STRAINED ELASTIC SOLIDS 461 



nite functions, small in value, of x', y', z'.) On this account 



•'(S) 



Xx' Sail = Xx' 51 , 



= Xx' ^ , ^x, 

 dx 



which on integrating by parts is equal to 



9 . dXx' 



-, (X., Sx) - ^ Sx. 



Hence 



Xx' dan dx'dy'dz' = — {Xx' 8x) dx'dy'dz' 



dXx' 



——r 8x dx'dy'dz'. 

 dx' ^ 



The first integral on the right hand side, which is an integral 

 throughout the volume of the soHd, can be transformed by 

 Green's theorem into an integral over its surface, viz., 



fa'Xx'dxDs', 



and in consequence we obtain the result [368]. (Will the reader 

 accept the truth of this transformation for the moment so as 

 not to interrupt the argument? We shall return in a moment to 

 Green's theorem for the sake of those unacquainted with it.) 

 In a similar manner 



/dx\ 

 Xr'-5ai2 = Xy' 8[ p. / j 



d 

 = Xy> —, 8x 

 dy 



d . dXy' 



= -, (Xy> ox) - -^ SX, 



