1922. No. 13. ON" A SPECIAL POI.YADIC OF ORDER H — p. 1 9 



ment of any sum of elementary polyads understand the sum of the space 

 complements of each polyad. Once this extension established, it follows 

 immediately that it must hold good for sums of all kinds of polyads, as 

 they always can be reduced to elementary ones. That is : we can lay 

 down the 



Definition (b). Bv the space coniplciuriit of a polyadic is understood 

 the sum of the space coiiiphiueiits of each of its polyads. 

 Or: 



= </> a^ a., .... cv + </■ (\ b., .... b^ + ... . 



Accordingly we get from any equation between polyadics a new equation 

 by inserting the sign </> (or X) in the same way in each of its terms on 

 both sides of the equation. 



And from this follows that the operation of forming the space comple- 

 ment obeys the distributive law because the indeterminate multiplication 

 does. Since we e. g. have : 



^jQ^ ^\c^,a. .... a^ + bib, .... 6/> + ... ) 



= \? ai a« . . . cip + iU\ bo . . . . bp + . . . . 



we know that those two equal polyadics (of order n + \) must also have 

 equal space complements, i. e.: 



I J J I 1^ X^ ^<\ a., . . . . ap 4- bi bo . . . . bp + . . . .) 



_ = Ü X' <\ ao . . . . ap + \? X^ bi b., . . . . bp + 



where s^p. 



§ 7. The space complement of a determinant of the form 



<Xy . . . . <\p 



a^ . . . . iXp 



Each row here consists of the same p independent vectors (p ^ n). 

 The multiplication being indeterminate (or general) the determinant is a 

 polyadic of the p^^ order. 



If we expand this determinant we get p ! terms (polyads). One of 

 them is the principal diagonal a^ a, .... Cïp, all the others are permuta- 

 tions of this term. And, by what is said above, we get the desired space 

 complement by taking the space complement of each of these terms and 

 summinsf. 



