542 
Proceedings of the Royal Society of Edinburgh. [Sess. 
it at once follows that the fictorplex (A) of (22) may be taken as a real one. 
[Thus from the above if ((p — x)i 1 = 0 both ((p — x)p — 0 and (<j> — x)cr = §.\ 
Hence q, i 2 ... . of (56) may be taken as real. 
We now have the very important facts that if </> is any real linity : — 
(1) (p'<p and <p<fi' are colinities ; (2) they are 'positive colinities; (3) cf>'(pp = 0 
if, and only if, (pp = 0, and (p<p'p = 0 if, and only if, <p' p = 0. We will prove 
the statements for only. 
So - j <t>' <$>p = S<£cr | <j)p = S <h'(f)(r | p- 
This proves (1). If <p'<pp = ap, which is always true for some non-evanescent 
real p, when a is a root of <p'(p, 
aSp\p = Sp\ <j>'cf>p = S cf>p | $p, 
so that a is positive or zero according as <pp is not or is zero. This proves 
(2) . If cpp = 0, then (p'cpp obviously = 0. If <pp is not zero, S (pp | <pp is not 
zero ; therefore S p | <p'<pp is not zero ; therefore <p'(pp is not zero. 
The square arrays corresponding to </></> and (jxp' [(55)] are familiar as 
those occurring in the two symmetric forms of the square of a determinant. 
If, then, from each diagonal element of a symmetric determinant so formed 
x be subtracted and the expression so obtained be equated to zero, not only 
are the roots of the equation real, but they are positive (including zeros). 
Thus J((p'(p), except as to the signs of its roots, is a real colinity without 
ambiguity. If we impose that the roots are all positive, or else that they 
are all negative, no ambiguity remains. These two special colinities may 
be denoted by ( + [read “ the plus radical of cj)<p ”] and ( — 
respectively. 
Skewlinities. — [I could write down several interesting formulae involving 
first and second order multenions which are analogous to quaternion 
formulae involving vectors. The most important statement of the kind for 
our immediate purpose is that though S a/3y is necessarily zero when a, /3, y 
are fictors, yet Spcocr is not in general zero, but is a combinatorial part of 
p, Co, or when p and cr are fictors and oo is a second order multenion. Thus 
Sputcr = Stocrp = Scr put — — Scrcop = — Spcrco = — S (opcr. 
In treating of skew fictorlinities and rotational fictorlinities it is the 
general second order multenion, co, with its \n{n — 1) scalars which appears 
as the analogue of a vector, rather than the general fictor, a, with its n 
scalars.] 
Putting in (11) J2S 2 /3|a = o) so that oo is any second order multenion 
we obtain that 
XP = S 1 o>p (58) 
is the most general type of skewlinity. It can easily be proved that 
