474 ON THE SEXTIC RESOLVENT EQUATIONS OF JACOBI AND KRONECKER. [950 
which, if in the first instance the figures are regarded as points, represent the twelve 
pentagons which can be formed with the points 1, 2, 3, 4, 5; each form in the right- 
hand column is derived from the corresponding form in the left-hand column by 
stellation, say we have 13524 = $12345, and so in other cases. 
A pentagon is in general reversible, but we sometimes consider it as irreversible 
(viz. we distinguish between the pentagons 12345 and 15432); when this is so, we 
write 15432 = R12345, and we have thus twelve new forms, in all twenty-four forms. 
The symbols R, S are such that R 2 =l, S 2 = R, RS = SR, $ 4 = 1. But for a reversible 
pentagon, there is no occasion to use the symbol R, and we have simply S 2 = 1. 
In a somewhat different point of view, we may for an irreversible pentagon write 
12345 to denote any one of the forms 12345, 23451, 34512, 45123, 51234, 
.K12345 „ „ „ „ 15432, 54321, 43215, 32154, 21543, 
and for a reversible pentagon 12345 to denote any one of these same ten forms. 
Each pentagon gives thus ten forms, viz. we have in all 120 forms which all 
are the different arrangements of the five figures. But further regarding the arrange 
ment 12345 as positive, then the forms in the left-hand column are each of them 
positive, and the ten forms derived from any one of these are each of them positive, 
that is, the forms in the left-hand column give all the 60 positive arrangements of the 
five figures: and similarly the forms in the right-hand column give all the 60 negative 
arrangements of the five figures. 
Taking 1, 2, 3, 4, 5 to denote any quantities, or say the five roots x 4 , x 2 , x 3y 
x 4 , x 5 of a quintic equation, we regard 12345, ..., as denoting functions of these roots: 
in particular, 12345 may denote a cyclic reversible function, the analogue of the 
reversible pentagon, or it may denote a cyclic irreversible function, the analogue of 
the irreversible pentagon. 
Jacobi’s resolvent equation. 
The most simple course is to take 12345 a cyclic reversible function of the 
roots x; a root of the resolvent equation is then 12345 —13524, = (1 — S) 12345, and 
the six roots are 
s, = (1 -S) 12345, 
z 2 = (l- S) 13254, 
¿3 = (1 - S) 24315, 
z 4 = (l- S) 35421, 
z 5 = (l -S) 41532, 
z 6 = (l- S) 52143. 
Here effecting on the roots x any positive substitution whatever, we permute inter 
se the roots z\ but effecting on the roots x any negative substitution whatever, then