JoLY — The Associative Algebra applicable to Hyperspace. 89 



are long, I do not print them here ; but there is no difficulty in deducing 

 the correct expressions for any special cases. Consider, for example, 

 a product J3(2)2'(3). As it is of odd order in the units, it is evident that 



Taking conjugates, 



•and hence V^i^p^^^q^s) = i (i?(2)2'(3) + S(3)P(.2,) = ^wSf3)P{2), 

 and '^mPmm = i {Pi^i-) - l{z)P{%)) = - ^(3j2'(3)i'(2> 



20. The functions q and 'Kq are not generally commutative. The 

 conditions of commutation are comprised in simple formulse which I 

 shall now give. 



For brevity, let q = q' -^ q", ^^^ ^q = q! - q", 



then qKq = q''^ - q"'^ - {q'q" - q"q'), 



and Kq • q ^ q''^ - q"'^ + {q'q" - q"q')- 



So the condition of commutation of q and Kq is 



q'q" - q"q' = ; 

 and when this is satisfied, 



qKq = Kq . q = q'"^ - q"'^. 

 Now, 



?Y' = (?(0) + ?(3)) (£(1) + 2'(2)) = (S'W^d) + ^(3)?(2)) + (5'(0,^(2J + q'MH)), 



in which the parts odd and even in the vector units are separated. 

 So the formulae of commutation are 



$'(o)$'(i) - 9.{i)lm = S'(2)$'(3) - 5'(3)2'(2). 

 ^nd q^^o)qi2) - qi2mo) = q^qm - qiz)qw, 



or ^(3) (s-w^d) - 2'(2)2'(3)) = 0, and V^o){qmq(2) - qam^)) = 0. 

 This last step follows from the last article, or directly, since 

 q'q" - q"q' = K(q'q" - q"q') ; 



this function involves only terms under the signs ^"(o) and F(3-. 



21. Por a quadratic or a cubic function q^o) is a scalar, and the 

 conditions become 



M3 - M2 = 0) and q^q^ - q^q^ = 0. 



These are identically satisfied for a quadratic, as q^ does not then exist, 

 or a quadratic is always commutative with its conjugate. 



