THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 179 
1l. Cubic Equations. The elimination of x between 
2 +ax’+be+c=0= (2), 
and y= (a) =pa +2°, 
gives y+ Ay?+ By+C=0, 
where 
A=-y=- (p>#+ zz") =ap —(a— 26), 
B= 2y, Pr=P VA Lz + PU Xz + Taj A 
= bp’ — (ab—3c)p—2ac+ 6, 
and 
C=- 9, YoYs=— 2 Lyk, p> +p” LH + PLA, Lz + 2 Xp 2s) 
=c(p*— ap’ + bp —¢) =— ex(—p). 
Again: — The evanescence of the symmetric product 
gives 
7(Y) =72( px + 2*)=0, 
or (a*—3b)p*-- (2a -7ab+9c)p+ at — 40? b+ 6ac+0?=0; 
and, if we make 
a@—3b=x, ab-9c=n, and 4’-3ac=p, 
this equation takes the form 
Kp’ — (2ak—r)p +a? «-ar+pu=0, 
Qax—r+ V2 — dep 
Pp = . 
2K 
Consequently A, B and C are known; and, since (Art. 7) 
G+-)=@)-¢ 
3f/ KR S 
ae y=-3+V(G)-C. 
Appropriating this root to y,, the corresponding formulz 
for y, and y; may be obtained by writing ¢ and é before 
the cubic radical; and, combining the two equations at 
the head of this article, we are conducted to 
(y+p*—ap+b)x—(p-a)y+c=0, 
pa _P-OYRe 
y+p(p—a)+6 
All the roots of the proposed cubic may be obtained by 
substituting y,, y, and ys successively for y in this expres- 
sion. 
VOL, XV. BB 
whence 
and 
