788 Dr. Chree on the Stresses in the Earth's Crust 

 and close to the surface 



7r' = T0 = -gph (7) 



§ 5. Gravitating spherical " Earth" consisting of a core of 

 radius b, density p + p and elastic constants m! 9 n\ and of a 

 crust or layer of density p, and elastic constants m, n, resting 

 on the core and bounded externally by a spherical surface of 

 radius a. 



The expressions for the displacements and stresses are as 

 follows : — 



In the core : 



6 lo m' + tt j 



£ =i(3m'-OA+ 15 5 ( ^ () 2*ip + p')V, 1 ■ • («) 



In the layer : 



rr =l(3m — n)B — 4nr- s G + t^ -27roV - — - -°- — , 



6 v y 15(ra + n) m + nd r 



66 = l(3m - w) B + 2nr~ 3 C + — — . 2-777) V — ~ -^— . 



dV y 15(m + «) r m + n6 r 



Here A, B, C are three arbitrary constants to be determined 

 by the surface conditions, and the gravitational force between 

 two masses p, and p! at distance R is taken as (fipu'/R 2 ) x 1. 



We shall suppose the outer surface r = a to be free from 

 force. If uniform normal pressure acts its effects are most 

 easily obtained separately. 



Supposing the outer surface r=a free from force, rr must 

 vanish over it, whence 



-K3m-^B-4na- 3 C = _ 5m ^ n 2 7rpV + ^ | ti^' - . (10) 



Over the common surface r = b, the radial displacements must 

 be the same for the core and layer, whence 



B + 3^C=A + |^^-|^? + ^^. (11) 



o m + n bin + u m-j-7i 



