Imperfectly Homogeneous Elastic Solid. 247 



dh xx dhy X dJj zx Q q _ A 



dL s + dL a + 'J^ /+ Sot _S m =0, ■ • • • (5) 



dx dy dz 



dl-ixz . d\j yz dh zz Q q _n 

 da? dy dz v y 



The existence of these internal stress-couples was indicated by 

 Professor Stokes in his review of MacCullagh's theory of 

 double refraction (Rep. Brit. Assoc. 1862). 



5. The laws of resolution of these internal forces and couples 

 may be readily investigated by considering the equilibrium of 

 an elementary tetrahedron, three adjacent edges of which are 

 dx, dy, dz. If X, Y, Z be the three forces, and L, M, N the 

 three couples which act on the oblique face per unit area of 

 the same, we shall have 



X = IS XX + mS yx + n$ zx , Y = ZS^ + m^yy + nB Z9J Z = . . . ,"] ,„. 



L = lL xx + mL yx + nL zx , M = ZL a , y + . . . , N= ...J 



Z, m, n being the direction-cosines of the normal to the oblique 

 face. These expressions satisfy the condition that the work 

 done by the external forces on the oblique face (dX) during 

 any indefinitely small arbitrary twist (8u, Sv, Bw, Sot^ 6V 2 , Sct 3 ) 

 shall be 



d2(Xfa + Yfa + Z8w + Lkar 1 + M8vr a + 'K$>Gr 3 ); . . (7) 



and, indeed, the values of X, Y, ... N might have been found 

 from this condition by the application of the principle of vir- 

 tual velocities. 



6. We shall now prove that, if the energy contain terms be- 

 longing to either of the classes (I., III.) or (I., IV.), the cor- 

 responding part of the stress will consist of a force and couple 

 satisfying equations (5). Let us consider any portion of the 

 solid, and let it receive infinitesimal arbitrary displacements 

 throughout its interior and over its boundary, and let 8W be 

 the work done by the forces arising from the action of the sur- 

 rounding matter, and 8E the increase of energy stored up in 

 the solid ; the increase of kinetic energy 8k will be 



the density being taken as unity for simplicity, and impressed 

 forces neglected for the same reason. By the principle of the 

 conservation of energy we have 



SW=SE + 3k; (8) 



