•294 PROFESSOR G. H, DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
18. The Integrals B/. 
Adopting the notation of the last section we have 
^[1/ (7.) = a -^h {v~ - 1 ) + C (W _ 1)2 ^ g _ 1)3 _|_ ... 
I^et n = - - , and k sin ib = sin v, so tlmt at the surface of the ellipsoid where 
K sill ^ 
xji = y, we hare y = /3. 
Then 
(t.) = a -{-h cot" X + c cot^ X + • • • 
Pf (di) = cot^ /3 c cot* ^ + . . . 
civ 
N 
and 
ow 
- 1)4 _ 
1 + 
d\fr 
1-/3 
a; = »,• (-o)]^ 
0 
\/ (1 — a:- sill- A) 
dv 
K \ sec X dr// 
r 0 
>'o 
1-/3 
Hence = k [a + h cot^ yS + c coH /3 + . . .)” | 
sec X dr/c 
0 [« + & cot- X + ^ cot* X- • d" ’ 
We have, in § 4 of the “ Pear-shaped Figure,” the rigorous expression of this 
integral for harmonics of the second order, viz. ; — 
, _ tcjl- 2g/) (1 - g/ sill" 7)- I d’(7) _ E (7) sill 7 cos 7 cos /3 1 ^ ^ 
2iz/F/«ii* 7 1^/ "^!?T(l-2Yshd7)j ’ ' ^ 
The values of i/q-, gl have been already given, and thus all the quantities involved 
are known. 
The two factors of (viz., p/ and A are given in approximate forms in 
“ Harmonics,” and therefore, if we made allowance for the different definition of 
adopted in that paper, we might calculate 'H/. The computations I made shoAved 
that the results obtained in that way would liaAm been sufficiently exact, but as it 
was clear that the approximation to the functions was not A-eiy close, and as the 
computation is tedious, it seemed better to find the A/ l>y quadratures. 
In order to do this I divided y or 69° 49' by 12, and took 5° 49'A^ as the common 
difterence, say S. I then computed sec X; ~\' 1) cot- X + C cot* X + • • • J 
sec X -F (« + cot'- X -P c cot* x • • •)' P**’ Amines of xjj = 0, S, 28 , . . . 128 or y. 
As a fact the first fiAm or six Amlues need not he computed because the early A-alues 
of the functions to be integrated are practically zero. The ordinary fbrmulte of 
