FIGUKE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
273 
— p civ = — (1 — cla- clz 
and the limits of integration are z = 1 to z = D. 
Therefore 
L\(r,-v-)pdv 
= TTp^ (I — \ez) [ 2 z — 2~ — Xe (2 — 2~ ^2''’)} ch da [|e®2 (I — \e2) dz da. 
. dV 
In this expression we neglect terms of the order and note that e’z^ is ol 
that order. 
Thus 
dV 
(Fo- lT)pdy = 77p2jje3[22-23-\e(2+22_|23)]tZ2do-+lpjJe22~ dzda {z=l to 0), 
= IV 1 - V daIp\^e^ da , 
the integrals being taken all over the surface of the ellipsoid. 
We must now consider the meaning of the integral i | (Fq — pdv. 
Let P be a point on J and Q a point in R on the same orthogonal curve. 
Let — JJ be the potential at Q of the density — p throughout R, and — Uq its 
value at P. 
Let 8 ]je the surface density of the positive concentration on J, IF its potential at 
Q, and IFq its value at P. 
The lost energy of the double system consisting of — p throughout R, and 8 
on J is 
ifFpc/c + i-j lF,8(/(T-i| F,8(/o--ij IFp 
do. 
This is equal to 
Consider the triple integral j j — IFq) pdv. Here dv = e (I — Xes) da ds ; also 
u,- TFq is not a function of s, and the limits of s are 1 to zero. Therefore 
■fi 
LJO 
e (1 — \es) p ds 
da. 
But [ e (I — Xes) p ds is equal to 8 the surface density of concentration. Therefore 
{([£/„ - Il/„] S </cr = f|f( U, - TU) p dv. 
VOL. CC.—A. 
2 N 
