SYSTEMS OF EQUALITIES AMONG THE BOOTS OF AN EQUATION. 
Table for the equal Roots of a Quintic. 
729 
2111 
221 
311 
32 
41 
5 
2111 
221 
311 
32 
41 
5 
1 
X 
X 
X 
X 
X 
X 
12.13.14. 
15. 
23. 
24.25.34.35.45 
O 
o 
o 
o 
o 
o 1 
o 
X 
X 
X 
X 
X 
12|13.]4. 
15. 
23. 
24.25.34.35.45 
O 
o 
o 
o 
o 
X 
o 
X 
X 
X 
X 
X 
12.13114. 
15. 
23. 
24.25.34.35.45 
O 
0 
o 
0 
o 
X 
o 
o 
X 
X 
X 
X 
]2.34|l3. 
14. 
15. 
23.24.25.35.45 
O 
o 
o 
o 
X 
X 
o 
X 
X 
X 
X 
X 
12.13.14 
15. 
23. 
24.25.34.35.45 
O 
o 
0 
o 
o 
X 
0 
o 
X 
X 
X 
X 
12.13.45 
14. 
15. 
23.24.25.34.35 
O 
o 
o 
o 
X 
X 
o 
o 
X 
X 
X 
X 
12.13. 24 
14. 
15. 
23.25.34.35.45 
O 
o 
o 
0 
X 
X 
o 
X 
0 
X 
X 
X 
12.13. 23 
14. 
15 
24.25.34.35.45 
O 
o 
o 
X 
X 
X 
o 
X 
X 
X 
X 
X 
12.13.14 
. 15 
23 
24.25.34.35.45 
o 
o 
o 
X 
o 
X 
0 
o 
o 
X 
X 
X 
12.13.14 
.23 
15 
24.25.34.35.45 
o 
o 
o 
X 
X 
X 
o 
o 
X 
X 
X 
X 
12.13. 14 
.25 
15 
23.24.34.35.45 
0 
o 
o 
X 
X 
X 
0 
o 
X 
X 
X 
X 
12.13.24 
.34 
14 
15.23.25.35.45 
o 
o 
o 
o 
X 
X 
o 
o 
X 
X 
X 
X 
12.14.23 
.35 
13 
.15.24.25.34.45 
o 
o 
o 
0 
X 
X 
o 
o 
o 
o 
X 
X 
12.13. 23 
.45 
14 
15.24.25.34.35 
o 
o 
X 
X 
X 
X 
o 
o 
X 
X 
X 
12.13.14 
15 
24.25.34.35.45 
o 
o 
o 
X 
X 
X 
12.13.14 
.23 
24 
15.25.34.35.45 
2111 
1 221 
1 311 
1 32 
41 
5 
o 
0 
o 
X 
X 
X 
12.13.14 
.23 
25 
15.24.34.35. 45 
o 
o 
o 
X 
X 
X 
12.13.15 
.24 
34 
14.23.25.35.45 
o 
o 
o 
o 
X 
X 
12.13.14 
.23 
45 
15.24.25.34.35 
o 
o 
X 
X 
X 
X 
12.15.23 
.34 
45 
13.14.24.25. 35 
2111 
221 
311 
32 
41 
5 
The two Tables enable the discussion of the theory of the equal roots of a quartic or 
quintic equation : first for the quartic : — 
5. In order that a quartic may have a pair of equal roots, or what is the same thing, 
that the system of roots may be of the form 211, the type to be considered is 
12.13.14.23.24.34; 
this of course gives as the function to be equated to zero, the discriminant of the quartic. 
6. In order that there may be two pairs of equal roots, or that the system may be of 
the form 22, the simplest type to be considered is 
14.24.34; 
this gives the function 
which being a covariant of the degree 3 in the coefficients and the degree 6 in the 
variables, can only be the cubicovariant of the quartic. 
7. In order that the quartic may have three equal roots, or that the system of roots 
may be of the form 31, we may consider the type 
13.14.23.24, 
and we obtain thence the two functions 
(“— W— rX/3— 
which being respectively invariants of the degrees 2 and 3, are of course the quadrin- 
variant and the cubinvariant of the quartic. If we had considered the apparently more 
simple type 
12.34, 
