XIII. On a new auxiliary equation in the theory of equations of the fifth order

Abstract

Considering the equation of the fifth order, or quintic equation, (*)( v , 1) 5 = ( v — x 1 ) ( v — x 2 ) ( v — x 3 ) ( v — x 4 ) ( v — x 5 ) = 0, and putting as usual fω = x 1 + ωx 2 + ω 2 x 3 + ω 3 x 4 + ω 4 x 5 , where ω is an imaginary fifth root of unity, then, according to Lagrange’s general theory for the solution of equations, fω 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. ( fω ) 5 , ( fω 2 ) 5 , ( fω 3 ) 5 , ( fω 4 ) 5 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 ( fω . fω 2 . fω 3 . fω 4 ) 5 , is determined by an equation of the sixth order; and not only so, but its fifth root, or fω . fω 2 . fω 3 . fω 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 equation may be called the Resolvent-Product Equation. But the recent researches of Mr. Cockle and Mr. Harley 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 1 , 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.