Elements of Algebra

CHAP. II. ADDITIONS. 505 Therefore the question is reduced to making the product of the quantities $p - \alpha q$, $p - \beta q$, $p - \gamma q$, &c. and $(p - \mu q)^2 + \nu^2 q^2$, $(p - \pi q)^2 + \rho^2 q^2$, &c. the least possible, when $p$ and $q$ are integer, positive numbers. Suppose we have found the values of $p$ and $q$ which answer to the minimum; and if we substitute other smaller numbers for $p$ and $q$, the product in question must acquire a greater value. It will therefore be necessary for each of the factors to increase in value. Now, it is evident, that if $\alpha$, for example, were negative, the factor $p - \alpha q$ would always diminish, when $p$ and $q$ decreased; the same thing would happen to the factor $(p - \mu q)^2 + \nu^2 q^2$, if $\mu$ were negative, and so of the others; whence it follows, that among the simple real factors none but those where the roots are positive, can increase in value; and among the double imaginary factors, those only, in which the real part of the imaginary root is positive, can increase. Farther, it must be remarked, with regard to these last, that in order that $(p - \mu q)^2 + \nu^2 q^2$ may increase, whilst $p$ and $q$ diminish, the part $(p - \mu q)^2$ must necessarily increase, because the other term $\nu^2 q^2$ necessarily diminishes; so that the increase of this factor will depend on the quantity $p - \mu q$; and so of the others. Therefore, the values of $p$ and $q$, which answer to the minimum, must be such, that the quantity $p - \alpha q$ may increase, by giving less values to $p$ and $q$, and taking for $\alpha$ one of the real positive roots of the equation, $Ax^m + Bx^{m-1} + Cx^{m-2} +$, &c. $+ V = 0$, or one of the real positive parts of the imaginary roots of the same equation, if there be any. Let $r$ and $s$ be two integer, positive numbers less than $p$ and $q$; then $r - \alpha s$ must be $< (p - \alpha q)$, abstracting from the sign of the two quantities. Let us therefore suppose, as in Art. 23, that these numbers are such, that $ps - qr = \pm 1$, the upper sign taking place, when $p - \alpha q$ is positive; and the under, when $p - \alpha q$ is negative; so that the two quantities $p - \alpha q$, and $r - \alpha s$, become of different signs, and we shall exactly have the case to which we reduced the preceding problem, Art. 24, and of which we have already given the solution. Hence, by Art. 26, the values of $p$ and $q$ will necessarily be found among the terms of the principal fractions converging towards $a$; that is, towards any one of the quantities, which we have said may be taken for $a$. So that we must reduce all these quantities to continued fractions; which