54 7 
1907-8.] Algebra after Hamilton, or Muitenions, § 8. 
It appears as if no misleading would occur by saying that R = q( )q~\ But 
consider the question, are the replacements R and K commutative ? 
KR(t 1 t 2 | = K(A. 1 X 2 ) = (X 1 X 2 ) _1 , 
RK(qt 2 ) = R.(t 1 t 2 ) _1 •= (X 1 A 2 ) _1 , 
,so they apparently are. But putting R = q( )q~ 1 , 
q(Ku 1 L 2 )q~ 1: =q.(i 1 i 2 )-\q-\ 
[In Quaternions these two right-hand members would be equal.] Now 
these two are not equal in general, as can be shown by particular cases. 
The apparent inconsistency arises from K being able to affect q. We lay 
down the following, then, as a rule governing our meanings. 
A replacement symbol R must be supposed not explicitly to involve 
muitenions of any kind without justification by examination. We must 
suppose R to denote an actual replacing of certain symbols by others, the 
new ones taking the place of the old for all purposes whatsoever. We 
must examine the new meaning of every single symbol, such as S c . At the 
same time the new meanings of formulae are capable of representation in 
the old dress ; and indeed this is the main object of the process of replace- 
ment. Thus in the fictor-to-fictor rigid replacement, the mainly useful 
feature is that certain fictors not originally called fictits were found to 
obey among themselves and scalars all the laws that the original fictits 
obeyed among themselves and scalars. 
With this warning I shall continue to call the rigid replacement q( )q ~ l , 
but shall refrain (when it is to be regarded as a replacement, and not as a 
multilinity) from using q( )q~ x in the equations. 
A replacement is a process of (1) replacing n specified original fictits by 
other multenion symbols, and (2) in the case of retroplacements writing 
every product of the new fictits in the reversed sequence of the original 
fictits. Laws (1) to (4), § 2 (in so far as they apply to the n fictits), are 
invariably to hold with the new meanings ; and law A, § 3, will only 
occasionally be violated. We may further impose that 
R(^i + x 2 q 2 +....) = aqR^ + a* 2 R q 2 + . . ( 1 ) 
I consider the following [(2) (3)] to be really involved in the above 
prescriptions, but place them as definitions in order to avoid a doubtful 
.argument, and also to render matters clear. 
R . p qr — R^.R^r, R r qr = R r ?\ R r q 
R(%) = (RS c )(Rg) . 
( 2 ) 
(3) 
