Elements of Algebra

CHAP. II. ADDITIONS. 503 Then $\mu \angle a$ $\mu' \angle \frac{p^0-aq^0}{aq'-p'} \angle \frac{1}{a-\mu}$ $\mu'' \angle \frac{aq'-p'}{p''-aq''}$ $\mu''' \angle \frac{p''-aq''}{aq'''-p'''}$ $\mu^{iv} \angle \frac{aq'''-p'''}{p^{iv}-aq^{iv}}, \text{\&c.}$ Where the sign $\angle$ denotes the integer number immediately less than the value of the quantity placed after that sign. Thus, we shall successively find all the values of $p$ and $q$ that can satisfy the problem; these values being only the correspondent terms of the two series $p^0, p', p'', p'''$, \&c. and $q^0, q', q'', q'''$, \&c. 26. Corollary 1. If we make $b = \frac{p^0-ap^0}{aq'-p'}$ $c = \frac{aq'-p'}{p''-aq''}$ $d = \frac{p''-aq''}{aq'''-p'''}, \text{\&c.}$ we shall have, as it is easy to perceive, $b = \frac{1}{a-\mu}$ $c = \frac{1}{b-\mu'}$ $d = \frac{1}{c-\mu''}, \text{\&c.}$ and $\mu \angle a$, $\mu' \angle b$, $\mu'' \angle c$, $\mu''' \angle d$, \&c. therefore the numbers $\mu, \mu', \mu''$, \&c. will be no other than those which we have denoted by $\alpha, \beta, \gamma$, \&c. in Art. 3; that is to say, these numbers will be the terms of the continued fraction, which represents the value of $a$; so that we shall have here $a = \mu + \frac{1}{\mu' + \frac{1}{\mu''}} + \text{, \&c.}$ Consequently, the numbers $p', p'', p'''$, \&c. will be the nu-