590 
PROFESSOR SYLVESTER ON THE REAL 
for general values of g, y. This will be accomplished most expeditiously by taking the 
resultant of the two derivatives of the above form, say U and V, where 
U =# 4 -J- 4 sx 3 y -f- 6aV 2 _y 2 -f- 4 ri 2 xy 3 + ^ 4 , 
V =.zx iJ f-^z 2 x 3 y-\-Wx 1 y 2j r^xy 3 -\-y i ; 
so that 
gU —V =6(s 3 — tf)x 2 y 2 -\-k{ztf— ri)xy 3 -\-(zri— l)y 4 =yP, 
-U+^V=(g^-l> 4 +4(^ 2 -g)^+6(^ 3 -g>y=r ! Q. 
Hence 
Resultant of (U, V)=^^x Resultant of (y 2 P, x 2 Q)= Resultant of (P, Q); 
where 
P=6(g 3 -^)f+4(g^-^-f(g^-l)^ 2 , 
Q= (gpj— l)# 2 + 4(??g 2 — s)xy -f- 6(fj 3 — z 2 )y 2 . 
Hence, calling A the discriminant of the original form, we obtain by the well-known 
formula for the resultant of two binary quadratics, writing for the moment 
P=(B, 4?jA, AXx, y)\ Q=(A, 4gA, B';$>, y)\ 
A = ( 4g A 2 — 4 jj AB') ( 4^ A 2 — 4g AB ) + ( A 2 — BB') 2 
= ( 1 — 1 6 g;?) A 4 + 1 6 (g 2 B + p? 2 B f ) A 3 — 1 6 g^BB' A 2 — 2 BB' A 2 + B 2 B' 2 . 
Plence writing zr\—^ g 5 — |- ? 7 5 = S, 
A=(l-16^-l) 4 +96(S-2^ 2 )^-l) 3 -72(8^+l)y+^-S)(^-l) 2 
+ 36 2 fe 3 +f-S) 2 . 
Let ^>—q 2 —y 3 =a, q—l=ja, so that 
S— 2q 2 =ff— q 2 +q 3 =(T+(p + l) 2 p. 
Then 
A=36V + 72(8^+9)y<r+96y<r+96(p+l)y-(16^+15)^ 
=1296^ 2 + (648y+672p 3 >+96/+176p 5 +81i) 4 , 
=i{108<r+27y+28y) 2 +729y+1584y+864^ 6 -(27y+28y) 2 }, 
or 
9A=(108(r+27y+28p 3 ) 2 +72y+80/. 
(9) Hence we see at once that A can be negative only when jp lies between 0 and 
—Ti) ? *• e • when zv\ (which is y)-fl) lies between 1 and y^. Accordingly when A is 
negative, g and yj must be both positive or both negative. The latter supposition may 
easily be disproved as follows : treating the equation A=0 as a quadratic equation in <r, 
in order that A may be capable of becoming negative, its discriminant in respect to a 
must be negative, and its value when c= — oo is positive. Now 
S=g 5 +„ 5 , p+l=fy, V=S— (jp+1) 2 — Q?+l) 3 ; 
so that when g and p? are real we have 
S>2Q»+1)*(“). i. e. <r>-( i > + l) 2 +2( i >+l)^-( i )+l) 3 
( 16 ) It is of course understood that (_p + l)^ is to he taken positive. 
