Material Stresses, and their formal solution. 237 



In obtaining the equAtions (yS) and (7), the equations (a) 

 have been made use of, but it remains to secure that those 

 equations are satisfied. 



By operating on the several equations (a) and substituting 

 from (/3) and (7), we find that 



is independent of x, //, and z. (S) 



Also by adding the three equations (/?), we find that 



— v (P + Q + R) + 2 p KW+ ~w "S? / = °- (e) 



It therefore follows that 



both V 2 (P + Q + R) and ^J + U + ^J . (g) 



are independent of a, y, and e, unless m = n. 



A condition is thus imposed upon the nature of the force- 

 system under which this stress-system can be maintained. 



The formal solution. 

 In what follows, 1 shall write 



P + Q + R = V 2 & pV^.V'tyi, f>V 2 = V 2 &, /oV 3 = V 2 c£ 3 , (1) 

 so that, from (/3) and (7) 



ra) r 3»i — ?i\Oj/ 0~ / O* 



p m(m + ?*,) 



>i(3m 



==2 ^n L _B 2 n 6 B 2 a, 





d.f 2 V B- 2 ' 



Q + n( W «) V ■ * " 3^fe + 5?/* + 2 ^ 



d 2 n 2 d 2 n 6 d 2 a 4 



=2 



d?/ 2 B.f 2 Be 2 ' 



m(m + n) 2m / d 2 d 2 \ , , 9 B 2 <fc 



n(om — n) r om — nXox* Of/"/ O" 



«(3m- n) 'dm — n\ox' 2 of' 



_oB a a 8 >a«, a- 



i> 



a*» V ; 



(2) 



