506 Proceedings of the Royal Society of Edinburgh. [Sess. 
If a v a 2 , .... a n are n given independent fictors, all multenions of the 
form 
5( X Q + aqcq -f x 2 a 2 + . . . .)(&' 0 + x\ a 1 + X 2 a 2 + . . 
are said to form a multiplex of order n, and it is called the multiplex 
cq a 2 ... . a n . The multiplex q * 2 . . . . * N of all the fictits is called the 
continent multiplex. 
[First N.B. — A fictorplex is a complex of given fictors ; but a multiplex 
is not defined as a complex of given multenions, as the last phrase would 
naturally mean all multenions of the form x 1 q 2 -\- .... where 
q v q 2 . . . . are the given independent multenions. Of course a multiplex 
is a complex of multenions, but it is not any such complex. Thus the 
continent multiplex i ± i 2 . . . . i N is a complex of all the 2 N_1 independent 
multits ; and the complex 1, q, i 2 is not a multiplex, whereas the complex 
1, i v i 2 , q i 2 is a multiplex. 
Second N.B. — The importance of distinguishing between the continent 
fictorplex (or multiplex) and fictorplexes (or multiplexes) of lower orders 
arises from the statement in law (3) that q z 2 . . . . % is a scalar, and the 
consequential statement in law (1) that N is odd. There are but 2 N_1 
independent multits in the continent multiplex, whereas there are 2 n 
independent multits in any other multiplex of order ni\ 
A product t i l" .... =v of fictits is called a multit, and unless the 
context implies the contrary it is to be understood that all the fictits 
i, l, ... . are different. [All like pairs can be got rid of by transpositions 
and use of the relations i 2 = scalar, t 2 = scalar, . . . .] 
The order of a multit is the number of constituent fictits in it (when 
they are all different as just prescribed). The part S a q of a multenion q 
depending on the a th order multits, is called the a th order part, or a th 
part of q. The zero th part S 0 q, being a scalar, will be called the scalar 
part, and will be denoted by S q. 
The following important theorem is intimately connected with the 
fundamental laws. If X p X 2 . . . . \ n be n multenions such that 
X 1 2 , X 2 2 . . . . \ 2 are all scalars differing from zero , and each pair is 
anti-commutative {that is, X 1 X 2 =— X 2 X 15 etc.)', then if n is even the 
2 n multiplicative combinations are independent ; and if n is odd they 
are independent unless the product W ... . X n is a scalar , and if it is, 
2 n_1 of the combinations {which may be taken either as the odd formed 
combinations or as the even formed) are independent. [An odd formed 
multiplicative combination means a product of an odd number of different Xs.j 
If 'Zxv= 0 where v is a multiplicative combination, put the relation 
2xv = 0 in the form q=r; where every term of q is commutative and 
