1883.] On the Figure of Equilibrium of a Planet, fyc. 163 



Any value from unity to an infinitely small value may be assigned 

 to Jc, that is to say, we may have any arrangement of density from 

 homogeneity to infinitely small surface density, but if h be greater 

 than -f the compressibility increases with the density, which is 

 physically improbable. 



The infinite density and infinite pressure, which occur in this 

 solution actually at the centre, may be avoided by imagining the 

 centre occupied by a rigid ^spherical homogeneous nucleus, of very 

 small radius Ba, and of density l/kda m ~ k) . 



We have to compare this solution with Laplace's. 



For this case Jc is not constant, and its surface value is fe. 



Let where ic is a constant, being the arbitrary constant intro- 



K 



duced in this solution ; and let 6 be the surface value of $■* 

 The solution is — 



e sin# 



w=- 



#sin<9 



And the mass inside of any radius a is —a u . 



dw 



h= ^_ 



3(i-a cot &y 



The length of the modulus at the surface is 1/(1 — cot 6) or 3&0 -2 

 of the planet's radius. 



en-h 



e — t — ■ 



£ 2 l-fe 



z 6(1 -ft) 

 C=[l-6(l-fe)0-2]fMa 3 . 

 C-A 1 



l-6(l-fc)0 



zo(e-»- 



The following table gives the numerical values of the solution, 

 together with columns for comparison with the results of Laplace's 

 theory, for various values of the ratio of surface to mean density. 



* See Thomson and Tait's « Nat. Phil.," 1883, § 824. 



