204-206] 
Binary Stars 
225 
unstable configuration, so that if the representative point once passes beyond 
B", on the line BB"D, it will not return but will describe some path such as 
BB"B in the plane through B. 
As the point B approaches J, the range of vibration which is possible 
without instability setting in becomes smaller and smaller and finally vanishes 
altogether, so that in the limit any disturbance, no matter how slight, causes 
the representative point to move permanently away from the line J'J. The 
path of this point is necessarily in the horizontal plane through J, and we 
know that the direction of this path initially is that of the tangent JL at J 
to the pear-shaped series JB". In other words, the motion is one in which 
a furrow of the type depicted in fig. 26 forms on the ellipsoid, and this furrow 
continually deepens. 
It seems likely that the furrow will deepen until the mass divides into two 
parts. If so, the motion, which must be in the plane JL in fig. 32, will end by 
the representative point coming to rest at some point such as L in fig. 31 on 
a series of configurations representing two detached bodies revolving around 
one another. Such configurations have been studied in detail by Roche and 
Darwin. We shall refer to the study of these configurations as the Binary 
Star problem, from its obvious application to the dynamics of binary stars. 
The Binary Star Problem. 
205. We proceed to search for equilibrium configurations of two detached 
bodies of masses M, M' revolving round one another in such a way that the 
whole system remains at rest relative to a system of axes rotating with angular 
velocity w in the plane of xy. 
Let the centre of gravity of the mass M be taken as origin, the line 
joining the centres of gravity of the two masses being taken as axis of x, 
R being the distance between the two centres. Then the axis of rotation, 
which of course passes through the centre of gravity of the two bodies, is 
M' 
x = -T 7 — R, y — 0. 
M + M * 
The centrifugal forces acting on the mass M may accordingly be derived 
from a potential 
% 0) 2 
(— 
M' 
-,R) +f 
.(205T). 
M + M' 
The remaining forces which have to be considered are the gravitational 
attractions of the masses, both on themselves and on one another. Our 
problem is to search for configurations in which both bodies can rotate in 
relative equilibrium under these attractions and the centrifugal forces resulting 
from their rotation. 
Roche’s Problem. 
206. The simplest problem occurs when the mass M', which we shall 
describe as the secondary, may be treated as a rigid sphere; this is the