Elements of Algebra

514 ADDITIONS. CHAP. II. $ \mu^{\pi+2\varrho} = \mu^\pi $, then $ Q^{\pi+2\varrho+1} = Q^{\pi+1} $, and $ P^{\pi+2\varrho+1} = P^{\pi+1} $; whence, also, $ \mu^{\pi+2\varrho+1} = \mu^{\pi+1} $, and so on. So that, if $ \Pi $ is any number equal, or greater than $ \pi $, and $ m $ denotes any integer positive number, we shall have, in general, $ P^{\Pi+2m\varrho} = P^\Pi, Q^{\Pi+2m\varrho} = Q^\Pi, \mu^{\Pi+2m\varrho} = \mu^\Pi $; therefore, by knowing the $ \pi + 2\varrho $ leading terms of each of the three series, we shall likewise know all the succeeding, which will be only the $ 2\varrho $ last terms repeated, in the same order, to infinity. From all this it follows, that, in order to find the least value of $ P = Ap^2 + Bpq + Cq^2 $, it is sufficient to continue the series $ P^0, P', P'', \&c. $ and $ Q^0, Q', Q'', \&c. $ until two corresponding terms, as $ P^\pi $ and $ Q^\pi $ appear again together, after an even number of intermediate terms, so that we may have $ P^{\pi+2\varrho} = P^\pi $, and $ Q^{\pi+2\varrho} = Q^\pi $; then the least term of the series $ P^0, P', P'', \&c. P^{\pi+2\varrho} $ will be the minimum required. 35. Corollary 1. If the least term of the series $ P^0, P', P'', \&c. P^{\pi+2\varrho} $ is not found before the term $ P^\pi $, then that term will be repeated an infinite number of times in the same series infinitely prolonged; so that we shall then have an infinite number of values of $ p $ and $ q $ answering to the minimum, and all discoverable by the formulæ of Art. 25, by continuing the series of the numbers $ \mu', \mu'', \mu''', \&c. $ beyond the term $ \mu^{\pi+2\varrho} $ by the repetition of the same terms $ \mu^{\pi+1}, \mu^{\pi+2} $, as we have already said. In this case we may likewise have general formulæ representing all the values of $ p $ and $ q $ in question; but an explanation of the method for arriving at this, would carry me too far; for the present, I shall only refer to the *Memoires de Berlin* already quoted, ann. 1768, page 123, &c. where will be found a general and new theory of periodical continued fractions. 36. Corollary 2. We have demonstrated (Art. 34), that, by continuing the series $ r', r'', r''', \&c. $ we ought to find consecutive terms with different signs. Let us suppose, therefore, for example, that $ r''' $ and $ r^{iv} $ are the first two terms, with this property. We shall necessarily have the two quantities $ p''' - bq''' $, and $ p^{iv} - bq^{iv} $, with the same signs, because the quantities $ p''' - aq''' $, and $ p^{iv} - aq^{iv} $, have from their nature different signs. Now, by putting in the quantities $ p^v - bq^v $, $ p^{vi} - bq^{vi} $, &c. the values of $ p^v, p^{vi}, \&c. q^v, q^{vi}, \&c. $ (Art. 25), we shall have $ p^v - bq^v = \mu^{iv}(p^{iv} - tq^{iv}) + p''' - bq''' $ $ p^{vi} - bq^{vi} = \mu^v(p^v - bq^v) + p^{iv} - bq^{iv}, \&c. $