SURFACES IN HYPERSPACE. 281 



(X(»-)) = ^iyiiXO\ XO-) = S,Ci,(A'('-)), (9) 



(ZC')) = Sfcmi7/,-Z(*'>, X^'^-) = ^uCikCn{Xi^'^), (10) 



and 



\-^^ J ^liH . ;mTJl^lT;2^2 . . . 1 1 mlm-^ > \^ '^ ) 



these systems X with upper indices are contra variant of orders 1, 2, 

 and VI, respectively. 



An important contravariant system is formed of the elements a^ 

 which are the cofactors of the determinant of the quadratic form (8) 

 each divided by the determinant of the form. We may prove this as 

 follows. Let tik be or 1 according as ^" ?^ k or j = k. Then, 



Substitute for a,/ from (6). Then 



Multiply by Cjs and sum over y; the expression reduces by virtue of the 

 fact that ^iCja-Yiq is zero unless s = q and unity ii s = q. (We have 

 therefore 



^l-qJiqCjaiapq) = {Ups) (12) 



which is a formula often used for reducing certain double sums to a 

 single term.) Hence 



2t2p7ip(aps)atfc = SyejfcC/s. 



Multiply by (a^g) and sum over s; then p = t alone gives something. 

 (We have then 



'^spjipiapi) (ttjg) = yit (13) 



which like (12) reduces a double sum to a single term.) Hence 



^iyuO-ik = '^jaijkCjsio-ta)- 



