Elements of Algebra

CHAP. II. ADDITIONS. 501 must be an integer, positive number, we may prove, that the quantity $\frac{aq^{iv}-p^{iv}}{p''' - aq'''}$ will necessarily be positive, and greater than unity, and that $\mu'''$ can only be the integer number immediately below the same quantity; and so on. It follows, 1st, that the quantities $p - aq$, $p' - aq'$, $p'' - aq''$, \ldots will successively have different signs; that is, alternately positive and negative, and will form a series continually increasing. 2dly, that if we denote by the sign $\angle$ the integer number, which is immediately less than the value of the quantity placed after that sign, we shall have, for the determination of the numbers $\mu, \mu', \mu'', \ldots$ $\mu \angle \frac{aq''-p''}{p'-aq'}$ $\mu' \angle \frac{aq'''-p'''}{p''-aq''}$ $\mu'' \angle \frac{aq^{iv}-p^{iv}}{p'''-aq'''}$. Now, we have already seen, that the series $q, q', q'', \ldots$ must terminate in 0; and that then the preceding term will be 1, and the term corresponding to 0 in the other series $p, p', p'', \ldots$ will be $= \pm 1$ at pleasure. For example, let us suppose that $q^{iv} = 0$, we shall then have $q''' = 1$, and $p^{iv} = 1$; therefore $p''' - aq''' = p''' - a$, and $p^{iv} - aq^{iv} = 1$; therefore $p''' - a$ must be a negative quantity, and less than 1, abstracting from the sign; that is, $a - p'''$ must be $\succ 0$, and $\angle 1$; so that $p'''$ must be the integer number immediately below $a$; we shall therefore know the values of these four terms, $p^{iv} = 1 \qquad q^{iv} = 0$ $p''' \angle a \qquad q''' = 1$ by means of which, going back through the former formulæ, we may find all the preceding terms. We shall first have the value of $\mu''$, then we shall have $p''$ and $q''$, by the formulæ, $p'' = \mu'' p''' + p^{iv}$, and $q'' = \mu'' q''' + q^{iv}$; from which we shall get $\mu'$, and then $p'$ and $q'$; and so of the rest. In general, let $q^\rho = 0$, then we shall have $q^{\rho-1} = 1$, and $p^\rho = 1$; and shall prove, as before, that $p^{\rho-1}$ can only be the