of a homogeneous fluid mass that revolves upon an axis. 149 
from which it is easy to infer that the most general value of 
y is thus expressed, viz. 
p' = A . p / 2 + B ( 1 — f 2 ) Sin. 2 */ + C ( 1 — y 2 ) Cos. 2 */. 
It deserves to be remarked, that the equations just found 
are the very same that result from the second condition of 
equilibrium, when, for R', we substitute a (1 -f- a .y'f and 
neglect the powers of u. 
Now, leaving out the evanescent terms, the two foregoing 
expressions will become, 
C = d — ij a 2 y + a 2 x . ffy' d\f d */ 
+ a 3 - {*///• c (2) dfdv' + f ( 1 - y ) } 
4 -it.y m ffy ' dpf.dw-fa 5 ffy' ■ C (2) d f d */ .- 
and farther, if we exterminate the integral containing 
from the first, we shall obtain, 
c=gy+j.a’u.ffyd,jd^ 
— a\ {r s -*y — 
4 vy — ffy' dv + 5 •ffy. dydw. 
The first of these equations proves that y is a function of p, 
only, and that the spheroid sought is one of revolution. The 
second is satisfied by putting, 
y=n 1-t* 2 ) 
/=/( 1-y 2 ): 
wherefore, by substituting these values, the first will become, 
Hence we finally get. 
