 
then, R' 2 = ~ ; and R'= -JU . 
s vs 
The value of Q being reduced to the first term of its expan- 
sion, we have, 
Q =//R ' = 
and hence, 
V (R) = Q + K . R*= k ' .// K R-. 
Let this value be substituted in the equation (A), and we 
shall obtain, 
R a = 
d Y dir' 
~S 
— c 
K + —( 1 — Y) 
MDCCCXXIII, 
