19^2. Xo. 13. ON A SPECIAL POLYADIC OF ORDER U—p. 



Here we write, as in § 9 (141: -I— ll'^ '(_/;•/ ^-fij)^djt. Further, 

 let the (// 21-rowed vector-determinant, formed from the f s after f) and f/ 

 have been stricken out, be denoted by Ty/, i. e. : Fjt is formed from the 

 f's similarly as Eji of § 9 (12) and (131 is formed from the e's. Then wc 

 can write the expression for O. in the verv simple wav: 





(6) 



the sum extending to all the f ^j possible combinations of /, /(y -^ /I. We 

 may think of these two sets as arranged in some definite order, for example: 



(A) 



' 1 J ' 1 3 ' ■-> 3 ' " — 1 « 



and then regard the sum as the Grassmaxn „inneres Produkt" of these 

 two ordered sets. Thus we realize that j, I here plavs the role of a single. 



not a double, index running trom 1 to == . 



\2} 2 



Let us consider a definite determinant Fji. We will expand it as 

 § 8 (13). Let us in the (;/ — 2)-rowed matrix of the \ectors of Fji, viz.: 



fBj 



y 1 1 /1 •> /1 " 



I'll J" ^> J>! '! 



[n — 2 rows), 



strike out the r'' and /-'' columns \r <^J\ and denote the determinant thus 

 obtained by ^;/.,-/. Thus we see that %/,,■/ is f/ie second /iiiuor of //ir 

 detcr})iiuaiit of the f's, obtained b}- striking out its y'*' and /'*' rows and its 

 r**^ and t^^ columns. 



Further, let us put, as we have done § 9 (12): 



(7) 



.Cr. . . ,e / 





= E,.,. 



Thus we have, by § 8 (13): 



(8) Fj! = 2- g, /,,/£-/ 

 and, accordingly : 



(9) ü,= I Ertl^ii'tdjl 



(rl) a I) 



