140 Mr . Ivory on the figure requisite to maintain the equilibrium 
b- + c 3 
-4- 
12 I 
1. 
A 2 » A 3 4 e l 
Put R a p a ; 6 8 +A= R 2 ( l — P 3 ) ; «*= k % x x* ; 
then, by substitution, we shall get, 
R 2 ( i + XV) = ** ( i + **)• 
Again, from the formula (A), we get 
V(R) + R 2 . — =C, 
o being the centrifugal force at the distance i from the axis 
of rotation. This equation must be made identical with the 
former one, and for this purpose we must substitute in it the 
value of V (R) reduced to a proper form. Now, M being the 
mass of the spheroid, we have 
and for the sake of abridging, if we put, 
A 
B 
>. — Arc. Tan. x 
: _ 
Arc. Tan. \ ■ 
i 4 a 3 
2 X 3 
and likewise attend to the values of a 3 and If -j- f, we shall 
get by the formula in § 6, 
^ rc - .. Tan -* — R», 8»((i 4V)B4(i 4x«)( A - B) .?} 
and hence, by substitution, the equation we are considering, 
when brought to the same form of expression as the first 
one, will become, 
R 2 
i + 
(i +**)(A. 
B) + A & ( I 4*p _£. 
v}=— 
— 77 J (i + ^) B “~ 
4* 9T 4 ^ 
(1 4 **) B 
By comparing the two equations, it will appear that the only 
condition necessary to make them identical, is this, vi#„ 
