20. The formulae for the cubic were obtained by J.L. Lagrange (1770-1771) from a different point of view. Upon examining and comparing the principal known methods for the solution of algebraical equations, he found that they all ultimately depended upon finding a "resolvent" equation of which the root is a + [omega]b + [omega]^2 c + [omega]^3 d + ..., [omega] being an imaginary root of unity, of the same order as the equation; e.g. for the cubic the root is a + [omega]b + [omega]^2 c, [omega] an imaginary cube root of unity. Evidently the method gives for L^3 a quadric equation, which is the "resolvent" equation in this particular case. For a quartic the formulae present themselves in a somewhat different form, by reason that 4 is not a prime number. Attempting to apply it to a quintic, we seek for the equation of which the root is (a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e), [omega] an imaginary fifth root of unity, or rather the fifth power thereof (a + [omega]b + [omega]^2 c + [omega]^3d + [omega]^4 e)^5; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity [omega], [omega]^2, [omega]^3, [omega]^4, viz. the values (a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e)^5, (a + [omega]^2 b + [omega]^4 c + [omega]d + [omega]^3e)^5, (a + [omega]^3 b + [omega]c + [omega]^4 d + [omega]^2e)^5, (a + [omega]^4 b + [omega]^3 c + [omega]^2 d + [omega]e)^5, any symmetrical function of these, for instance their sum, is a 6-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange's resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory.