478 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



negative, the result will be of no interest except as proving stability for smaller 

 values of c. 



21. It may be well to take a general survey of the equations before giving special 

 calculations. For simplicity we again consider two layers only, core and crust. From 

 (75) and (76) it is clear that, when a = or K = K (involving a- = </), the values of 

 and ft vanish. Broadly speaking, the more distinct the core is from the crust, tl it- 

 larger a and ft are. Equation (78), of course, differs only from the corresponding 

 equation previously found, by the presence of the terms a. and ft. The effect of these 

 terms is seen ou noticing that, in the notation already used, , (K'C, a.) is of the form 

 <f> (K'C a). Thus, to allow for the effect of the core on the term M, (K'C, a), we have to 

 leave the algebraic part of the function unaltered, but to change all the trigono- 

 metrical arguments from K'C to K'C a. Speaking very broadly, the general effect on 

 the graph of , (cf. fig. l) is a compromise between leaving the graph unaltered and 

 moving it bodily a distance a. along the axis. Similar statements apply to the graph 

 of . Thus, while rotation as before is represented by flattening the graph of u t in 

 fig. 1, the presence of a core is represented by a distortion of the graphs which may, 

 with some truth, be thought of as bodily movements parallel to the axis. These 

 bodily movements may cause new intersections between the graph u t and the other 

 graphs, and the points of intersection will represent points of bifurcation at which 

 the symmetrical configuration will become unstable. 



No Rotation. 



22. The case that may properly be inspected first is that of no rotation. The 

 equations reduce to 



^u l ( K 'a,a) = ^u l ( K a),. ..... . . (79) 



, ( K 'a, oL)u m+ ,( K 'a, ft)} =0- {!+, (**),,+,(<**)}, . . . (80) 

 and, the equation for points of bifurcation, 



u n ( K 'c, -ft) = Ul ( K 'c,a) ..... . . . . (81) 



When n = 1, it is seen that ft = a. is a solution of (80), and must therefore ( 20) 

 be the only solution. To verify that ft = a is a solution, replace ft by a in (80) 

 and it becomes 



. . . .... . (82) 



which is seen to be identical with (79) (cf. equations (53) and (74)). Equation (81) 

 now reduces to an identity, so that every configuration is formally a point of 

 bifurcation. The interpretation is, of course, the same as that of 11, the displace- 

 ment for which n = 1 is a rigid body displacement, and so requires no force to 



