FIGURE AND STABILITY OF A LIQUID SATELLITE. 195 
The sum of these last four expressions is equal to —dV/dr, and therefore equal to 
jf(o 2 di/dr. 
Now let 
cjV 3 = f7rpa 3 (l + £), 
so that £ represents the correction to Kepler’s law of periodic times on account of 
the ellipticity of the two bodies. Then we have 
£ = 
3 r ^+i)+Y(i+hl + 3 
10 r 2 
+ k 2 Kc (-J^2 + ^5 + — + 3 ) + 
+ 
2 Or 4 
9 
8 Or 6 
k K K“ k 2 
56r 4 
1 
^ ( “t + “2 + 3 ) ( Ui + “ ^ ° 
(— + AL _l o 
\K 4 K 2 
48r b 
K? + l + l +5 ) + ^(l i+ J'‘ + J 5+5 
+ ^2 + Ja + “a + 5 ) + -. + WWV + —, + i + 5 
K V k 4 K V k 2 K 
-1W W (1) ©/ (|tt) ) (£>2, s even) 
+ 1 11 S ^ {1 + M 5 + ...1 (all harmonics) 
r* 2 2t+ 1 sm /3 L 14k 2 r 2 J v ' 
(24) 
When (r/k) is developed in powers of 1/r, its first term is one in r (t+1) ; hence 
0 
¥ — <2t/ begins with r _ \ Now /* s will be determined from terms in the potential of 
the ellipsoid E of the i th order of harmonics, and will therefore involve r _(i+1) . 
Therefore in the series contained in the last term but one of £ each term is of order 
r -( 2 i+i)' gi nce the lowest value of i is 3, the term of lowest order in this series is one 
in r~\ and as I shall not attempt to evaluate £ beyond r~ & the whole of this series is 
negligible. 
Again, since (f/) 2 is of order r~ 2l ~ 2 , and since r~ 2 occurs as a factor, each term is of 
order r~ 2l ~ l . Thus the lowest term is of order r~ w and is negligible. # 
It follows that the only sensible part of £ arises from the portion of V denoted 
(eE) 2 , and the last two terms of (24) may be erased. 
We next consider the parameter ff, and, since I does not involve it, the equation 
reduces to dV/df { s = 0; or, since V only contains fl in the part denoted (vv), it 
becomes 8 ( vv)/dff = 0. 
This gives, for 2, s even 
fi — ~i (2f + 1) 
X®/ &/-A* 1 - 
¥ 
3 Xr 
COS y 
sm" 
1 + 
3 (3 + k 2 ) ¥ 
14k 2 
+ 
(25). 
Since this formula contains \ in the denominator, it would appear at first sight as if 
* It is proper to remark that the terms retained in £ are really of higher orders than they appear to he. 
I recur to the neglected portions of f hereafter in § 23. 
2 c 2 
