A FIFTH MEMOIR UPON QUANTICS. 
528 
[15G 
where (1) is the quadric, and (2) is the discriminant, which is also the quadrinvariant, 
catalecticant, and Hessian. 
And where it is convenient to do so, I write 
(1) = V, 
(2) = □. 
93. We have 
(0 C , - 06, 0«5®, y) 2 o=u, 
which expresses that the evectant of the discriminant is equal to the quadric; 
(a, b, c$dy, —d x yU=4f[J, 
which expresses that the provectant of the quadric is equal to the discriminant; 
(a, b, c§bx + cy, —ax — by)* =□£/’, 
which expresses that a transmutant of the quadric is equal to the product of the 
quadric and the discriminant. 
94. When the quadric is expressed in terms of the roots, we have 
or 1 U = (x- ay) (x - ¡3y), 
a~ 2 □ = — | (a — /3) 2 ; 
and in the case of a pair of equal roots, 
a~ x U =(x — ay)*, 
□ =0. 
95. The problem of the' solution of a quadratic equation is that of finding a 
linear factor of the quadric. To obtain such linear factor in a symmetrical form, it 
is necessary to introduce arbitrary quantities which do not really enter into the solution, 
and the form obtained is thus in some sort more complicated than in the like 
problem for a cubic or a quartic. The solution depends on the linear transformation 
of the quadric, viz. if we write 
so that 
then 
(a, b, c$\x + yy, vx + py) 2 = (ab', c'^x, y) 2 , 
a' — (a, b, c$\ vf, 
b' = (a, b, c#\, v§jl, p), 
c' = (a, b, c$jl, p)\ 
a'c — b' 2 — (ac — b 2 ) (\p — yv) 2 , 
an equation which in a different notation is 
(a, b, cjoc, y) 2 . (a, b, c\X, Y) 2 - {{a, b, c$x r y$X, Y)\ 2 = □ (Yx-Xy) 2 ,