FIGURE AND STABILITY OF A LIQUID SATELLITE. 
205 
Now from (25) of the “ Pear-shaped figure,’ 
K 
& = - 2 , . 
sm y J U A 
sm 
A[1 = ^ y r ^ a , = KCOS^p 
sin y Jo A sm 2 y Jo 
x _ kcos 2 (3 U sin 2 y , 
y L A 3 /* 
When the A’s under the integral signs are expanded, all the terms of the series 
involve integrals of one of two types. If we write 
fl = 
the types are 
U sin 2 ”y j 
J_ log, 1+sill >' 
sm y cos y 
1 
2 n— 2 
tan : 
2 « — 2 
y ~ jT> 
2 n— I 
(2n—2) (2n — 4) 
■ / _ (2n— 1) (2n 3)...5 , 2 , / \»+i(2h— 1)...3 ^ 
+ ( } (2n-2)(2«-4)...2 tan r + (_) (2w-2)...2 ^ 1) , 
tan 4 y + 
. 2«, 
U sm y ; 
' - t i-j r ’ y- 
In r 
■-— -rtan 2 ' ( 2 y+ • ^ ^ 
2 n(2n—2) r 5 
tan 
2«-4 
tmv| ~ r 
2n(2n—2){2n — 4) 
(2n-l)...3 
2n (2?i — 2). ..2 
y-... 
(n-i) 
As it is not quite obvious what interpretation is to be put on these formulae for the 
smaller values ot n, I may mention that when n — 0, 1, 2 respectively, the first 
integral is sin y ; sin y (fl— l); sin y []r tan 2 y—§ (fl — 1)], and the second is sin y fl; 
S4n 7 [i tan" 7~h (^~ 1)] 5 sin y tan 1 y — ■— tan 2 7 + iry (U— 1)]. For larger values 
of n the interpretation is obvious. 
If we use these integrals and write 
^ (Sfi-Afi) = k ' 2 [o-o-o-^ + o-^-a-gK' 6 ...]! 
K | 
S ^ n ^ (Si — Afi) = T () — T 1 K r - + T 2 K' i — T- i K m ... 
we find 
• (37), 
°" 0 — s [2 tan 2 y + 3 — (3 + sin 2 y) fl] 
crj — 3 'V []' 3 _ tan y 3 - tan" y— 5 + (5 + sm" y) fl] 
°*2 — To ‘2 4 [If tan 8 y tan 1 y + f— tan" y + 7 — (7 + sm" y) fl] 
03 = iWe [fttan 8 7 -tM tan 8 7+fttan 4 7~ 4 tan 2 y-9 + (9 + sin 2 y) O] 
T 0 = [ — 3 +(3 — sin 2 y) fl] 
u = if [t tail 2 y+ 5 — (5 — sin 2 y) fl] 
r 2 = iff [if tan 4 y-f tan 2 y - 7 + (7 - sin 2 y) fl] 
t 3 iroTTf [toV tail 8 y jf tan 4 y + 2 tan" y + 9 — (9 — sin" y) II] 
u = [- 3 \- 5 - tan 8 y - -^5 tan 8 y + f tan 4 y - 1 tan 2 y - 11 + (11 - sin 2 y) tl] . 
( 38 ). 
