538 Proceedings of the Royal Society of Edinburgh. [Sess. 
minants each of the n t]x order as a determinant of the same order. To 
prove (31) we have 
[M] = (Q'hYnWn [( 26 )] 
= [(13)] 
[(26)] 
■ WM [( 26 )> ( 30 )]* 
By putting ^ = in (31) we get 
M[^> _1 ] = [i] = i 
or [^] = M| (32) 
We may also note here that the roots of the f(<p) n-tio, are f(gi),f(g 2 )r 
.... where / is a rational integral function of and g v g 2 , ... . are the 
roots of the ^-tic. For, first, if 0 is given by (22) its ^-tic must be 
(x — a) A (x — b) B . ... as we see by putting \ v X 2 , .... /ul v ... . for cq, a 2 , ... * 
in (21). But ( Octonions , p. 159), 
= Ai/(a),/(<£)A 2 = X. 2 f(a) + X/', .... 
where \" e belongs, like A' e , to the fictorplex \ v \ 2 , ... . \ e , etc. Hence the 
roots of the f(cp) ^-tic are f(a) occurring A times, f(b) occurring B times, 
etc. 
It will be seen that this result may be put 
[/W - ®] = {f(9i) ~ x}{f(g 2 ) -x} ) 
when [<f>-x] = (g 1 - x)(g 2 -x) ... . ) 
From (32) and (33) we easily deduce that 
M = W (33a) 
where h is any integer positive or negative. 
Before proceeding further with fictorlinities it is desirable to ask and 
answer the question — How far are the results hitherto obtained capable of 
simple extension from fictorlinities to multilinities ? The answer is that 
all the numbered equations of this section excepting (19), (20), (25) are 
thus capable of simple extension. The feature of (19), (20), and (25) 
incapable of the extension is the interpretation of Safl/3? (or of a$») 
except when c is 0, 1, n — 1, or n. There is no simple close analogy in 
multenion multiplication to even so simple a fictor product as a/3 of only 
two fictors. 
I will content myself with merely stating (the proof is simple) the 
principles of extending the meanings of the other equations. 
