228 
SIR G. H. DARWIN ON THE 
In order to find the minimum moment of momentum for a given value of X, I 
compute l, m, n, R, S, T, U, and assuming several equidistant values of r compute 
values of /x 2 x V —When the coefficients are computed we very easily find the 
value of r corresponding to the minimum. 
When that value of r is found, we are in a position to compute the axes of the 
two ellipsoids. 
For values of X less than ^ the results found in this way would be satisfactory, 
and for X = ^ they are, I think, adequate. Even for the case of X = 1 the result is 
not very remote from the truth, for whereas the correct result for the minimum of 
angular momentum is r/a = 2 638, the result derived from this approximate method 
is r/a = 2‘51. But it would have been impossible to foresee that the result would be 
as good as it is. 
Part II.—Numerical Solutions. 
§ 20. Roche’s Infinitesimal Ellipsoidal Satellite in Limiting Stability. 
We require to find the form of an infinitesimal satellite (so that X = 0) revolving 
in a circular orbit about a spherical planet. When this problem is solved we shall 
be able to see how far the solution will be affected when we allow the spherical 
planet to become oblate under the influence of a rotation of the same speed as that of 
the revolution of the infinitesimal satellite. This last is what I have called the 
modified form of Roche’s problem. 
The planet being spherical and X being zero, the small terms £, e, y vanish, so that 
our solution becomes rigorous. 
The angular momentum of the planet’s axial rotation is to be omitted, and the 
satellite being infinitesimal the momentum of its axial rotation is zero. Thus the 
moment of momentum of the system varies as the square root of the satellite’s radius 
vector, and minimum momentum coincides with minimum radius vector. 
The solution of the problem has been obtained in two ways : first by Legendre’s 
tables of elliptic integrals, and secondly by means of the auxiliary tables given above. 
In the first method, 1 knew with fair approximation by various preliminary compu¬ 
tations the values of k and y which lay near to the required solution. Now there is 
a certain function of k, y, say /(sin -1 /c, y), which vanishes when the ellipsoid is a 
figure of equilibrium; accordingly I computed by means of Legendre’s tables the 
following eight values of /(sin -1 k , y) for integral degrees of sin -1 k and y :— 
f(77°, 57°) = +0-0000878, /(77°, 58°) = -0-0000624 
/(78°, 59°) = +0-0000724, f(7 8°, 60°) = -0-0000785 
/(79°, 61°) = +0-0000562, /(79°, 62°) = -0-0000939 
/(80°, 63°) = +0-0000408, /(80°, 64°) = -0'0001046 
(probably the last significant figure in each of these is inaccurate). 
