SURFACES IN HYPERSPACE. 287 



the relations S,- Ct>7,s = 6rs between the elements Cir of any non- 

 vanishing determinant ] dr \ and the elements jig obtained by dividing 

 the cofactor Cg by the determinant, we see at once that the elements 

 ,X'<*) are the cof actors of iXj divided by \i\r\,- — and similarly from (21') 

 the elements iK'p are the cofaetors of {X*^^ divided by | iX^^) |. These 

 relations are also reciprocal, i. e., the elements iX^^^ and »Xr are respec- 

 tively the cofaetors of ,X'p and iX'^*"^ divided by the determinants | ,X'p | 

 and liX''*")]. Hence by summing the other way, namely upon the 

 index r we may get the relations 



S..iXr.yV(^) = €iy, S..iXV/X(^) = e,,-. (22') 



12. A standard form for systems. If we have a contravariant 

 ?i-tuple iX^"") and any covariant system Xr we may form by composition 

 the n invariants 



a = 2,Zr.iX('-). 



These equations may be solved with the aid of the reciprocal w-tuple. 

 For, by (21'), 



Hence 



Xs = 2iCt.iX'g . (23) 



Any system Xa is therefore representable as a linear function of iX'g 

 with invariant coefficients. In like manner 



Z(«^ = SiCi.iX'(«), ci = S.ZC-^Xr. (23') 



In general for systems of any order we may write 



A-n T2 • • rk ^^ ^tl 12 • • ik ^h 12 • • ik-ii.^ ri-t2^ r2 • • • i/fc^ rfc j 



Cii 12 • • ik ^ •"ri T2 • • rk -^ n T2 • • rk-il^ • h^ ' . . . ii^h , \^0 ) 



and 



Y(ri r2- -rjt) = V. . . f, . . . . \'(rO . \'('-2) ■ X'C'"*) 



Cii t2 • • ti = ^n r2 • • rkX'''''-^ ' ' ^ •U^r2-i2^r2 • • ik'^rk • (,2o ) 



Any system of order 7n is linearly dependent, with invariant coeffi- 

 cients, on the product system of the wth order made up of the X"s. 



