Proceedings of Royed Society of Edinburgh. [sess. 
O O 1 
or 
o o o 
2A 2 B 2 Tv* - TjEKG-2ABK- - 2ABCL2G 2 E 2 
- 22 ACB 2 K 2 G 2 + 2ABE 2 K 2 G 2 + 8ABCL-DEKG = 0 ; 
O 
where 2 indicates cyclical summation, four terms being always 
o 
included in the sum, except in the case of 2G 2 E 2 , which is used 
to stand for G 2 E 2 + D 2 K 2 , and not for G 2 E 2 + D 2 K 2 + E 2 G 2 + K 2 D 2 , 
as it might well do. 
7. In Sylvester’s original paper he explained, in passing, that 
his process of solution had been suggested to him from a considera- 
tion of the problem of finding the discriminant of the ternary 
quadric 
Ax 2 4 - By 2 + C z 2 + 2 A 'yz + 2B ’zx + 2C 'xy , 
and in my paper above referred to the nature of the relation 
between the two problems is attempted to be made clear. It is, 
therefore, not strange that the solution of the preceding paragraph 
should be due to a similar suggestion. 
If the quaternary quadric 
Af, 2 + B$ 2 + C4 2 + U 2 + D(f 2 + E^4 + + Gif, + H£ 2 £ 4 + 
be the product of two linear factors, 
( a l^l d" $f‘2 d" y& > ( a 2^1 d P 2^2 d" 72^3 d" ^2^4) 5 
the ten equations 
a l a 2 = A j a l@2 d" a 2@l — » ^i 72 d* 27l ^ 5 
P 1 P 2 = » a l72 + a 27l = F J PAl + AA = H , 
7i 72 = C , a 1 ^2 + a A = dr , yA + yA = K , 
8 1 8 2 = L > 
will hold, and as consequences of these, the 10 principal minors of 
the determinant 
2A D F G 
D 2B E H 
F E 2C Iv 
G II K 2L 
will vanish. That is to say, we shall have 
(1) an equation connecting A, B, C, ,D,E,F, , 
(2) „ „ A, B, , L, D, , , G, H 
(3) „ „ A, , C , L, , , F, G, 
