Imperfectly Homogeneous Elastic Solid. 253 



nentsof the vector product of (.i', y, z) (£, 77, J) ; hence X 2 . . . XY 

 are cogredient with x 2 . . . xy, and consequently the coefficients 

 i K44 . . . K 36 are also cogredient with x 2 . . . xy. It is possible, 

 therefore, so to choose the axes of coordinates that the last 

 three terms shall vanish ; and the expression of J will then re- 

 duce to 



J=-cc(Bc + Cb-2T>d)-{3(Ca + Ac-2fte) 



- 7 (Ab + Ba-2¥f). (13) 



Suppose, then, that the axes are so chosen, and let us de- 



d?u 

 termine the values of the accelerations -r-*-, . . . due to J, com- 



mencing with the variation of the coefficient of y ; we have to 

 put therein 



A =£> B =g> *-i(£+|). 



_ 9 d / sr 1 _ d 2 io d 2 v j __ d 2 u d 2 w 



dx dx dy, clx dz 1 dy dz dx dy 



- d 2 u cPw cPw d 2 v 



dxdz dx 2 dy 2 dy dz 



On substituting these values and integrating properly, we 

 finally obtain for S(Ab + Ba — 2F/*) a series of surface-terms 

 with the addition of 



-( ihul- - d f - cfA cp¥ ) hv( da df d " B d ^ s 



Jv I \dx dy dy dz dxdz) \dy dx dxdz dy dz, 



\ dx dy dx dy dx 2 dy 2 ) ) ' ^ < 

 But 



_ dA d¥ _ cPu 1 d /du dv\ _ d-& z 



dy dx dxdij 2 dx\dy clx J di 



dy dx dx dy 2 dx \dy clx) dx 



„ /du dv\ _ 

 dx) 



dB__dF_ d 2 v 1 d (du dv\ _ fe, 

 dx dy dxdy 2 dy \dy dx) dy 



Substituting these, and inserting for a, b, f their proper values, 

 expression (14) may be reduced to 



We may now take together all the three terms of J in (13), 

 when we shall clearly obtain for the three accelerations d 2 ^, 



