HISTORICAL SKETCH. 
5 
in conjunction with his formula (4), provided a method for expressing any 
symmetric function of the roots in terms of the coefficients; but the 
method is a rather cumbersome one, as may be judged from the following 
comparatively simple application, the calculation of S a\a.\al : 
A = $ 4 
and 
$*•$, 
B = A 
$4 + 3 $4 + 2 $ 
ry rr i rr n T n rt 
.O4 *^3 O 4 * ^2 ^3*^2- 
3 + 2 
Satalag 
l = A -B + 2C 
— $ 4 • $3 " $ 2 $2 ' $ 4+3 " 
= Saf • • 2a? — 2a? 
p _ a r $ 4 + 3+2 1 
^ o’ cr rr 
LO4 * |J3 * OoJ 
$3 ’ $ 4+2 $4 ' $3 + 2 + 2*S, 
-’ 4+2 
2a?-2a? 
'4+3+2 
V„6_y„4.y a 5 + 2 V a 9 
-’4 
ta? 
3+2 
at 
(5) 
During the very year (1771) that Waring announced his formula (5), 
Vandermonde followed it by actually calculating the values of various sym¬ 
metric functions of the roots of an equation in terms of the coefficients, 
publishing the results in the form of tables. 
A very pretentious work, 3 with the harmless title “A Collection of 
Examples, Formulae, and Exercises on the Literal Calculus and Algebra,” 
but announcing that the author, Meyer Hirsch, had “discovered the gen¬ 
eral solution of equations,” appeared in Berlin in 1808. In this work 
symmetric functions, their calculation, and their applications—including a 
method for their use in calculating the resultants of pairs of equations—- 
occupy an important position; and in an appendix are to be found the 
tables themselves up to and including the tenthic (that is, the table giving 
the values of the symmetric functions of an equation of the tenth degree). 
In his tables he arranges the coefficients according to the dictionary 
method, or rather according to the reverse of the dictionary method. For 
example, for the tenthic he uses the equation in the form 
x n — Ax n ~ l A Bx n ~ 2 — • • • —Ix n ~ 9 A Kx n ~ 10 — • • • =0 
where the positive integers a, b, . . . t, are all different and 
A=S a - Sb- . . .St 
B 
S(i\b ^ Sfi 1 c ^ 
A So- 
lSa-S b ' S a -S, 
1 r *s<x+&+ 
~ lS a -S b - 
Sa+b+c+d 
| Sb+c | 
Sb-S c 
c 
D = A 
*^a+6+c - ^a+b+d , 
s7 + AAsV-Sa 
S a -Sb-S c -S d 
+ 
n r> \ r S a +b * Sc ■ :i 
BB=A [s a .s b -s c ss d + ---\ 
■pi _ 4 j _ *S>a+fr+c+<i+e 
~[s~-s b -s^ d -s < 
L * 
f - a [s7- 
Sb-S c -S d -Se 
$a+b+c+d+e+/ 
+ 
+ 
Sb ■ S c ■ Sa ■S e -S J 
Then 
Zx«x 2 b . . . x n ‘=A-B + 1 • 2C—1 • 2- 3D+1 • 2- 3 -4E+1- 1 • BB 
-11- 2BC—1 -2-3 • 4- 5E+1 • 2- 3-4 - 5-6G+1 ■ 1 • 2 - 3 BD 
-1 • 1 ■ 2 - 3-4RE + 1 • 2 - 1 • 2CC-1 -21 ■ 2-3CH-1- 1 • 1 BBB 
+ 1-1-1-2 BBC+- • ■ 
Each letter introduces into a coefficient which it enters a factor as follows: 
B,— l C, 1-2 D,-1-2-3 E, 1-2-3-4 F,- 1 • 2 • 3 • 4 • 5; etc. 
When any of the exponents become equal, it is necessary to divide the quantities A, B, C, D, etc. by the 
factorials of the appropriate numbers. This theorem was independently discovered by Gauss in 1816. 
3 English Translation by Ross, London, 1827. 
