SURFACES IN HYPERSPACE. 301 



The covariant derivative of the covariant system formed by the 

 composition of a contravariant system of order m and a covariant 

 system of order 7n -\- p may be written 



-^1112- • -ipt = 2,i,-2 . . . im^ilil • • • iphh ■ • ■ Jp<l^^"'^" ' ' '"^ 



+ Sy,y, . . . ,-„.,Zi.i, . . . i^i,,; . . . y,„ F^"''^ " " " '■'»^)a,, (40) 



There is a dual proposition for the contravariant derivative of a con- 

 travariant system formed by composition of a covariant system of 

 order p and a contravariant system of order m + p. 



A special case of importance is the differentiation of the invariant 

 which arises from the composition when the orders of the covariant 

 and contravariant systems are equal. We have, from (40), 



If we write for F^'i'^ • ■ • J-"^^ its value 



•* ■^1112 • • • tmr"' "• "■ I* ■» 1112 • • • Xmr > 



j 



we may sum over the i's combining the a's with the X's; then, with 

 proper change of indices, 



It = ^nn • • . ;. \X nn ■ ■ • imtY^^'^' " " " ^•"•) + X^nn ■ ■ ■ M y .^ .^ . . . .^^] . (41) 



20. Relative covariant differentiation. — Covariant differentia- 

 tion is a process which derives from a covariant set of order m another 

 covariant set of order m + 1 containing the derivatives of the elements 

 of the first set and certain derivatives of the coefficients of the quad- 

 ratic form, namely the Christoffel symbols. We may obtain a co- 

 variant set of order m + 1 from one of order m in other ways, without 

 the use of Christoffel symbols but with the aid of the functions which 

 define an n-tuple and its reciprocal. 



Let us express Xr in terms of the X's as a basis (§12). 



