284 



WILSON AND MOORE. 



.like order and conversely 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, 



,(<y) 



'^ = ^kia^'^^a^'^^aki, 



an = Hkicikiaija^''^^ 



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-0, 



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 = AX°. 



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\ 



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



•"1112- ■ •lm-^tlt2- • ■tm-' ■"llt2 • •lm'";U2 • • ]m llljl Ytili- ■ • 



'YimimK'^ }1J2 • ' • im) ^klki • • ktfiilkl'^iiki, • • • (^imkm\-^^ "* )> 



and this reduces to S/i/j. .. ,„(Xyi/2. ..;„,) (5^^""" ' ''"0; 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 



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



Z{nn • • • ip) = T . . . T'Cn J2 • • • iphh ■ ■ ■ im)y . . . (^Ci\ 



We may combine a contravariant system of order m + p with a 



