6 
THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
and his arrangement from right to left begins 
K IA HB HA 2 GC GBA GA 3 , etc. 
He arranges the functions according to the number of parts in their par¬ 
titions, that is, according to the number of a’s composing the individual 
terms (3 2 21, which when inclosed in brackets signifies Safa^a|a 4 , is called 
a four part partition of 9), beginning with the least. Thus for the tenthic 
his arrangement begins at the top with 
[10] [91] [82] [73] [64] [5 2 ] [81 2 ] [721], etc. 
This plan gives his table the triangular form with the principal diagonal 
elements each unity, as the following reproduction of his fifthic indicates. 
lO 
Qq 
CO 
<NJ 
O 
<N 
O 
oq 
kq 
[5] 
1 
-5 
+ 5 
+ 5 
-5 
-5 
+ 5 
[14] 
1 
-3 
-1 
+ 5 
-1 
-5 
[23] 
1 
-2 
-1 
+ 5 
-5 
[1 2 3] 
1 
-2 
-1 
+ 5 
[ 12 2 ] 
1 
-3 
+ 5 
[ 1 3 2 ] 
1 
-5 
[l 5 ] 
1 
A half century later Brioschi announced a differential equation , 4 of 
use in calculating symmetric functions, as follows: 
Consider equation (1) and let s K = \K~\ = 2af; then 
Its application may be illustrated by the calculation of [321]. Write 
[321] = CxCIq + c 2 a 5 a 1 + c 3 a 4 a 2 + c 4 a 4 a 2 + c 5 aj + c (> a 3 a 2 a i + c 7 a% I 
Next express [321] in terms of the s’s by means of (5) thus 
[321] — S 3 S 2 S 1 SgS^ s 4 s 2 S 3 T 2sg II 
then apply ( 6 ), making K = 6 
—|c 1 = 2 , or c 4 = —12 
4 M v Brioschi: Annales de Tortolini, Rome, 1854. 
