i8 



AI.MAK NÆSS. 



M.-N. Kl. 



Let lis fiirtlicr consider tlu- set /',, k.^ .... /.7, with all its po.ssible 

 pci iiiiilatioiis ; 1< I y/.i, //,■_> . . . .//,■,, h<' any sik li pi 1 iiiiiiaiioii. Wc then first 

 observe that 



</■ <?//., ^il.2 ■ 



± </■ c/.-, e/.-2 . . 



e/', 



where + or — is tu l)c chosen aceordinf^ as the set y'/^y'/ta . . . . y'/t . is 

 an even or otld pci nuitation of the /''s (s. § 5 (b)l. 



Let us now consider those /> ! terms in (3) which are of the form: 



V/.-1 V/,-; 



• • ^//.../'ly^-, ^'•2.//,3 



'h'Jk, 



i. e. : all those />! terms which contain the same unit vectors, viz. 



e/,'i tin .... C/t^,, 



in all possible order. We will take the space complement of each of those 

 p\ terms and then sum. By what is said above, we get: 



^ i^ ^.ik, ^Jkz ■ ■ '^Jkf, 'hu-^ ^-ij;n ■ ■ ■ "PJk^ = <^ «^-^' ^''-- 



ex-, 



±^l7>i.,«2 



Jk\ ''■2Jk2 



"PJk. 



ip tkx ex-2 ... ex- 



(6) 



Ol k\ o I k2 



C?i k 



= (__ ^^i"'P+^) + ---" + Ik- 



Op k\ Op ki . 



Ci . . . /k, . . . e„ 



e« 



<^pkp 



C?! k\ O^ k2 



O, k. 



Opk\ Opki 



Opk. 



Therefore: The sum of the space complements of all terms in (31 is 

 equal to the sum of all possible terms of this kind, i. e. : the sum for all 

 possible sets of the k's, k\ <Z k.^ .... /7,. And, by (2), this shows that 

 the sum is equal to ^Z- i\ a.> .... up. 



Now let Pj and P.> be two different forms of the same polyad of the p^^ 

 order (i. e.: two equivalent polyads, P^ = P.J and thus giving, when expressed 

 by elementary polyads, the same form P,. Then 



(7) 



accordingly : 

 (8) 



</- Pi = </> P. and (7' I <^ P., = </> P, 



</> P, - </- P., 



That is: the space complement of the vectors o( a polyad (we will sa}', 

 shorter: of a polyad) is independent of the particular form in which the 

 latter is expressed, which is the desired result. 



This can always be applied to any sum of elementary polyads which 

 can be summed up to a single polyad, but, strictly speaking, not to a sum 

 of such polyads in general. But what we have found above very naturalh' 

 leads to an extension of our definition, such that we bv the space comple- 



