1922. No. 13. ON A SPECIAL POLYADIC OF ORDER ;/ — />. 



33 



If now !"• be any vector, and 1^=0- 0, then : 



(7) ( fP* • (P) ■ V = 0,* • ( • ^y) = <P,* ■ 'C'. 



B:;t f ,* f, is equal to the idemtactor it", and only if, 0," • I"»' ^V. 1. e. the 

 transformation 0,* must be the iiii'rrse of ø, v and its matrix accordingly 

 the inverse of the matrix of 0, Hence the matrix of ø*, being the con- 

 jugate of that of 0,*, consequently is : 



(Al 



^ 



I./1 



1/1 



F^ 



1/1 



F„n 



1/1 



whereby the validity of (41 is shown. 



From (al follows immediately that the Ergänzungss3'stem of y.i is con- 

 jugate to W, (where y.,- is the conjugate system of the f's). 



We also ha\-e as in 5., 



(81 



r; • tv,. = r; • av, = - f„ • w., = \/\ . 



The dyadic determined by the tv's, the Erg(ïiiziingS((v(nfic, is in S.^ 

 given by the following determinant, the primary system being a, b, C : tt 



1,9) 



i i f 

 aX bx CX 



a b c 



i!.u. 



X c — c X bl — etc. 



As we see deriving from a (somewhat special) determinant-triadic by 

 taking — as the crosses indicate — the vector product of the two last vectors 

 in each of its triads. 



In the analogous way we can derive the Erganzungsdyadic /i =e,lV, 

 in S« by means of the space complement. It is readily shown that: 



(10) 



O =r 







- .> 



where the space complement is to be derived of the last ;/ ^ 1 vectors in 

 each of the polyads of the polyadic, represented by the determinant. 

 (101 can also be written: 



t Usually in literature denoted by 1?~ '. 

 ft Almar Næss, loc. cit. § 13 ([land (2), and § 12 (4). 



Vid.-Selsk. Skr. I. M.-N. Kl. 1922. No. 13. 



