1922. No. 13. 



>X A SPECIAL FOLVADIC OF ORDER // — p. 



n 



Then 



(2) 'p «1 ao . . . . (Xp 



^*! — 1) 



(" -/>—!) 



1 



. Zkx . 



■ e^ 



. e. 



Ci; . 



Opi . . . apk\ . . . apki ■ 

 Ci . . . e'^, . . . C;, 



e« 



/- 



e„ 



. e,2 . . . . e^ 



/> ■ 





Op k, 



^1 Am Oy kz 



(J pi 



(h A-, 



Op ki Op kz . . . Op kp 



The /k- indicates that the first determinant is formed from the rest of 

 the unit vectors after C/ti ikz ■ ■ ■ ■ ^k^ have been stricken out. The sum 

 is understood to be taken for all possible sets of the ^"s. On the other hand, 

 we can express the indeterminate product a^ Clo .... ci/) as a sum of 

 elementary polyads b}- putting a,- = Cya/y (sum fory from 1 to //, / = 1 , 2 . . . . />) 

 and multiplying according to the distributive law : 



(3) 



a^ a.. 



dp - ^ e^l e;2 . . . . C;^ a^j, a._j2 .... Gpj^ 



j\j., . . . . jp here denotes any set of /> integers in a)iy order picked 

 out of the numbers 1,2. . . . ;;, and the sum is to be taken for all 

 possible sets of the y"s. 



Now let k^, ko . ... kp as before he a set of /> integers picked out of 

 1,2. . . . n such that k^^ <C k.^ -^ . . . . kp. Then we have : 



(4) (j, tkx Zk2 



e>t. 



Ci . . . . e*, 











e^, . 



e„ 











If we expand this according to the determinants of the first ii — p rows, 

 we notice that all but one of the plain complements of these (// — />l-rowed 

 determinants vanish, the non-vanishing plain complement having the value 

 one (each element in its principal diagonal is one, all the others zero). 



Thus we s:et : 



(H - p-r \)~ — »-r ^k 



(5) (y e.ti Zk2 . . . . Up = (— 1) 1 ' 



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



ei . . • . ,e k- 

 ei 



e„ 



e„ 



