533 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
p being any fictor of a given fictorplex of order n, (pp, a fictorlinity is 
defined as a function of p such that 
0(/3 + y) = </>/? + 4>y always (1) 
and that <pp is a fictor of the given fictorplex. It follows that 
xcp/3 = (p(x/3) always (2) 
The general form of cp may be made to depend on n given fictors /3 V ... . 
according to the first of the following, from which the second is deduced, 
<pii = fii, <pi 2 = (S 2 , ) 
<pp = /3 1 Sl j ~ 1 p + /3 2 Sl 2 ~ 1 p + . . . . J 
The following is a fictorlinity in all cases, and if the number of pairs of 
fictors (/3 lf cq), (f3 2 , a 2 ) .... is ^ or more it is a general form 
pp = ft 1 Sa 1 \p + P 2 Sa 2 \p+ . . . . = 2/3Sa|p . . (4) 
We may apply eq. (22), § 6, to either p or <pp thus 
<pp=<Pt$t\p=m<t>p ( 5 ) 
The equation 
Sp|0<x — Scr\<p'p ..... (6) 
if true for all fictor values of p and cr, defines <p ' ; and <p' so defined proves 
to be a fictorlinity ; it is called the conjugate of cp. 
[If Law A is assumed, (6) may be replaced by Spcpo- = Scr(p / p, but the 
two equations do not mean the same when Law A is not assumed. In the 
latter case eq. (6) is better than the alternative, since if <p is given by 
<£q = aq'q + x 2 i 2 + x 3 l 3 + . . . . 
du 2 = ajf'q + . . . . 
<pi 3 ~ Xj -f- . . , . 
then, according to (6), to get <p' from 0 we have merely among the scalars 
xfi x 2 , .... to interchange rows and columns. The alternative S p<pa — 
Scrtp' p produces no such simple statement among n 2 given scalars.] 
From (6) and (5) 
Soj <p'p = Soj£S£j0 p = So-|(£Sp|0£). 
Since or is arbitrary we have [(10) § 2] 
= or 0' = 2S( )|0{ .... (7) 
which gives cp' explicitly in terms of cp. Conversely, (6) follows when cp' 
is defined by (7). 
From either (6) or (7) 
</>'p = 2aSp]/3 when 0p = ]£/3Sp|a . 
(8) 
