FIGURE AND STABILITY OF A LIQUID SATELLITE. 
219 
Since 
o.m = 22il)! r_L + i±li_L_ 
v '' ' 2£+ 1 ! \y +1 2.1 !»'! (2i+3)i 
i + 3 
+ . . . 
by differentiation, and by expansion of (v 2 — l) s/2 in powers of l/v 2 we obtain 
ouv) = 
kl ^ ‘ ^ 2 i +1 ! v i+1 
' (■ i+ 2 )(i+l) + s 2 
2 (2i+ 3) z^ 2 
Accordingly, in order to find the second term to the degree of approximation 
adopted, it is merely necessary to multiply the leading term by 
(i+ 2) (a+l) + s 2 
2 (2^ + 3) v 2 
In order to find the leading term explicitly we have to insert for the q s their 
values, and after some tedious reductions I find 
«.*(!) = {I +*A (S+*)+*A‘ [-2T+S’ (**+8) 
+ 2S(6i-l+2^) + 2i(3 l +l)-^]}x{l+t±|iL±ii±i!^J . . (61), 
where 
* _ Ki_±jQ T _ (^+1) (^ + 2) 
5' 2 — 1 ’ s 2 —4 
This formula fails for the cases of 5 = 0 and s = 2, and these cases have to he 
treated apart. Following a parallel procedure I find 
<& 2 ( t) = ! ^ ( 1 + (2 + 20 + yj ¥ /3 0 2 [29X 2 + (144^ + 298) S-*48i —40]} 
jl + ( i ± . 2 ) _ ( i ± l ) + 4 . (62). 
1 2 ^+3) r 2 J v ’ 
2 a (ei) 2 ^ +1 
2^+1! r i+1 
{1 ~ (»-!) + rh-AH’ (» -1) ( 7i 2 - Si - 6)} 
1 + 
(i+ 2 )(i+ 1) Zrl /ggx 
2(2i + 3) r 2 J ' 
The values for t less than 3 are not required, and when i = 3 these formulae are 
found to agree mutatis mutandis with those of the last section. 
It is pretty clear from general considerations that the higher inequalities 
corresponding to harmonics other than the zonal ones must be very small. I 
have, in fact, computed the third tesseral harmonic inequality (i = 3, s = 2), and 
find that it is so very minute compared with the third zonal inequality (i = 3, s = 0) 
as to be negligible. Accordingly it appeared to be a waste of time to develop 
formulae for any other than zonal inequalities for values of i greater than 3. Thus 
of the formulae just determined the only one of which actual use is made is (63). 
2 f 2 
