310 
ON A NEW AUXILIARY EQUATION IN 
[268 
equation may be called the Resolvent-Product Equation. But the recent researches of 
Mr Cockle and Mr Harley ( x ) 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 (x lt x 2 , x 3 , x 4 , x 5 ) of 
the given quintic equation by 1, 2, 3, 4, 5, and putting moreover 
12345 = 12 + 23 + 34 + 45 + 51, &c. 
(where on the right-hand side 12, 23, &c. stand for x x x 2 , x,x 3) &c.), then the auxiliary 
equation, say 
№ 1)*=0. 
has for its roots 
</>! = 12345 - 24135, 0 4 = 21435 - 13245, 
0 2 = 13425 — 32145, 0 5 = 31245 - 14325, 
0 3 = 14235 - 43125, </>« = 41325 - 12435, 
and, it follows therefrom, is of the form 
(1, 0, C, 0, E, F, 1)* = 0, 
where G, E, G are rational and integral functions of the coefficients of the given 
equation, being in fact seminvariants, and F is a mere numerical multiple of the 
square root of the discriminant. 
The roots of the given quintic equation are each of them rational functions of 
the roots of the auxiliary equation, so that the theory of the solution of an equation 
of the fifth order appears to be now carried to its extreme limit. We have in fact 
0 i08 + 020 4 + </>30 5 = (*$#], 1) 4 > 
0102 + 0304 + 0508 = l) 4 , 
0105 + 0203 + 0408 = l) 1 , 
0103 + 0206 + 0305 = (*][#4> l) 4 , 
0104 + 0205 + 0306 = (*fe, l) 4 , 
where (*$#!, I) 4 , &c. are the values, corresponding to the roots x lt &c. of the given 
equation, of a given quartic function. And combining these equations respectively with 
the quintic equations satisfied by the roots x x , &c. respectively, it follows that, con 
versely, the roots x x , x. 2 , &c. are rational functions of the combinations 0i0 fi + 0204 + 0 3 0 5 , 
0i02 + 0304 + 0506> &c. respectively, of the roots of the auxiliary equation. 
1 Cockle, “ Researches in the Higher Algebra,” Manchester Memoirs, t. xv. pp. 131—142 (1858). 
Harley, “ On the Method of Symmetric Products, and its Application to the Finite Algebraic Solution of 
Equations,” Manchester Memoirs, t. xv. pp. 172—219 (1859). 
Harley, “ On the Theory of Quintics,” Quart. Math. Journ. t. in. pp. 343—359 (1859).