546 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Thus qz 3 , i 2 i 3 , and thereafter qq, * 2 q, etc., may be got rid of from 
'Exu, etc.] 
We have seen that, as is well known, every fictorlinity can be expressed 
uniquely as the sum of a colinity and a skewlinity. Similarly, every 
fictorlinity <fi can be expressed as the product and also as the product 
xV' where \Js is a colinity and x is a rotational linity. This is not unique, 
as \fr is any value of in the first case, and is any value of *J. </>'</> 
in the second (except possibly as to the sign of one root). It is unique if 
we impose that J-(p'<p or shall be the positive value when possible, 
and shall be the negative value when, by reason of perversion, this is 
necessary. This can be proved from the facts (1), (2), (3) above stated for 
<p'(j) and <pcj). 
A similar analysis of a real <p into a real product or xV'* where \fr is 
still a colinity but now x is a skew linity cannot be made in general ; that 
is, <f) must for this satisfy special conditions. For if p is any one of the 
undeviated fictors of (and there are always n and sometimes an infinite 
number), S p(pp would have to be zero. Hence for some complete set of 
fictits we should require S npL to be zero. Thus if 0 = 1, although there are 
imaginary fictors for which p 2 = 0, there are no real non-evanescent ones. 
8. Replacements. — The fictorlinity replacement. — Replacements 
are subdivided into proplacements and retroplacements, that is, every 
replacement is either a proplacement (Rgr = RgRr) or a retroplacement 
(Rqr = RrRg). 
A very important class of replacements is that of unireplacements 
(R 2 q = q). Some of these are proplacements and others retroplace- 
ments. 
We may use the following notation. Any replacement may be denoted 
by R ; proplacement by R^ ; retroplacement by R r ; uni-replacement by 
R m ; uni -proplacement by R Mp ; uni-retroplacement by R wr . These symbols 
will be understood and not explained on each occasion of using them. 
Other replacement symbols will be explained as required. K, P, Q will be 
exclusively used as explained in § 3. 
Before proceeding to details, I will make an important but decidedly 
fine distinction, partly to render my meanings clearer, partly to warn the 
reader that it is necessary to walk warily. 
Let R stand for the rigid replacement. That is, among other things, if 
Ai = qi Y q~ l , X 2 ~ we shall say that q is replaced by \ and i 2 by X 2 , and 
we shall denote this by 
A.-^ — A*2 = 
