1922. No. 13. 



O.V A SPECIAL POLYADIC OF ORDER ;/ — p. 



(6) 



■^b Xc; b* = |^cXa: c* = p^ a X b 



where | ^¥\ designates the determinant of the matrix of '/-'. Each vector in the 

 reciprocal system is thus determined as a vector product of two vectors. 

 We will carry out the analogous operations in two-space (unit vectors 

 being i and j). Assuming given two vectors a and b in S.,, we determine 

 two others a* and b* such that 



(7> 



As we have 



and by putting : 



a* a ~ b* b = ii -f ) j. 

 a = i <7i + j a., and h = \ b^ + \ h.^ 



we easily get : 



(8) a* - 



where the two vectors 



I I 



^1 b. 



b* 



I 1 



^1 ^2 



t 1 



b^ /;, 



, etc. must be considered to be quite ana- 



logous to b X C, etc. above. I. e. : each of the corresponding vectors in the 

 two-dimensional case derives only from one of the primary vectors, by an 

 operation given by (8). 



If therefore a generalization of the vector product also shall cover this 

 operation as a particular case, it is readih' understood that the generaliza- 

 tion cannot exacdy be limited to a quandty deriving from two vectors 



i i 



onlv. On the other hand, we cannot verv well characterize e. g. 



bi b., 



which is completely determined by b alone, as a "product" of b. It 

 seems merely to be accidental that the number of vectors in the ana- 



i i f 



logous quantity in S^, viz.: 



, is two, and it may be questioned 



b, b, b, 



whether the term "product" is a proper name for the quantity also in 

 this case. As a matter of fact, the idea that the vector product cannot 

 naturally be characterized as a product of its two vectors is not new. It 

 has been set forth for example by E. W. Hyde. 



§ 4. The Space Complement. 



Our view^ point in the following is to consider the vector product as being 

 a particular case of a (somewhat special) polyadic that can be derived from 

 anv number (^;/l of independent vectors in S„ by means of the following 



