300 PROFESSOR 0. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
The coefficient of e® is positive and the pear is stable, provided that 
492-2074e < -0688539, 
or 6 < -00014. 
Inspection of the table of numerical results shows that the zonal harmonic terms 
contribute by very far the larger portion of the sum. Now the sixth zonal term was 
[ 6 , 0 ] + 
W 
= -00002835 + -00003118 = -00005953. 
This is about ^ of the critical total -00014. The j^ear is then stable unless the 
residue of the apparently highly convergent series shall amount to 2 ^ times the 
contribution of the sixth zonal term. Such an hypothesis appears profoundly 
improbable, but I have thought it expedient to make a rough determination of the 
contribution of the eip’hth zonal harmonic to the sum. 
O 
If we take k as equal to unity, Sg = (/x) (iTg {(f)) = Pg (/x), and we easily see that 
the formulse ( 8 ) reduce to 
3 0 
^cos^^Jo) 
_ G cos^Yj-Dj-D 
~ TT sin 7 Jo Jo 
D (1 - 1 - siid 7 sill- 0) cos 6 
0 (1 — siii-7siir 
{Sgf SgcW (/(/), 
cos 0 
1 — sill- 7 sill- 0 
{Sg)\SgcWd(f). 
In these integrals cf) only enters through {S^)^ or (p-)]'^ [C 3 (^)]k 
Now 
. , ip, , 1-3 , , 1.3.5 
Hence 
f\Cg{<j))Yd<f) = iu 
j 0 
a 
Wg =: f A” cos"^ y [ 
J I 
.4 ' ‘2.4.6 
= ^ttK, where K = -1452. 
>( 1 +siffi7Siffi^)cos^ 
0 (1 - Slid 7 Slid 0f ^ 
Ps = 
3 A’'cos-7rD cos ^ 
ein 7 
In these integrals 
-^8 (p) — li'A [6435 sin® $ — 12012 sin® 6 + 6930 siid 0 — 1260 siir 6 + 35] 
[Pg (p)]' = a — /3 COS' 6 -\- y cos^ 0 — S cos® 0, 
where a, j8, y, S have known numerical values. 
The integrations may of course be effected rigorously, but it seemed far easier to 
determine them by quadratures, I therefore computed the values of the functions to 
