409] 
OF EQUAL ROOTS OF A BINARY QUARTIC OR QUINTIC. 
303 
to each other they will also be = 
¥ 
21 ' 
The equations may then be written 
ac - b\ ad - be, ae + 2bd - Sc 2 , be - cd, ce - d 2 
a , 2b , 6c , 2d , e 
= 0, 
and the ten equations of this system reduce themselves (as it is very easy to show) to 
the seven equations 
(A, B, G, D, E, F, G) = 0, 
which, as above mentioned, are the conditions for the root system 22. 
8. It may be added that we have 
ABODE EG 
where it is to be noticed that the four equations having the left-hand side = 0, give 
B : C : D : E : F proportional to the determinants of the matrix 
— 3c, 
a 
• ) 
6c, ., — a 
— d , 
Sc, — b 
-e , 
. , + 3c, — b 
the determinants in question contain each the factor c, and omitting this factor, the 
system shows that B, C, D, E, F are proportional to their before-mentioned actual 
values. 
Article Nos. 9 to 15, the Quintic. 
9. For the quintic function 
(a, b, c, d, e, fjoc, y)\ 
the condition of a root system 41 is that the covariant, [B =] No. 14, shall vanish, 
viz. we must have 
A = 2 (ae — 4bd + 3c 2 ) = 0, 
B = af—Sbe+2cd =0, 
(7 = 2 (bf — 4ce + 3d 2 ) = 0. 
10. The condition of a root system 32 is that the following covariant, viz. 
[SA 2 B — 25(7 2 , = ] 3 (No. 13) 2 (No. 14)-25 (No. 15) 2 ,