410 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Computing from this formula, I find the following values of the moment of 
momentum in the case where the masses are equal, when 
£ = T, (jJ V X -468 
P=h X -472 
P=h X -482 
Now I find by a numerical investigation* that, if we imagine a mass of fluid equal 
to %Trb 3 rotating in the form of a Jacobian ellipsoid of three unequal axes, then, when 
the momentum is (-§ 7 r) ? 6 5 X '392, the axes of the ellipsoid are 1*8986, 0*81136, 0’649&; 
and, when the momentum is (f7r) ? b~° X '644, the a,xes are 3T36&, 0'5866, 0\5456. 
It seems probable, then, that the Jacobian ellipsoid of mass f 77-fr 3 becomes unstable, 
at least as soon as when the moment of momentum is somewhere about (f 7 r) 3 b b X '5. 
It may be worth mentioning that the greatest moment of momentum for which the 
ellipsoid (of mass f 7 r// J ) is stable, when it is a figure of revolution, is (^ 7 r) 3 6 5 X '3038. 
§11. On the Conditions under which the two Masses may be close to one another. 
If at any point on the surface of either mass the sum of the tide-generating and 
centrifugal forces is greater than gravity, it is obvious that equilibrium cannot subsist. 
It is also clear that, if this condition is to be found anywhere, it will be at that point 
of the smaller mass which lies nearest to the larger mass. Hence, in order that the 
system may be a possible one, we must satisfy ourselves that at that point gravity of 
the body itself exceeds the sum of the tide-generating and centrifugal forces. 
To determine the limitations of size and proximity of the smaller of the two 
masses to a high degree of approximation would be very laborious, and we shall, 
therefore, content ourselves with a rough investigation, to be explained below. 
We shall now find approximations for the shapes of the two masses and for then- 
potentials. 
The radius-vector of either mass and the potential may be expanded in powers of 
a/c and Ajc , and a term involving c" in the denominator will be referred to as being 
of the n th order. 
Now the term of the highest order which can be included without the introduction 
O 
of great complication is the 7th, and we shall content ourselves with that term. 
The expressions for the various parts of the potential have been developed above, 
but it may be observed that the terms involving the first order of harmonics may be 
* ‘Roy. Soc. Pror.,’ vol. 41. 1887, p. 319. 
