﻿278 Mr. R. F. Gwyther on an Analytical Discrimination 



Note. — If, in the ordinary notation for strains, we give 

 a, 6, c each one-half of" the usual value given to it, strains 

 would follow the same laws of composition and resolution as 

 stresses, and would therefore have the same differential 

 operators. In this paper, I shall use a, b, c in this sense- 

 that is, one-halt' of their usual. value. 



Lemma B. 



Except when we have reasons for keeping the expressions 

 quite genera], it will suffice to limit the arbitrary stress- 

 system to such stresses as have the co-ordinate axes as their 

 principal axes. . " 



If in equation (2) the elements of stress, P, Q, etc., 

 are made zero, the set of equations will then be recognized 

 as indicating that 1 = e, 2 = f, 3 = g, ty 1 — a, ^—b, ^ 3 = c, 

 where e, f, g, etc. are elements of strain arising from some 

 arbitrary displacement. 



Hence on the right-hand side of equations (2) we may 

 always replace 0\ by 0\ — e, 6 2 by 02— f, 0s hy B — g 9 "^i by 

 "ty\ — a, yfr 2 by ty 2 --b, yfr s by t|r 3 — c. Consequently we may 

 eliminate several sets of three functions, such as a/^, ty 2 , 

 and i^ 3 , when some displacement is possible which makes, 

 say, ^*i = a, ^ 2 = b, ^ 3 = c. 



Hence 





p-_^_^3 etc 



3y3* 



, etc., 



which is the form given by Airy, is a quite general form of 

 solution, although for the purpose of this paper the full form 

 given in (2) is requisite until we have decided upon some 

 particular set of axes. 



2. The choice of a vector to represent the displacement, 

 and the descriptive criterion of elastic stress. 



The mechanical stress has been represented in terms of an 

 arbitrary stress-system, and it is possible and desirable to 

 represent the displacement in terms of a similar stress- 

 system. 



For this purpose I form a subsidiary stress-system, 



