5 08 Proceedings of the Royal Society of Edinburgh. [Sess. 
which may be taken as (a) the odd formed multits, or (b) the even formed 
multits, or (c) the mnltits not containing some one assigned fictit, such as 
h or i N . 
(3) We may take as a definition that B a q = S N _ a g. 
(4) The J(N + 1) odd parts of q are independent, and the J(N + 1) even 
parts are independent. 
For the multiplex q i 2 ... . i n where ^<N we have : — 
(1) If vs = q i 2 . . . . i n , when n is odd vs like a scalar is commutative 
with every multenion of the multiplex ; when n is even vs behaves like a 
fictor in commutation with multenions, and in that vs 2 is a scalar differing 
from zero. [When n is odd, vs, unlike a scalar, is independent of 1, and 
whereas x is commutative with S OJ vs is not ; when n is even, multiplica- 
tion by vs changes the orders of the parts of q in a quite different way 
from multiplication by a fictor. Thus vsS a q is of the (n — <x) th order, 
whereas aS a g (a a fictor) contains parts of orders a ± 1 only.] 
(2) The multiplex is a complex of 2 n independent multenions, which 
may be taken as the 2 n multits. 
(3) The n -\- 1 parts of q are independent. 
Whether n is equal to or less than N, there are n C a multits of order a. 
Returning for a moment to the general theorem above, note that 
X v X 2 . . . . A n were not confined to a given multiplex such as that of 
q i 2 . . . . i n . In considering the rigid replacement below they will be so 
confined. Does this confinement produce any noteworthy effects ? None 
that I have been able to find. For instance, if n is odd it may still be true 
that \ \ . X n is a scalar even when q i 2 . . . . i n is not ( i.e . n =# N), as we 
see by putting 
A-i q, A . 2 ... . A n _! = L n _ u AJ qq • • • • L n—i 
(or, more generally, if we put \ n = \ 1 \ 2 . . . . X n _ x ). 
The reader will perceive the bearing of this negative result when he 
studies the unfinished investigation of the rigid replacement below. 
The proofs of the following important statements are simple and are 
left to the reader. 
v 2 = scalar . . . . . ( 1 ) 
uv = scalar if, and only if, v' — zhv. In other words, Sci/ is or is not zero 
according as v and v are not or are indentical ; and when they are 
identical, Svv =vv'. 
Putting, then, q = 'Exv we get by operating by Sa -1 ( ), 
x = Sr -1 ^, or 
q = 
( 2 ) 
