CRITICAL AND SUPPLEMENTARY. 



209 



This condition means that if the curves be supposed given in the form 



p=fir,a) (92) 



where a is the parameter of integration, whose value corresponds, for 

 instance, to the total radius or the central density, then the modulus H 

 deduced therefrom must be a function of p only, independent of a. Equa- 

 tions (3), (4), (5), (15) show this to be equivalent to the condition that 



shall be a function of p only, or in terms of the functional determinant 



m 



dp 

 dr 



dp 

 dr 



dp 

 da 



=0 



(93) 



in which is put, as before. 



r» = 47r 





pr^ dr 



By differentiation and elimination of m this fundamental condition can be 

 reduced to a partial differential equation for p, in rather cumbersome form. 

 Trial shows that (74) is not a solution for any manner of dependence of p^ 

 and c on a; but (25) is a solution, if p^ be a function of a, and q a numerical 

 constant. 



Another special solution of (93) in simple form is found to be 



<==a^(l+^ayr^ 



) 



jM= const. 



for which 



m =^ Ttr^a^ (^ "'"T ^*!^^^) 



2 



(94) 



(95) 



so that for various values of a it would give the density curves for masses of 

 different dimensions, if the substance were to satisfy the condition 



dp I. - 



h = 



47rfc 

 5]f? 



(96) 



With such a substance the compressibility at various densities would be such 

 that not only would there be a definite limit to the dimensions which the 

 mass could attain, but this would be reached with a finite mass, and beyond 

 that point any further addition of strata at the surface would result in an 

 actual decrease in size, as appears from the following analysis. 



If p^, pi, r^ be the mean and surface densities and total radius at any 

 chosen epoch, and a, the corresponding value of a, then 



/0,;»=ai'(l+c) 



(97) 



