STRAINED ELASTIC SOLIDS 499 



ei, 62, ... et from this article or from [418], [419] of Gibbs: 



ee = 611^12 + 621622 + 631632, 

 61 = 611^ + 621 '^ + e3l^ 



62 = 612^ + 622^^ + 632^ 



In making the approximations we take as usual 623, 632, 631, 613, 

 612, 621 to be very small compared to en, 622, 633; but the three 

 latter quantities do not now approximate to unity, as formerly, 

 but to 7*0, since in the unstressed state at temperature t, there 

 exist elongations of amount ro as compared with the state of 

 reference. Hence the approximations now must involve re- 

 placing 66 by ro(6i2 + 621), 61 by ro^ 62 by ro^ 

 Hence 



_^ 612 + 621 



Thus 



ro 



Xy a j 



R = — #= J'o 



/e ' 612 + 621 



As we are assuming that the range of stress and strain is 

 covered by Hooke's law it is also true that 



Xy ~\~ 8Xy 



R = To 1 — ; : ' 



612 + oei2 + 621 



where SXy is a small change of shearing stress produced by a 

 small change Sen in the coefficient 612, and thus 



8Xy 

 ^ = ^0 ^ ' 



06X2 



This corresponds to Gibbs' equation [449] but with the ro on 

 the right hand side of the equation, not on the left. The 

 symbol ro can be obtained on the left if 66 is taken as the approxi- 

 mation to /e (which is the case when change of temperature is 

 not involved since en and 622 are then approximately unity) ; for 



