460 RICE 



ART. K 



(See Gibbs, I, 71-74.) The object of the method is to ehminate 

 certain of the variations from the condition of equihbrium so as 

 to leave in it only those variations which are independent of 

 each other and are therefore completely arbitrary in their 

 relative values. Those variations which can be regarded as 

 arbitrary are the displacements of the points in the solid and 

 on the surface arising from the arbitrary variation of strain in 

 the soHd, and also the thickness of the layer of material deposited 

 on or dissolved off the soUd. The object is partly attained by 

 the time we reach equation [367] and the steps are fairly 

 obvious; but in addition to bx, by, bz and bN' we have also the 

 nine variations ban, ba^t, . . . baas. But as we have seen these 

 are not independent of each other since straining only depends 

 on six functions of an, a^, . . . ass- The step from [367] to 

 [369] actually eliminates them all and replaces them by varia- 

 tions bx, by, bz for points in the solid and on its surface. Gibbs 

 is very brief at this point, and to elucidate the step made in 

 [368] we shall have to make a short digression. The point 

 P'{x', y', z') in the reference state is displaced to P(x, y, z) 

 during the strain an, ai2, . . . 033- The additional strain ban, 

 bai2, . . . bas3 displaces it still further to Psix -\- bx, y -\- by, 

 z + bz). Hence the variation in the value of an, i.e., ban or 

 b(Jdx/dx'), is equal to 



b{x + bx) dx 



dx' dx' 



Thus 



\dx') ~ dx' 



bx. 



Similarly 



<5)= 



a 



—,bx. 

 dy 



(Note that x, y, z are definite functions of x', y', z' and x + bx, 

 y -\- by, z -{- bz are also definite functions of x', y', z' slightly 

 different in value from the former; thus bx, by, bz are also defi- 



