SURFACES IN HYPERSPACE. 301 



The covariant derivative of the covariant system formed by the 

 composition of a contra variant system of order m and a covariant 

 system of order r/i + p may be written 



7 • , = E • • X ■ ■ ■ ■ ,y(JU2- ■ -im) 



^xui- • -ipt ^nn • • • im-^im • • • tpiin ■ • • ipt'- 



4-S- • ■ X '• ■ • ■ Yihh- • ■ims)f, . (AC\\ 



r •"ni2 • • • jm«-^m2 • • • tpjin • • • jm^ "'st y*^J 



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), 



■"3132 • • • ;mS-^ 3132 • • • 3m-* "s* • 



If we write for F^'i'^ " " • ''«*) its value 



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

 proper change of indices, 



7, = 2-- • FY- • • ,y(3i32- ••;■«.) -U F('i'2- • -J^^y- ■ • ,1 (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 7i-tuple and its reciprocal. 



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



