268] 
309 
268. 
ON A NEW AUXILIARY EQUATION IN THE THEORY OF 
EQUATIONS OF THE FIFTH ORDER. 
[From the Philosophical Transactions of the Royal Society of London, vol. cli. (for the 
year 1861), pp. 263—276. Received February 20,—Read March 7, 1861.] 
Considering the equation of the fifth order, or quintic equation, 
(*$>> l) 5 = {v - ®i) (v - x 2 ) (v - x 3 ) (v - O (v - x 5 ) = 0, 
and putting as usual 
fco = x x + cox 2 + 0) 2 X 3 + 0) 3 X 4 + Q) 4 X 5 , 
where to is an imaginary fifth root of unity, then, according to Lagrange’s general 
theory for the solution of equations, /&> is the root of an equation of the order 24, 
called the Resolvent Equation, but the solution whereof depends ultimately on an 
equation of the sixth order, viz. 
(/")*, (/" 2 ) 5 > (/" 3 ) 5 > (/« 4 ) 5 
are the roots of an equation of the fourth order, each coefficient whereof is deter 
mined by an equation of the sixth order; and moreover the other coefficients can be 
all of them rationally expressed in terms of any one coefficient assumed to be known; 
the solution thus depends on a single equation of the sixth order. In particular the 
last coefficient, or 
(/«./*./«»./«-)», 
is determined by an equation of the sixth order; and not only so, but its fifth root, or 
fco ./or ./&) 3 ./<U 4 , 
(which is a rational function of the roots, and is the function called by Mr Cockle 
the Resolvent Product), is also determined by an equation of the sixth order: this