266 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
Now 
Vcp civ 
(It rW ( 14 > 
A ¥1 (±V J + 2 (|; + /') a ><i.,sv 
n 8 
^‘”0 V ^*’0 / 
jip 
^'0 \ ^’0 / 
T j + ieV.ao'P + 2 
(Oi 
4>i 
. 3^Y—Y r 
>S? 
4-% (e^ — 4- f-"\ 13 ' 
2 -p 2 W - -r./, y 
Let us integrate these three lines separately. 
First integral 
8 ~ (£f eg 3 <l>.s;, + -JeV^gol^ + S [e^ +/•) a,-*,S'.- [ Ic.% + XfiS:] cW # 
= 3 
5 + {M 
Second integral 
+ V^crMo^ + V (e3 ^ + f ;) 
{^{SY^ + 2ScftS,SndecIct> 
= s'fYfT 
'• 0 ^ '‘ 0 / L 
,4 Cxll + 
+ 2 Se:/:%"/ 
Third integral 
M“ f h y cos” 13 COS" 7 
* \ / ®hi /3 
<'13:,TAi; + 2A-g+/‘')B 
'3 
Sf 
{c”" {S.,f + 2 Sc//N3 ^/} d<^ 
All the terms, excepting tlie first of the first integral, are of the fourth order, and 
in them we may put equal to unity. 
Therefore 
3P Ik 4 
CH = 3 ^ e^%4., 
^ (I \ '' 0 / 
+ 1 
3P 
id 
l^u(o- 2 )^ + S(2aV + 15fpf)^j 
+ Sey;'[4(a/ + ^3)a.; + (i3f + 2i33)pr] + 2:^(/?)^^/c/>/ . . . (ir) 
