Manchester Memoirs, Vol. Ixii, (191 7) No. 1 



I. The Specification of Stress. Part V. 

 By R. F. Gwyther, M.A. 



(Received and read May 8th, Tgiy.) 



On the Formal Solution of the Elastic Stress 

 Equations. 



There is little gained by introducing the bodily forces, which 

 will be supposed to be conservative. It is also intended that "re- 

 sistances to acceleration " should be included among " forces " in 

 cases of motion and these will be different in different problems. 

 I shall therefore leave the forces to be supplied as required; in 

 other words, I omit " particular integrals," and deal with " com- 

 plimentary functions " only. Solutions will only be of real 

 interest when they are shown to satisfy the surface traction 

 conditions of a special problem, but the knowledge of a general 

 formal solution may lead to the solution of particular cases, and 

 in any case the possibility of such a proceeding is necessary to 

 my argument. 



The set of equations with which I propose to deal are: 



V *J>+ _i^_ ll(P+Q + R) = 0, 

 3m — n §x- 



with two similar equations, 



V *S + -J™— J>L(P+Q + R) = 0, 

 yn = n 8y8z 



with two similar equations (1), 



and V*(P+Q+R) = (2). 



We therefore Have, in the first instance, 



P±-^— x^-(P+Q + R) = Q, 

 2,m - n fix 



yn - n 8y 



R + —^— Z -^(P+Q + R) = ^r, 

 ■yn - n 8z 



2(3*1 -n) { y 8* 8y) 



T+ . m ( «-§-+*!} (/>+Q + ig)-$', 



2(3« - n) \ 8x 8zJ 



