THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 181 
The evanescence of the symmetric product gives 
my) =Ts(p2+a*)=0, or 
(a3 — 4ab + 8c)p* — (3a* — 14a? b + 20ac + 86° -82d)p” 
+ (3a° — 16a* b+ 200° c+ 16ab? — 32ad —16bc)p 
— (a Gat b 4+ 8a® c+ 8a? b°— 8a? d—16abe + 8c”) =0, 
a cubic in p. ~ 
Again: — The biquadratic in y may be put under the 
form (Art. 8) Fett 
(y+4Ay-x) =(q)-D: 
or phe ieee ery 
Finally : — Combining the equations 
v+aa2+ba*?+cx+d=0, 
and vtpe—-y=0. 
by the method of the highest common divisor, we find 
(ay -—2py —p>+ap*-bp+ojaty + (P- ap+b)y+d=0, 
ya yt (r—apt+bjy+d 
(Qp—a)y+p*-ap*-+bp-e 
and, as in cubics, all the values of z may be evolved. 
14. Of course y admits of an infinite variety of con- 
structions equaily available. For instance, if we give to y 
another form, and assume 
y=pet+a”; 
then, eliminating « between this equation and the pro- 
posed biquadratic, we are conducted to an equation in y 
for which 
or 
A=apt9, 
a pie, GO Ad ab 
B=bp'+ —{—Pt+@ 
Ce = Bee SG ; ae +7) 
2bd — —h-2d P—2 1 
deep 2 b ad gt b 
D=dp'- 
d a Pa 
and the evanescence of the symmetric product now gives 
