1907-8.] Algebra after Hamilton, or Mnltenions, §§ 3, 4. 513 
reversate respectively. [I use the mental phrase “self R-placed” for a 
multenion q such that ~Rq = q where R is any replacement.] 
4. The rigid replacement q( )q~ 1 . Generalisation of fictit. — We 
must here make a brief digression on the meaning of q~ Y , the reciprocal of 
a given multenion q. We will suppose that q is not a scalar. In a given 
multiplex of order n there are 2 n independent multits only. Hence the 
2 W +1 multenions 
1, i b 4*- • • • <Z\ 
where k — 2 n cannot be independent. Let, then, q m be the one with lowest 
index which is not independent of those with still lower indices. Thus 
q m + Wq™* 1 + . . . . + h {m ~ l) q + h {m) = 0 . . . (1) 
is an identity satisfied by q, and there is by hypothesis no identity of lower 
degree so satisfied. When h {m) is not zero we have 
© 
pq = qp = 1 
where 
p = - + h'qWm* + .... + h (m ~ l) ). 
Now when pq — qp = l for any multenion p, there is no second multenion 
p' such that eitfier p'q = 1 or qp' = 1 ; e.g. if p'q — 1 we have (p —p')q = 0, 
and therefore (since qp = 1), 0 = (p —p')qp =p —p', that is, p' =p or p' is not 
different from p. When, therefore, h {m) is not zero there is one and only 
one multenion q~ x such that qq~ 1 = l, and for this multenion we also have 
q~ x q = 1. Thus in this case there can be no doubt as to what the reciprocal 
q~ x of q must mean. 
When h (m) is zero let (1), the minimum degree identity, be put in the 
form 
/fa) •2“ = 0 (2) 
where /( 0) is not zero and a is a positive integer not zero. Since (2) is the 
minimum degree identity f(q) . q a ~ l is not zero. Thus putting 
P = [/fe) • ( “ g) a-1 ]//(0) = Lt. ^- w (l + xq)- 1 . . • (3) 
£=co 
we have 
pq=qp = o (i) 
and on account of the limit in the third member of (3) we see that p is thus 
made to depend in a definite way on q. 
We see from (3) that p satisfies the quadratic identity 
jt> 2 = p or p 2 = 0 
( 5 ) 
according as a is equal to or greater than 1. 
VOL. XXVIII. 
33 
