472 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



throughout. But to represent natural astronomical conditions there is no question 

 that K ought to increase on passing from the centre to the surface, thus representing 

 a mass in which the temperature is highest inside and falls towards the surface. 



We are in this way led to study the question of stability when K is a function of r. 

 It would be difficult to say precisely what function ought to be chosen if we were 

 trying to represent natural conditions as faithfully as possible. It appears, however, 

 that no continuous function will lead to equations which admit of integration. The 

 only case which appears to be soluble is that in which the matter, before rotation, 

 may be treated as if formed of a series of different layers, each being homogeneous 

 and at a uniform temperature in itself, but the temperature varying from layer to 

 layer. To represent this we take different values of K in the different layers, K being 

 smallest in the interior. 



There is no limit to the number of layers which can be treated analytically, but 

 the assumption of a great number of layers naturally leads to highly complicated 

 formulae which are capable of conveying their meaning only after laborious numerical 

 calculations. Both in order to obtain comprehensible results and to simplify the 

 argument, the layers will, in what follows, be supposed to be only two in number. 

 They may conveniently be referred to as the core and the crust. It will be found 

 possible to generalize the results obtained so as to apply to any number of layers. 



16. We accordingly suppose that there is an interior core of radius a, in which 

 the coefficient of compressibility has the uniform value K, and that outside this is the 

 crust of external radius c, in which the coefficient is K. It is again necessary to 

 suppose the rotation to be so small that w 2 may be neglected. 



As in 3, the density /> must satisfy 



o ... ...... (47) 



throughout the core, and the same equation with the appropriate value of K throughout 

 the crust. The most general solution of equation (47) is 



_ l ^(r)} ..... (48) 



ZTT o 



Iii the former problem all the terms in B 71 could be omitted because p had to be 

 finite at the origin. In discussing the solution for the crust these terms must be 

 retained. The solution can, however, be put in a more concise form. 



Let the constants A n , B n be replaced by new constants C n , 6 n given by 



A,, = C n cos 6 H , B n = C sin d n , 

 and let us introduce a function J B+ i/ a (x, 9) defined by 



J +v.(*. 0) = *.+/,() cos + J_ (n+1/2) (x) sinB ...... (49) 



