494 
IHE. A. CAYLEY ON THE SYZVniETEIC EENCTIONS 
And comparing with the value o£ the left-hand side, we see that this expression may be 
considered as the generating function of the symmetric functions of (a, /3), xiz. the 
expression in question is developable in a series of the symmetric functions of (jr. ?/). the 
coetficients being of course functions of h and c, and these coefficients are (to given 
numerical factors ])res) the symmetric functions of the roots (a, /3). 
And in general it is easy to see that the left-hand side of M. Boechardt’s formula is 
equal to 
[0]+[i](i)(i)'+[2](2)(2)'+[r](rXi7+&c., 
where (1), (2), (1^), &c. are the symmetric functions of the roots (a, |3, ... k). (1)', (2)', 
(1^)', &c. are the corresponding symmetric functions of y, . . u), and [0], [1], [2], [1^]. 
&c. are mere numerical coefficients; viz. [0] is equal to 1.2.3...W, and [1], [2], [I'J, &c. 
are such that the product of one of these factors into the number of tenns in the corre- 
sponding symmetric function (1), (2), (1^), &c. may be equal to 1.2.3...??. The right- 
hand side of M. Boechaedt’s formula is therefore, as in the particular case, the gene- 
rating function of the symmetric functions of the roots (a, |3, ... yS), and if a convenient 
expression of such right-hand side could be obtained, we might by means of it express 
all the symmetric functions of the roots in terms of the coefficients. 
Tables relating to the Symmetric Functions of the Roots of an Equation. 
The outside line of letters contains the combinations (powers and products) of the 
coefficients, the coefficients being all with the positive sign, and the coefficient of the 
highest power being unity ; thus in the case of a cubic equation the equation is 
x'^-\-hx^-\-cx-\-(l—{x—a){x—^){x—y) = ^. 
The outside line of numbers is obtained from that of letters merely by writing 1, 2. 3.. 
for b, c, d.., and may be considered simply as a different notation for the combhiations. 
The outside column contains the different symmetric functions in the notation above 
explained, viz. (1) denotes 2a, (2) denotes 2a^ (1^) denotes 2a/3, and so on. The Tables 
(a) are to be read according to the columns; thus Table II(«) means i^=l(2)-j-2(l)‘h 
The Tables (b) are to be read according to the Imes; thus Table 11(5) means 
(2)=-2c+15^ (P)= + lc. 
II 
(3) 
( 21 ) 
(P) 
(3) 
( 21 ) 
(P) 
III(«). 
3 
12 
d 
be 
— 1 
—1 
— 3 
— 1 
-3 
-6 
III(5). 
3 
12 
d 
be 
b^ 
-3 
+ 3 
— 1 
+ 3 
— 1 
— 1 
1(a). 11(a). 
1 
2 
II 
b 
II 
c 
(1) 
— 1 
(2) 
+ 1 
(H) 
+ 1 
+ 2 
1(5). 
11(5). 
1 
2 
1' 
= 
b 
= 
c 
(1) 
— 1 
(2) 
— 2 
+ 1 
(IT 
+ 1 
