626 
PROFESSOR SYLVESTER ON THE REAL 
and consequently, if we call I the resultant in respect to x , y, we have 
4-1 =p, 45 gW(<r — §)(r — a)(q — r 2 ), 
and 
I 2 = fi 90 gVV((r — q ) 2 (r — <r) 2 (g — r) 2 
== P' 3 °( (r ?) 2 ( r ff ) 2 ( ? t) 2 L 2 . 
(43) Thus we see that the two quantities G, I 2 , which are both rational integral 
functions of the degree 36 in the coefficients of F(x, y ), cannot one vanish without the 
other, at all events when L is not equal to zero. This is sufficient to show that they are 
identical to a numerical factor pres, whatever L may be, zero or not zero( 38 ), and con- 
sequently that the quantity called G, proved to be positive upon the supposition of L 
not being zero, must also remain positive when L is zero, because it is in fact the square 
of a rational function of the coefficients. But we may also prove this independently 
by virtue of the supplementary reduced form ait 5 -\-c>euv 4 -\-fv 5 applicable to the case of 
L zero. 
For when L=0, G becomes JK 4 ; so that the condition “G not negative” implies 
simply that J is positive unless K vanishes. 
Now the canonizant, when it does not vanish, i. e. when e is not zero, contains v 2 u as 
a factor, and, its coefficients being real, u, if are both of them necessarily real functions 
of x, y. Consequently J, which by definition is — 4 X discriminant of quadratic cova- 
riant, becomes — 4^ 10 X discriminant of au(eu-\-fv) in respect to u, v, which =y J 10 a 2 f 2 , [h 
being real. Consequently J is positive, since the reality of u, v implies that of a, e, f. \ 
when e is not zero. When e is zero u, v may be either real or imaginary ; for u 5 -\-v 5 may 
be real whether u , v be real or conjugate imaginary functions of x, y ; but in that case 
K, which is found by operating twice upon the Hessian with a canonizant turned into an 
operator, vanishes, since then all the coefficients of the canonizant vanish ( 39 ). Hence 
the rule that G cannot be negative is seen to be true, whatever L may be. 
or any two of its quantities u, v, w, are interchanged, such interchange having the effect of introducing as a 
multiplier the 5(2m+l)th power of the determinant of substitution ( — 1). Hence (£2) is of the form 
which again is of the form 
so that £2 is of the form 
(§ — <r)(<r— v)(r— f)E(f, cr, r), i. e. T ) ; 
(L)* 
(I).F((J), (K), (L)) 
(L)2m-8 
I.F(J, K, L) 
L ^-8 
Hence since, as before, £2 cannot become infinite when L=0, and since, furthermore, I does not vanish (for if 
so then G, which is I 2 , would vanish) when L=0, £2 must be of the form Ll>(J, K, L). Q. E. D. 
(33) For if Q 2 =KF for an indefinite number of systems of values of a, b, c, cl, e, f, of which Q, I are rational 
integral functions, Q 2 and KI 2 must he absolutely identical ; this of course is the case when Q 2 and KI 2 , as proved 
in the text, are known to he identical for all values of a, b, c, d, e, f which do not make L zero. 
(39) j n thg more general form au’’ + 5euv i +fv 5 , taking y—l. The canonizant is aehcv 2 ; this squared and 
