of a homogeneous fluid mass that revolves upon an axis. 139 
and as this quantity is to be valued at the limit, or when 
x — h, we have = 0. Wherefore the expressions of 
the attractive forces are reduced to, . — • 
, — J ^ r - ; that is, to the partial fluxions of V(r) , supposing 
that x is independent of the co-ordinates, a, b, c. 
The oblate ellipsoid of revolution, corresponds to the sup- 
position, e 2 == e' 2 ; and, in this case, we get, 
d x 
■ d x 
(* 2 + e 2 ) 5 
r ) — 2 ^ • f x a + e 2 a . I'M ^ + 
the equation for finding h , the limit of jc, being, 
a 2 ■ + c 2 
IF F+l* — 
And, when the attracted point is in the surface of the ob- 
late spheroid of revolution, h is equal to k ; and, if we put 
x= -j> we g et > 
Y / r \ 3 M Arc. Tan, x 
' ' 2 k ' x 
3_M 
2 k 3 
x — Arc. Tan. x 
(b' + c‘) 
3 M 
4 A 3 
Arc. Tan. x 
x 3 
x 
1 + x 2 o 
7. It may not be improper to apply the foregoing solution 
to find the relation between the figure and the velocity of 
rotation in the case of an oblate ellipsoid of revolution. As 
it has been proved that the supposed figure will satisfy one 
of the conditions of equilibrium,* nothing more is requisite 
than to employ the other condition, namely, that contained 
in the equation (A), to determine the relation sought. 
Let k be the semi-axis of revolution, and V k 2 -|- e 2 , the 
radius of the equator ; if a, b, c be three rectangular co-ordi- 
nates of a point in the surface, a being parallel to k, the equa- 
tion of the figure will be, 
* Section 5. 
