14 
iv( (5R? + R 2 ) 2 — 20R* | =^(* I — Xt) (c) 
and 
R l ='2a r % r and R 2 = -'2 l a r xl-\-'2bx r x s 
But does the square root of the discriminant of a quintic 
admit of being put under such a form ? At present I can- 
not assent to the proposition that Hargreave’s process suc- 
ceeds in solving the quintic.” 
Mr. Kirkman’s method does not differ essentially from 
that employed by Vandermonde in his Essay on the Resolu- 
tion of Equations in the Memoirs of the French Academy 
for 1771. Both start from the same identical equation, 
and both, proceeding by involution, seek to express one root 
of the given equation by means of a finite combination of 
radicals and rational symmetric functions of the roots. Such 
functions, it is well known, can be expressed rationally in 
terms of the coefficients.* The following illustrations of the 
method may not, perhaps, be altogether devoid of interest. 
The Quadratic. Let OCfy t )C\ be the roots of the equation. 
x 2 + bx + c = 0 ; 
then 
2x 0 = %x + (x Q - x x ) 
2 _ 
= 2x + (x 0 — Xi) 2 
= 2# + (XV - 2x 0 x 1 ) i 
= - b + (b 2 - 4 cf ; 
# Vandermonde, in the Memoir above referred to, gives tables of such 
functions up to ten dimensions in the roots of an equation of any degree, ex- 
hibiting their values in terms of the simple symmetric functions 2A, HiX 0 X^ 
&C. The last-named functions are respectively equal to — 6, c, 
—d, (fee. in the equation 
x n + bx n ~ l + cx n ~ 2 4- dx n ~ z 4 - &c. = 0 ; 
so that the tables at the end of Meyer Hirsch’s Algebra which give the sym- 
metric functions of the roots in terms of the coefficients, may be immediately 
deduced from Vandermonde’s, by simply changing the signs of the numerical 
coefficients in the tables if odd dimensioned functions. I may here mention 
also that Professor Cayley, in a Memoir on the Symmetric Functions of the 
Roots of an Equation, {Phil. Trcms. for 1857), has joined to Hirsch’s tables 
another set, giving reciprocally the expressions of the powers and products of 
the coefficients in terms of the symmetric functions of the roots. 
