282 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
§ 14. The Lost Energy of the System; Solution of the Frohlem. 
If the several contributions to the energy be examined, it will be seen that if i, the 
order of harmonics in fS^L is odd, there is no term with coefficient ef:^ in E; this 
follows from the fact that the w and p integrals vanish for the odd harmonics. Hence, 
as far as concerns the odd harmonics, E involves fi only in the form {ff. The 
condition that the pear shall l^e a level surface is that E shall be stationary for 
variations of the /’’s and of e. It follows that when i is odd ft is zero. We may 
therefore drop all the odd harmonics, inclusive of f^ and it is clear that the term 
— in E (given in (20)) vanishes to our order of approximation. 
For the sake of brevity, I adopt a single symbol for the coefficients of the several 
kinds of terms in E Therefore let 
^0 — ^3 [i (<^ 2 )^ + + 5 (^iW + Bfpi') , i 13/, 
2 9 i 2 (pi 
sni^ ^ 
25 / = 23 / 0 ,/ + (B/ + 2B3) P/ - P 
C/ = (33 - 3/) <(,/, 
(1 + COS® /3) sin /3 
a = 
10 cos /S cos 7 ’ 
6 = i.3 - V./ + (f - % + ir>,„), 
c = L(f)2, 
a = where 3 / = f/©/. B/ = 0 / = 
((Pq CIVq UVq 
With this notation 
1/2 r 00 00 ^r»2 3 
E = I - + 2^ Bt(fft ~ t Ct {fff + f ^ (a + + r/i - \ff) ■ 
L 2 2 ■^'TTp J 
Let us now inake E stationary for variations of e and ft. 
First, by the variation of any ft excepting f^ and /y, we have 
/i* 
f s 
(29). 
On eliminating all these we have 
]? _ 3 
I/® r/ 
X ^If\ 
+ 2 '-^1;' j e* + 2B,e% + - C, {f.f - W (//) 
So) 
+ ./, d" ^fz — '^ft) f • 
iirp 
