521 
1907 - 8 .] Algebra after Hamilton, or Mnltenions, §§ 4 , 5 . 
Pi = Vi -1 + 1 = (A -1 + q)q -1 
and q _1 is a fictor, viz. 1 2 .q. Let p{ifpf ~ 1 = l Ui 2 so ^2 i s a fictor. Thus 
pp 2 = p 2 p v and therefore 
p 2 = (<£ 2+1 )p 1 = ^tg- 1 (Xgftg- 1 + l)i?! = (x 2 + = fictor product. 
Let P2I3P2P 1 = /U 3 so that /x 3 is a fictor. Thus #> 2 q = p z p 2 , and therefore 
p 3 = (<£3 + l)p 2 = X 3 ^ 2 t 3 - 1 +p 2 = + l)p 2 = (X 3 + ^)pf x p 2 = fictor product. 
This reasoning is evidently general, so that p n is an even fictor product. 
[The “ circular variation ” from (q, i 2 , i 3 , ... . i n ) to 
(q cos # + t 2 sin 6, - q sin 0 + i 2 cos 6, i 3 , .... if) 
is effected by the operator )q~ 1 , where g = (qCOS Jd-hqsin Jd)q _1 . This 
fact may be used to establish the properties of the rigid replacement 
when the fictit-replacements are fictors.] 
We will henceforth understand the meaning of fictit to be any fictor X 
such that X 2 = i 2 , and the meaning of set of Jictits to be any set of fictors 
X-p X 2 , . . . . which satisfy ( 20 ), i.e. 
\ = PhP~\ X 2 =P l 2P~\ ( 2 1 ) 
where p is any fictor product whatever. 
[We could, of course, mean by “ set of fictits ” the multenions X v X 2 . . . . 
of ( 9 ), but since the replacement of S ft would thereby be not S a itself, I 
think this undesirable.] 
We may now assume from chapter iv. of Octonions that any given 
fictorplex cq a 2 . . . . a n may be expressed as a complex of fictits i t i 2 ... . i n , 
and therefore that any given multiplex oq a 2 . . . . a may be expressed as 
a multiplex based on the fictits q i 2 . . . . i n . [I am sorry to observe that 
the attempted proof of the first italicised statement of § 30 of Octonions is 
quite unsound. The theorem is true, and can indeed be proved by ordinary 
algebra. Also the proof is sound for the only case we require in the 
present paper.] 
It may be remarked that if Law A, § 3 , is not assumed (or rather if it is 
not assumed that the signs of if, if, . ... if are alike), a -1 where a is a 
fictor may be infinite. Thus if if = — if, 
(q + iff = 0, (q + q) -1 = co . 
5 . Fictor Products, Complements (including Grassmann’s progressive 
and regressive multiplication). 
Let q a stand for a multenion of order a belonging to a given multiplex 
