that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in fact, solvable by radicals; but this is not here material). If, however, a, b, c, d denote the roots r, r^2, r^4, r^3 of the special equation, then the expression becomes r^4 + r^3 + r + r^2 + 6(1 + 1)+12(r^2 + r^4 + r^3 + r) + [omega] {4(1 + 1 + 1 + 1) + 12(r^4 + r^3 + r + r^2)} + [omega]^2{6(r + r^2 + r^4 + r^3) + 4(r^2 + r^4 + r^3 + r)} + [omega]^3{4(r + r^2 + r^4 + r^3) + 12(r^3 + r + r^2 + r^4)} viz. this is = -1 + 4[omega] + 14[omega]^2 - 16[omega]^3, a completely determined value. That is, we have (r + [omega]r^2 + [omega]^2 r^4 + [omega]^3 r^3) = -1 + 4[omega] + 14[omega]^2 - 16[omega]^3, which result contains the solution of the equation. If [omega] = 1, we have (r + r^2 + r^4 + r^3)^4 = 1, which is right; if [omega] = -1, then (r + r^4 - r^2 - r^3)^4 = 25; if [omega] = i, then we have {r - r^4 + i(r^2 - r^3)}^4 = -15 + 20i; and if [omega] = -i, then {r - r^4 - i(r^2 - r^3)}^4 = -15 - 20i; the solution may be completed without difficulty. The result is perfectly general, thus:--n being a prime number, r a root of the equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0, [omega] a root of [omega]^(n - 1) - 1 = 0, and g a prime root of g^(n - 1) [equivalent] 1 (mod. n), then {r + [omega]r^g + ... + [omega]^(n - 2) r^g^(n - 2)}^(n - 1) is a given function M0 + M1[omega] ... + M_(n - 2)[omega]^(n - 2) with integer coefficients, and by the extraction of (n - 1)th roots of this and similar expressions we ultimately obtain r in terms of [omega], which is taken to be known; the equation x^n - 1 = 0, n a prime number, is thus solvable by radicals. In particular, if n - 1 be a power of 2, the solution (by either process) requires the extraction of square roots only; and it was thus that Gauss discovered that it was possible to construct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting developments in regard to the theory were obtained by C.G.J. Jacobi (1837); see the memoir "Ueber die Kreistheilung, u.s.w.," _Crelle_, t. xxx. (1846). The equation x^(n - 1) + ... + x + 1 = 0 has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by N.H. Abel (1828), and since called Abelian equations, viz. for the Abelian equation of the order n, if x be any root, the roots are x, [theta]x, [theta]^2 x, ... [theta]^(n - 1)x ([theta]x being a rational function of x, and [theta]^nx = x); the theory is, in fact, very analogous to that of the above particular case.