156 Compressible and Non- Homogeneous Masses [OH. vn 



On the positive part of the #-axis, 



( M \^ 

 so that dfl/dx vanishes for a value x = # > where # = ( STO] -R- It is readily 



seen that 9fl/3a? vanishes for an equal negative value of #. Thus in this 

 special case of M'/M = oo it appears that the two critical equipotentials 

 coincide in one equipotential ; this is readily found to be given by 



and is the curve drawn thick in fig. 31 ; for values of H above this the equi- 

 potentials are closed curves, for values of H below this, the equipotentials 

 are open at both ends. By a rough quadrature it is found that the volume 

 contained by this critical equipotential is that of a sphere of radius '72^ . 



158. If the primary, before distortion, was a sphere of radius r , the 

 limit of statical stability, in the case M'/M = oo , will be reached when it is 

 distorted to the shape of the thick curve in fig. 31. Thus it is reached when 

 M' approaches to a distance R such that 



This critical value of R may be put in the form 



. ........................... (408). 



This may be compared with the critical value of R found in the incom- 

 pressible problem (p. 46), namely 



*r 9 ........................ (409). 



Similarly for the special case of M' 2M, the critical value of R has been 

 seen to be given by 



which may be written in the form 



(410). 



This is still closer to the critical value in the incompressible problem, as 

 given by equation (409). It must however be remembered that equation 

 (409) applies strictly only to the special case of M'/M = oo , since the equation 

 was obtained by neglecting all terms beyond the second harmonic term in 

 the tide-generating potential. When the secondary is at a distance of only 

 2'87r away from the primary, the third harmonic terms may not legitimately 



