JoLY — The AsHOciative Algebra ajjplicable to Hyperspace. 95 



As a preliminary, it should be noticed that, if ^ is a product of an. 

 odd number of vectors, it is of the type 2^'.\) +1^(3)5 and a product of an 

 even number of vectors is of the type jt?(o) + ^(2). 



Collecting from Ai-ts. 20 and 25, the .^and Zformulse of commu- 

 tation are contained in the equations 



S'(0)2'(l) - iWii.0) = 2'(2)2'(3) - S'(3)2'(2), 



2'(0)2'(2) - ?(2)2'(0) = 5'(1)2'(3) - ?(3)5'(1; = 2'C3)$'(1) - 2'(l)?(3) = 0, 



2'(0)S'(3; - $'(3)5'(0' = 5'(2}$'(1) - $'(1,2'(3)- 



Next, equating to zero the odd parts of 

 (?(o) + 2'(3.)' - (S'd) + $'(2))', and of (2'(o; + ^a.)' " (^'(2) + q{z)\ 

 the formulae ^'(oj^'d) + q(^i)q{o) = 2'(2)2'(3) + 2'(3)S'(2) 



2'(0)?(3) + 2'(3)2'(o; = ^(2>^(1. + qi)q[% , 



which have been already given, are recovered. 



Adding and subtracting corresponding pairs of both sets, all the 

 relations hitherto deduced are comprised in these folloTving formulae : — 



2'(0)2' 1; = 2'(2;2'(3)J 2'(l)2'fO) = 2'(3)2'(2) ; 



2'{o)2'(3) = $'(2)2'(i)j 2'(3,2'(o) = SwS(2) ; 



2'(o)2'(2) = 2'(2)2'(o)5 2'(i)2'(3. = 2'(3)2'(i)' 

 Trom these, it is evident that 



S(o) {Sw - 2'(3)') = 0> and q^^ ($-(1/ - ^'(3)') = 0, 

 and g-d, ($'(0)'' - qizf) = 0, and qi^^ {q^^^^ - q^^^) = 0. 



I shall now examine the signs of the scalar parts in the squares of 

 the functions qfj,), g'^i, q{2), and g'^j, when these functions are supposed 

 to be real. 



Por 2^m a product of m unit vectors, PrrJpm = (-)'"? and is positive 

 when m is even. Taking in turn m = 0, 1,2, and 3 (mod. 4), and 

 remembering the nature of the characteristic /, it is found that ^(qj^ 

 and f;^^ are positive, and p(xf and pf^f are negative. It is evident that 

 the same law governs the signs of the squares of the more general 

 functions q^^^, q^i), q^2): and $0), so that Sq(^i\^ and - Sq^^f have the same 

 sign, and also Sq^^f and - Sqroj^. Hence, it follows that the equations 

 lately written can be satisfied only by having 



S'.o) = q'.i) = 0; or s'vi. = I'X = ; 



at least, when the functions are real. 



