dx 



Imperfectly Homogeneous Elastic Solid. 251 



of the form f J, where 



V J = K u Aa + K l2 Ab + . . . + K 66 F/, 

 K n . . . K 66 being constants. The variation of J, 



SJ = K n (A8a + fl8A)+ ...; 

 and each of the terms of this sum may be separately treated 

 by the method given in art. 6, and applied to find the corre- 

 sponding parts of the stress-components and of the equations 



of motion, and thence the value of -,-5-. But we may avoid 



d 2 d 

 much labour in finding _-=-, if we observe that the variation of 



° dt 1 ' 



8a can contribute nothing towards it. For 



and we may put 



\dz 

 which is equal to 



dBv dBiv 

 dz dy 



Thus, on integrating by parts again, we see that C ABa con- 

 tains, besides surface-integrals, the volume-integral 



(Su. - — 8w 7 > 7 ) dx dy dz. 



JjJ\ dxdz dxdy) J 



Hence it appears that the part which this term contributes to 



-7-H is — K n —. — r-, and the part contributed to -=-s- is K n -= — 7-. 

 dt 1 dxdz 7 L dt 2 dxdy 



d 2 6 

 Taken together they contribute nothing to -=-j. The same 



thing is true for the terms due to 8b, 8c, 8d, Be, Bf. 



Writing, therefore, for A, B, . . . F their values, we find that 



f dv chv\ 



JM( 



-^ dBu - v fdBiv d8v\ \ 



d?u 

 Integrating by parts, and forming the equations in -j^r, 



d?d 

 we obtain the following equation in -rp-, 



+ . 



^=^{K n a^K 12 b + ... + K 1 J} 



{K 61 « + K 62 6 + ...+ K 66 /} 



dxdy 



