520 
Proceedings of the Royal Society of Edinburgh. [Sess. 
(n-l)(w-2) 
[at present this requires X n 2 = i 2 = ( — 1) 2 ], and that when it is satisfied 
there are but 2 n_1 independent multiplicative combinations of \ v X 2 , . . . . X n . 
Now if g _1 were finite there would necessarily be 2 n such, corresponding to 
the 2 n independent multits. But if we impose that X x X 2 .... X n is not a 
scalar when n is odd, or if we impose that n is even, I cannot say whether 
q~ l is necessarily finite.* 
Note added after conclusion of the paper ( Anno 1906). — I find that 
q — \{sr[( 15) and (17)] does not render q so arbitrary as I had supposed. 
If q, q' are two values of q , then q = xq', where when n is even x is a scalar, 
and when n is odd x is of the form a -{-bus where a and b are scalars. 
Thus q{ )q ~ 1 involves 2 n —l or 2 n — 2 scalars according as n is even or odd. 
I merely indicate the proof. Unity, and when n is odd tjs also, is com- 
mutative with every multit. This statement is not true of any other 
multit, but instead we have the rather unexpected simple statement v any 
such other multit is commutative with half or 2 n ~ 1 of the multits (includ- 
ing v and unity), and anti-commutative with the other half. Hence 
2~ n 1vpv~ 1 = Sp or Sjp + S n p .... (19) 
according as n is even or odd. The statement that q = xq' now follows 
from (17). 
The particular case of the rigid replacement when the replacements 
X p X 2 , . . . . (of the fictits q, , ... .) are fictors is specially simple and 
possesses none of the unintelligibilities of the general case ; though when n 
is odd we may still have to “ pervert ” the system (that is, change the sign 
at will either of a single selected X or of every X). 
(10) becomes 
A 1 2 = X 2 2 = .... =t 2 , SX 1 X 2 = 0 = SA l A 3 , etc. . . , (20) 
We will show that if we put r = l in (15) and (17) q becomes an even fictor 
product. We have already seen that if p is a non-evanescent fictor product, 
whether even or odd, two great simplicities result : (1) p~ x is never infinite ; 
(2) p.S a r.p~ 1 = S a (prp~ 1 ), so that in particular pap~ x is a fictor when a is. 
We proceed by showing in succession that 
(<£i + l)l=Pi, (<£ 2 + lK=l>2, . • . (</>„ + 1 K-i=A = 2^ 
are even fictor products. 
* I am now (1908) able to show that when n is even q~ l is finite. If g _1 = oo let 
qqo= : 0=q Q q where q 0 is not zero. Since qp—p'q we have that qpq 0 =0 for all multenion 
values of p. In the Supplement below, between equations (44) and (45), it is proved that 
when n is even this is never true when both q and q 0 differ from zero. Hence q~ l is finite. 
The case of n odd still remains incomplete. qpq Q may be zero for all multenion values 
of p when n is odd, as we see by putting q— 14- w\/(® 2 ), q 0 = 1 - ts s](t3 2 ). 
