1922. Xo. 13. ox A SPECIAL POLYADIC OF ORDER « — />. 



and, accordingly, for igi we get an expression completely analogous to (10) 

 ivitli change of sign. It we put /'3 instead of k.^ we ha\-e change of sign 

 once more [tivo columns to the left of k^ stricken out) and so on. If we 

 then sum all those p space complements of the type which we get from 

 that fixed set of the k\ which we have considered, we arrive at the fol- 

 lowing expression : 



»/p-L I 



(- 1)"^ '(//-/.Cl- 



e,;-2 ^ki Cav 



e/.-2 ex-3 ■ 



e-t. 



eA-2 



C/ti Ca.? Ca-^ 



Ca-, eA-3 ^k- 



(11) 



e/t-3 



ex-i ex-2 ^H .... e*. 



Cam ex-2 0/1-4 



^k. 



etc. 



= (- n"'"^'(;/-/.)!ü 



Cam e/t-2 . 



C/i-i e/t-2 . 



e^-. 



Therefore : 

 2) V X" - F^if a, a, . . . ap) = — (-- \)"^{n — All v • - 



ex-. 



Cä-, 



Op k\. . ■ 

 <^1 Ä-1 «1 /•2 



^p /il lip k2 



Ck^ I I <h ^-1 



^'1 /^. 



f7r/^. 



^1 /i-. 



(7r/t-, 



e/t-, 



• f^-pl r'p^-' 



f^n/i'. 



where the sum is to be taken for all possible sets of the Ä''s. 

 We now onlv have to show that : 



(13) 



<\ 



^\ 



. ar 



e^i 



• e/t„ 



Câ-i 



C/t'. 



^1/tl 



. ^1 /t„ 



^'p k\ 



This formula follows from elementary properties of vector determinants, 

 and is well known in literature. For the sake of completeness we shall 

 also give this last step of the proof. 



We put a, = eyrt/y, and inserting this in 1131 (left side) we get for the 

 principal diagonal: 

 (H) Zejaij^ÇjOoj . . ■ .2ZjOpj y =1,2...." 



while the other terms in the expansion are all possible permutations of this 

 one. We carry out the multiplication. Let one term thus obtained from 

 the principal diagonal be : 



(II 



Cr/i Ca.. 



^ap (^i «1 ^o n-i 



a pap 



but to this there is a corresponding one in each of the permutations of 

 the diagonal; i. e. : a term consisting of the same vectors and scalars m 



