FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
275 
dn = 
■ cos^ ^ cos^ 7 /1 1 
dr. 
Also to the order, zero — n — pyX. 
Since — n is what was denoted in § 9 l)y e.s, the element of volume is 
■ (I -h dndcr, and this is equal to 
or 
[1 - 
Vo 1 ~ " I + 
+l_2r7 
AdUd \Ad“ ' Fd 
■cos- (3 cos- 7/1 1 
— 20 - 
AdUd VAx^ ' r, 
But hy (5) the element of volume is 
da dr, 
da d.T. 
Vo 
■ r cos^/3 cos- 7 / 2 ^ ^ 
Adud Ud rd " ^ 
Equating coefficients of x in the two exj^ressions we find 
cn,s- V cos- 7/1 I 
A = . .■,n -T m. + 
y^AdFd \Ad ' r, 
(23). 
§ 11 . The Energy 37 r/>'^| e®(] — \d)da. 
From ( 22 ) and (23) we have 
To" 
cT {S^ + {S^ Si + le 
cos- j3 cos^ 7/1 1 
AdUd \Ad ' Uv 
. + 26^ (S')^' 
\ ' r( 
So that 
e3(i_Xe)= -Pq3 
Again from ( 6 ) 
eHS,r 
no) n 
AdUd 
1 
\ A 21'' 2 I 2 \ A 2 ~ T'' 
,Ad ■ Uf 
3(7 
0 2 3 7 ■20 ens~ Vcm^'Y cW (- 14 ) M^/kT . ^ ^ la 11 
sV j g = -A-7“(,,j <=»« P r sm H d 0 < 14 . 
(V 
Therefore 
172 //. \5 
■| 7 rpA^ (1 — Xe) c/cr = — ^6 E id) ^ T ^ 
o \A/ 
Cl>(S3d 
L *^1 1 
,3 7s 
AdFd 
COS' V cos^ 7 
1.1' 
