212 GEOPHYSICAL THEORY UNDEE THE PLANETESIMAL HYPOTHESIS. 



obtained from (105) by differentiation and elimination of x, which is admis- 

 sible, since r is a monotonic function of x. 



Moreover, the variations in horizontal and vertical dimensions of a 

 given element of the mass are proportional to the variations of r and of 



dv 



-r— respectively as functions of a. The condition of no distortion demands 



1 dr 

 therefore that — -^ shall be independent of a, which gives 



r dr r 



, =0 (109) 



80 that P may be written in the form 



P = -rQ(a) (110) 



showing that, to satisfy the conditions named, a differential accretion at 

 any epoch must depress each particle an amount proportional to its dis- 

 tance from the center, but so that the factor of proportionality depends in 

 an indetermined way on the momentary total dimensions of the body. 

 The last equation reduces (107) to 



'■^-^^+^p=o (111) 



dr Q da 



of which the general solution is 

 where 



P=^^'(0 (112) 



/ 



^=logr+ Qda (113) 



The mass within the radius r is then 



m=4:7t<p{0 (114) 



provided that in the function <p, whose derivative is the arbitrary function 

 <p' in (112), the additive constant be chosen suitably, which the finiteness 

 of the mass would show to be possible. 



It remains to impose the condition that the substance have a definite 

 compressibility. Equations (112) and (114) yield 



i^ = _4^fc^j5|_, (115) 



p dp ip"—6(p 



and this must be a function of p only. This is equivalent to the condition 

 tliat if y-^f' 



then r~'^0 and r"~V' niust be dependent functions oi r, !^\ or in terms of the 

 functional determinant 2(p /»' 



-i =0 



