300 



WILSON AND MOORE. 



For a contravariant system of higher order the process is similar 

 and the result is as follows : 



X{uvw) = s.a^*"'^ 



dxs \ ( V \ ( u 



(37') 



and similarly in general. 



The partial derivatives of a contravariant set may then be obtained 

 by solution. For, 



dXr i V 



(38) 



S.X("-)a™ = ^4^ + 2, (x^-O V '1 + Z('"> \^'l]' (380 



dXr 



u 



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

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

 find 



Oral 



dttrs 



dxt 



dttrs 



dxt 



\r t). \s t 



Oms j Y + arm ] 



( 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 Urs , no formal proof is necessary. 



The covariant derivatives of a product of covariant factors follows 

 the rule of ordinary differentiation. For example, 



{XrXsJt — XrtXs-T XrXst, 



(39) 



smce 



(XZ.),=^^^^-S, 



dxt 



XmXa ] [ ~r XrXm ] 



( m ) ( m 



idxt 



Zim-^n 



r t 



m 



X3 + 



dXa „ ,, \S t 



dx 



— 2/nt-^n 



m 



V- 



