130 Mr, Ivory on the figure requisite to maintain the equilibrium 
. And the solution of the problem is now reduced to show that 
this formula is similar to the equation 
R 2 — 
When this is done, the relation between the figure of the 
fluid mass and the given rotatory velocity will be found by 
making the two expressions of R and R' coincide, so that 
both shall belong to the same surface. 
In the first place, the integral in the numerator of the value 
of R 2 is a function of k , k', k " ; and as any value may be as- 
signed to C, the whole numerator may be regarded as an 
arbitrary quantity. The denominator is therefore all that 
remains to be considered. Now, 
K = — ~ 4 // Log. R' x CT'dv! d 
ST 
and, since Log. R' = Log. k — \ Log. S, we shall get, 
K = — - + i// - Log- Sx C (,) d (.' dtf. 
because, 
Again, 
Log. k x ff d p/ d-sr = 0. 
(2) I d \( 1 ■— y a )» 3 
C W == — . 
2.4 
dy 2 - 
1 . 
2 » 
and, y = V 1 — jit* • V 1 — p/ 2 . Cos. ( m — ; 
or, if we write m,n,p, for p., ^ \ — f. Sin. tzr, V 1 — ^ . Cos. 
we shall obtain, 
y = m fjL -j- n V 1 — jtt /2 . Sin. sr 4 p V' 1 — p/ 2 . Cos. ts : 
‘ST, 
and hence, 
K = - T + -Jf- (i r) s • v 
4. { -L ( 1 _ p/ ■ 2 ) Sin. V — ~ J Log. S. d^dsr' 
{±(1^-) Cos. 2 ®-' - -L | Log. S. 
4 3 njf — \d V 1 — p/ 2 . Sin. Log . S. dy! d-sr 
4 3 mp ff — f V i—p 2 . Cos. zs* Log. S .df d-sr 
3 npj f — ( 1 — jw.' 2 ) Sin. w Cos. 37' Log. S.dp d-a*. 
