224 
SIR G. H. DARWIN ON THE 
momentum, either of /x, or of /x 2 , as the case may lie. Such a solution is a figure of 
bifurcation, and of the two coalescent solutions one has one more degree of instability 
than the other. If one of the two is continuous with a stable solution, and if. 
moreover, m the passage to the undoubtedly stable solution it passes through no 
other point of bifurcation, one of our two solutions is secularly stable and the other 
unstable. 
Now, two liquid masses revolving about one another orbitally at an infinite distance 
are undoubtedly stable, and such a case is also continuous with one of our solutions. 
Further, Schwakzschild has proved that Roche’s ellipsoid has no point of bifurcation 
from first to last, and as this is true of one such ellipsoid, it is true of two.* Hence 
we conclude that the minima of g 1 and g 2 will afford figures of limiting stability. 
§19. Approximate Solution of the Problem. 
It is clear that spherical harmonic analysis is applicable to the case when the two 
liquid masses are widely distant. When they are so much deformed by their 
interaction that that method becomes inapplicable, good results may be obtained from 
the formulae of the last sections by means of development in powers of sin y, and it is 
this plan which is especially considered in the present section. 
It appears from § 1 that when one of the masses is small compared with the other 
(X small), the configuration of limiting stability for the problem of figures of 
equilibrium occurs when the two masses are very far apart. As X increases, that 
configuration corresponds with diminishing radius vector. It seemed then probable 
that at least some of the solutions might be found by means of these series, and if 
this were so it might, in many cases, prove unnecessary to follow the same laborious 
procedure as in finding the limiting stability of Roche’s ellipsoid. This view was 
found to be correct, and I therefore think it well to record the methods by which the 
developments may be obtained, without however giving the full details of the very 
laborious analysis. 
When the masses are far apart, the terms denoted e and rj in the equation for k ' 2 
and in that for a 3 /A are small, and they must be neglected in the developments. 
Writing for brevity g = sin y, we may prove that 
H log e 1+ sin Y = 
Sin y cos y 
o 2n 
v y 
o 2n+ 1 ’ 
tan 2 y = -1 + 2 g 2 ’ 1 , tan 4 y = 1 + 2 (n- 1) g 2 ’ 1 , 
tan" y = -1 + 2 ^— q g 2n , tan 8 y = 1 + 2 ^— T ) ( n ^ ( n & c< 
^ * o 1.2.3 
Hence the developments may be obtained of the functions 
^0, ^”l5 ^*2 ••• 7*05 ^*1? To - • • 5 
Schwarzschild, “ Die Poinear<$sche Theorie des Gleichgewichts.” ‘ Neue Annalen der k. Sternwarte 
Miinchen,’ Band III, 1896. 
