578 
PROFESSOR CAYLEY ON THE CONDITIONS EOR THE EXISTENCE 
the condition for three equal roots, or, say, for a root system 31, is that the quadrinvariant 
and the cubinvariant each of them vanish, viz. we must have 
\—ae— 4^+3c 2 =0, 
J — ace — ad 2 — b 2 e -f 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, 5C, 10D, 5E, F, GJx, yf= 0, 
we must have 
A= add — octbc J r 2b' 3 =0, 
B = a?e +2«^-9ac 2 +66 2 c = 0, 
C= abe~3acd-\-2b 2 d =0, 
D — —ad 2 -\- b 2 e =0, 
E =— ade-\-%bce — 2bd 2 =0, 
F = — ae 1 — 2bde-\-9c 2 e — Qcd 2 =0, 
G=-be* +Zcde-2d 3 =0. 
3. But the condition for the common quadric factor is 
a , 
U, 
3c, d 
b , 
3c, 
3 d, e 
a, 3 b, 
3c, 
d 
b , 3c, 
3 d, 
e 
and the determinants formed out of this matrix must therefore vanish for (I, J)=0, and 
also for (A, B, C, D, E, F, G) = 0, that is, the determinants in question must be syzy- 
getically related to the functions (I, J), and also to the functions (A, B, C, D, E, F, G). 
4. The values of the determinants are — 
1234 = 3 X 
1235 = 3 X 1 
1245= 
1345 = 3 X 
2345 = 3 X 
+ 1 a 2 ce 
— 1 a-de 
-1 aV 
— 1 abe 2 
+ 1 ace 2 
— 3 d 2 d 2 
+ 4 abce 
+ 2 abde 
+ 4 aede 
— 1 od 2 e 
— 1 aide 
+ 1 abd 2 
+ 9 ac 2 e 
-3 ad 3 
- 3 
+ 14 abed 
— 3 ac 2 d 
— 9 a.cd 2 
+ 1 b 2 de 
+ 14 bede 
— 9 ac 3 
-3 b 3 e 
— 9 Idee 
— 3 bde 
- 8 bd 3 
- 8 b 3 d , 
+ 6 b~c 2 
+ 2 b-cd 
+ 8 Idd 2 
+ 2 bed 2 
— 9 c 3 e 
+ 6 ddd 
5. The syzygetic relation with (I, J) is given by means of the identical equation 
= — 6I.HU+9J.U, 
6*y, 
X 4 - 
a , 
u , 
3c , 
d 
b , 
3c , 
3 d , 
e 
a. 
3 b , 
3c , 
d , 
3c , 
U , 
