302 



WILSON AND MOORE. 



Now differentiate with respect to Xs . Then. 



^-X'r _ diX'r.^ dCi , 



ZiiCi — f- ^i~— jA r 



dXs dXs dXs 



We next Observe that jX'r is covariant and that 



dCi _ ^ dCi di/t 



(42) 



dx, 



-'I 



dijtdxs 



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

 corresponding to dci/dxg. If we introduce the new X"s, we have 



■^iT— lA ,■ — ^itu I T ) \if^ u) 7— - • 



dXs \^Ht/ dXsdXr 



Hence the set of terms '^i.iX'rdci/dXg is covariant of order 2. 

 Now, replacing in (42) the invariant Ci by wi.iX('>X< and transposing, 

 we have as a covariant set of order 2, 



Z„ = ^^ -S.Z.2,.,X(0^^^'^ 



dx. 



dx.. 



(43) 



(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 



A r 



iyCij.tX'r.jX's , with Cij = SpgZpg.iX^P^/X^^^. 



Differentiate and transpose, 



dXrs _ ^ d-iX'r . / _ ^ ., dj\ s _ ^ dCij , , 



^ij^ij ^ J" s —< ij' ij-ii^ T '^ij lA r-jA g 



dxt dxt 



dxt 



dxt 



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

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

 If now we replace Cj,- by its value and if we note that Sj.jX^«^,-X'g = 

 Cs 3 by (21'), we see that 



Xrst = ^ - S,Z,.S,.,X(^) ^-^^ - ^pXr.-^i.iX^"' ^^ (43') 



dxt 



dxt 



dxt 



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 -\- \. 



