/ [lambda] + 2s[pi] [lambda] + 2s[pi] \ x = [root n][rho] ( cos ----------------- + i sin ----------------- ), \ n n / which has the n values obtained by giving to s the values 0, 1, 2 ... n - 1 in succession; the roots are, it is clear, represented by points lying at equal intervals on a circle. But it is more convenient to proceed somewhat differently; taking one of the roots to be [theta], so that [theta]^n = a, then assuming x = [theta]y, the equation becomes y^n - 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course = 1). And the original equation x^n - a = 0 is thus reduced to the more simple equation x^n - 1 = 0; and although the theory of this equation is included in the preceding one, yet it is proper to state it separately. The equation x^n - 1 = 0 has its several roots expressed in the form 1, [omega], [omega]^2, ... [omega]^(n - 1), where [omega] may be taken = cos 2[pi]/n + i sin 2[pi]/n; in fact, [omega] having this value, any integer power [omega]^k is = cos 2[pi]k/n + i sin 2[pi]k/n, and we thence have ([omega]^k)^n = cos 2[pi]k + i sin 2[pi]k, = 1, that is, [omega]^k is a root of the equation. The theory will be resumed further on. By what precedes, we are led to the notion (a numerical) of the radical a^(1/n) regarded as an n-valued function; any one of these being denoted by [root n]a, then the series of values is [root n]a, [omega][root n]a, ... [omega]^(n - 1)[root n]a; or we may, if we please, use [root n]a instead of a^(1/n) as a symbol to denote the n-valued function. As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations. _Algebraical Equations._ 11. The equation is x^n - p1x^(n-1) + ... [+-]p_n = 0, and we here _assume_ the existence of roots, viz. we assume that there are n quantities a, b, c ... (in general all of them different, but which in particular cases may become equal in sets in any manner), such that x^n - p1x^(n - 1) + ... [+-] p_n = 0; or looking at the question in a different point of view, and starting with the roots a, b, c ... as given, we express the product of the n factors x - a, x - b, ... in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c.... In either case we have p1 = [Sigma]a, p2 = [Sigma]ab, ... p_n = abc ...; i.e. regarding the coefficients p1, p2 ... p_n as given, then we assume the existence of roots a, b, c, ... such that p1 = [Sigma]a, &c.; or, regarding the roots as given, then we write p1, p2, &c., to denote the functions [Sigma]a, [Sigma]ab, &c.