374 HITCHCOCK. 



to the common subscript j. If several rows are alike the fact may be 

 denoted by an exponent, as in (61). Thus in (60) the left side might 

 have been written [<pii A 7 ,- •, (p p :Aj, (A/) n-p ], Where, however, the. 

 notation implies merely n polyadics (instead of w 2 ) the square bracket 

 signifies the ordinary determinant of the n 2 elements by the scheme of 

 (11). Thus instead of [an] we might write [Ay] without change of 

 meaning. Unless the contrary is stated, all subscripts are to run 

 from 1 to n. 



10. Forms of the Scalars (36) which show their Polyadic 



Character. 



Up to this point have been noted three distinct methods of actually 

 calculating a scalar of the form (36), namely 



1°. Multiply the double polyadics term by term and take scalars 

 according to definition. 



2°. Form a sum of cubic determinants. 



3°. Form a polyadic determinant by operating on any set of 

 linearly independent polyadics and divide by the determinant of the 

 set. 



In all these cases the double polyadics have behaved formally as if 

 their antecedents and consequents were vectors in space of n dimen- 

 sions, instead of polyadics each of which is a sum of products of K 

 vectors in space of N dimensions, where n = N K . Each of these 

 methods applies with no change whatever if K = 1 so that n = N and 

 the polyadics become actually vectors, — with the obvious exception 

 that the multiple dot product, indicated by a colon, becomes ordinary 

 or single dot product. While I am not aware that any of these three 

 methods has actually been employed by writers on vector analysis in 

 N dimensions, yet it can hardly be said that the individual steps do 

 not occur, in principle at least, in the writings of Hamilton, Gibbs, or 

 Grassmann. For example, the definition of these scalars is equivalent 

 to a double indeterminate product followed by inner product. In the 

 case K == 1, N = 3 they were introduced by Gibbs as scalars of double 

 cross products. In the same case they were expressed by Hamilton 

 both by summations which are equivalent to double multiplication, 10 

 (though never so regarded by him), and also as quotients equivalent 

 to our method 3°. It is true that none of these writers expressed the 

 scalars as sums of determinants whose elements are extensive magni- 

 tudes, use of which was first made bv Cayley, nor by cubic determin- 



