5. We shall now give the result of the preceding investigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall transform it into another, in which the coefficients are integers, by supposing z = 1/4 v. Thus the equation p p^2 - 4r q^2 z^3 + -- z^2 + -------- z - --- = 0 2 16 64 becomes, after reduction, v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0; it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are 1/4 a, 1/4 b, 1/4 c, so that our rule may now be expressed thus: Let y^4 + py^2 + qy + r = 0 be any biquadratic equation wanting its second term. Form this cubic equation v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0, and find its roots, which let us denote by a, b, c. Then the roots of the proposed biquadratic equation are, when q is negative, when q is positive, y = 1/2([root]a + [root]b + [root]c), y = 1/2(-[root]a - [root]b - [root]c), y = 1/2([root]a - [root]b - [root]c), y = 1/2(-[root]a + [root]b + [root]c), y = 1/2(-[root]a + [root]b - [root]c), y = 1/2([root]a - [root]b + [root]c), y = 1/2(-[root]a - [root]b + [root]c), y = 1/2([root]a + [root]b - [root]c). See also below, _Theory of Equations_, S 17 et seq. (X.) V. _Theory of Equations_. 1. In the subject "Theory of Equations" the term _equation_ is used to denote an equation of the form x^n - p1x^(n - 1) ... [+-] p_n = 0, where p1, p2 ... p_n are regarded as known, and x as a quantity to be determined; for shortness the equation is written [f](x) = 0. The equation may be _numerical_; that is, the coefficients p1, p2^n, ... p_n are then numbers--understanding by number a quantity of the form [alpha] + [beta]i ([alpha] and [beta] having any positive or negative real values whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting [root]-1. Or the equation may be _algebraical_; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers. 1. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.) _Real Theory_. 2. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any); that is, p1, p2, ... p_n are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that a^n - p1a^(n - 1) ... [+-] p_n = 0, or say, [f](a) = 0.