

582 Mr. H. Jeffreys on the Compression 



But the mass is unaltered. Hence we have the equation of 

 continuity 



2a+^-(ra)~3»V = (1) 



V being supposed known throughout the earth, this is a 

 differential equation to determine a, subject to the boundary 

 condition that a is zero at the centre of the earth. Thus for 

 any shell a is determined by the changes of temperature 

 within that shell. If the shell simply expanded independently 

 of the interior, the radius would increase by rnV instead of 

 by ra, so that the excess r(a — nV) is due to stretching. 



Denote a— nV by k, and substitute for a in (1). . . (2) 



Then the stretching is given by the equation 



d n 3\ 3<M n V) 



— (Jcr 6 ) = —r d ~^k — - 

 dr y J dr 



whence 



* = _4t>cn-v^ r (4) 



(3) 



r z J Br 



Now consider the changes that must take place in a short 

 time dt. If the integral stretching be K, then k — ~z—dt, 

 and for V we must write ^—-dt. Hence 



ot 



f~ L 4H("%>- 



If now c denote the radius of the earth, and oo the depth 

 of a point below the surface, then r=c— x. As the tem- 

 perature changes only extend downwards through a small 

 fraction of the radius of the earth, we can neglect the square 

 of x/c. Finally then 



f=("?)f('-^("£h 



or, integrating by parts and again neglecting [ocjcf, 



c } x 



dt 



<3V 



-dt 



n-^r-dx. 



(5) 



Let now rc = e4-e'V, where e and e' are two constants. 

 Consider first the case of a distribution finite in depth. 



Put AX 2 /2* = «, S-« = /9. Then « = 398°, /9 = 802°. 



