EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
421 
ellipsoid and the dumb-bell; except, perhaps, in the case where the two bulbs pass on 
to two masses of a definite ratio. 
M. Poincare’s work seemed so important that this paper was kept back for a year, 
whilst I endeavoured to apply the principles, which he has pointed out, to the dis¬ 
cussion of the stability of the two masses. The attempt, which is given in the 
Appendix, is apparently abortive, on account of the imperfections of spherical harmonic 
analysis when applied to bodies which depart considerably from the spherical shape. 
We must, therefore, leave this complex question in abeyance, and merely point to 
the Appendix as an example of the method which must almost certainly be pursued 
if this problem is to yield its answer to analysis. 
Allusion has just been made to the imperfection of spherical harmonic analysis, and 
this brings us naturally to face the question whether that analysis may not have 
been pushed altogether too far in the computation of the figures of equilibrium under 
discussion. This question is considered in § 9, and a rough criterion of the limits of 
applicability of this analysis is there found. From this it appears that even in the 
cases of figs. 2 and 3 the result must present a fair approximation to correctness. 
The criterion, indeed, appears to be such as necessarily to give too unfavourable 
a view of the correctness of the result. 
The rigorous method of discussing the stability of the system having failed, certain 
considerations are adduced in § 11 which bear on the conditions under which there is 
a form of equilibrium consisting of two fluid masses in close promixity. It appears 
that there cannot be such a form with the two masses just in contact, unless the 
smaller of the two masses exceeds in mass about one-thirtieth of the larger. 
If we take into consideration the fact that the criterion of the applicability of 
harmonic analysis is too severe, it appears to be worth while to find to what results 
the analysis leads when two masses, one 27 times as great as the other, are brought 
close together. The numerical work of the calculation is omitted, since the numbers 
can only represent the true conclusion very roughly; but the result is illustrated 
graphically in figs. 6 and 7, Plate 23. These figures can only serve to give a general 
idea of the truth, but the form into which the smaller mass is thrown is so remarkable 
as to be worthy of attention. The deep furrow round the smaller mass, lying in 
a plane parallel to the axis of rotation, cannot be due merely to the imperfection 
of the solution; and it appears to point to the conclusion that there is a tendency for 
the smaller body to separate into two, just as we have seen the Jacobian ellipsoid 
become dumb-bell shaped and separate into two parts. 
In this paper, indeed, we have sought to trace the process in the opposite direction, 
and to observe the coalescence of two masses into one. The investigation is comple¬ 
mentary to, but far less perfect than, that of M. Poincare, who describes the series 
of changes which he has been tracing in the following words :— 
“ Considerons une masse fluide homogene animee originairement dun mouvement 
de rotation ; imaginons que cette masse se contracte en se refroidissant lentement, 
