FIGUKE AND STABILITY OF A LIQUID SATELLITE. 
217 
Each term in these expressions is integrable, and the limits of integration are 
x = sin y to 0 . After integration a must be put equal to 1 / q , and since the result 
will only be true as far as K n we must put for <f its approximate value f (1 — |V 2 —§/c /4 ). 
When these processes are carried out, and the formulae combined, we find 
& = TJThrr .i -!+f (1 + i^+fi-c' 4 )n-l(l + o V-**«)A 
Af 
— iV ' 2 sec 2 y ( 1 + ff/c /2 ) + -AV 4 sec 4 y 1 . . . (59). 
q Sill y 
In the coefficient and in Aj J (or 1 —^ 2 sin 2 y) it is as easy to use the rigorous value 
of q 2 as the approximate one, so it may be well to repeat that 
q 2 = f[l+ K 2 -y(l-fK 2 + K 4 )], and fl = log, — -. 
sin y COS y 
If a process parallel to that adopted for finding % :>J were adopted in the case of A,-, 2 , 
it would lead to a divergent series, but fortunately a much simpler process is 
available. 
In the case of i = 3, s = 2 , we have q 2 = 1 — |-K /2 + f«: /4 ... , so that 
Aj 2 = COS 2 y [1 +i/c' 2 (1 -f/c' 2 ) tan 2 y]. 
Accordingly we are now able to expand l/Afi in powers of « / 2 tan 2 y. 
We have then 
h = At: [1 -«' 2 (1 -}*") tan 2 y+f / 4 tan* y ... ], 
Af COS y 
1 1 
A cos y 
;— [ 1 — |k ' 2 tan 2 y + f K n tan 4 y ... ] . 
Whence 
= Sy [ 1 ~ f A 2 (1 - tan 2 y + ^V 4 tan 4 y ... ]. 
Now 
fy q lr . b v 
j„ d r = y ft r~A tan 2 7 +£f (n-i)], 
sin r » tan * r-j*i ta ’ 4 * r+Ar* tan 2 r-Aftl («- 
I, ^ A = si ”y ft*an» r-A tan* y + A 2 ! tan* tan 2 y + (ft - 1 )], 
where 
n 
1 i 1 + sin y 
■ -f0g e --- L , 
Sill 
y 
cos 
y 
VOL. CCVT.—A. 
2 F 
