4 
THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
A function of several quantities which is not changed when any two 
of the quantities are interchanged, such as Xa x a 2 , is called a symmetric 
function of the quantities. Such a function of the roots of the equation 
(1) has been designated by inclosing the exponents in brackets, expressing 
the repetition of a number by an exponent: thus 
2afafa|a 4 a 5 a 6 
would be written [3 2 2 l 3 ]. 
In particular, a symmetric function of the form 
[K] = a* + af -f • • • • +a 
K 
n 
is called an elementary symmetric function. 
Newton in his Cambridge Lectures, Arithmetica Universalis (published 
in 1707), adds the following relations between the coefficients and the ele¬ 
mentary symmetric functions of the roots of equation (1) 
[2] = 2cx 2 = a? A A • • • + a 2 — a? — 2a 2 
[3] =2a? = a? + a!+ • • • +a 3 = — a? + - 3a 3 _ ^ 
[4] = 2ai = a 4 -|- 0-2 A ■ ■ ■ A ot-n = — 4ct 2 &2 A 2n2 A 4cqci3 — 4a 4 
etc. 
These results were utilized for the calculation of resultants of pairs of 
equations as early as the middle of the eighteenth century by Euler and 
Cramer, who were interested in the problem of the number of intersections 
of two curves. 
Toward the close of the eighteenth century Waring published a for- - 
mula 1 giving the value of any coefficient in the expression for [K] in terms 
of the coefficients: for example, the coefficient of a 4 a|a 2 , written (4 1 2 2 1 2 ), 
in [10] is 
, 1v+a+ » -10-(l + 2 + 2-l)! 
^ ; 1! 2! 2! 
or —60 
(4) 
It is to be noticed that this formula is a generalization of Newton’s for¬ 
mulas (3). 
A more important contribution made by Waring to the subject was 
a reduction formula 2 by which any symmetric function [x Y x 2 . . . x n ] could 
be expressed in terms of elementary symmetric functions. This formula, 
1 In general, the coefficient of a, A ia 2 A 2 . . . a,/n in [X], as given by Waring, is 
where J = b + • ' +*» 
See Waring: Miscellanea Analytica de Aequationibus Algebraicis et Curvarum Proprietatibus, Cambridge, 
1762. 
2 The following is Waring’s statement of the formula (see Waring: Meditationes Algebricae: also M. 
O. Terquem: Nouvelles Annales, 1849). Let 
*Sa=*i a + a: 2 a + • • ‘ + x n a 
Sb=X 1 b +Xj> + ■ ■ ■ +X n b 
S t =X/ +X 2 ‘ + ■ ■ ■ +xj 
