The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows:--Because vz = -(1/3)q, therefore z = -(1/3)q X 1/v = -(1/3)q / [root 3]A; hence v + z = [root 3]A - (1/3)q / [root 3]A; thus it appears that the three values of y may also be expressed thus: y = [root 3]A - (1/3)q / [root 3]A y = [alpha][root 3]A - (1/3)q[beta] / [root 3]A y = [beta][root 3]A - (1/3)q[alpha] / [root 3]A. See below, _Theory of Equations_, SS 16 et seq. IV. _Biquadratic Equations_. 1. When a biquadratic equation contains all its terms, it has this form, x^4 + Ax^3 + Bx^2 + Cx + D = 0, where A, B, C, D denote known quantities. We shall first consider pure biquadratics, or such as contain only the first and last terms, and therefore are of this form, x^4 = b^4. In this case it is evident that x may be readily had by two extractions of the square root; by the first we find x^2 = b^2, and by the second x = b. This, however, is only one of the values which x may have; for since x^4 = b^4, therefore x^4 - b^4 = 0; but x^4 - b^4 may be resolved into two factors x^2 - b^2 and x^2 + b^2, each of which admits of a similar resolution; for x^2 - b^2 = (x - b)(x + b) and x^2 + b^2 = (x - b[root]-1)(x + b[root]-1). Hence it appears that the equation x^4 - b^4 = 0 may also be expressed thus, (x - b)(x + b)(x - b[root]-1)(x + b[root]-1) = 0; so that x may have these four values, +b, -b, +b[root]-1, -b[root]-1, two of which are real, and the others imaginary. 2. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form, x^4 + qx^2 + s = 0. These may be resolved in the manner of quadratic equations; for if we put y = x^2, we have y^2 + qy + s = 0, from which we find y = 1/2{-q [+-] [root](q^2 - 4s)}, and therefore x = [+-][root]1/2{-q [+-] [root](q^2 - 4s)}. 3. When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the _Petersburg Commentaries_, and afterwards explained more fully in his _Elements of Algebra_. We have already explained how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form, y^4 + py^2 + qy + r = 0, where the second term is wanting, and where p, q, r denote any known quantities whatever. That we may form an equation similar to the above, let us assume y = [root]a + [root]b + [root]c, and also suppose that the letters a, b, c denote the roots of the cubic equation z^3 + Pz^2 + Qz - R = 0;