Manchester Memoirs, Vol. Iviii. (1914), No. \%. 3 



applicable for cases of rest or of motion, and including 

 the case of stability of equilibrium. The expressions lor 

 the elements of stress found in this manner are complete 

 solutions, but it is thought convenient to express separately 

 and in as simple a manner as possible the relations between 

 the elements of stress and the elastic constants after elimi- 

 nating" any reference to displacement or strain. 



The Elastic Stress relations. 



The relations are readily obtained by the method of 

 this series of papers, but they can also be verified from 

 the fundamental stress equations and the six geometrical 

 elastic stress relations, and by this means a test of 

 accuracy of working may be attained. 



I state in the first instance the typical relations in 

 cartesian coordinates in the form in which I obtained 

 them. 



These typical forms are as follows : 



{ {yn - n)(fn + ny ^ ') 



Altering the form of these relations in order to obtain 

 a further simplification, I write them as follows : 



yii-H ^ ^ ' yn-ndx ^ ' 



These are the stress relations the correctness of which 

 can be otherwise verified. It will be noticed that the 

 invariant P+Q + R appears in every term on the right- 

 hand side. 



