1907-8.] Algebra after Hamilton, or Multenions, § 2. 509 
Similarly, 
a = ...... (3) 
Sqq = 'Zxx'v 2 when q = hxv, q = ^x'v .... (4) 
S qq = S qq always ....... (5) 
S qq' . . . .... q (n) q always . . . . (6) 
Saa/ = when a = %xl, a = %Xi . . . (7) 
Saa' = Sa'a always ....... (8) 
(8) is, of course, a particular case of (5), and (5) a particular case of (6). 
If r and p are an arbitrary multenion and fictor respectively ; then, if 
p, q are multenions and a, /3 fictors, 
p = q when Spr = S#r ..... (9) 
a = /3 when Sap = S/?p ..... (10) 
It is convenient to define here the conjugate K q of q, the reversate 
Q q of q , and an associate P q of q (for which I have not thought it 
necessary to provide a name). If v is a multit of order a, Qv is defined as 
a multit with the same fictits written in the reverse sequence; Ku is 
defined as 
v 1 ; and P^ is defined as (-)V Qq, 
K q, and P q are then 
defined by 
Qq = IxQv , = IxKv, P q = 'ZxPv when 
j—H 
P 
X 
7* 
II 
Cjh 
The following may now be proved : — 
Q 2 = 1 , K 2 = 1, P 2 = 1 
. . . . (12) 
QK = KQ, QP = PQ, KP = PK . 
. • . . ( 13 ) 
P( 2 r) = PgPr 
. . . . ( 14 ) 
Q(qr) = QrQq, K (qr) = K rKq . 
. (15) 
K and Q are called retroplacements on account of the sequence change of 
(15), P is called a proplacement, and all three are called replacements. 
They are further called uni-replacements on account of (12). It will be 
seen that all these three symbols are commutative [(13)]. Replacements, 
in general, will be considered later. These three are by far the most 
important. QP and KP are easily shown to be uni-retroplacements and 
QK (like P 2 = Q 2 = K 2 = 1) a uni-proplacement; that is, a product of uni- 
replacements is itself a uni-proplacement or uni-retroplacement respectively 
according as the number of retroplacements in the product is even or odd. 
The following equations are true (and the similar equations also where 
P, Q, K are replaced by any product of powers thereof)* — - 
S.^Pr = S.rP^, S.^Kr = S.rKg, S.qQr = S.rQg . . (16) 
Also 
S.^Kg' = ^xx=S.qKq when q = %xv, q . . (17) 
