l6 AI. MAK NÆSS. M.-X. Kl. 



§ 6. The space complement regarded as a function of the 

 indeterminate product of its vectors. 



I)\' the elemc-nt.'iiy law for addition of determinants, we get: 



( 1 .1 (a + b + C + . . . . ) X Ü - a X v? + b X V -f c X t? -r . . . . 



The combination of vectors in the space complement is thus evidently 

 in this case distributive, which — according to Gihbs's general view of multi- 

 plication — might justify the consideration of the space complement as a kind 

 of product of the two vectors of which it is formed. 



Clearly it is immaterial whether t) in (I ) is post- or pre-factor. 



As we have not yet deiined wbat we understand by the space complement 

 of a complete polyadic (i. e. : a sum of polyads) we cannot rightaway extend 

 (1) to the case when we instead of a + b + C . . . . etc. have a sum of 

 polyads. In order to obtain a meaning to (1) also in this case, we proceed 

 as follows : 



The space complement : 



</> i\ a, . . . . a^> 



can be considered as a function of the polyad of the p^^ order 



i\ a. .... a/> 



i. e. : as a function of the indeterminate product of the same p vectors. 

 This is in accordance with the fact that the scalar and vector product of 

 ordinary vector analysis are considered to be special functions of the corre- 

 sponding dyad. 



Firstly it is then necessary to show : (a) The space coniplcnient of the 

 vectors of a polyad is independent of the particular form in which the polyad 

 is expressed. 



It is sufficient to prove that if the polyad <X^ a,, .... a/.' is reduced 

 into a sum of elementary polyads, and if we derive the space complement 

 of each of these and sum, this sum is equal to the space complement of the 

 primary polyad. 



Let us expand the space complement (i. e. the determinant) according 

 to the (;/ — ^/))-rowed determinants of the first n—p rows. Let k-^, k.^ ..../> 

 denote any set oï p numbers picked out of the set 1, 2, . . . . ;/, such that: 



k^<k.,<. . . .<kp 



' p <^)i and the Cl's are independent vectors ; if not, the theorem is true, 

 but trivial. 



