4 Gwyther, Specification of Stress. 



No complete solution of any special case is given, and the surface 

 traction conditions have not been considered at all. I may> 

 however, say that we are in possession of the equations from 

 which the stresses in a. rigid body, whether in a definite state of 

 motion, or in a definite position of equilibrium, are to be found. 



The next step is to- introduce Hooke's Law, which I shall 

 for the purpose of this paper state as follows : — 



In an elastic body the vector, of the first differential 

 coefficients of which the elements of the stress at a point are 

 functions, is the displacement of that point. 



Accordingly the elements of stress having been found, we 

 are to determine the displacement from a=Sjn; b—Tjn: 

 c=U/n. 



If the body is in motion, the displacement will be a func- 

 tion of the time as well as of the point, and we may deduce 

 the velocity and acceleration of the displacement. Whether 

 the (body is at rest or in motion, we may proceed to consider 

 the displaced or strained condition of the body, and to^ deduce 

 corrections for the stresses, and thence again for the displace- 

 ment, if such procedure were desirable. 



The method of procedure here indicated appears to conform 

 with methods which have proved useful in other fields, and by 

 deferring the notion of a displaced position until the first measure 

 of the stress has been made, and by doing away with the idea 

 of a " natural " state of the body in which it is free from stress, 

 the tendency is in the direction of simplification. The whole 

 change may be described as consisting (i) of introducing a 

 general condition as affecting all stresses as preliminary to intro- 

 ducing Hooke's Law, instead of making it appear to be a 

 consequence pf Hooke's Law, and (2) of making the stress 

 equations fundamental instead of putting the displacement equa- 

 tions in that position. 



APPENDIX A. 



On General Stress-strain Relations. 



In the latter part of this series I have pointed out that, in as 

 much as the displacement is eliminated in forming the stress 

 equations, these equations apply to stresses which have the 

 general character of elastic stresses, although they may not 

 satisfy the specific requirements. In this section I propose to 

 examine the results of a hypothesis that the nine elements of 

 a stress may be functions of the nine first differential coeffi- 

 cients of the components of some vector. 



The method I, shall employ is one that I have already 

 made use of in a paper read before the Society, 1 to which J 

 venture to refer the reader. 



" 1 Manchester Memoirs, Vol. ix. (1895), No. 3. 



