409] 
ON THE CONDITIONS FOR THE EXISTENCE &C. 
301 
unless, there is satisfied either the relation for the existence of three equal roots, or 
else the relation for the existence of two pairs of equal roots; or the relation for 
the existence of the quadric factor is compounded of the last-mentioned two relations. 
The relation for the quadric factor, for any value whatever of n, is at once seen to 
be expressible by means of an oblong matrix, giving rise to a series of determinants 
which are each to be put = 0; the relation for three equal roots and that for two 
pairs of equal roots, in the particular cases n = 4 and n = 5, are given in my “ Memoir 
on the Conditions for the existence of given Systems of Equalities between the roots of 
an Equation,” Phil. Trans, vol. cxlvii. (1857), pp. 727—731, [150]; and I propose in the 
present Memoir to exhibit, for the cases in question n = 4 and n = 5, the connexion 
between the compound relation for the quadric factor with the component relations 
for the three equal roots and for the two pairs of equal roots respectively. 
Article Nos. 1 to 8, the Quartic. 
1. For the quartic function 
(a, 6, c, d, e\x, y)\ 
the condition for three equal roots, or, say, for a root system 31, is that the quadrin- 
variant and the cubinvariant each of them vanish, viz. we must have 
I — ae— 4 bd + 3c 2 = 0, 
J = ace — ad 2 — 6 2 c + 2bcd — c 3 = 0. 
2. The condition for two pairs of equal roots, or for a root system 22, is that 
the cubicovariant vanishes identically, viz. representing this by 
(A, B, 5G, 10D, 5E, F, G\x, y) 6 = 0, 
we must have 
A = a 2 d — 3 abc + 2 b 3 = 0, 
B = a?e + 2 abd — 9 ac 2 + 66 2 c = 0, 
C = abe — 3 acd + 2 b 2 d = 0, 
D = — ad 2 + b-e = 0, 
E — — ade+ 3bce — 2bd 2 = 0, 
F — — ae 2 — 2bde + 9c 2 e — 6cd 2 = 0, 
G = — be 2 + 3 cde — 2 d 3 = 0. 
3. But the condition for the common quadric factor is 
a, 
36, 
3c, d 
b, 
3c, 
3d, e 
a, 36, 
3c, 
d 
b, 3c, 
3d, 
e 
and the determinants formed out of this matrix must therefore vanish for (/, J) = 0, 
and also for (A, B, G, D, E, F, G) = 0, that is, the determinants in question must be 
syzygetically related to the functions (/, J), and also to the functions (A, B, G, D, E, F, G).