302 WILSON AND MOORE. 



Now differentiate with respect to Xg . Then. 



dXr ^ dik'r , ^ dci , 



= SiC,^^ + S,pi,X'r. (42) 



OXs dXs dXs 



We next observe that iX'r is co variant and that 



dCi _ dci dijt 



— — ^t • 



dXs di/tdxs 



As Ci is an invariant, dci/di/t is the expression in the new variables 

 corresponding to dci/dxs . If we introduce the new X' 's, we have 



- iA r — -litu I - I \ii^ u) -— T 



dXs \dyt/ dxsdxr 



•V 



i'^ r 



Hence the set of terms 'Li.ik'rdci/dxs is co variant of order 2. 

 Now, replacing in (42) the invariant d by S^.i-X^'^Xt and transposing, 

 we have as a covariant set of order 2, 



Xrs=^-^ -2,Z,S,.,X(')^^ (43) 



dXs - axs 



(We may verify directly that Xrs is a covariant set of order 2 by 

 transforming it.) 



If we had a set of order 2 expressed in terms of the basis, we find 



Ars = SjjCij.iX r-;A g , With Cij = j^ pqX pq.iK^^' jK-'^' . 



Differentiate and transpose, 



dxt oxt oxt oxt 



The right hand member forms (for all different values of /•, s, t,) a 

 covariant system of order 3; so also must the left hand member. 

 If now we replace Cf/ by its value and if we note that Sj.yX^^^X'a = 

 esq by (21'), we see that ■ 



X., = ^ - 2,Z,32,.,X(-) ^-^ - SpZ.pS,..X(^) ^ (430 

 axt oxt oxt 



is a covariant system of order 3. And in like manner we could 

 form from a system of order m a covariant system of order m + 1. 



