264 
TABLES OF SYMMETRIC FUNCTIONS OF ROOTS. 
[829 
In the new tables we have a property in regard to the sums of the numbers 
in a line : viz. except for the last line of each table, where there is only a single 
number +1 or — 1, this sum is always =0. I have given in the several tables on 
the right-hand of each line, the sums for the positive and the negative coefficients 
separately: thus V (6), line 1, the number +375 means that these sums are +375 
and — 375 respectively, the sum of all the coefficients being of course = 0. The 
property is an important verification as well of the original tables (6) as of the new 
tables derived from them ; and I had the pleasure of thus ascertaining that there was 
not a single inaccuracy in the original tables (6). 
The symbols in the left-hand outside column of each table denote symmetric 
functions of the roots a, /3, y, ...; 5 = 2a 5 , 41 = 2a 4 /3, &c.: and the tables are read 
according to the lines: thus in table V (6), 
5 (= 2a 5 ) = (5/+ 25be + 50cd - 1006 2 d - 1506c 2 + 3006 3 c - 1206 5 ), 
41 (= 2a 4 /3) = obe — oOcd + 20b 2 d + 906c 2 — 606 3 c), &c. 
II (6) 
III (b) 
- 2 
r- 6 
c 
6 2 
= 
d 
be 
b 3 
-2 
+ 2 
±2 
3 
-3 
+ 9 
-6 !±9 
1 
+ 1 
+ 1 
21 
+ 3 
-3 
±3 
— 
l 3 
-1 
1 -1 
IV (6) 
+■ 24 
= 
e 
bd 
c 2 
b*c 
6 4 
4 
-4 
+ 16 
+ 12 
-48 
+ 24 
31 
+ 4 
- 4 
-12 
+ 12 
±16 
2 2 
+ 2 
- 8 
+ 6 
± 8 
21 2 
-4 
+ 4 
± 4 
l 4 
+ 1 
+ 1 
i 1 
I— 1 
^ g 
be 
cd 
bH 
be 2 
b 3 c 
b 5 
5 
-5 
+ 25 
+ 50 
-100 
-150 
+ 300 
-120 j ±375 
41 
+ 5 
- 5 
-50 
+ 20 
+ 90 
- 60 
±115 
32 
+ 5 
-25 
+ 10 
+ 40 
- 30 
± 55 
31 2 
-5 
+ 5 
+ 20 
- 20 
± 25 
2 2 1 
— 5 
+ 15 
-10 
± 15 
21 3 
+ 5 
- 5 
* 5