474 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



while the cmter surface becomes 



XP, ........... (56) 



Let us suppose the densities in the two layers to be 



r-'''J i ,(,r)P. . . . (57) 



wT 



in the core, and 



, = ^ C r ~ l/Jjl ^ . . (58) 



in the crust. The boundary between the two layers must clearly be an equipotential, 

 and therefore a surface of constant pressure and density. . Let a-, </ be the densities 

 at this boundary in the core and crust respectively. 



On replacing r by a + frS B + /3P 2 in equations (57) and (58), the values of p must 

 become a- and or respectively. This leads to the relations 



* ~ 5 - = A a- 1/s J Vi ( K a), .......... (59) 



(60) 

 (61) 



h (_ _ = c -. h . + . <f a.. < ,a. - (62) 



\ 27T/ (cJ^(/ca, a) 



From similar analysis applied to the outer boundary, if <r is now the density at 

 this Ixnmdary, 



< fi- il >j, l ,( K t c,*),. . . . .-;.-' : ;: . .' . (63) 





- 



K J 3 /., (KC, at.) 



(64) 



Similar equations, of course, connect the coefficients which depend on the rotation. 

 The value of V m at a point on the sphere > = c can now be written down, as in 

 11, and is found to be 



V = 4-7r 

 '" " c 



~ 7T + ^ {e 1 '' J.,,^, -)-a '^J 3 ,(/, -a)} 



