STRAINED ELASTIC SOLIDS 437 



if we denote this "strain-energy function" by W(fi, . . . /e) it 

 follows that 



_dW dW 



If then each Xr is a linear function of /i, ... /e, as experiment 

 shows to be approximately the case for isothermal small changes, 

 it follows that W must be a quadratic function of the six 

 variables /i, . . . /e- Now such a quadratic can only involve 21 

 numerically different coefficients; thus 



W = hCufi" +^66/6^ 



+ C12/1/2 + Cie/i/e 



+ C23/2/3 + C2G/2/6 



+ C34/3/4 . . . + Cirjaf^ 

 + C45/4/5 + Cicfif^ 

 + C^efhfe, 



and so it appears in assuming that the various stress-constitu- 

 ents satisfy equations such as 



Xr = Crlfl . . . + Criflj , 



that 



This justifies the statement made above that in the cases where 

 there are linear isothermal stress-strain relations, there are 

 at most 21 elastic constants. 



In the arguments that follow, however, we shall require no 

 such restriction as to the nature of the relations between stress- 

 constituents and the strain-coefficients. Actually these relations 

 also involve the temperature. Moreover, if we are going to 

 follow Gibbs' reasoning we shall have to realize his somewhat 

 different treatment of the stress-constituents from that outlined 



