Elements of Algebra

506 ADDITIONS. CHAP. II. may easily be done by the methods elsewhere taught, and then deduce the converging fractions required: after which, we must successively make $p$ equal to all the numerators of these fractions, and $q$ equal to the corresponding denominators, and of these suppositions, that which shall give the least value of the proposed function will necessarily answer likewise to the minimum required. 29. *Scholium* 1. We have supposed that the numbers $p$ and $q$ must both be positive; it is evident that if we were to take them both negative, no change would result in the absolute value of the formula proposed; it would only change its sign in the case of the exponent $m$ being odd; and it would remain quite the same, in the case of the exponent $m$ being even: so that it is of no consequence what signs we give the numbers $p$ and $q$, when we suppose them both of the same kind. But it will not be the same, if we give different signs to $p$ and $q$; for then the alternate terms of the equation proposed will change their signs, which will also change the signs of the roots $\alpha$, $\beta$, $\gamma$, &c. $\mu \pm \nu \sqrt{-1}$, $\pi \pm \varrho \sqrt{-1}$, &c. so that those of the quantities $\alpha$, $\beta$, $\gamma$, &c. $\mu$, $\pi$, &c. which were negative, and consequently useless in the first case, will become positive in this, and must be employed instead of the other. Hence, I conclude, generally, that when we investigate the minimum of the proposed formula, without any other restriction, than that of $p$ and $q$ being whole numbers, we must successively take for $a$ all the real roots $\alpha$, $\beta$, $\gamma$, &c. and all the real parts $\mu$, $\pi$, &c. of the imaginary roots of the equation $Ax^m + Bx^{m-1} + Cx^{m-2} + \text{, \&c. } + V = 0$; abstracting from the signs of these quantities; but then we must give the same signs, or different signs, to $p$ and $q$, according as the quantity we have taken for $a$, had originally the positive, or the negative sign. 30. *Scholium* 2. When among the real roots $\alpha$, $\beta$, $\gamma$, &c. there are some commensurable, then it is evident that the quantity proposed will become nothing, by making $\frac{p}{q}$ equal to one of these roots; so that in this case, properly speaking, there will be no *minimum*. In all the other cases, it will be impossible for the quantity in question to become $0$, whilst $p$ and $q$ are whole numbers. Now, as the coefficients $A$, $B$, $C$, &c. are also whole numbers, *by hypothesis*, this quantity will always be equal to a whole number; and, consequently, it can never be less than unity.