STRAINED ELASTIC SOLIDS 465 



from zero. It appears that in such case £"11 approximates to 

 unity since 623 is small and 1 + 623^ differs but little from unity. 

 Similar statements are true of jE'22 and £'33, while E23, E32, etc., 

 all approximate to zero for similar reasons. On examining the 

 modified equations (38) it will appear that in the event of such 

 coincidences Xx' approaches to Xx, Xy' to Xy, Xz' to Xz. We 

 thus illustrate in another manner Gibbs' conception of gradually 

 bringing not only axes of reference but the two states into coin- 

 cidence. But it will be realized on a little thought that even if 

 we have the states approximating to coincidence, but not the 

 axes, the considerations just raised do not hold; for then an, 

 an, ... 033 involve not only the actual elongations and shears 

 but also the direction cosines of the axes OX, OY, OZ with 

 reference to OX', OY', OZ' which change with any reorientation 

 of the former relative to the latter. In consequence an, 

 an, ... ass do not approximate to unity in general even for 

 slightly separated states, and An, An, ■ ■ • ^ss do not tend 

 to the values which are the limits of £"11, £'12, . . . Ess. 



Gibbs' own proof may now be clearer to the reader. From 

 [355] 



dev' dev' 



Xy' = ~ — and Yx' = ~ — 

 oax2 0021 



Under the conditions of coincidence assumed ai2 approaches en 

 and a2i approaches 621 in value. Hence the limit of Xy is 

 dev/ den and that of Yx' is 967/9621 since under these circumstances 

 ev ' approaches ev. Now actually ev is a function of /e, and /e 

 becomes in the limit 612 + 621- Since therefore in the limit 



and 



it follows that Xy which is the limit of Xy is equal to Yx which 

 is the limit of Yx'. The reference in Gibbs to the difference 



