_Case_ 4. The expression a + bx + cx^2 may be transformed into a square as often as it can be resolved into two parts, one of which is a complete square, and the other a product of two simple factors; for then it has this form, p^2 + qr, where p, q and r are quantities which contain no power of x higher than the first. Let us assume [root](p^2 + qr) = p + mq; thus we have p^2 + qr = p^2 + 2mpq + m^2q^2 and r = 2mp + m^2q, and as this equation involves only the first power of x, we may by proper reduction obtain from it rational values of x and y, as in the three foregoing cases. The application of the preceding general methods of resolution to any particular case is very easy; we shall therefore conclude with a single example. _Ex._ It is required to find two square numbers whose sum is a given square number. Let a^2 be the given square number, and x^2, y^2 the numbers required; then, by the question, x^2 + y^2 = a^2, and y = [root](a^2 - x^2). This equation is evidently of such a form as to be resolvable by the method employed in case 1. Accordingly, by comparing [root](a^2 - x^2) with the general expression [root](g^2 + bx + cx^2), we have g = a, b = 0, c = -1, and substituting these values in the formulae, and also -n for +m, we find 2an a(n^2 - 1) x = -------, y = ----------. n^2 + 1 n^2 + 1 If a = n^2 + 1, there results x = 2n, y = n^2 - 1, a = n^2 + 1. Hence if r be an even number, the three sides of a rational right-angled triangle are r, (1/2r)^2 - 1, (1/2r)^2 + 1. If r be an odd number, they become (dividing by 2) r, 1/2(r^2 - 1), 1/2(r^2 + 1). For example, if r = 4, 4, 4 - 1, 4 + 1, or 4, 3, 5, are the sides of a right-angled triangle; if r = 7, 7, 24, 25 are the sides of a right-angled triangle. III. _Cubic Equations_. 1. Cubic equations, like all equations above the first degree, are divided into two classes: they are said to be _pure_ when they contain only one power of the unknown quantity; and _adfected_ when they contain two or more powers of that quantity. Pure cubic equations are therefore of the form x^3 = r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube root of each side of the equation. Let us consider the equation x^3 - c^3 = 0 more fully. This is decomposable into the factors x - c = 0 and x^2 + cx + c^2 = 0. The roots of this quadratic equation are 1/2(-1 [+-] [root]-3)c, and we see that the equation x^3 = c^3 has three roots, namely, one real root c, and two imaginary roots 1/2(-1 [+-] [root]-3)c. By making c equal to unity, we observe that 1/2(-1 [+-] [root]-3) are the imaginary cube roots of unity, which are generally denoted by [omega] and [omega]^2, for it is easy to show that (1/2(-1 - [root]-3))^2 = 1/2(-1 + [root]-3).