23(5 
SIR G. H. DARWIN ON THE 
The agreement seems to be as close as could be expected when five-figured logarithms 
are used. 
Certain terms in £, as expressed in (24) of § 10, were neglected on the ground that 
they are of higher order than those retained. But it appears from the approximate 
solution in §19 that the coefficients of the terms retained are themselves small, so 
that we are really only retaining terms of the same order as others which are 
neglected. 
The most important of the neglected terms in £ is 
- $3 (1) C 3 (i 77 ) ' r ^ (j)j ’ 
and this is a term of the seventh order. It seems, therefore, well to compute this in 
one case, and see how large a proportion it bears to the whole value. 
Since 
¥ 
7k 2 i 
,2.,4 
i+io ii_ 
1+ 9 K,- 2 
■U © 3 7 
ch 
7 ¥r s 
Using this value and the other approximate values given in § 14, where f 3 is 
determined, I find that the neglected term is 
4 Al 1 /I | a J2 1 5 /4\ U / 1 , 2 5 
7l, 1+ 9 
(7r _ a 1 _ JL cos 2 y f 3 (3 + K 2 ) J£ 
3 1 3Xr 3 sin 3 /3\ 14 k 2 r 2 
The numerical value of this, for the case of Roche’s ellipsoid in limiting stability 
when X = 07, is found to be +0'0016. Now, the value of £, as computed from the 
terms retained, was found to be 0’0677. Thus the neglected term is about one 42nd 
of the whole. The neglect then seems fairly justified. 
I thought it worth while to discover how far the modification of Roche’s problem, 
whereby the larger body is ellipsoidal, affects the result. I find that whereas it makes 
but little difference in the solution for any single assumed value of y, it does make 
a sensible difference in the incidence of the minimum of angular momentum, and' 
therefore of limiting stability. Thus, when X = 0‘5,1 found in one of my preliminary 
solutions for Roche’s modified problem that limiting stability occurs when r/a = 2'49 
(the more correct value is 2‘484). but when the larger body is a sphere it occurs 
when r/a = 2 - 35. Thus we see that ellipticity in the larger body induces instability 
at a greater distance than if it were spherical. This might have been conjectured 
from general considerations. 
