517 
1907-8.] Algebra after Hamilton, or Multenions, § 4. 
Law (3). A x 2 = = 9. 1 \ ( ST X ~ l \ 
A x A 2 . . . . A n = qt Y q~ x .... qi^q~ l = 2 • • • . In< 2 -1 = *p2 • • • • % 
since the last is a scalar. Hence AA 2 • • . . A N is also a scalar. 
Law (4). A 1 A 2 + A 2 A 1 = (/t 1 t 2 ^“ 1 + g't 2 t 1 g'“ 1 = 0. 
Law A, § 3, states that i-f = i 2 — . . . . ±1. Since X 1 2 = < 1 2 etc. we have 
A 1 2 = A 2 2 = .... = ± 1 . 
The second statement [(2)] of the enunciation given in italics obviously 
follows from the above ; first for a multit v since 
1^2 . ... l c is replaced by q( qq .... L c )q~ x , 
when in it we replace q by qi 1 q~ 1 , i 2 by qi 2 q~ l , etc. ; and second for a 
multenion r since 
r~%xv is replaced by q%xv.q~ l 
when in it we replace v by qvq~ x . 
We are now in a position to establish the following converse of the 
enunciation. [This converse is unsatisfactory in that the words “ in 
general ” appear. Their meaning will be obvious from the immediately 
following enunciation, and also in the proof.] 
If Ap A 2 , . . . . A ?l are n non - evanescent multenions of the given 
multiplex q/ 2 . . . . i n such that for each A 
A 1 2 = A 2 2 = . = 
and for any two V (10) 
AjA 2 A 2 Aj = 0, etc. j 
then there is a real multenion q, in general * which satisfies eq. (9) [pro- 
vided we are permitted to change the sign of but one of the A’s, say AJ. 
I propose the following alternative enunciation to explain the meaning 
of the words “ in general,” the proviso in square brackets remaining. 
“ If the multenions \ . . . . \ n satisfy (10) there is always a real finite non- 
evanescent multenion q, such that qp —p'q where p is any multenion and 
p' is what p becomes when in p, q, i 2 ... . i n are replaced by \ v A 2 . . . . \ n ; 
and when q~ x is not infinite there are 2 U independent values of p'.” 
What I have failed to establish is that q~ l is necessarily finite. I would 
ask readers who are interested to think over this problem. I suspect that 
it is finite, or rather that it is so if we impose some further condition such 
as that when q* 2 . . . . i n is not a scalar, neither is W • • • -A B a scalar, or 
such as that n is even. 
I will at first suppose thaf we are permitted to change the sign of any 
A, and will show subsequently that this permission is only required for 
any one A, and even for that only when n is odd. 
* Invariably when n is even , in general when n is odd , by the next footnote below. 
