557 
1907-8.] Algebra after Hamilton, or Multenions, § 9. 
n is odd the multiplex may when t^ 2 = + 1 be identified with the continent 
multiplex, but not when vs 2 — — 1. 
We now have what seems to me a very remarkable result. Except in 
the anomalous case, the effect of making the replacement k times is to 
change q to B k q. In the anomalous case the result of making the replace- 
ment once, twice, etc. is to change q to 
Eg, B q, Vq, . . . . 
whereas the effect of operating by B, B 2 , B 3 . . . . is to change q to 
B q, IvB$, B q, KBg, .... 
Addition to § 9, April 1908. — Since in the last case B q consists of even 
order terms, we have 
KBg = QB^ = PQB^. 
The anomalous case can be avoided by use of the imaginary, and I am 
inclined to regard the following as the standard complementary replace- 
ment. Let 
\n even] Ctj = (t 2 ) 4(n f 2) tD't 1 \ 
[n odd] Cij = Ttj [ (32) 
where £7 = ^ i 2 . ... i n J 
These give Cq = Bq, except in the anomalous case, when they make a 
pure imaginary. The effect of making the replacement C successively k 
times is the same as what is denoted by C* where C is treated as a 
multilinity. Also without exception 
[ n even] C 2 = P, C 4 = 1 
[w odd] C 2 = C 
(33) 
The minimum degree identity satisfied by C is that given in (33), 
except for n — 0 and n — 2, when we have 
C - 1 = 0, (C- 1)(C 2 + 1) = 0 
respectively for this identity. 
(11) becomes 
(n odd) Cq = (q 0 + q 2 + g 4 + ) ) ( 
+ J(G‘ 1 )-'R{qi + qz + q b + ....)) 
and (12) remains unaltered, with C in place of B and (* 2 ) i(n+2) in place of x. 
At the end of the supplement it is recommended that the continent 
multiplex be ignored, and that instead it be replaced by treating of the 
even order multenions of an odd order multiplex. 
(7) above furnishes us with the fundamental expansion of q in such a 
treatment, and shows us how to treat such a multenion as generated from 
the n independent (n— l) th order units Wi v tsi 2 , .... 
