286 PROFESSOR G. II. DARWIX ON THE STABILITY OF THE PEAR-SHAPED 
Again, Hg is given by (41), and the successive H;'' are given by the general 
formula (37). 
Again (38) and (39) give 
= Gu°o - 
cos” /3 cos” 7 
(cos‘^ yF + sin^ yE), 
n° = ^ Gnl - H'nl + 
4 cos” /? cos” 7 
— F- 
o -*■ ? 
and by successive applications of the formula (37) we find the successive values of 
Uf, Uf. 
It is convenient also to have the series of n_ 2 , T_o integrals. These are to he 
found fi'om 
U% = - sin- / 3 n 5 "'+", sin- + cos' 7 X 5 " . . (42). 
The T integrals may apjoarently be derived by a similar set of formulae, but since 
at each step we divide by k'^, a small Cjuantity, all accuracy is rapidly dissij)ated. 
Although we may safely derive one series of T integrals from a preceding one, we 
cannot so derive a succession of series, and it becomes necessary to find new formulae. 
In order to determine the T integrals, consider the group of integrals 
cos*' (p 
If we write f = 
cos 7 tan p 
cos /3 ’ 
^ 2 m — 
COS 7 . 
a = - 7 , we find 
cos 13 
1 _ r -t 
iOs3«'-l7Jo 
COS /d cos®"' ^ 7 J 0 (1 + P) 
whence, by some easy integrations, 
Ul = 
TT 
2 cos /3 cos 7 
TT 
Cf 4 = , '' ' (sec^ ^ + sec'^ 7), 
^ 4 cos/3 cos 7 ^ ^ 
m — ,7 [sec'^ /3 A sec^ 7 + f (sec^ ^ + sec^ 7 + sec- /3 sec- 7) + II. 
lb cos/3 cos 7 f ' 6\ r- \ /I // I J 
1 . 
On expanding - in powers of k we see that 
i . o 
7'2« 7T2>! I l.,,'3 7T2'/l+2 I UJ-pZn + i I 
^ 2 m ^ 2 m “l 2 ^ ^ 2 m “I o ) ^ ^ l~ 
2.4 
When ni = 0 the U integrals are easily determined. 
