284 WILSON AND MOORE. 



like order and con\ersely to every contravariant system corresponds 

 a covariant system of like order, with the systems occurring in dual 

 pairs. In particular the systems an and a^''^ are dual since, 



The results of this section may again be put in matrical form and 

 gain in brevity. We set X° = X-A~^ or X° = A~^-X, it matters 

 not which, since A is self conjugate. Then, 



X° = X-A-i = [(X)-M-i]- [MM:(A-i)] = (X).(A-i)-Mc= 



M.(X).(A-i), 



which shows that X° transforms contravariantly. The terms X^*'^ 

 may be treated as a symbolic product X°Y° and the result is that 

 XY:A~'A~^ is contravariant, etc. The dual is obtained by writing 

 X= X°-A = A-X°. 



10. Composition of systems. If we have any two systems X, 

 Y, of order m, one covariant, the other contravariant, we may form 



7 = S- • X- ■ y(»l'2- -im) (■\-7\ 



■* — 'ni2 • • •tm-'*-»l'2 • • •^m•' • K'- ' ) 



This system / contains only one element and is invariant. For 



^>112- • ■ tm-^1112- • • tm-* '"«lt2 • •lm'"3U2 • ' Jm Ttiyi Tl2;2 • • • 



TtntJmV-^ Ji;2 • • • im) ^kiki • • kn^ilklCi2k2 • • • Ct„fc„(,A '" j, 



and this reduces to S,i/2. .. ,„,(Xyi/2. ••/„) (F^""' • •'"'»)); because 

 when summed on i the right hand member gives something only when 

 k = j, and then gives 1. 



In like manner if we have a system X of order m + p and a contra- 

 variant system Y of order m, we may get a system 



7 . = y. . . Y y(;ij2- -im) (-[0.) 



■^1112 ■ • • tp — 'nj2 • ■ • 3m -^1112 • • • »p;i;2 • • • ;m -* > \^^/ 



of order p, which is covariant. By a similar definition, 



7lili2---ip) = -^ . . . T'(tli2- • • ipJU2- ■ • Jm)y. . (■\Q\ 



^ -'jU2---3m-^ ■'jU2---Jm' \^^ ) 



We may combine a contravariant system of order ?n + p with a 



