6 Gwyther, Specification of Stress. 



If we limit ourselves to the case when the elements of stress 

 are linear functions of the first differential coefficients, we shall 

 obtain the elastic 'stress-strain relations, with the addition of 



I do not propose to complete the general solution nor to deal 

 with invariant and covariant functions generally, nor with the 

 application to fibrous or crystalline bodies by the employment of 

 constants related to determinate directions in the body, and on 

 that account affected by the operators (2d 9,.,, (2 3 . 



Instead of dealing wiith the general solution of the differ- 

 ential equations containing higher powers of the differential 

 coefficients, I shall take only one case, that put forward by 

 Lord Kelvin, dealing with the quadric in the natural state of a 

 material which becomes a sphere in the strained state. 



According to the method of this section the left-hand side of 

 the equation to< this quadric should be a covariant expression. 

 Writing the expression las 



Ex 2 + Fy 2 + Gz* + 2Ayz + 2Bzx + 2 Cxy, 

 the conditions of covariancy as affecting the coefficients are seen 

 to be that 



1 \8F 8GJ V J 8A SB gC" 



with two similar 'expressions. 



From the formation of the equation of the quadric from 

 expressions such as 



8x I 8y 8z 

 we find 



E* = ze + e- + \(b- + c-)- (bn - eg) + ^ + '("', 

 etc. 



A* = a + \bc + %a(f+g) + (g -/)« + ¥& - b - li , 

 etc. 

 and we can verify that the condition of covariancy is satisfied. 



Now we can form expressions for the elements of stress to 

 the second degree in the first differential coefficients, which 

 will also conform iwith the geometrical conditions in the Theory 

 of Elasticity, by writing 



F= \{m - n){E + F+G) + nE, 

 etc., 



etc. 



*These expressions are, algebraically, partly of the first order and partly of the 

 second order, but in estimating them arithmetically it is to be noted that £, 77, f, 

 are not necessarily small and may be large. 



