934 Proceedings of Royal Society of Edinburgh. [sess. 
‘ determinant ’ as connected with a single function, — that is to say, 
Gauss’ original use of the term, — and the substitution of the term 
* discriminant ’ in its place. The introduction of a new word, it is 
explained, is for the purpose of avoiding the obscurity and con- 
fusion which arises from employing the same word in two different 
senses, and “ ‘ discriminant ’ because it affords the discrimen or 
test for ascertaining whether or not equal factors enter into a 
function of two variables, or more generally of the existence or 
otherwise of multiple points in the locus represented or 
characterised by any algebraical function.” * 
Cayley, A. (1851, end). 
[On the theory of permutants. Cambridge and Dubl. Math. 
Journ., vii. pp. 40-51 : Collected Math. Papers , ii. pp. 
16-26.] 
The second part of his paper of 1843, as we have seen, Cayley 
devoted to the consideration of a class of functions obtainable 
from the use of m sets of n indices in the way in which a deter- 
minant is obtainable from only two sets. The general symbol 
used for such a function was 
fV 
°1 
T 1 ' 
P2 
°"2 
r 2 • 
< Pn 
T n • 
• • 
this standing for the sum of all the different terms of the form 
-- V — 5 
A x • • • • x A 
Pn o- 8l r h • • • Pr n <?s n r 
* Apropos of this happy coinage, Sylvester adds in a footnote the general 
remark : — “ Progress in these researches is impossible without the aid of clear 
expression ; and the first condition of a good nomenclature is that different 
things shall be called by different names. The innovations in mathematical 
language here and elsewhere (not without high sanction) introduced by the 
author, have been never adopted except under actual experience of the em- 
barrassment arising from the want of them, and will require no vindication to 
those who have reached that point where the necessity of some such additions 
becomes felt.” The truth of the remark is not appreciably diminished by the 
occurrence of the word ‘ meso-catalecticism ’ in another footnote two pages 
further on. The year of the paper (1851) was for Cayley and Sylvester a year 
teeming with fresh ideas as well as with fresh words. 
