Elements of Algebra

502 ADDITIONS. CHAP. II. integer number immediately below $a$; so that we shall have these four terms, $p^{\rho} = 1$ $q^{\rho} = 0$ $p^{\rho-1} < a$ $q^{\rho-1} = 1$; we shall then have $\mu^{\rho-2} < \frac{aq^{\rho}-p^{\rho}}{p^{\rho-1}-aq^{\rho-1}} < \frac{1}{a-p^{\rho-1}}$ $p^{\rho-2} = \mu^{\rho-2}p^{\rho-1} + p^{\rho}, q^{\rho-2} = \mu^{\rho-2}q^{\rho-1} + q^{\rho}$ $\mu^{\rho-3} < \frac{aq^{\rho-1}-p^{\rho-1}}{p^{\rho-2}-aq^{\rho-2}}$ $p^{\rho-3} = \mu^{\rho-3}p^{\rho-2} + p^{\rho-1}, q^{\rho-3} = \mu^{\rho-3}q^{\rho-2} + q^{\rho-1},$ and so on. In this manner, therefore, we may go back to the first terms, $p$ and $q$; but it must be observed, that all the succeeding terms, $p', q', p'', q'', \ldots$ possess the same properties, and serve equally to resolve the problem proposed. For it is evident, in the preceding formulæ, that the numbers $p, p', p'', \ldots$ and $q, q', q'', \ldots$ are all integer and positive, and form two series continually decreasing; the first of which is terminated by unity, and the second by 0. Farther, we have seen that these numbers are such, that $pq' - qp' = \pm 1$, $p'q'' - q'p'' = \mp 1, \ldots$ and that the quantities $p - aq, p' - aq', p'' - aq'', \ldots$ are alternately positive and negative, and at the same time form a series continually increasing. Whence it follows, that the same conditions which exist among the four numbers $p, q, r, s$, or $p, q, p', q'$, and on which, as we have seen, the solution of the problem depends, equally exist among the numbers $p', q', p'', q''$, and among these, $p'', q'', p''', q'''$, and so on. Therefore, beginning with the last terms $p^{\rho}$ and $q^{\rho}$, and going back always by the formulæ we have just found, we shall successively have all the values of $p$ and $q$ that can resolve the question proposed. 25. As the values of the terms $p^{\rho}, p^{\rho-1}, \ldots$ $q^{\rho}, q^{\rho-1}, \ldots$ are independent of the exponent, $\rho$, we may abstract from it, and denote the terms of these two increasing series thus, $p^0, p', p'', p''', p^{iv}, \ldots$ $q^0, q', q'', q''', q^{iv}, \ldots$ so that we shall have the following results, $p^0 = 1$ $q^0 = 0$ $p' = \mu$ $q' = 1$ $p'' = \mu' p' + 1$ $q'' = \mu'$ $p''' = \mu'' p'' + p'$ $q''' = \mu'' q'' + q'$ $p^{iv} = \mu''' p''' + p''$ $q^{iv} = \mu''' q''' + q''$ $\ldots$ $\ldots$