I 
EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
413 
Thus the criterion of the possibility of equilibrium is that 
C = 1 - 
43 A s + 13 a 3 
" 12c 3 
should be positive. 
But the radius-vectors of the poles are 
(80) 
+ 
and, similarly, 
+ 
9. 
6 
Therefore 
r + R = a + A + j^[(7 
+ 5-4W + 
a + A) A* + (a + 7A) a *J + T AW (A + a) 
II AW (a + A) + j^AW (a* + A*). 
Now the interval between the two masses is c — (r + R); hence, if the two masses 
are just in contact, 
c — a + A + [(7a + A) A" + (a + 7A)a° J -f ^A~a~ (A -f- a) 
+ ~ 5 ^ ^ a ° ( a + A) -f A 3 a 3 (a 3 + A"). . (81) 
In order, then, to test whether equilibrium is still possible when the two masses are 
just in contact, it is necessary to determine c from (81); and then, substituting in (80), 
find whether C is positive or not. 
The solution of an equation 
c — a + 7s + A + ,5 + 2 + 77 > 
and the determination of 
C= 1 — 
can only be performed by trial and error. 
Now suppose that the solution is c 0 -J- 8c, where 8c is small ; and that 
