SURFACES IN HYPERSPACE. 283 



9. Dual systems.^* Consider next the system of the first order 



Z(») = S,-a(''^ Xj, (15) 



formed of a system X of the first order and the contravariant systems 

 of the second order a^''\ We shall prove that this system .Y*'' is 

 contravariant of the first order, — which will justify the use of upper 

 indices. Carry out the transformation above; then, by (10) and (5), 



(X^->) = Sya('')(X,) = SyS,m.T//a('=')2pCpyZp. 



The sum taken over j requires / = p. Hence by (12), 



and the theorem is proved. We thus have, associated with every 

 covariant system, a contravariant system relative to the quadratic 

 form (8). 



If we proceed in a similar manner for systems of the second order, 

 we may construct, 



X^'i^ = 2,za(*)a('')Z,i. (15') 



This likewise is seen to be a contravariant system. In general if 

 we have a covariant system of order ni, we may define a contra- 

 variant system of equal order by the equation 



X(ili2 . . .Im) ^ V .^ .^ _ ^ _ ,-„a(^i»W^"'2\ . .a(^'"''''»)Zyi,-i. . . ,„. (15") 



Moreover this relation is reciprocal; for we may pass back to the 

 original system by the formula, 



^fcifc2 • • • fcm = ^hh ' ■ ■ iv^aiikiai^ki • • •aimkmX^^'^-' ■ ■ ^'"^- (16) 



To prove this we have merely to substitute from (15") in (16), taking 

 j = k, i = I, and use the fact that 2y a^^^^aijXi = Xj. Thus to every 

 covariant system of order m corresponds a contravariant system of 



14 Ricci uses the term reciprocal systems in place of dual systems and there 

 are advantages in this use ; but we have preferred to reserve the term reciprocal 

 for sets of systems, thereby following the notation of Gibbs in his vector analy- 

 sis. The term dual suggests itself strongly in connection with a quadratic 

 form. 



