SURFACES IX HYPERSPACE. 281 



(ZC-)) =-2,^,-,Z(^ Z(») = ^iCiiiXii)), (9) 



(ZC-'^) = ^.mo'i/^^*^ X^'^^ = ^kiCikCn^X^"'^), (10) 



and 



(Ydih ■ • im)\ z= y, ■ ■ . -V ■ -v ■ -v • • xTC'i" • • ''») HTt 



V-^ / ■^;i;2 . ;mT;itiT;2i2 , . . Tjmtm-^ > v-^-^y 



these systems Z with upper indices are eontravariant of orders 1, 2, 

 and m, respectively. 



An important eontravariant system is formed of the elements an 

 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 €ifc be or 1 according a.sj^ k or j = k. Then, 



Substitute for atj from (6). Then 



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

 fact that 'EjCjajjq is zero unless s — q and unity ii s = q. (We have 

 therefore 



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

 single term.) Hence 



2tSp7tp(aps)aiA; = 'SjejkCjs- 



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

 (We have then 



2,p7,p(aps) (ats) = 7i< (13) 



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



