154 Compressible and Non- Homogeneous Masses [CH. vn 



Let us suppose that the tidal forces originate from a secondary mass M ', 

 which may be treated as a point at a distance R. If r' denotes distance 

 from this mass, the whole tidal potential is M' /r', but of this a part is 

 effective only in producing the acceleration M' IR 2 of the primary, and this 

 part may be taken to be a potential M'xjR 2 , where the axis of x is taken 

 to be the line joining the two masses. This part of the potential of the 

 second mass must be supposed neutralised by the acceleration of the axes to 

 which the primary is referred, so that the effective tide-generating potential 

 may be taken to be 



v _M' M'x 



VT ~~~ ~ ( } - 



The value of H is now 



M M' M'x 



(406), 



and the boundary of the primary must be one of the surfaces H = constant. 



On parts of the #-axis which lie between the two masses, we put r = a, 

 r = R x, and find 



M 



It is easily found that dfl/dx vanishes once and once only on this part 

 of the #-axis. On parts of the a?-axis which lie outside the two masses, 

 between x > = and x = - oo , the first term in dfl/dx must be taken to be 

 + Mjx\ and it is easily found that in this range also dfl/dx vanishes once 

 and only once. 



Each of the points at which 9O/9# vanishes on the axis of jj is a point 

 at which one of the equipotentials intersects itself, and so represents a 

 possible transition from closed to open equipotentials. But it is readily 

 shewn that, except in the limiting case in which M'/M is infinite, the 

 equipotentials first open out at the intersection which lies between the 

 two masses. 



Thus the arrangement of equipotentials is as follows : For the highest 

 values of H the equipotentials are spheres round the nucleus M. As fl 

 decreases these give place to elongated but still closed figures which persist 

 until H reaches a critical value fl,, which is the value at the point at 

 which dfl/dx first vanishes. After this the surfaces are open at the end 

 towards the secondary until H reaches a second critical value O 2 , for which 

 dfl/dx vanishes on the negative axis of x. For still lower values of H the 

 equipotentials are open at both ends. 



