620 
PROFESSOR SYLVESTER ON THE REAL 
The Hessian of <J> may be obtained without difficulty under the form 
rsu 3 v 3 + stv 3 w 3 + trw 3 u 3 or ru 3 v 3 -|- pv\v 3 + aw 3 u 3 ( 32 ). 
Operating upon this with 
wli-l (4~D)\ 
\dv du \du dvj / 
we obtain + C<r), where 
and as we know that this quantity must be of the form h(K)-f- f/-(J) 2 , we have /k<= 0, 
h=— 72 ; so that, denoting the operator corresponding to the canonizant by T, and the 
Hessian by H, we have (K)=— y^PHO ( 33 ). This gives a ready practical method for 
finding the discriminant of a general quintic F by means of the identity D= J 2 +^T 2 H, 
where D is the discriminant, H the Hessian, T the canonizantive operator, and J the 
quadratic invariant of F in respect to its own variables. 
(40) If now we suppose the determinant of u, v in respect to x, y to be where ^ 
is by hypothesis a real quantity, and if we call the 
Quadratic invariant in respect to x, y . . — J J, 
Discriminant of primitive „ „ . D, 
Discriminant of the canonizant „ . . — -^-L, 
we have obviously 
J 2 -j-r 2 — 2g<r— 2 qr — 2 <rr), 1 
K==|& so g<rr(g+ff+r), D=J 2 — 128K, [invariants of <E>. 
L =^ 30 ? Vr 2 , ) 
This applies to the case where the reduced form is <F, i. e. where the roots of the cano- 
nizant are all real, and consequently where — L is negative, i. e. L positive. 
When L is negative and the reduced form is <3> y , then, since the determinant of u p v, 
in respect to u t v is i, we have 
J = — ^ 10 (g 2 +<J' 2 +’' 2 — 2g<r — 2 qr — 2 <rr), 1 
K= D=J 2 — 128K, [invariants of 
L=-rtVr 2 , J 
L K 
By means of the ratios p’ ^ it is obvious that in either case alike the ratios of §, <r,r 
( 32 ) It will be the quadratic invariant of nt 3 g 2 +s2/ 3 ij 2 +fcc/ 3 5 2 with respect to r h Z+v)+Z being zero; just as 
the quadratic covariant of $ is the quadratic invariant of ru^+svtf + twtf with regard to the same variables. 
This latter is in fact rsuv -f stviv + trwu. 
( 33 ) The intervening covariantie form of degree 3 in the variables and 5 in the coefficients, viz. TH<fi, will 
easily be seen to be 
rstf(y?v — w 2 ) + str 2 (v 2 w — vw 2 ) + trsr(w 2 u — ww ) . 
