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

 3. Completion of the discrimination. The metric criterion. 



The remaining requirements of the stress-strain relations 

 may be written 



\o% oyJ 



p +Q+B =(3»-„)g4; + ^). . . (6 ) 



which are to be completed from the values in (2) and (3). 

 The first two only confirm the descriptive criteria of (4) 

 and (5). The last leads to 



dj/0^ a^os d#dy J 





•im - n r 2 , „ » . f a^, d 2 2 ?i 2 



dyo^ d^o^ oocdy J J 

 and therefore to 



(3m + 7i)V 2 (0i + 2 + 3 ) 



lor a«/ 2 oz 2 oylbz d#d- d^'dyJ 

 or 



(m+n)S7 2 <j) 



( 0« 2 0«/ 2 O^ di/Oz 0*'02 0*02/ J 



• • • (7) 

 which is the metric criterion and completes the dis- 

 crimination sought for. 



This can be integrated, and gives <p in terms of the %'s 

 and ^s_, of which we can always take the ty's to be null, 

 when desirable. 



This completes the investigation for Cartesian coordinates 

 under normal tractions only and with no inertia terms. 

 With the conditions in (4), (5), and (6), the equations (2) 

 give the general elastic stress- system, and these con- 

 ditions discriminate an elastic stress-system from any other 

 mechanical stress-system. 



