248 Prof. C. Nivcn on the Theory of an 



and if SE be now separated by integration into surface-terms 

 and a term of the form ff T (P&t + QSv + H8w)dx dy dz, we 

 may find at once both the components of the stress over the 



boundary and the equations of motion. 



j 



If we consider the term in E due to A -—. the result may be 



ay 



most conveniently exhibited thus : — Let d% be any element of 



the surface, I m n the direction-cosines of the normal at that 



point, and let J 2 , [ v denote integrations over the surface and 



throughout the enclosed solid respectively ; then 



8. f Ap=((A m Sv 1+ d p . ISu+i^.nBv-i^mBiv) 



Jv d v JA d y d v d v J 



J V V dxdy 2 dydz 2 df ) 



The corresponding parts of the stresses are 



_d^\ o _ L dA o __i^r T _ A 

 xx ~ dy' °' y ~ 2 dy ' yz ~ 2 dy ' yx ~ ' 



the others being zero. 



These evidently satisfy equations (5) ; and the same thing 

 may be proved true for each of the terms arising from multi- 

 plying A, B, . . . F by differential coefficients of «r 1? <bx 2 , vr%. 



Theory of an Isotropic Solid. 



7. We shall now find the form of the energy when the sub- 

 stance is isotropic. This is done by considering what inva- 

 riants arise by combining the different groups with group I. 



(1) Group I. gives of itself the two invariants, 



A + B + C, D 2 + E 2 + F 2 -BC-CA-AB. 



(2) Group III. gives the invariant a + b + c, which, how- 

 ever, is identically zero. Class (I., III.) contains the invariant 



Aa + Bb + Cc + 2Y>d + 2E<? + 2F/". 



(3) There is no invariant in class (I., IV.), and none in 



class (I./V.) ; for in\ . . . are cogredicnt with xyz, and -=— . . . — 



are cogredient with f x 2 . . . %osy. 



The energy corresponding to the element-volume is there- 

 fore equal to 



{X(A + B + C)' 2 + ^(D 2 + E 2 + F' 2 -BC-CA-AB) 



+ v(Aa + Bb+ ...2Ff)}dxdydz. . (9) 



