Elements of Algebra

524 ADDITIONS. CHAP. II. And the fourth fraction, the tenth, \ldots will give the values of $p$ and $q$; which will therefore be $p = 39, q = 7$, or $p = 6225, q = 1118$, \ldots In this manner, therefore, we may regularly find all the values of $p$ and $q$, that will make the given formula $= -3$, the least value it can receive. We might even have a general value, which would comprehend all these values of $p$ and $q$. Any person who has the curiosity may find it by a method which we have elsewhere explained, and which has been already noticed (Art. 35). We have just found, that the minimum of the quantity proposed is $-3$, and consequently negative; now, it might be proposed to find the least positive value, that the same quantity can receive: we should then only have to examine the series $P^0, P', P'', P'''$, \ldots in the two cases, and we should see that the least positive term is $5$ in both cases; and as in the first case it is $p^{iv}$, and in the second $p'''$, which is $5$, the values of $p$ and $q$, that will give the least positive value of the quantity proposed, will be $p^{iv}, q^{iv}$, or $p^x, q^x$, or \ldots in the first case, and $p''', q'''$, or $p^{xi}, q^{xi}$, \ldots in the second; so that we shall have, from the above fractions, $p = 83, q = 11$; or $p = 13291, q = 1762$, \ldots or $p = 11, q = 2$; $p = 1843, q = 331$, \ldots We must not forget to observe, that the numbers, $\mu, \mu', \mu''$, \ldots found in the above two cases, are no other than the terms of the continued fractions, which represent the two roots of the equation $9x^2 - 118x + 378 = 0$. So that these roots will be, $7 + \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3} + \ldots$ $5 + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3} + \frac{1}{1} + \frac{1}{1} + \ldots$ expressions which we might continue to infinity merely by repeating the same numbers. Thus, we perceive how we are to set about reducing to continued fractions the roots of every equation of the second degree. 41. Scholium. In volume XI. of the New Commentaries of Petersburg, M. EULER has given a method similar to the preceding; but deduced from principles somewhat different, for reducing to a continued fraction the root of any integer number, not a square, and has added a Table, in which the continued fractions are calculated for all the