150] SYSTEMS OF EQUALITIES AMONG THE ROOTS OF AN EQUATION. 467 
the types at the foot of the Tables are complementary to themselves, but I have, not 
withstanding this, given the complementary type in the form under which it naturally 
presents itself. 
4. The Tables are: 
Table for the equal Roots of a Quartic. 
211 22 31 4 
211 22 31 
211 
22 
31 
4 
23 
24 
34 
o 
o 
o 
o 
23 
24 
34 
o 
o 
o 
X 
23 
24 
34 
o 
o 
o 
X 
14 
23 
24 
o 
o 
X 
X 
23 
24 
34 
— 
— 
— 
14 
23 
34 
211 
22 
31 
4 
14 
24 
34 
Table for the equal Roots of a Quintie. 
25 
25 
25 
25 
25 
25 
24 
24 
25 
25 
34 
34 
25 . 34 
24 . 25 
25 . 34 
24 . 25 
34 
34 
34 
34 
24 . 34 
23 . 25 
25 
25 
34 
34 
24 . 34 
23 . 25 
24 . 25 
14 . 24 
35 . 
35 . 
35 . 
35 . 
35 . 
34 . 
35 . 
35 . 
35 . 
35 . 
35 . 
35 . 
34 . 
34 . 
35 . 
35 . 
35 . 
35 . 
34 . 
25 . 
2111 221 311 32 ! 41 5 
2111 
221 311 
32 
41 
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; 
59—2