Elements of Algebra

504 ADDITIONS. CHAP. II. merators, and $q'$, $q''$, $q'''$, \ldots the denominators of the fractions converging to $a$, fractions which we have already denoted by $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, \ldots (Art. 10). So that the whole is reduced to converting the value of $a$ into a continued fraction, having all its terms positive; which may be done by the methods already explained, provided we are always careful to take the approximated values too small; then we shall only have to form the series of principal fractions converging towards $a$, and the terms of each of these fractions will give the values of $p$ and $q$, which will resolve the problem proposed; so that $\frac{p}{q}$ can only be one of these fractions. 27. *Corollary 2.* Hence results a new property of the fractions we speak of; calling $\frac{p}{q}$ one of the principal fractions converging towards $a$, (provided they are deduced from a continued fraction, all the terms of which are positive), the quantity $p - aq$ will always have a less value (abstracting from the sign), than it would have, were we to substitute in the room of $p$ and $q$ any other smaller numbers. 28. *Problem 2.* The quantity $Ap^m + bp^{m-1}q + cp^{m-2}q^2 + \ldots + vq^m$, being proposed, in which A, B, C, \ldots are given integers, positive or negative, and $p$ and $q$ unknown numbers, which must be integer and positive; it is required to determine what values we must give to $p$ and $q$, in order that the quantity proposed may become the least possible. Let $\alpha, \beta, \gamma, \ldots$ be the real roots, and $\mu \pm \nu \sqrt{-1}$, $\pi \pm \rho \sqrt{-1}$, \ldots the imaginary roots of the equation $Ax^m + Bx^{m-1} + cx^{m-2} + \ldots + V = 0$, then we shall have, by the theory of equations, $Ap^m + Bp^{m-1}q + cp^{m-2}q^2 + \ldots + vq^m =$ $A(p - \alpha q) \times (p - \beta q) \times (p - \gamma q)\ldots \times$ $(p - (\mu + \nu \sqrt{-1})q) \times (p - (\mu - \nu \sqrt{-1})q) \times$ $(p - (\pi + \rho \sqrt{-1})q) \times (p - (\pi - \rho \sqrt{-1})q)\ldots =$ $A(p - \alpha q) \times (p - \beta q) \times (p - \gamma q)\ldots \times$ $((p - \mu q)^2 + \nu^2 q^2) \times ((p - \pi q)^2 + \rho^2 q^2)^*\ldots$ * Because $(p - (\mu + \nu \sqrt{-1})q) \times (p - (\mu - \nu \sqrt{-1})q) = p^2 - 2p\mu q + \mu^2 q^2 + \nu^2 q^2 = (p - \mu q)^2 + \nu^2 q^2$, and the same with the others. B.