300 



WILSON AND MOORE. 



For a contravariant system of higher order the process is similar 

 and the result is as follows : 



X{.uvw) = 2,a( 



sw) 



dxs \ [ V ) ( u 



, (37') 



and similarly in general. 



The partial derivatives of a contravariant set may then be obtained 

 by solution. For, 



dXr ( V 



(38) 



2.Z("-)a™ = ^-^ + 2, ("z^"') I ^ J I + Z('^) I ^ J I ) • (38') 



19. Properties of covariant differentiation. If we apply (36) 

 to the set Urs of the coefficients of the quadratic differential form, we 

 find 



ttrst 



dCtrs 



dxt 



dOrs 



dxt 



( m ) ( m 



r t 

 s 



s t 

 r 



= 0, 



as follows from (32) and (32'). Hence the first covariant derived set 

 of ttrs vanishes identically. The same may be proved of the first 

 contravariant derived set of a^^^; but as the set a^'"*^ is the dual of the 

 set ttrs f no formal proof is necessary. 



The covariant derivatives of a product of covariant factors follows 

 the rule of ordinary differentiation. For example. 



yXrXsJt — AriZg-}- ArAsi, 



(39) 



smce 



(Z.Z,), = ^^^^-2, 



dxt 



dXr 



dxt 



XmXa ] f + A rXm ] 



( m ) ( m 



— Z/tmAt) 



r t 



m 



W 



z,+ 



dXs ^ is t 



ox ( m 



\] 



Xr. 



