468 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



first point of bifurcation for an incompressible mass, instead of being given by 



o 2 



= '400, is known to be given by the widely different value ^ = '1871. 



2?rfr 



Our analysis has nevertheless proved rigorously the point which is really most 

 important, namely, that there can be no point of bifurcation at all for quite small 



Q 



values of - t . At the same time, since the question of when and how a rotating mass 



first becomes unstable is one of considerable importance, I have attempted to obtain 

 a more reliable investigation than the preceding. I have found that the accuracy is 

 not greatly improved by taking the analysis as far as squares of w 2 /27r<7, while the 

 labour of working with a general power series would be appalling. I have, therefore, 

 reluctantly been compelled to give up hopes of carrying the rigorous solution of the 

 problem further in this direction, but have thought it worth while to examine the 

 analogous problem for rotating cylindrical masses. All the essential physical features 

 of the natural three-dimensional problem appear to be reproduced in the simpler 

 cylindrical problem, so that it seems legitimate to hope that an argument by analogy 

 may not lead to entirely erroneous result. 



Cylindrical Masses in Rotation. 



14. The fundamental equations are, of course, the first two of the equations 



already written down in 3. The third equation does not occur, since -'- = 0. The 



oz 



equations have, as before, the integral (13) leading to the differential equation (15) 

 for p. 



The most general solution possible will be 



= -+ Z A n J n (^)cos(^-e), ....... (40) 







in which r now stands for \/(x 2 +y a ). No matter how great the rotation, there is 

 always a special circular solution 



r), ......... (41) 



this being analogous to the spheroidal figures of equilibrium investigated in 12. 

 Let us examine the deformed solution 







p = ^-+AoJ (/cr)+A B J B (/cr)cosn0, ....... (42) 



/TT 



in which A B is supposed small, but there are no restrictions on the value of . If 



2?r 

 the free surface p = or is supposed given by (cf. equations (21) and (30)) 



r = a + b cosn6, 



