Imperfectly Homogeneous Elastic Solid. 255 



But since 



dx iff/ dz 

 this must be (to admit of two transversal waves) of the form 



\ l dx *dy 3 dzJ\ ( (x + dij ^ dz> 



whence the following series of relations : — 



= N u = N u = N u = N al = N 23 = N 25 = N n = N 32 = JST 36 , 



N 1 =N 11 ==N 26 =N 35 ,^2=N 2 2=N 16 ==N 34; N 3 =N 15 =]Sr 24 = 



and the energy takes the form 



J v { N x ( A< + F^ + E«rJ) + N 2 (F^' + B< + D<) 



+N,(E< + Dfli + 0<)}. (19) 



Now A ... F are cogredient with x 2 ... xy, and w' 1? st[ 2 , ct' 3 

 with 0, 77, 5 ; "thus the coefficients of N 1? ]ST 2 , N 3 are cogredient 

 with x(j-x + vy + £0? y(f * + W + &)> z iS x + W + £0 — that is to 

 say, with lines. The energy in an element of volume appears, 

 therefore, of the nature of the component of a force along a 

 line, like the potential of a small magnet in a magnetic field. 

 It may be shown, however, that the energy given by (19) 

 resides wholly at the surface of the body. h\ fact, let 



and let vs 2 = tx\ + tx\ + wj ; then the total energy of the solid 

 arising from terms of this class is 



£ {X X (A\ + FyL6 + Ev + i^ 2 ) + N 2 (F\ + B/* + Dv + i m^ 2 ) 

 + N 3 (E\ + D//, + Cv + \ nv 2 ) \ . (20) 



Such terms therefore, if they existed, could not affect the in- 

 ternal motions of the solid. 



Terms of Class (I., Y.). 



11. We now come to the terms formed by the combination 

 of groups I., V. Some of these are already complete inte- 

 r/A ■ 

 grals, such as A -7—, the integral of which throughout the 



solid =^\ % mA 2 . Moreover it can be shown that, in every 

 case, any term of the class (I., V.) may be resolved into a 

 series of such surface-terms coupled with terms of classes 

 (I.,IIL), (I., IV.). The proof of this proposition will be given 



