AND THE NEBULAR HYPOTHESIS. 



new function being defined by 



H.(X,0)= -Alog{x-'^J.. l J X ,0) } = J^i^l. , 



J.-'/.u, fl) 

 Equations (73) and (74) now become 



<r '~2~ ff ~~ 



K 



1 \ / n+l\*^*/! \'*/ 



while (72) becomes 



so that points of bifurcation are given by 



Ul (A-,a). . '. ..... (78) 



Again, if the rotation may be treated as small, there will invariably be a change of 

 stability at these points of bifurcation, since vjC n changes sign on passing through 

 one of them. 



20. It is at once clear that the method can be extended to a mass consisting of 

 any number of layers the only difficulty occurs in the numerical computations at 

 the end. At each boundary between two consecutive layers there will be equations 

 of continuity precisely similar to (75) and (76), while the final value of '. will be 

 given by an equation exactly similar to (77), which it will be seen involves only 

 quantities associated with the outer boundary. 



The procedure in any particular case will be to start, so to speak, with the 

 innermost core of the system. Equation (75) is linear in cos a and sin a, so that 

 tan a is uniquely determined. Leaving out of account systems in which the densitv 

 is, in any part, negative, this will be found to l>e adequate to determine a uniquely. 

 Equation (76) now becomes a linear equation in cos ft and sin ft, from which ft can le 

 determined uniquely. Tn this way. passing from layer to layer, we can determine 

 the various values of a, ft for the different layers. Finally, the a's and ft'n la-ing 

 known, equation (78) can be regarded either as an equation for w* or as an equation 

 for c, i.e., it can be regarded either as determining the highest rotation for which 

 the symmetrical configuration is stable for a given value of , or as determining the 

 largest value of c for which the mass is stable under a given rotation. If the value of 



-^ obtained by the first method is not small, the result will be inaccurate ; if the 

 2*xr ( 



value for < obtained by the second method is so great that the density is in places 



