578 Proceedings of the Royal Society of Edinburgh. [Sess. 
that \ 4c+2 \ ic+1 = z*y 0 e( ) is commutative with all the rest. It is equally easy 
to verify that 
1 = V = - V = A 3 2 = • • • • = - W= - W • • (41) 
In other words A x , X 2 . . . . obey all the laws of fictits and form a set of 
precisely the same type as (37). 
Since the number of As is even (whether n = 2c or 2c + 1), their 
multiplicative combinations are independent by the first theorem of the 
paper in § 2 above, and the number of these combinations is 2 2 -. Thus 
between them they must furnish the most general multilinity of (34). 
We have now established that a multilinity may always be looked upon 
.as a multenion, and that A b X 2 , .... X 2?l are a set of fictits from which it 
may be constructed. 
Conversely, it is obvious that a multenion belonging to an even order 
multiplex may be regarded as a linity, and in particular as a multilinity 
in a multiplex of half its own order. Since a multenion of a (2m — l) th 
order multiplex may always be regarded as one of a 2m th order multiplex, 
any multenion may be regarded as a linity. But the multenions of an 
odd order multiplex do not form a self-contained system of linities ; they 
form half the members of such a self-contained system. 
I have thought (37) the best to start from, but it is desirable temporarily 
to suppose (37) replaced by 
e / 2 = x i} e 2 2 = X 2 , . . . . , e 2c 2 = x 2c , e 2 = X 
where every x is an arbitrary constant scalar differing from zero. Retain 
(38) and (40) to define m 0 , X 1? X 2 , . . . . The modifications of (39) and (41) 
are not wanted here, but are quite easy to write down if required. In this 
general case the conjugates, in a linity sense , of X v \ 2 , ... . would by 
{36) be A-l 1 , Xg 1 , .... where 
X a 1 = X a - 1 = KX a , 
where K has its original defined meaning, in a multenion sense, when 
applied to the multiplex \ ,\ 2 . . . . These two meanings, then, of the 
conjugate of a multenion are identical. Either may be taken as the 
definition, and then the fact that the other applies is a theorem, true, but by 
no means obviously true. 
We will now again suppose (37) and (41) to hold, but we will cease to 
speak of X l5 X 2 . . . . as the fictits of the multiplex \ \ ... . Instead we 
will suppose q, i 2 , .... to be the fictits where 
*1 " 2 /lAl> L 2 = y^2i • . • l 2 m~y'2m^2m • • • (42) 
