541 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
From the above it appears that the theory of multilinities is virtually 
included in that of fictorlinities. We shall therefore henceforth only 
consider fictorlinities, and shall assume the truth of any transformation 
into multilinity form that we may require. 
As this section has extended far beyond my desire, I must content 
myself with a mere summary of other properties, and proofs thereof, of 
linities. 
To refer to ordinary algebra the following notation will be used, 
</> = 
r a i b i • 
& 2 • 
| cpu 1 = a 1 u 1 + b 1 u 2 + 
means < cpu 2 = a 2 u 2 + b 2 u 2 + 
(55) 
and the rows and columns of [eq . . . . ] will be called the rows and columns 
of (p . Thus the rows and columns of <p' are the columns and rows of cp 
respectively. 
There are three specially important kinds of real linities. (1) Colinities, 
<p' = 0. Columns the same as rows. Colinities the roots of which are all 
positive or all negative will be called positive or negative colinities 
respectively. (2) Skewlinities, <p' = — <p. Columns are the rows with sign 
changed so that the elements of the principal diagonal are zeros. (3) 
Rotational linities, <p'<p — 1. A rotational fictorlinity is a rigid replacement 
A rotational multilinity is in general not a rigid replacement. 
Colinities. — A set of fictits can always be found such that 
<£q = Cqii, <p L 2 = a 2 L 2’ * • * • c f )L n = a n L n • • • . (56) 
when <p is a colinity ; where a v a 2 , ... . are all real. The n-tio, and the 
discriminant are 
(cf> - aj)(<£ - a 2 ) . . . .(<p-a n ) = 0, [<p] = a Y a 2 . ... a n . . (57) 
Proved from (22) thus : — \\ is zero since SA 1 | <p \ 2 = S\ 2 | <p\ ; \' 2 is zero 
since S\ 3 \(p \ 1 = S\ 1 \(p\, SA 3 |<pA 2 = SA 2 j0A 3 ; etc. The A th order fictorplex 
may be expressed as that of A fictits X 2 , ... . Similarly for /ul v ju 2 , ... . 
From SX 1 \(piu 1 = SfjL 1 \(pX 1 and a 4= 6 we get SA 1 | y a 1 = 0. Lastly, a x of (56) 
is real, for putting 
(^1 = X + y “1)> t i = p4cr^/(— 1) 
where xyy are real scalars and p, a real fictors, we get from 0 = ( ( p — a 1 )i 1 
that 
(<£ - x) P = - y<r, ( 9 - x)(t = y P , 
and now from 
S/o|(<£ — x)cr — S(Tl(<^> — x)p 
we get y — 0, since Sp\p and S<r|o- are both positive [(18) § 2]. From this’. 
