OF ROTATING COMPRESSIBLE MASSES. 
189 
36. We turn now to the evaluation of AE,. The potential of the whole mass at 
an internal point, V,-, is given, as in equation (20), by 
V; — Po (Vd 0 + f AV; (l) + •••) — {po~<r) (E, : + eAE t + ...). 
We have also 
V 2 V ; . = —4tt/ 3 = — 47rp 0 + 47T (/0 () — 0-) ( 2^- +eP u + e 3 Q 0 + 
a 
whence, by comparison 
V’(AE.) = -4xP, = -4 T (t w . + 2m w v + N z , + 22 wJ+ g; 2! 
\a ac 
c 8 a 4 
c 4 / 
(139) 
We may assume 
AE, = — 4tt + Sk. 2 w l z 2 4 3 Jc s w 2 z i + & 4 z 6 + & 5 w 4 + 2 k 6 iv 2 z 2 + k-^ + 2 k 8 w 2 + 2k»z J ], 
in which equation (139) necessitates the following relations between the coefficients 
i 36/CJ + 6&2 — —r> 
a 
^ 48ii; 3 + o(yk :i = 
2 m 
a i c i 
12&3 + 30&4 = 4 , 
<r 
16i 5 + 4i 8 = %■ 
a 
8k 6 +12L = 
G 
2k s T ktj — 0. 
By comparison with equation (123) 
= \abc 
LJ A . - ~ I A .- ^I w + yfc f 64 t H a . +32 ^5 H a . c + 24 ^ H a ,c 
a 2c \ a arc c 
K = i abc 
2k 8 = \abc 
O rt T _ P_ \ _1 ' T 
A 3 2 1 A 3 2 “2 1 A-C 
a c 
^ I A .+i 5 - A ( M h Ha-+ 32 H 4 , c +24 ^ H i0 , 
c \ a a c c 
By direct computation from these values 1 find 
4 k x 
abc 
44 
abc 
= 0-019734 (y-2)-0-021633, 
= - 0-007359, 
8 k 
abc 
8 
0-017022. 
2 c 
VOL. CCXVIII.-A 
