﻿166 Mr. C. Chree on the Local Alteration of 



of p and jt/, while pb and p c are obtained at once by writing 

 b for r and c for r respectively in (7). 



Carrying out the necessary reductions we find 



*^ = %$$-% [ 3A («'- w ) ^ u + 4n '« 3 + 4n (*' - *) t" 8 -* 3 )}^ 



+ 4n(A 1 -*){n 1 (3* + 47 i )^-3i(n 1 -n)(c 3 -^)}^^], . (] 



£ ^' = nS^S^[ 3/i;(ni - re){ * l(34+4 " )aS ~ 4n(il ~* )(a3 -* 3) ^ 



+ 4n(* x - *) {^(3* + in) V + 3*(% - n) (a 3 - 6 3 ) \ P *~ ^'"" j ■ . (1 

 where, for shortness, 

 Ife^*»(3*+4»)(3A 1 +4» 1 ) 



+ («)[(3* + 4n) { 3 -^=^ + 4 -^g=^ } 



+ MMK* -*)(« -n) (<8 ~ ( ^r y) ]- • • (18) 



The quantity II is essentially positive. 



To facilitate reference it will be well to record explicitly 

 the changes in the strains, dilatation, stresses, and stress- 

 difference in the two layers. In (b . a . a) we have 



*4- «h^tt+i?). • • • p» 



oA= *^.£, • • • • - • (2D 



^--)= K^P 1 )' < 22 > 



S( &) = ^_^l + 4 £), .... (23) 



SS=±^»^i-l? (24) 



The existence of the cr Dmmon factor fyb-^r-p will be 

 noticed. 



