:er. [950 
950] ON THE SEXTIC RESOLVENT EQUATIONS OE JACOBI AND KRONECKER. 475 
the twelve 
the right- 
column by 
reversing the signs of all the roots z, we permute inter se these reversed values. 
Thus effecting on the root z 1 the negative substitution 12, it becomes (l-$) 21345, 
which is 
irreversible 
= (1 — $) $23514 = (& — $ 2 ) (that is, S — 1 or — (1 — $)) 23514 = — (1 - S) 41532 = — z 5 ; 
is so, we 
and similarly for the effect of the same substitution 12 upon any other of the roots z. 
■four forms, 
i reversible 
1. 
It follows that any rational symmetrical function of the roots z is a two-valued 
function of the coefficients of the quintic equation, viz. it is a function of the form 
P + Q Va, where A is the discriminant and P, Q are rational functions of the 
m write 
coefficients of the 'quintic equation. 
>34, 
>43, 
is. 
In particular, if 12345, ..., are rational and integral functions of the roots x, then 
for any rational and integral function of the roots z, we have P and Q rational and 
integral functions of the coefficients, and any rational and integral homogeneous function 
of the roots z, according as it is of an even or an odd degree in these roots, will 
which all 
be of the form P or of the form Q VA ; the resolvent equation is thus of the form 
ie arrange- 
of them 
m positive, 
mts of the 
(1, pVa, g, pVa, e, FVA, G\z, 1) 6 = 0, 
where A is the discriminant, and B, C, D, E, F, G are rational and integral functions 
of the coefficients of the quintic equation. 
0 negative 
The most simple form of the function 12345 is that employed by Jacobi and 
for which 12345 = 12 + 23 +34 + 45 + 51, where 12, ..., denote x 4 x 2 , ..., respectively. 
Xi, x 2 , x 3 , 
lese roots: 
ie of the 
lalogue of 
For comparison with Kronecker’s equation, it is proper to take 12345 a cyclic 
irreversible function of the roots x\ we have then 
12345 + 15432 = (1 + P) 12345, 
>n of the 
.2345, and 
a cyclic reversible function, and the roots of Jacobi’s equation will be in the first 
(or since Kronecker writes x 0 , x lt x 2 , x 3 , x 4 instead of x 1} x 2i x 3 , x 4 , x 5 , in the 
second) of the following two forms, say 
z l = (1 + P) (1 - 8) 12345, 01234, 
z 2 = ( 1 + P) (1 - 8) 13254, 02143, 
z s = (l +P)(1 -S) 24315, 13204, 
z 4 = ( 1 + P) (1 - S) 35421, 24310, 
s 5 = (1 + P) (1-$) 41532, 30421, 
z 6 = ( 1 + P) (1 - S) 52143; 41032, 
viz. in the first form the terms are 12345, ..., and in the second they are 01234, ..., 
but the theory is in no wise altered by this change of form. 
iute inter 
ever, then 
60—2