THEIR APPLICATION TO THE SOLUTION OF EQUATIONS, 173 
veniently consult those works, I propose to give, in the 
first section, a brief exposition of Mr. Cockle’s method 
(with alterations), and to indicate its application to the 
solution of the lower equations. 
Section I. 
The Method of Symmetric Products. 
1. Let 2, 2, 23,-+-+”, be any n symbols, and let X be 
a linear unsymmetric function of these symbols, such that 
X HM + Q Lyt Ay Uy +++ + Gy 9 Ly +n 2p, 
where the n—1 constants a@,, d,:++d,_, are arbitrary. 
When x is less than 5, these constants may be so distri- 
buted and determined as to render the product 
Tn+(#), or X, Xp X;---X,4, 
(or, when »=2, X*) symmetric with respect to the symbols 
21, Ly, L3,+++x,. When n is equal to, or greater than 5, 
the symmetry is in general unattainable; but in seeking 
to satisfy its conditions we are conducted to significant 
results. The product, 7, ,(v), which may be called the 
symmetric or resolvent product, according as it is or is not 
symmetric, plays an important part in the finite algebraic 
solution of the equation of the » th degree. 
2. When n=2, we have 
{m(7) P=X?= (a, + q 2); 
and the condition of symmetry is aj=1; consequently 
a,=1, or —1; and, since by definition X is unsymmetric, 
the former of these results must be rejected. Hence 
a= -1, and 
{1, (x) }?= (ay — a)? = (ay + ag)? — 4a, 2p. 
3. When n=8, assume 
X=2\ 4+ 4 &2+ dy Xs, 
X,=4+ Ay &y,+Q Xs, 
and (av) =X, X,. 
Then the conditions of symmetry are 
@ @=1, and a?+a=a,+m%; 
