1922. Xo. 13. ON A SPECIAL POLYADIC OF ORDER ;/ — p. 



Hence : 

 (51 <"-2aXb = (« — 2l!2' 



e,- ey 

 e, e/ 



bi bj 



= (// — 2) ! 2' [e, C; Uli bj — (ij bi) + e, e, iOj b, — a, bj\\ 

 with the restriction tliat / <Cy. But this sum is, of course, equal to the si m 



of all terms of the form 



(A) 



/ and /—1,2. . . . // 



or: 



(;/ — 2)! ZitjCij 



a b 



(6) <" 2a X^ = 1"^ -I! [^^^ — b a) = itt— 2\\ 



which also follows from 15) directly, by § 8 (131. 



From this we easily get the space complement of 0; , viz. : 



(7) <" 2 ,. = \u — 2) ! (<P — (Pc) 



The equation (6) can be considered as a particular case of a formula for 

 the space complement of the space complement of a set of any number of 

 vectors less than // (say p). 

 We found ^ 8 I A) that: 



(8)<r aj . . . dp = 2'(— 1) 



II P - ~- — K, 



e^.-.-CA-, 



e„ 



^i 



e« 



«1 A-, .... ^1 k. 



(Jpki 





Now we will derive the space complement of this quantity. We 

 notice that : 



Ci . . . . z\, . . . . e„ 



= (// - p-\ <"-/'ei. . .,e\- . . . e« 



(9 



ei 



e„ 



This can be developed according to § 14 (1) by putting //—/) instead 



0Î p and, accordingly, 2".^^. . instead of 2"/.. whereby the final 



2 1 1 



sign of (9) will be (-^ 1 ) 



(- ir" 



Vve get from (8 1 : 



-^^^-A-. Noticing th 



at 



(- \r' ^{- \f 



(101 <"-rl<rai. . .(Xp)^[n—pV.:L\—\S 



iip-^p 



en .... Cav 



Cla 



et. 



Op kl . . . ^pAy 



which by §8(1 3) gives : 



(II) <"-p(<r ay ... . dp) ^ (ii — p)'. (— II 



a, . . 



;"-r-i)p 



