576 Proceedings of the Royal Society of Edinburgh. [Sess. 
That this is true for every multenion r appears when it is noted that 
|(1 + Ii) kills every multit containing q and leaves every other multit 
unchanged. Hence 2~ r '(l + RX1 + I 2 ) • • • • kills every part of r except the 
scalar, which it leaves unaltered. In (35) put 
ii=P. tl( ) h -^ h p( ).cr\ 
Thus Sr is expressed as Hum' + HvTrv' where u, v, u', v' are multenions. 
Applying this to S v~ l q we get the second of (34) as the general form of (pq 
whatever he n. When n is even we may put P = tss{ )vs~ l and get the first 
of (34). 
The following is required below. The root sum of the multilinity 
v( )v is zero except when each of the two multits is unity , and if n is odd 
when each is m ; in these two cases the root sum is 2 n and 2 n m 2 respectively ; 
the root sum of the multilinity uP( ),v is in every case zero. This may 
be proved by putting 
< f>v j = av ^ + bv 2 + . . . . 
<pv 2 = a'v 1 + b’v 2 + . . . . 
and examining the scalars a,b' ... . in the principal diagonal when 
cp = v( )v and when <p = uP( ) v ; each such scalar is zero or ±1 ; the root 
sum is a-\-b'-\- . . . . 
Law A will be violated in certain cases below. In all cases, however,, 
the definition of cf as follows, 
SrKcf>q = SqKcp'r, 
leads to the fundamental property that 
-a, b , . . 
-a, a\ . . 
when 
cf> = 
a, b', . . 
then <// = 
b, b\ . . 
and also gives 
<f>'q = ^(Kp.q.Kp) + 2P(Kp 0 .g.Kp 0 ' ) 
when cf>q = %pqp + ^p^qp^ f 
or more simply: — the conjugate of p( )p' is K p.( ).K p' and p>q = ¥q is 
self -conjugate. [The last is true of any unireplacement when regarded as 
a multilinity. ] 
We will now prove that a multilinity belonging to a given multiplex 
of order n may always be itself regarded as a multenion belonging to a 
multiplex of order 2 n, and we will at the same time find a set of fictits 
belonging to this higher order multiplex. Conversely, it will appear that 
a multenion may always be regarded as a Unity. 
