582 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
It is a decided convenience, of course, to be able to put the i, j, k of 
quaternions equal to q, i 2 and qq respectively, but my main reasons are 
purely algebraic. 
In an even order multiplex the product tjs is, except for the value of w 2 , 
virtually a fictit. We cannot make ts 2 = i 2 without using x /( — 1) or 
making i 2 depend on the order of the multiplex. But we can make 
w 2 = (mi 1 ) 2 , namely by taking i 2 = — 1. Thus the n fictit complements 
and the product qq .... (or what is the same when, as at present, n is 
even, the product tsl v tsi 2 .... 7Si n ) are not only all anti-commutative, but 
are also in harmony as to their squares. As a consequence, as we should 
expect, we can very simply identify without introducing J( — 1) the 2 2m 
even order multenions of a multiplex of order 2m + 1 with the complete 
set of 2‘ 2w multenions of a multiplex of order 2m. Thus if ei, e 2 , . . . . e 2m+1 
are the fictits of the former and e 2 =....=— 1 ; if we put 
L 1 = e l e 2m+l) h = e 2 c 2m+lJ ■ • • ’ l 2 m “ e 2m c 2m+l» 
and therefore 
t, l t 2 • • • • hm e l e 2 • • • • e 2m 
the identification is obviously effected and q 2 = ....=— 1. 
If S, S-p S 2 . . . . have their usual meanings with regard to q, i 2 , . . . . and 
S (0) , S (1) , S (2) .... the corresponding meanings with regard to e l5 e 2 , . , 
and if I be the operator negativing e 2m+1 , it is easy to show that 
so that 
S = S (0 „ Si = J(1 -I>S (2) , S 2 = 1(1+I)S (2) 
S 2c -i = i(l - I)S (2C) , S 2 = J (1 + I)S(Jc) 
S(0, = S, S (2 ) = S x + S 2 , S {4 , = S 3 + S 4 . . . . 
The following convention should, I think, be adopted not only in the 
present method but in all allied methods. It is derived from the i, f k of 
Hamilton and the e v e 2 , .... of Grassmann. Whenever primitive units 
are to satisfy the condition i 2 = — 1 consider i v i 2 ... , as permissible 
symbols to denote them ; whenever they are to satisfy l 2 = + 1 consider 
e v e 2 , .... as permissible ; but never consider the converse permissible. 
Thus 
- 1 = q 2 = if — 
+ 1 = q 2 = e 2 2 = . . . . 
When we want i 2 = ±1 we can use q, i 2 ,.... 
It is just as well that in this first paper the sign of i 2 has been left 
ambiguous, because it was desirable to test fully what assumption seemed 
best. 
I would also recommend another convention, namely, utterly to ignore 
the concept of a continent multiplex. The multiplicative combinations of 
