

JoLY — The Associative Algebra applicable to Hyperspace. 87 



It is thus proved as a theorem that the conjugate of the product of 

 any two functions of the units is equal to the product of their conju- 

 gates in the inverse order, or that Kpq = Kq^Kp. Of course the effect 

 of K on to, the product of all the units, must not he overlooked. It 

 may be agreed to replace co by a scalar,^ at least when the number of 

 units is odd ; this is not the kind of condensation referred to here, 

 and ambiguity in the present Paper is avoided by retaining the special 

 symbol co for the product of the units. In this article, and elsewhere 

 throughout this Paper, the word condensation simply refers to the 

 degradation of a product (such as «i«2^'i) to a simpler expression (such 

 as 4). 



As a simple example of the conjugate of a product, gKq is always 

 its own conjugate, and so is the generally different expression Kq . q. 



17. The second new characteristic of operation is /, which inverts 

 the order of the units in any product, but without changing their signs. 



Thus, /e>2 . . . 4 = 4.4_i . . . i^i^ = (_)(--i) + ('»-2H. . . + i.j\4 . . . 4 



= (-)i'"''"-\4...^., 

 and if q^ is homogeneous and of order we, 



^m == ^m, if in = 0, or 1 (mod. 4), 

 and Iq„, = - q„„ if m = 2, or 3 (mod, 4). 



Just as in the case of the conjugate, the inverse- of a condensed product 

 is the inverse of the uncondensed product, and, taking account of co, in 

 general, 



I.pq = Iqlpi 1 {qlq) = qlq, and 1 {Iq -q) = Iq -q- 



^ The general consideration of w is given by Cliiforcl. It is briefly this : — 



im • ilh . . .in = im ■ W = (— )''"^£U . im, 



where im is any one of the n units. Thus, when n is odd, oo is commutative with any 

 (linear) vector p, or oip = pw, and indeed more generally cnq = qw, where q is any 

 function of the units. But when n is even, aip = — poo. These properties sharply 

 separate spaces of odd and even dimensions. Again, 



«2 = (-) J" (»-i) ww' = (-) i» ("*i), if co' = inin-i . . . hh ; 

 or ft)2 = + 1 for n = 0, or 3 (mod. 4) ; while w^ = - 1 for w = 1, or 2 (mod. 4). 

 For n = 2, Clifford says, "Here u has clearly the properties of a unit vector" 

 (Collected Works, p. 401). The present writer prefers to regard the w of even 

 space as a scalar of a new kind, perhaps applicable to the measurement of angles as 

 the ordinary scalar is applicable to the measurement of lengths. 



2 Perhaps the name "Keverse" would have been better, as "Inverse" and 

 "Eeciprocal" are usually synonymous. 



