549 
1907-8.] Algebra after Hamilton, or Multenions, § 8. 
As we may remember by aid of the rigid replacement we have for 
replacements in general: — (1) RS fi ( ) is not equal to S C .R( ); (2) R 2 is not 
equal to unity ; (3) RR' is not equal to R'R. 
When it is proposed to find whether some operator X is a replacement the 
principal questions to answer are : — (1) Is (Xq) 2 a scalar differing from zero ? 
(2) Is XqXq = - XqXq ? (3) Is Xqr equal to XgXr or else to XrXq ? 
When X has been shown to be a replacement R the more important 
questions are : (1) What is the new meaning R q in the old dress, q being 
any multenion ? (2) Are the new multits equal in number to the old, or 
are they only half as numerous ? (3) What is the interpretation of RS C ? 
The unireplacements which merely negative fictits (that is, the uni- 
proplacements) depend on n independent such replacements, viz. on 
I 1? I 2 , . . . ., which negative q only, q only, .... For to negative q, q, q 
(say) we have to make the replacement I 1 I 2 I 3 - Thus corresponding to 
each multit there is one definite such replacement, namely, the one that 
negatives the fictit constituents of the multit, and no others. Thus P 
corresponds to the multit tu, and this is the main reason for calling it P. 
Also there are 2 n — 1 such replacements. These replacements I made some 
use of in the early stages of the work, and their properties are very simple, 
so I have described them. I think they would be occasionally useful, for 
instance, in a calculus of motors, or if attention were directed to a set 
of commutative second order multenions ccqtp 1 + y uq 1 + .... (and their 
products) where x,y ... . are arbitrary scalars. Nevertheless, as they do 
not (except P) treat different fictit sets impartially, they have no very 
extended applications. Moreover, in the same sense that the rigid replace- 
ment may be said to be q( )q~ 1 J the replacement PI X may be said to be 
q( )h 1 - Ii is what we have called a perversion with respect to q. By 
three-dimensional geometry (plane mirror reflections) we see that two 
such perversions with reference to fictits not belonging to one set (successive 
reflections in mirrors not perpendicular to one another) produce a rotation. 
Thus we see the necessity of restricting R wp to one set of fictits if (5) is to 
hold universally. 
We are now about to consider the most general replacement which has 
the property that fictits are replaced by fictors, and we shall call it the 
fictorlinity replacement. The fictors which thus replace the fictits do not 
necessarily belong to the fictorplex qq . ... i n which is replaced. They 
may form any fictorplex of order n whatever. Clearly all the replacements 
hitherto considered except rigid replacements which are incapable of being 
expressed as e w ( )e~ u> (that is, except such rigid replacements as are not 
commutative with S c ) must be included in the fictorlinity replacement. 
