AND THE NEBULAR HYPOTHESIS. 46 1 



The boundary being a surface of constant pressure must also be a surface of 

 constant density, say o-. The equation of the boundary is accordingly 



, z)ao6\l, = <r ..... .... (8) 



The whole mass inside this boundary may be regarded as composed of coaxial rings 

 of matter as follows. Inside the figure of revolution / (m, 2) = <r, we suppose there 

 to be a series of rings of density given by (7), while the surface inequality can be 

 regarded as represented by the presence of rings on this figure of revolution of 

 density proportional to cos \[r. 



On integration the potential V m at any external point is seen to be of the form 



osi/' ........... (9) 



where xo> Xi are functions of w and 2 only. 



Suppose now that the surface is so nearly spherical that spherical harmonic analysis 

 may be used with reference to it, then, since V m is a solution of LAPLACE'S equation at 

 all external points, and is also of the form (9), it must be of the form 



(10) 



where M = cos 6, and P, 1 (/x) is the usual tesseral harmonic P, (/u). Moreover, since 



t*\j 



the centre of gravity of the mass is supposed to coincide with the origin, AI must 

 vanish. 



We have, from equation (3), if V = V m , 



v = 







, z) cos + +' {/(, z)} - 



at all internal points. Equating these two expressions, we must have at the 

 boundary 



or, neglecting small quantities of the second order, 



Hence either /, (w, z) vanishes at the boundary or is of at least the second order of 



harmonics. 



It follows that if there can be a configuration of equilibrium which dil 

 figuration of revolution P =f.(w, 2), by a displacement proportional to the first 



con 



