552 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The middle of these three transformations is merely a particular case of 
the general statement in multilinities 
Sg|xr — SHx'#. 
{<p} furnishes us with yet another standard form of h {c) , the general 
coefficient in the <p ?i-tic, namely 
■ * • ( 23 ) 
for by (16) 
■ 2S(^,a)Pi| ) [(26) §6] 
and this last is the form of h (c) given two lines below (23) § 7. 
(17) shows at once that {<p} may always be regarded as a replace- 
ment, namely one which replaces the definite set of fictits which are not 
killed by <p'<p (or (p [§ 7]) and are undeviated by cp'cp. For let these fictits 
be i v i 2 . . . . L n and let cp = e (a J(<p'(p){ ).e~ w . Then since {$} i x = <pi v etc., the 
conditions are satisfied: (1) that ({</>bi) 2 , etc. are all scalars differing from 
zero; (2) that ({<p} i v {<p} i 2 = — W^W l v etc. ; is not in general 
equal to {<p}qr, as we see by putting cp = x. In this respect, then, the 
multilinity {cp} is not strictly a replacement. 
Moreover, from (1) we see that the most general replacement which 
replaces fictits by fictors is that which replaces any fictor by the most 
general linear fictor function of itself. Hence {<£} is the most general 
proplacement of the kind; and {^} and Q{<^>} together form the most 
general replacement of the kind. 
The replacement here considered consists of first replacing the fictits of 
a given set by (non-zero) scalar multiples thereof ; and then superposing an 
arbitrary fictor-to-fictor rigid replacement e w ( The second operation 
in general changes the multiplex operated on to a second multiplex of the 
same order ; but this is not contrary to the conditions we have laid down 
for the constitution of a replacement. 
Law A will only be retained if the multiplication of each fictit is by 
dt 1 ; hence the most general type of unireplacement that converts fictors 
into fictors is that defined above. 
It may be shown that the replacement may always be effected by first 
making an arbitrary rigid replacement. This is done by operating on the 
undeviated fictits of <p<p'. 
I may put here what perhaps should have been placed in the previous 
section. <p'cp and <p(p' have the same n-iio,. Putting (p = e Ui \p-( )-e~ U} = x x ! r 
where ^ is a colinity, x — X _1 an( ^ 
<£'</> = <£</>' = x-V^-x -1 = x-^-x -1 - 
