AND THE NEBULAR HYPOTHESIS 





maintain it. There is, of course, nothing o f the nature of a change of stability, for 

 ?</(}, instead of changing sign, remains permanently zero. The consideration of 

 n = 1 is of no value except that it provides a check on the result of a rather involved 

 series of analytical processes. 



When there is homogeneity hetween core and crust the non-rotating system has 

 l>een found to he stable for all displacements. To examine whether this is altered by 

 the presence of the crust, it is natural to test first the extreme case in which (In- 

 difference between the core and crust is as great as possible. Let us make th- core 

 so hot that its density is zero, so that * has to be zero in order that the internal 

 pressure may be maintained (cf. equation (12)). 



Putting a- 0, equations (79) and (80) reduce to 



= 0. 



or, by equation (74a), 



J. fc (*', -a) = 0, J. + . fc ('o, ft) = 

 whence (equation (49)) a, /3 are given by 



= ; tan ft = - 



(83) 

 (84) 



(85) 



The values of a, /3 corresponding to a few values of *a are given below 



KCI. 



ft- 



n = 2. 



n = 3. 







2 



2 21 



16 39 



46 _>.' 



S6 9 



The case which is most favourable to the occurrence of points of bifurcation with 

 positive values of P is when <r u falls to zero at the outer boundary. Let us accordingly 

 examine this case. We have (equation (63)) 



J 1/3 (/c,a) = (86) 



so that . 



KC = r-a. 



