﻿504 Mr. R. F. Gwyther on Conditions for 



0i> 02> 03, %i, % 2 , %s in terms of 1? 2 , 6> 3 , ^ b i/r 2 , i/r 3 similarly 

 thus : — 



0i = ^ii#i+X 12 2 + ^13^3 + 2^i 4 ^i + 2X 15 ^ 2 + 2X 16 i|r 35 etc., 



and ^j = X 14 0! + X 2 4# 2 + X 34 3 4- 2X 44 ^i 4- 2X45^3 + 2X 46 ^ 3 , etc., 



which include all of the 21 constants. 



The next step would be to substitute these values in the 

 mechanical values of the elements of stress in (1) and equate 

 to the elastic values found as above from (3) and (2). 



We should thus obtain six independent differential equa- 

 tions in 6 h # 2 , #3, ^r l9 yfr 2 , ty 3 . But these quantities are not 

 independent, and we are at liberty to put each of the 

 -^-functions equal to zero, and thus get six independent 

 equations in three quantities, leading to conditions which 

 I do not pursue. 



Instead, 1 shall follow the method used in my earlier 

 paper and be guided by the form of equations (1). 

 Accordingly I determine the values of 0i, </> 2 , 03 by 

 selecting the terms affected by ^/'dy'dz in S, by ~d 2 /'dx'dz 

 in T, and by d 2 /d#dy in U, both in their mechanical and 

 elastic expressions. 



We thus obtain 



01 = (^24 + ^34)^1 + X 44 (0 2 + #3) + ^-45^3 + ^4 6 ^ 2 , 



02 = (^15 + X 35 )^ 2 + A, 45 ^3 + X 55 (#3 + #l) + ^56^1 j 



03 = (Xi6 + X26)^3 f" ^46^2 + ^56^1 + ^6e(#l + #2), . (5) 



thus completely connecting the two sets of functions. A 

 first condition to be applied is that iij 0i = O, and therefore 



(X 3 4 + X 2 4)(6 > 3 -6> 2 ) + 2(X34-X 24 )((9 2 +l93) 



+ (X 33 — A 22 )^i + (X 36 — X 26 4- 2X45)^2 + (^35 — X 25 — 2X 46 )i|r 3 = 0, 



... (6) 



with two other identities, whose consequences we can infer 

 hj symmetry. 

 Thus 



Xj 1 = X 22 = X 33 = X (an invarian t) , 



X24 = X 3 4 = X15 = X25 = X16 = X 2 6 = 0, 



X 14 + 2X56 = X25+ 2X46 = X 3 e 4- 2X45 = 0. . . (7) 



Also, as there is no term in fa or ga in 2 V, the coefficients 

 of d/dX 24 and of ~dl~du in X^ are separately zero, and 



X 2 3 = X — 2X 44 , Xi3 = X — 2X55, Xi2 = X — 2\qq. . (8) 



