507 
1907-8.] Algebra after Hamilton, or Multenions, § 2. 
every term of r is anti-commutative with \ ; that is, where in q, X 1 occurs 
in every one of the odd formed combinations, and in r, A x occurs in every 
one of the even formed. Thus 
Hence 
Multiplying by 
A x g = 2'A 1 = A x r = r\ v A x r = — rA r 
\ x q = A = 0. 
q = r = 0. 
Hence in the supposed relation A 1 occurs in each of the even formed or 
else in each of the odd formed combinations ; and similarly for X 2 . . . . X n . 
Let \ v X 2 . . . . occur in the even formed, and X' 2 . . . . in the odd 
formed combinations. The supposed relation now becomes 
/ 0 (Ai, A 2 , . . . .) =/i(X i, A 2 , . . . 
Since \ 2 , ... . each occur in every even formed combination, 
fo = x A^ Ao . . . . A 2ft , 
and similarly 
* • ’ * ^ 26 + 1 , 
where x and y are scalars (that is, f 0 and f x each contain but one term). In 
the relation f 0 —f v multiplying by \ X 2 . . . . X 2 « we get 
scalar = product of odd number of X p X 2 . . . . 
The scalar is not zero because the square of the product is not zero. All 
the X’s must be present in the product, because if X c were not present it 
would be anti-commutative with the product, that is, with a non-evanescent 
scalar. Hence, when n is even no such relation can hold. 
If n is even, then, the 2 n multiplicative combinations are independent. 
If n is odd, similar reasoning shows that the 2 n_1 multiplicative 
combinations of \ v X 2 , . . . . X n _ x are independent ; and since in this case, 
when X x X 2 . . . . \ n — zs is a scalar, every odd formed combination becomes 
even formed by multiplication by w, the odd formed combinations are 
independent, and again the even are independent. 
[In the above X p X 2 , . . . . are not hctits in general ; they are not even 
fictors, but multenions. Thus v is not a multit in general. We shall now 
again confine v to meaning a multit.] 
If, as a particular case, we put 
Ai = q, A 2 =t 2 , .... 
we have for the continent multiplex q i 2 .... i N : — 
(1) N must be odd, since t 2 ....% = scalar. 
(2) The continent multiplex is a complex of 2 N_1 independent multenions, 
