In the most simple case, n = 5, the equation to be solved is x^4 + x^3 + x^2 + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r^2, r^4, r^3. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r^4 and r^2 + r^3 respectively;--these being such that the symmetrical functions P1 + P2, P1P2 are rationally determinable: in fact P1 + P2 = -1, P1P2 = (r + r^4)(r^2 + r^3), = r^3 + r^4 + r^6 + r^7, = r^3 + r^4 + r + r^2, = -1. P1, P2 are thus the roots of u^2 + u - 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r^4 for P1, r^2 and r^3 for P2; the symmetrical functions of these are then rationally determined in terms of P1 and P2; thus r + r^4 = P1, r.r^4 = 1, or r, r^4 are the roots of u^2 - P1u + 1 = 0. The mode of division is more clearly seen for a larger value of n; thus, for n = 7 a prime root is = 3, and the arrangement of the roots is r, r^3, r^2, r^6, r^4, r^5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r^6, r^3 + r^4, r^2 + r^5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r^2 + r^4, r^3 + r^6 + r^5 respectively; in each ease the symmetrical functions of the periods are rationally determinable: thus in the case of the two periods P1 + P2 = -1, P1P2 = 3 + r + r^2 + r^3 + r^4 + r^5 + r^6, = 2; and the periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r + r^2 + r^4 = P1, r.r^2 + r.r^4 + r^2.r^4 = P2, r.r^2.r^4 = 1. The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, x^(n - 1) + x^(n - 2) + ... + x + 1 = 0 (the roots a, b, c ... being the several powers of r, the indices in geometrical progression as above), showed that the function (a + [omega]b + [omega]^2 c + ...)^(n - 1) was in this case a given function of [omega] with integer coefficients. Reverting to the before-mentioned particular equation x^4 + x^3 + x^2 + x + 1 = 0, it is very interesting to compare the process of solution with that for the solution of the general quartic the roots whereof are a, b, c, d. Take [omega], a root of the equation [omega]^4 - 1 = 0 (whence [omega] is = 1, -1, i, or -i, at pleasure), and consider the expression (a + [omega]b + [omega]^2 c + [omega]^3 d)^4, the developed value of this is = a^4 + b^4 + c^4 + d^4 + 6(a^2 c^2 + b^2 d^2) + 12(a^2 bd + b^2 ca + c^2 db + d^2ac) +[omega] {4(a^3 b + b^3 c + c^3 + d^3 a) + 12(a^2 cd + b^2 da + c^2 ab + d^2 bc)} +[omega]^2{6(a^2 b^2 + b^2 c^2 + c^2 d^2 + d^2 a^2) + 4(a^3 c + b^3 d + c^3 a + d^3 b) + 24abcd} +[omega]^3{4(a^3 d + b^3 a + c^3 b + d^3 c) + 12(a^2 bc + b^2 cd + c^2 da + d^2 ab)}