FIGURE OF EQUILIBPJUM OF A ROTATING MASS OF LIQUID. 
The relationship between the successive U integrals is clearly 
Ui. = mA - \ cor 
/c - sin" 7 /c" 
2 ) 1-2 
I now write for brevity 
X — cos IB, y — cos y, z = sin y, \ — 
It appears that we may put 
1 . 3 . . . ~ 1 
To'‘ 2.4. . .2ft 
1 2..+ 1 ,3^ 
1 +z’ 
1.3 211 T 1.2 ft -f- 3 
_ y 
P x + y 
T-'" = — B 4- 
* ixy ^ 2(1 + z) 2.4... 2n ^ ^ 
2' 2n + 2" '2.4 2ft + 2.2ft + 4 
, 1 2ft + 1 ,2 1 
6 n . ■ • ^>3 
+ . . . 
TT 1.3... 2 ft — If ,1 2ft + 1 ,2 . 1.3 2'ft + 1.2ft + 0 . 
2A 2ft F 2.2ft + 4 ■ G 
2(1 + z) 2.4... 2ft ^ V'o 2 2ft + 2 ""2' 
TT 1.3... 272- — 1 
Tr=4-c’,,.+ 
TT 
1.3...2ft-l 
^3 
2(1+ z) 2.4... 2ft 
By considering in detail the cases where n = 0, I find 
A-q — 1 , Ojq — 1 ; 
^0 = 1 + 1 (1 “ 2p + 2p2), 5^^ = 1 _ i\; 
Jj U fJ ’O 
F = ]-^ + \i + + + ]) +1 
= 1 + (1 — 2p + 2p^) d- — (1 — 4p + — -3^p® + Ip^), 
Cq = 1 — 8^ + 
By some rather tedious analysis, it may be proved, by considering the manner in 
which each T is derivable from the preceding ones, that 
1.3 ... 2ft - 3 
■^2)1 1 o _ ( O ( 0„ O fi~ -d.2ii_2 ) H~ C) 2j2. _ 2^^ 
1 — 2p V2.4 ... 2ft - 2 
1.3 ... 2ft - 3 
<^2)1 — 1 4-^ + 
2\2 
1 - 2 \ 
ft 
^2,1 = 
1 - 2/7 ^ 
2ft — 1 
2 (1 - pf 
■ Ctor, 
2 ) 1-2 h 
1.3... 277 
P"^2n-2 + ^.2 ^2,1-2 H- 2.4 ... 2ft - 2 
d(l -d) 
^2n — 
1 - 2X 
227\3 
(1 X)2a2H - 1 ^>2); 
2 ) 1-2 
