THE THEOET OF EQUATIONS OF THE FIFTH OEDEE. 
267 
and to explain the formation of a set, consider the symbols 1, 2, 3, 4, 5 as belonging to 
five points. We may Avith these five points form in all (^. 1 . 2 . 3 . 4=)12 pentagons, and 
the symbol 12345 of any pentagon may of course be read backwards or forwards from 
any point (12345=:23451=&c.=15432=&c.) without alteration of its meaning. Now 
attaching to each arrangement of the five numbers a sign, + or — , according to the 
ordinary rule of signs, 12345 being as usual positive, the arrangements 12345, 23451, 
&c. . . 15432, &c., which belong to the same pentagon, have all of them the same sign; 
and we may consequently connect with each pentagon the sign + or — ; there are, in 
fact, six pentagons with the sign + and six with the sign — ; and to each positive 
pentagon there corresponds a negative pentagon, which is derived from it by stellatim, 
viz. to the positive pentagon 12345 there corresponds the negative one 24135, and so 
for the other positive pentagons. The above-mentioned system of equations 
?;i=12345-24135, (p, =21435-13245, 
. (^,=13425-32145, (^,=31245-14325, 
©3=14235-43125, (^6=41325-12435, 
in fact exhibits the six positive pentagons, each accompanied by its stellated negative 
pentagon, and the formation of the system of equations is thus completely explained ; 
the order of arrangement of the pairs inter se (or, what comes to the same thing, the 
order of arrangement of the suffixes of the (p’s) is wholly immaterial. 
10. The six pairs of pentagons, or, what is the same thing, the (p’s, correspond to each 
other in pairs in a fivefold manner, quoad the numbers 1, 2, 3, 4, 5 respectively ; thus, 
quoad 5, the pairs are p, and p^, p, and p,, p, and pg, or say 1 and 4, 2 and 5, 3 and 6. 
The relation is best seen by means of the positive pentagons ; thus, quoad 5, in the 
pentagons 12345 and 21435, the points adjacent to 5 in the one of them are the points 
2, 3, and in the other of them the complementary points 1, 4; and so in the other cases. 
The fivefold correspondence is shown by the symbolical equations 
1=16, 24, 35, 
2 = 12, 34, 56, 
3=15, 23, 46, 
4=13, 26, 35, 
5=14, 25, 36, 
which, in fact, indicate the combinations of the p’s which correspond to the several roots 
of the quintic. 
11. It is proper to notice that the right-hand sides of the last-mentioned equations 
contain all the duads formed with the six numbers 1, 2, 3, 4, 5, 6, each duad once, and 
once only. There are in all six such synthemes of duads, viz. 
2 o2 
