178 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
9. Thus for the lower equations, the evanescence of the 
symmetric product conducts to a solution. If therefore 
the result of the elimination of 2 between 
w+ ax" 4 ba"? +--+ +8a4+t=0, 
and y —W(v)=0, 
where yf is rational, be represented by 
y+ Ay” *+ By”? + ---+8y+T=0, 
and yy be so constructed that 7,_,(y) may vanish, y, and 
consequently x, will become known. 
The expression 77,_,(p+) does not contain p, and the 
relation 
y=(v)=pte 
leads to illusory results; but if we assume 
y=p(v)=pet+ 2, 
we can reduce the solution of the general biquadratié to 
that of a cubic, and the solution of the general cubic to 
that of a quadratic. For since p only enters the product 
T,-.(px+a"*) to the degree which y attains in 7,_,(y), 
that is, to the (n—1)th degree, and 7,(pa2+a2?) and 
7;(pe+a*) are symmetric with respect to 2, it follows 
that the equations 
1(px+a2?)=0, and 7;(pa2+2*)=0, 
are respectively of the second and third degree in p, and 
that their solution gives p, a symmetric function of 2, in 
terms of a, 6, &c. 
10. For quadratics this method must be modified. It 
is impossible to construct y so that m(y) may vanish 
without being led to nugatory results. But in this case 
the symmetric product yields an immediate solution. For 
(Art. 2) 
™(#) =*(a- a) = V(x, +2)? — 4a, a; 
or a — B= > Va —Ab. 
and &@+%=- a, 
whence the usual solution 
