162 Compressible and Non- Homogeneous Masses [en. vn 



special case of M'fM = oo ; in the more natural but less simple case in which 

 M'/M is finite, the boundary of the atmosphere will be a distorted pseudo- 

 spheroid : a conical point will develop at one end only, and matter will stream 

 out only from this end, which is the end nearest to the tide-generating mass.) 

 Of the matter which streams out, some will fall into the tide-generating mass, 

 and some will fall back on to the primary. The general effect may be thought 

 of as the creation of an outer atmosphere in which the subsequent motion will 

 take place. With a still further increase of tidal forces, the nucleus will attain 

 the critical shape shewn in fig. 32, the retained atmosphere now being reduced 

 to about a tenth of vy. After this the motion both in the nucleus and the 

 atmosphere will be dynamical ; the motion of the nucleus will be the same as 

 that already considered in 118 126 except in so far as this may be altered 

 by the presence of a resisting outer atmosphere. 



The Double-Star Problem 



163. To form a double-star problem on Roche's model, suppose we have 

 two masses M, M' rotating in steady motion at a distance R apart with 

 angular velocity o>, each body being so highly condensed that the whole mass 

 of each may be supposed concentrated at its centre of gravity. Let the line 

 joining them be taken for axis of a?, the centre of the primary being origin. 



The value of H is readily found to be 



and the point at which dl/dx = is given by 



The value of co is given by the usual relation o> 2 jR 3 = ( M 4- M f ), whence 

 it appears that equation (422) can be expressed in the symmetrical form 



where y = R x. The graphs of the two similar 

 functions in brackets are of the shape shewn 

 in fig. 33, whence it appears that the root of 

 the equation is x = OP where P is so chosen that 



M x PS = M' x PS'. 



There is therefore one and only one root of 

 equation (422), and the critical equipotential in- P 'x 



tersects the axis of x at a point distant OP from 

 the origin. If the volume of either component of the double star is greater 



x=R 



