488 Mr. J. Prescott on the 



Now allowing for the convention of signs in gravitation 

 potential we have 



W=-V-WP 2 , (19) 



Therefore, 



Now in the elastic equations the stresses expressed in 

 terms of r and 6 are stresses in the element which was at 

 (r, 6) before strain, and is at (r + e, + rj) after strain. 

 These stresses are in equilibrium with the applied^ forces at 

 (V + e, 6 + rf). Now every term that contains 6 in W is a 

 small term and consequently the difference between 6 and 

 (6 + rj) can be neglected. In the third term in V, the only 

 place where the difference between r and {r + e) is not 

 negligible, account has already been taken of this difference. 

 Therefore the expression on the right-hand side of (20) is 



3W. 



the proper expression for -^- in (13). 



Substituting from (15), (16), and (20) in (13) we get, 

 after dividing by sin 20, 



" DHa?< A >-«"} -5{S<*>- 6 £"''}] 



c 



(21 



This is the differential equation which e has to satisfy ; 

 and when p is given as a function of r the equation 

 determines e as a function of r. 



The Boundary Conditions. 



We have yet to find the boundary conditions from which 

 the constants are to be determined. The two conditions are, 

 that the shear and the tension at the earth's surface, where 

 r = a, are both zero. It should be remembered that r is not 

 the distance from the centre of a particle of the strained 

 sphere, but the distance of the particle before strain. Con- 

 sequently, r — a exactly for a surface particle. 



