AND THK NEBULAR HYPOTHESIS. 475 



in which the rotational terms proportional to the second harmonic are omitted and 



U B is given by 



TT C r *./ A 



U = " {(' '' .1 i IrV ftl n* + ll T / ' O\"\ " *' T / \ 



' I + /i \* *-') M/ "+/, \f , ~F*){ T fl J 11 ('() 







The value of V at / = r is, from equation ( l .;). 



,. (67) 



whence, evaluating V-V W and picking out the coefficient of 8., we find as the value 



of v n at r = c, 



As before, the points of bifurcation, if any, are given by i\ = 0. 



18. It is now necessary to consider the boundary conditions which must be 

 satisfied at the junction of the two layers. The condition of continuity of material, 

 i.e., that the inner surface of the crust shall coincide with the outer surface of the 

 core, has been expressed in equation (55), b and ft being the same for both core and 

 crust. There is an equation of continuity of pressure expressed by 



13 a 

 & ~ <r au 





(69) 



which <r ix) is now used to represent the density associated with zero pressure. 

 Finally, there is a condition of continuity of normal force and this requires careful 

 discussion. 



Let M! denote the mass of matter actually forming the core and let V, denote it 

 potential at any point outside the core. Let M., denote the mass which would 

 replace the core if the solution (58) for the crust were extended to the centre and let 

 V 2 denote the potential of this mass at any external point. It will Ixi noticed that if 

 solution (58) were extended to the origin, it would give an infinite density p at the 

 origin and also an infinite value of V in virtue of equation (o). On the other hand, 

 it is readily found, by direct integration, that V, the potential of the mass M,, in 

 finite at every point, including the origin. It follows that V can only be the 

 potential of this imaginary arrangement of matter when it is acted on by certain 

 external forces of which the potential becomes infinite at the origin. Let V, represent 

 the potential of these forces. The value of V, is readily found, for it must satisfy 



V a V, = and must coincide with V or with -^ to within an additive constant at 



K 



the origin. Thus V 3 is the limit of the right-hand side of equation (67) when r = 0, 



3 P "2 



