180 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
12. If, in place of 
y= (wv) =pet 2, 
y=¥(2)=(pt+2)"; 
and eliminate z between this equation and the proposed 
one, there results an equation in y for which 
we assume 
3p*—2ap+b 
A=- y= 
Bhs: Sop 
-3p+a 
B=> SS 
Nn Y2 x(—p) 
and 
C=-n y= 
1/273 x(—p) 
The condition 77,(y)=0 now gives 
Kp’ —Ap+ w=0, 
rt V2 = 4A 
or SSS 
2K 
Hence, as before, A, B, C and y are known, and the roots 
of the proposed cubic are 
2=Y,'—P, %=Y_—p, and 2,=y;*—p. 
13. Biquadratic Equations. If we eliminate w between 
a +aa*>+ ba’+cx+d=0, 
and x +px-y=0, 
there results 
y'+Ay?+By’+Cy+D=0, 
where 
A= y=—(pla+ 22°) =ap— (a — 26), 
B=Sy, =P UM, Ly +PUAj y+ VA @ 
= bp” — (ab—38c)p -2ac+ 6+ 2d, 
CH=-ly) Yo Yg=— (PVA, Wy Let pay Ly Wz +pBaj #3 xs 
+ Sa} x3 23) 
= cp* — (ac —4d)p” — (8ad — bc)p + 2bd-c’, 
and 
D=YoYsYs=% Vp Xq@(P+PLL +P LM, Uy + PUA Ly Ly 
+ 2 22 23 U4) 
= d( p* — ap* + bp” — cp+d). 
