1887.] Dr T. Muir on the Theory of Determinants. 
499 
general come to be classified according to the number of values they 
are able to assume in certain circumstances. 
At the outset of it the writings of Lagrange, Vandermonde, 
and Kuffini are referred to ; tbe fact is recalled that the maxi- 
mum number of values which a function can acquire by inter- 
changes among its n variables is 1.2.3 .... n\ also that when 
the maximum is not obtained, the actual number must be a factor 
of the maximum; and then proof is given of the very notable 
theorem that the number of values cannot be less than the greatest 
prime contained in n without being equal to 2. It is pointed out 
likewise that functions capable of having only two values are 
known from Vandermonde to be constructive for any number of 
variables. For example, the number of variables being three, 
a v a. 2 , a B , all that is needed is to form their difference-product 
(<x 3 — a 2 ) (a 3 — a x ) (a 2 — af) 
or 
afa 2 + afa Y + afa 3 - (a 3 2 a 1 + afa 3 + afaf) , 
when it is found that either of the parts 
afa 2 + afa Y + afa 3 , 
or 
afa Y + afa 3 + afa 2 , 
is an instance of a function capable of only two values by permu- 
tation of the variables ; the result indeed of any permutation being 
merely that the one function passes into the other. Further, the 
whole expression 
afa 2 + afa x + afa B - (« 3 2 cq + afa B + afa 2 ) 
is another example, the difference between the two values which it 
can assume being however a difference of sign merely. As a refer- 
ence to the title of the memoir of November 1812 will show, it is 
functions of this latter class which Cauchy there considers. 
At the commencement he contrasts them with functions which 
suffer no change whatever by permutation of variables, that is to 
say, sigmmetric functions : and, noting the fact, afterwards ascer- 
tained, that the new functions consist of terms alternately + and - , 
and that were it not for this alternation of sign they would be 
symmetric functions, he decides to extend the term “ symmetric ” 
to them, and having done so, seeks to distinguish them from ordi- 
