505 
1907-8.] Algebra after Hamilton, or Multenions, § 2. 
obtained by putting n = 2. We thus suppose quaternions based on two 
fictits, i, j ; the third (vector but not fictor), k, being defined as the fictor 
product, ij .] [April 1908. — For final views regarding (1) and (2) see end 
of Supplement.] 
Fundamental Laws of Multenions. 
(1) There are given an odd number (N) of primitive units (here called 
fictits), i p i 2 . . . . , besides the unit scalar 1. [Sometimes, below, the 
scalar 1 is classed as a multit (that is a fictit product) and sometimes as a 
unit apart ; I think the context will always serve to prevent inconvenience 
which might result from the ambiguity.] These are such that all the laws 
of ordinary algebra except the commutative law for multiplication apply 
to the expression (here called a multenion) 
^(x 0 + x 1 l 1 + x 2 l 2 + . . . . )(x 0 + x\l 1 + x 2 i 2 + . 
where E has its usual significance, and every x is a scalar. 
(2) Scalars are commutative not only with each other but with fictits. 
[It at once follows, and will always be assumed without proof or reference, 
that scalars are commutative with multenions.] 
(3) if, if, . , and also the product qt 2 .... of all the fictits, are all 
scalars differing from zero. 
(4) In a product i ii" .... (here called a multit) of fictits, if two adjacent 
different fictits be interchanged the product changes in sign, but otherwise 
remains unchanged. [It is by reason of the word “ different ” that a multit 
is not a combinatorial product of its constituent fictits.] 
(1), (2), (3), (4), and not what follows, are regarded as our fundamental 
laws. 
The multenions q v q 2 ... . are said to be (1) independent or (2) not 
independent according as (1) Exq is only zero (x v x 2 ... . being scalars) 
when every x is zero, or (2) Exq = 0 for some values of x v x 2 . . . . not 
all zero. 
An expression of the form Exi is called a fictor. If cq, a 2 , .... a n are 
n given independent fictors, all fictors of the form 'Ey a are said to form a 
fictorplex of order n, and it is called the fictorplex cq a 2 . . . . a n . [As this 
is liable to be confused with the fictor-product cq a 2 ... . a n , which 
is a multenion, it is safest always to use the word “ fictorplex ” when 
cq a 2 . . . . a n is used in the latter sense.] The fictorplex q i 2 . . . . i of all 
the fictits is called the continent fictorplex ; and it is best to understand 
by “ fictorplex,” unless the context obviously implies the contrary, always 
a fictorplex of lower order than N. 
