256 Prof. C. Niven on the Theory of an 



for two of these terms ; and we may observe that it follows 

 from this that the interior parts of these terms may be sup- 

 posed to have been already combined with those belonging 

 directly to classes (I., III.), (I., IV.). The existence, there- 

 fore, of the terms of the class at present under consideration 

 in no way affects the internal movements of the solid, though 

 it does affect the specification of the stress to which the solid 

 is everywhere subject. For in addition to the forces and 

 couples which have been already considered in art. 6, we must 

 also imagine stress whose proximate effect is a variation in the 

 state of strain of the parts of the solid adjacent to the plane at 

 which it acts. T proceed now to prove the proposition men- 

 tioned in this article by taking two terms of the class, viz. 



JA-r and I A -=— . It is unnecessary to trouble the reader 

 v dz Jv dz 



with the details of the reduction ; and I shall therefore merely 

 state that the former of these integrals is equal to 



j s (- dk+ ad, + 5 m ) + j;(-a^- F 5 + d5> 



and the latter to 



Each of the other terms of this class may be treated in the 

 same manner ; and we may therefore consider the proposition 

 stated above to be established, and proceed to the consideration 

 of the equations (18). 



Solution of the Equations of Motion. 



12. Before proceeding to the general equations (18) it will 

 be convenient to consider, first of all, equations (10). By 

 differentiation we obtain from these latter the following: — 

 dVi 2 d§ 



-&-<&**>- 3? 



where 





1 ax l ay ' a, 



2 az: 



