SURFACES IN HYPERSPACE. 283 



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



Z(^) = 2,-a(^'^ Xi, (15) 



formed of a system A' of the first order and the contravariant systems 

 of the second order a^^'^. We shall prove that this system X^^^ is 

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

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



(X(^)) = Sya(^'-)(A^-) = SyS,m-.7z/a'*')S^c,yZp. 



The sum taken over j requires I = ]). 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^'^^ = Zkia'^'^a^i'^X.i. (150 



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

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

 variant system of equal order by the equation 



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

 original system by the formula, 



Xklk2 • ■ .km = '^hh ■ • ■ Uahkiai2k2 • • -(hrnhmX^^'^-'- ' " ''">. (16) 



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

 j = k, i = I, and use the fact that S,- a^^'^aijXi = Xi. 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. 



