408 
PROFESSOR G. H. DARWIN ON FIGURES OF 
It thus appears that as the bodies recede the accuracy increases with great rapidity, 
and in the two cases considered last it is hardly necessary, from a physical point of 
view, to consider greater accuracy than that attained. 
It must be remarked, however, that this way of estimating the degree of inaccuracy 
must necessarily give much too unfavourable a view. 
If we have a single mass of fluid departing considerably from the spherical form, it 
is clear that the potential computed on the hypothesis of a layer of surface density on 
the true sphere will come to depart largely from the potential at the surface of the 
fluid. If, however, we compute the potential of such a mass at points a little remote 
from the surface, the approximation will be much closer. Now, where there are two 
masses, as in our problem, the potential at the surface of either mass consists of two 
parts, one due to the mass itself, the other due to the other mass. As regards the 
first of these two parts, the above criterion is applicable, but as regards the second 
part it gives too unfavourable a view. 
Now in the case of the single mass the deformative forces due to centrifugal force 
are considerably vitiated by computation at the spherical surface instead of the true 
surface, whilst in the case of the two masses the tide-generating forces are computed 
with greater accuracy than is shown by the criterion. 
Under these circumstances it has appeared worth while to give another figure 
below, which, judged by the criterion, would be no approximation at all. 
The reasons for giving this figure will be stated when we come to it. 
§ 10. To find the Moment of Momentum of the System. 
Rotating figures of equilibrium are classified according to the amount of moment of 
momentum with which they are endued. It is, therefore, interesting to determine the 
moment of momentum of the systems now under consideration. 
We must begin by finding the moments of inertia of the two masses. Let 87, Si, 
denote the moments of inertia of the shells of zero mass lying on the mean spheres of 
radii A, a. 
Then 
Si = jj (y 2 + z 2 ) ( 7 - — a) a? dz&, 
where = sin 6 d6 dcf), and where the integral is taken throughout angular space. 
Now 
y + 2 = t a + W 
w 0 
* r~ 
and r — a is the sum of a series of harmonics. Then, in consequence of the properties 
of harmonic functions, we need only consider the harmonics of the second degree in 
r — a, and 
