[ 263 ] 
XIII. On a New Auxiliary Egiiation in the Theory of Equations of the Fifth Order. 
By Aethue Cayley, Esq^., F.R.S. 
Eeceived February 20, — Bead March 7, 1861. 
CoNSiDEEiNG the equation of the fifth order, or quintic equation, 
{*Jy, lY={v—x,){v—x^){v—x^){v—x^){v—x,)=^, 
and putting as usual 
fu = X 1 “j~ C0X2 “h “h OJ^X^ 60 * x^^ 
where u is an imaginary fifth root of unity, then, according to Lageange’s general 
theory for the solution of equations, foo is the root of an equation of the order 24, 
called the Eesolvent Equation, but the solution whereof depends ultimately on an 
equation of the sixth order, viz. 
(/»)% (/“•)’. (Pf. ipr 
are the roots of an equation of the fourth order, each coefficient whereof is determined 
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 .fcA .fao^ ./^^ 
(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 equation 
may be called the Resolvent-Product Equation. But the recent researches of Mr. Cockle 
and IMr. Haeley* show that the solution of an equation of the fifth order may be made 
to depend on an equation of the sixth order, originating indeed in, and closely connected 
with, the resolvent-product equation, but of a far more simple form ; this is the auxiliary 
equation referred to in the title of the present memoir. The connexion of the two 
equations, and the considerations which led to the new one, will be pointed out in the 
sequel; but I will here state synthetically the construction of the auxiliary equation. 
Representing for shortness the roots (.r,, x^, ^ 3 , ^ 4 , x^) of the given quintic equation by 
1, 2, 3, 4, 5, and putting moreover 
12345 = 12+23+34+45 + 51, &c. 
* CocEXE, “Eesearches in the Higher Algebra,” Manchester Memoirs, t. xv. pp. 131-142 (1858). 
Haelet, “ On the Method of Symmetric Products, and its Application to the Finite Algebraic Solution 
of Equations,” Manchester Memoirs, t. xv. pp. 172-219 (1859). 
Haelet, “ On the Theory of Quintics,” Quart. Math. Journ. t. iii. pp. 343-359 (1859). 
