155] 
A FOURTH MEMOIR UPON QUANTICS. 
519 
and 
but the invariant 
is by the operations 
and 
reduced respectively to 
and 
o!d a + b djj + c'd c + d'dd; 
ac' —2 bb' -F ca' 
ad a > + bd v + cd c ‘ 
a'd a + b% + c'd c 
2 (ac — b 2 ) 
2 (a'c' - b' 2 ). 
75. For two qualities of the same odd order, when the quantics are put equal to 
each other, the lineo-linear invariant vanishes; but for two quantics of the same even 
order, when these are put equal to each other, we obtain the quadrinvariant of the single 
quantic. Thus the quadrinvariant of the quadric (a, b, effx, yf is 
ac—b 2 ; 
and in like manner the quadrinvariant of the quartic {a, b, c, d, e§x, yf is 
ae — 4 bd + 3c 2 . 
76. When the two quantics are the first derived functions of the same quantic 
of any odd order, the lineo-linear invariant does not vanish, but it is not an invariant 
of the single quantic. Thus the lineo-linear invariant of 
and 
is 
(a, b, c$cc, y) 2 
(b, c, d$x, y) 2 
(ad — 2 be + cb=)ad — be, 
which is not an invariant of the cubic 
(a, b, c, d\x, y) 3 . 
But for two quantics which are the first derived functions of the same quantic of 
any even order, the lineo-linear invariant is the quadrinvariant of the single quantic. 
Thus the lineo-linear invariant of 
and 
is 
(a, b, c, d'fyoc, y) 3 
(b, c, d, e\x, yf 
(ae - Sbd + 3c 2 — db =) ae — 4<bd + 3c 2 , 
which is the quadrinvariant of the quartic 
(a, b, c, d, e\x, yf.