SURFACES IN HYPERSPACE. 287 



the relations Sy CtrTts = ^rs between the elements Cjr of any non- 

 vanishing determinant ] c,> | and the elements jis obtained by dividing 

 the cofaetor Cis by the determinant, we see at once that the elements 

 jX'^*^ are the cof actors of jXs divided by | fXr |, — and similarly from (21') 

 the elements iX'p are the cofactors of {K'-p^ divided by | iX^^) |. These 

 relations are also reciprocal, i. e., the elements ,X(p) and ,Xr are respec- 

 tively the cofactors of iK'p and iX'^""^ divided by the determinants | tX'p | 

 and |iX''''^|. Hence by summing the other way, namely upon the 

 index r we may get the relations 



2..iX..yX'('-) = eij , S..iX'..;X('-) = eij. (22') 



12. A standard form for systems. If we have a contra variant 

 ?i-tuple tX*'"' and any covariant system Xr we may form by composition 

 the n invariants 



Ci = ZrXr.iX^'-l 



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

 For, by (21'), 



Hence 



Xs = l^iCi-iya • (23) 



Any system Xg is therefore representable as a linear function of fX's 

 with invariant coefficients. In like manner 



Z(«^ = SiCi.iX'(^ Ci=^rX^^\\r. (23') 



In general for systems of any order we may write 



Xri r2 - - rk ^ ^tl 12 • • U ^h h • • ik-h'^ ri-i2^ r2 • • • t*^ rfc j 



On n • . U = Sn r, • • rk X. r, . . u-nX^^'^- i.'^^''^ ■ ■ ■ i^^'", (23") 



and 



X(n r, ■.rk) = Sii i, . . i, Ci, i, . . u.i.X'^'"^^ i2V(--) . . . uX'('-«=) , 



Cu h ■•ik = -n r2--rk X"-' '' ■ • '"^•n^n-i-.K, ■ ■ U^r. • (23'") 



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

 cients, on the product system of the 7?ith order made up of the X"s. 



