580 PROFESSOR CAYLEY ON THE CONDITIONS EOR THE EXISTENCE ETC. 
B : C : D : E : F proportional to the determinants of the matrix 
d , — 3 c, , 
—e, . , 6c, 
a 
. , — a 
—d , 3c, — b, 
—e, . , -j-3<?, — 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, fjx, y)\ 
the condition of a root system 41 is that the co variant, Table No. 14, shall vanish, 
viz. we must have 
A=2(«e-4^-f-3c 2 ) = 0, 
B= af—3be+2cd = 0, 
C =2(bf— ice + 3(f ) = 0. 
10. The condition of a root system 32 is that the following covariant, viz. 
3 (No. 13) 2 (No. 14) — 25(No. 15) 2 , 
shall vanish, where 
No. 13 = [a, b, c, d, e,f\x, y ) 5 , the quintic itself. 
No. 14 = ( 
ae 
of 
bf 
-4 bd 
-3 be 
— 4 ce 
1 
+ 3 c 2 
+ 2 cd 
+ 3 d} 
0 . 15 = ( 
ac 
3 ad 
3 ae 
of 
3 bf 
3 cf 
df 
— 3 be 
3 bd 
+ 7 be 
+ 3 re 
-3 de 
—e- 
— 6 <r 
— Scd 
-6 d 1 
11. The developed expression of the foregoing function is as follows. — 
