﻿of Elastic Stresses in an Isotropic Body. 279 



indicated by 



fl.+fl 2 + fl 3 a Ol +Oi + O* n 0l + 0* + Oz n 



2 u 2 ' 2 3 ' 



This subsidiary system may be described as comple- 

 mentary to the primary stress-system, in the sense that 

 the two together form a hydrostatic pressure whose intensity 

 is one-half of the sum of the principal stresses or one-half 

 of the First Invariant of the primary stress-system. 



I shall form the assumed components of the displacement 

 from the elements of this subsidiary stress in the manner of 

 forming a force-system from a stress-system. 



Thus, I shall write 



2nw^-2^-2^ + ^ { 1 + 2 -0 s ). . . (3) 



Forming the values of S, T, U from these on the elastic 

 stress-strain hypothesis we have 



s = a^i ay* aVs BVi_BVi 



'dy'dz ~dx~dy ~dx~dz ~dy 2 "dz 2 ' 

 etc. 



On equating these to the values for the same elements given 

 in (2), we find they require 



V s f = 0, V 2 t 2 =0, V s f 3 = 0. . . (4) 



Since {0 h 2 > $3» ^i> ^2? ^3} ac ^ on transformation 

 of coordinates as elements of stress, it follows that the 

 system must consist of a hydrostatic pressure and a general 

 stress-system, each of the elements of which is a Spherical 

 Harmonic. That is, 



WWe V 2 *i = 0, V 2 % 2 =0, V 2 %3 = 0. . . (5) 



This is the descriptive criterion of an elastic stress- 

 system. 



