28 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
My value of Ak 2 having been seen to be — 0 - 079156cj 2 , it follows that Darwin’s 
ought to be 
A ¥ = 0-103906/ = 013317e D 2 
so that his value of k 2 as deduced from my calculations ought to be 
k 2 = 0-8441 (1 + 0-15 777e D 2 ). 
In point of fact the value actually given by Darwin was 
k 2 = 0-8441 (l + 0"157786e D 2 ). 
It appears, then, that Darwin’s moment of momentum agrees exactly with mine, 
as it ought, except for the difference introduced by the different values we have taken 
for n". But, besides showing this, the calculations just given provide a check on the 
accuracy of the computations of both of us. Although our figures, as far as the 
second order, have been calculated by very widely different methods, their moments 
of momentum have been found to agree very closely. # 
21. The moment of momentum in the cylindrical problem was announced in my 
two-dimensional paper to increase with increasing e 2 . 
On repeating the computations of this paper, I find that the coefficients in the 
equation of the surface were correctly given, but the final computation of k 2 was 
erroneous.! The corrected formula becomes 
¥ = k 2 (l —0"1 G79e 2 ). 
The Stability Criterion. 
22 . We have now found for the pear-shaped figure of equilibrium, 
2 
- — = 0‘14200 (l + 0"05227e 2 ) 
27 rp 
k 2 = 0-8441 (l — 0"09378<3 2 ) 
* It may perhaps he added that before I had discovered this check on my computations, I had calculated 
the mass of my pear-shaped figure by direct integration, using the method of § 19. The total mass ought 
of course to come to exactly M. I find that the terms in a 2 , a /3 ... result in an increase of mass 
0 - 0689519Me 2 , while those in L, M, N ... s balance this with a decrease — 0'0689514Me 2 . 
t The error is in the very last stage of all; the value of 4y on p. 95 ought to read 
|y = 20-25a 6 -118-93a 4 + 41-071a 2 + 229-51, 
and this leads to the formula 
k 2 = Jc 0 2 (1 - 167 -8802) = A'o 2 (1 — 0 ■ 16788e 2 ). 
I have also calculated k 2 by the method of § 19 of the present paper, and found 
k 2 = /t 0 2 (1 - 0- 16791c 2 ). 
