﻿Analytical Discrimination of Elastic Stresses. 275 

 tractions only are, in Cartesian coordinates, 



"d.v B*/ "dz 

 3.1* By B~ 



B* By + d~^ "' W 



and they are identically satisfied by the values 



B- 2 By^ByB*' 



B^l 



3. 



Q = 



R_ ^.cW, 



*"" V 3*" 



T = 

 U = 



B^'B- 



b 2 3 , 2 avs 



B* 2 B#B* 







3#By' 



B^h _ j^h 

 d« a B«By 



B 2 ^ 3 



b 2 ^ av, 



B^B* B3/ 2 



B 2 f 3 

 3y3-~' 



B 2 # 3 BVi B 2 ^ 2 

 B#B# B#B# ByB^ 



• • • (2) 



These contain six arbitrary and general functions, and 

 form the general solution of (1). 



Lemma A. 



These six arbitrary functions have the same mode of 

 resolution on transformation of axes as the elements of 

 stress. 



For proof I use the method employed in the subject 

 of differential-invariants. 



Imagine the axes of coordinates to be rotated about their 

 own positions by the infinitesimal amounts co z , e» y , o) z , and 

 consider the consequent changes in whatever quantities 

 we may be considering — components of a vector, elements 

 of a stress, etc. These changes will be linear functions of 



60 x , ftty, (O z . 



T2 



