§ I. Introduction. 



1 he object of this paper is to develop some of the chief properties 

 of a special determinant polyadic, deriving — by the definition given in 

 § 4 (a) — from any number of independent vectors, and which we shall 

 call their space complement. From the definition will be seen that the 

 vector product of ordinary vector analysis is nothing but a special space 

 complement. It is further our object to show that the equations expressing 

 characteristic properties of the space complement, from a formal point of 

 view can be regarded as generalized vector product formulae, and thus 

 formally the space complement ma}' be considered to be a kind of a gene- 

 ralized vector product. 



As will be known, by the vector product of two vectors is in modern 

 tensor anah'sis usually understood the skew symmetric tensor which is 

 determined by the same two vectors. This tensor is of the second order 

 in any space. But as in Sg only three of its six components are independent 

 quantities, there may in this case be associated with it a vector whose 

 components are those three quantities taken in a definite order. But this 

 tensor, which in Sg is different from, but representable by, the vector 

 product of classical vector analysis, can hardlv from a formal point of view 

 be characterized as a generalization of the latter. In fact, it only means 

 an old name on a new and different quantity. It is, of course, in this con- 

 nection of perfect indifference whether or not this new quantitv (the tensor) 

 is a more suitable or convenient representation of those physical phenomena 

 which formerly were represented by the vector product. 



Notwithstanding that the language and conceptions of vector analysis are 

 always used in the sequel, it may equally well be regarded as dealing with 

 (an extended) algebra, the unit vectors playing the rôle of positional sym- 

 bols, and their Gibbsian indeterminate products — to which any polyadic 

 can be reduced — only being new positional symbols. A few of our 

 theorems concern properties of matrices only, as for example § 12 (al, quite 

 independent of vector analysis notations and conceptions. 



Rather often reference is given to the writer's paper on triadics, where 

 a few of the theorems are worked out for the three-dimensional case. 



