480 MR. .1. H. JEANS ON GRAVITATIONAL INSTABILITY 



And in virtue of (86), the equation giving points of bifurcation (going back to 

 equation (72)) is 



. ^TTC ' On y / / n\ IQQ\ 



"2n+l *' " '' l 



so that points of bifurcation of order w are given by 



J._ Vf (ir-a, -^8) = (89) 



When n = 2, this becomes 



tan (TT 0. + /3) = TT OL ; 



when n = 3, it is 



3 (TT a 



% / > 



tan (,-_# = 



On treating these equations numerically it is found that they can never be satisfied 

 We conclude that the non-rotating mass is stable for all displacements, subject, of 

 course, to the condition that the density shall be everywhere positive. 



Slow Rotation. 

 23. We consider next the stability of a rotating mass of the type under 



2 



consideration, in which we are limited to - - being small compared with the density of 



^7T 



the main mass. If we suppose that a-, the density of the core at its outer boundary, is 



a 



equal to , we shall have a case somewhat artificial of course in which the density 



2?r 



of the core is very small compared with that of the crust, and in which the equations 

 are not too complex to admit of treatment. 



o 



We accordingly assume that a- = , and the equations (75), (76), and (77) (or (72)) 



2?r 



reduce to the same equations as in the case of no rotation (equations (83)). Thus a, ft 

 have the same values as before, being given by the table on p. 479. 



If we suppose that at the outer boundary of the crust the density falls to the small 



o 



value <r = , then the value of c, the radius of the outer boundary, is, as before, given 



2tTT 



by equations (86) or (87), and the value of v n is still given by equation (88). Thus the 

 analysis is exactly the same as in the case of no rotation, and there are no points of 

 bifurcation. 



It follows that, when <r does not have this special value assigned to it, the only 

 hope of finding points of bifurcation rests upon the gravitational tendency to instability 

 which arises from the presence of the small layer of crust in which p has a value less 



o 



than . Let us pass at once to the examination of the extreme case in which o- g = 0, 



iTT 



