133-135] The Double-Star Problem 185 



this more general problem must in its general features be very similar to that 

 occurring in the simpler problem of Roche. 



135. The general nature of the motion can be seen from considerations 

 precisely similar to those brought forward in considering the- dynamical 

 motion in the rotational problem ( 133). 



Let POP' (fig. 27) represent the series of configurations possible for the 

 satellite, the branch PO being stable, 



the branch OP' being unstable, and |P P'J 



the point representing the configura- \ j 



tion of limiting stability. * ' 



When the configuration is repre- A - 



V 



V 



sented by a point such as A on the >yC C' x x ,/ 



stable branch, a small displacement ^ f 



will result in stable oscillations 

 through some small range A' A A". 



When the representative point is at C the range of stable oscillations is 

 very small, and an oscillation of range greater than C'CC" will be un- 

 stable. Finally at any oscillation at all will result in an unstable motion 

 which will initially be represented by motion in a direction 00 ', and so 

 will consist of an elongation of the ellipsoidal figure of the satellite. 



The tracing out of this motion must present a problem very similar to 

 that already discussed in the tidal problem in 118 131 ; unfortunately 

 the presence of rotation makes it impossible to obtain exact results. But a 

 good deal of the motion is disclosed by a study of general principles. 



The radius of the orbit is determined by the same equation as it would 

 be if the whole mass of the satellite were concentrated at its centre of mass. 

 The satellite may be thought of as consisting of two halves H and H', the 

 former being nearer to the primary than the centre of mass and the latter 

 further away. If it were not for the presence of H', the half H would be too 

 near the primary for a circular orbit to be possible under the prescribed 

 rotation ; equilibrium is maintained by the gravitational pull from H' which 

 neutralises part of the attraction of the primary on H. Similarly it is only 

 the gravitational attraction of H which makes a circular orbit possible for H'. 



When the configuration reaches limiting stability at the point 0, a rapid 

 elongation of figure begins, and this lessens to gravitational attraction be- 

 tween H and H'. The immediate result is that H is drawn in closer to the 

 primary, while H' is driven further away. At first this motion is only another 

 representation of the elongation of the figure of the satellite, but it is clear 

 that this elongation cannot continue for ever a long thin filament of matter 

 must be unstable under all conditions. Thus the satellite must before long 



