575 
1907-8.] Algebra after Hamilton, or Multenions, §11. 
(1 ) when ft = (p and v — v ; and (2) when ft = — <p , a = 0 , v =( — ) g ~\; and 
where in all other cases the sets a v a 2 , . . . . , /3 V (3 2 , . . . . are independent. 
The most important of these cases, in multenion applications, we will 
partially consider without proving (33). This is when ft — (p and v = — v ; 
the sets mentioned are then independent ; it easily follows from (33) that 
not only do the roots then occur in pairs of equals, but (p satisfies an 
equation of degree only half as great as usual. It will be noticed that 
this necessitates that n the order of the fictorplex must be even. This is 
explained by the fact, which can be proved easily, that it is impossible for 
the conditions v —v~ x — — v to be satisfied when n is odd. Indeed, it was 
the peculiarities of behaviour qKq and gPQg in (12) and (14) respectively 
that led me to (33). 
To prove what has just been mentioned independently of (33), we first 
show that when ft = (p, v = — v there will in general be \n different roots 
occurring in equal pairs. If 
<f>a = aa, <p(3 = b/3 , etc. 
then <pp = «aS/)|a + b/3Sp\/3 + . . . . 
and therefore ftp = adSp\a + bf3Sp\/3 + . . . . 
Since <p = ft, 
<pva = vfta = vftd — Clvd. 
Thus not only does cp — a kill a but it also kills va. Now a cannot be a 
multiple of va, for Sa|a = 1 and Sa|r -1 a = 0 because v is skew. Hence the 
root a occurs twice, and — a kills the fictors of a second order fictorplex 
.(namely a, va). 
When the root a occurs more than twice, take a second i/-self-conjugate 
linity ft such that (p + xft has \n different roots. Thus (p + xft satisfies 
an identity of degree \n, and this holds up to the limit x = 0; the statement 
is therefore true of <p. 
We will now make some statements about multilinities. The following 
is not actually required below, but very directly bears on what is to come. 
The most general form of a multilinity (p is given by 
(n even), <pq =p x qp^ +p 2 qp 2 ' + .... = %pqp ) 
(n odd), = + i 
The fact that when n is odd Hpqp' is not the most general form is 
connected with the other fact that then vs is commutative with q. 
To prove (34), first put [(2) § 2] q^XvSv^q and therefore (pq = '2<puSv~ 1 q . 
Then transform S v _1 q by aid of the following, 
Sr =2-%L + !,)(!+ 1 2 ) . . . . r . 
(35) 
