750 Proceedings of Eoyal Society of Edinburgh. [sess. 
Of the three memoirs, the first and third, like Jacobi’s third and 
second, do not at present require attention. A slight reference to 
the first — on alternating functions — is, however, necessary, because 
Cauchy, unlike Jacobi, makes determinants a special class of alter- 
nating functions, and it is therefore of importance to see the exact 
position he assigns to them. It will he remembered that in 1812 
he partitioned symmetric functions into permanent and alternating, 
and made determinants a class of the latter; that is to say, his 
scheme of logical relationship was 
( (a) Determinants. 
( (a) Alternating < 
f (A) Symmetric < ( 
Functions < ( (&) Permanent 
((B) 
The memoirs we have now come to indicate a departure from this, 
both verbal and substantial. The change is made too without any 
reason being assigned ; indeed, there is not even a word to imply 
that any change had taken place. Alternating functions are, as in his 
Cours d’ analyse, put on the same level as symmetric functions; 
the term permanent is dispensed with; a new entity, alternating 
aggregates , is introduced; what were formerly called determinants 
are made a class of these alternating aggregates ; and for the name 
determinant resultant is substituted. The scheme of relationship is 
thus transformed into 
' (A) Alternating 
Functions - 
(B) Symmetric 
(a) Alternating Aggregates 
w 
JM 
l 08) 
Resultants. 
1(C) 
Neither scheme, we must at the same time remember, is really as 
simple as here indicated, being complicated by the fact that a 
function may be alternating in more than one way. This is brought 
out much more explicitly and clearly in the present memoirs than 
in that of 1812, as the following quotations will show. We have 
first of all (p. 151), an alternating function of several variables. 
“ Une fonction alternee de plusieurs variables x, y, z, ... , 
est celle qui change de signe, en conservant, au signe pres, la 
meme valeur, lorsqu’on echange deux de ces variables entre elles.” 
