530 
A FIFTH MEMOIR UPON QUANTICS. 
[156 
97. The Jacobian (7) may also be written in the form 
y\ - yx, x* 
a, b , c 
a', b' , c' 
The Resultant (6) may be written in the form 
a, 
2b, 
a , 2 b, 
c, 
a', 
2b', 
a', 2b', 
d, 
and also, taken negatively, in the form 
4 (ab' — a'b) (be' — b'c) — (ad — a'c) 2 , 
which is the discriminant of the Jacobian; and in the form 
4 (ac — b 2 ) (ad — b' 2 ) — (ad — 2 bb' + ca') 2 , 
which is the discriminant of the Intermediate. 
98. We have the following relations: 
(a, b, cjb'x + dy, - a'x - b'y) 2 = - (a'd - b' 2 ) (a, b, c $#, y) 2 
+ (ad — 2bb' + ca) (a', b', c'Qx, y) 2 , 
(a', b', c' ][bx + cy, - ax- by) 2 = + (ad - 2 bb' + ca') (a , b , c \x, yf 
-(ac-b 2 ) (a', b', c'\x, y) 2 , 
and moreover 
(ac — b 2 , ac' — 2bb' + ca', a'c' — b' 2 ][U', — U) 2 
= — {(ab' — a'b, ad — a'c, bd — b'c\x, y) 2 ) 2 , 
an equation, the interpretation of which will be considered in the sequel. 
99. The most important relations which may exist between the two quadrics are: 
First, when the connective vanishes, or 
ac' — 2 bb' + ca' = 0, 
in which case the two quadrics are said to be harmonically related: the nature of 
this relation will be further considered.