FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
25 
The moment of inertia of the pear-shaped figure about the axis of rotation, say 
M ¥, will be given by 
p (x 2 + y 2 ) dx dy dz 
where the integral is taken throughout the volume of the pear-shaped figure. We 
have, by our choice of the coefficient s, ensured that the volume of the pear-shaped 
figure shall remain always equal to that of the original ellipsoid, so that we have 
M = - 7 rpdbc, 
o 
and therefore 
¥ = 
_3_ 
47T 
jj 0 2 +r) 
dx dy dz 
abc 
( 101 ) 
Let us transform to co-ordinates x', y', z\ given by 
j _ x 
a 
x = -, y = v, z 
/_ y 
V 
r _ 5 
c 
( 102 ) 
so that the critical Jacobian ellipsoid is reduced to a sphere of unit radius, and the 
pear-shaped figure is reduced to a distorted sphere. With this transformation, 
equation ( 101 ) becomes 
Id = jj! (a 2 x' 2 +b 2 y' 2 ) du r dy’ dz', .(103) 
where the integral is taken throughout the figure bounded by the surface 
.|/ 
'b 
x r i x' 2 
x ' 2 + y ' 2 + z ' 2 — l-Ve— (a^r + +y dr+K 
a \ a b “ c“ 
+h 2 
Lx n , M y ri , 
-1—I- TT~ + • • • + 5 
a b 
= 0. 
(104) 
Let r 2 be written for x' 2 + y /2 + z' 2 , and let us further put 
x' = rx, y’ - ry, z’ = rz, 
so that x, y, z, are co-ordinates on a sphere of unit radius. Equation (104) becomes 
2 i ,x x , 0 y , z"\ , x/c 
r — 1 + 6? * — (a —o + /3 7 ~p + y-o) + C'T— 
a\ ar c 2 a 
+ie 2 
"LxV MyV 
a* 
Id 
+... +s 
= 0 . 
. (105) 
Let us suppose that r, the radius vector to the boundary of this distorted unit 
sphere is given by 
r = 1 +cf+e 2 g + e 3 h+ .... 
VOL. ccxvn.— A. E 
