STABILITY OF THE PEAR-SHAPED FIGURE OF EQUILIBRIUM. 
At the surface of the ellipsoid v = v Q , xp = y, and we shall, as before, write 
sin /3 = k sin y. At the surface of the ellipsoid we have then 
v = 
In this notation 
cosec f3, A. t 2 = 1 —q x sin 2 y, A 2 
A/ A,* \ _ VA/...A/ 
i sin 2 Q/ “ sin 2 ” xp 
= cos 2 /3. 
A consideration of the eight types of harmonics shows that they may be written as 
follows:— 
Type. 
EEC, 
= 
II (kV- ? ,>) 
i 
= cosec 2 ” xp n (A/), 
i 
EES, 
P»«*M = 
( K V-K 2 f (kV- i )* n («¥-(]/) 
i 
= cosec 2 ”* 2 xp A cos i// n (A/), 
1 
ooc, 
P 2n+ , 2t+l (v) = 
(/cV-i) J n(KV-^ 2 ) 
i 
= cosec 2 ” +1 xjj cos xp n (A/), 
oos, 
= 
(kV 2 —k 2 )^ n (fcV- ^ r 2 ) 
i 
= cosec 2 ” +1 i// A n(A/) ? 
i 
OEC, 
^p2« + T*(^) = 
n (i< 2 v 2 —q x 2 ) 
i 
= cosec 2 ” +1 x/j n (A/), 
i 
OES, 
P 2 » +3 » = 
KZ, (kV-k^ (kV-I)' n { K v-q x 2 ) 
= cosec 2 ” +3 xfj A cos xjj II (A/), 
i 
EOC, 
P* + .* +1 (*0 = 
^(«V-i) J n( K V-^ 2 ) 
i 
= cosec 2 ” +2 xp cos xfj n (A/), 
i 
EOS, 
^2«+2 2tTl (v) = 
Kr (k 2 p 2 — K 2 )* n (kV-^/) 
= cosec 2 ” +2 xp A II (A/). 
Using and ^ generically for any one of these and for the corresponding function 
of the other kind, we have 
dv 
(v 0 ) (v 0 ) = [P (^o)] 2 [ 
Or, changing the variable of integration to xp, 
a =*w MI 
dxfj 
(IWN 
To effect the integration the reciprocal of the square of ^ must be expressed in 
partial fractions. Inspection of the eight forms of functions shows that 
1 1 II II n I 
+ r + i-4- n + S 
B r 
{IWT / *V-k> g kV-1 h kV T A x l(K.V-q x 2 ) 2 icV-g/J 
with appropriate values of f, g, h, A x , B x to be given hereafter. 
In every case but that of OES some or all of f, g, h are infinite. 
VOL. CCVIII. — a. c 
