470 



MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



but with the important difference that the present equation is true for all values of 

 w 3 , without limit. The points of bifurcation are given by v n = 0, or 



'W > '\- I \ (AR\ 



/,(*), I 4 ) 



.-'/, M^ 1 - 



which again is exactly analogous to the former equation (38). The graphs of the 

 functions % M>/ 2 , . . . will be found to lie as in fig. 2, and we may again imagine that 



0-8 - 



0-6 - 



0-4 - 



0-Z - 



JC=0 



123 



Fig. 2. Graphs of the functions 



points of bifurcation are sought by flattening the curve u\ tt until it intersects the 

 other curves. 



It is clear that, under all circumstances, the first curve to be intersected will be 

 the curve u> tl , corresponding to a displacement proportional to cos 20. Thus, as 

 before, when the circular form becomes unstable, it gives place to a form of elliptic 

 cross-section, which is stable. Moreover, the smaller *a is the lower the value of 

 M^JTTO- for which the circular form becomes unstable. 



These results are true without any regard to the value of w 2 , so that they confirm 

 the results stated, but not rigorously proved, in 5. The numerical calculations 

 which follow will make the matter clearer. 



If p denotes the mean density of the rotating mass, the total mass per unit length 

 is given by 



P r dr = 



4-7T 



+ AO 





