Elements of Algebra

482 ADDITIONS. CHAP. I. 16. Let us, therefore, consider the series $\frac{A}{A'}, \frac{B}{B'}, \frac{C}{C'}, \frac{D}{D'}, \ldots$ in which the fractions are alternately less and greater than the quantity $a$, and which, it is evident, we may divide into these two series: $$ \frac{A}{A'}, \frac{C}{C'}, \frac{E}{E'}, \ldots $$ $$ \frac{B}{B'}, \frac{D}{D'}, \frac{F}{F'}, \ldots $$ of which the first will be composed of fractions all less than $a$, and which go on increasing towards the quantity $a$; the second will be composed of fractions all greater than $a$, but which go on diminishing towards that same quantity. Let us therefore examine each of those two series separately. In the first, we have (Art. 10, and 12), $$ \frac{C}{C'} - \frac{A}{A'} = \frac{\gamma}{A'C'} $$ $$ \frac{E}{E'} - \frac{C}{C'} = \frac{\varepsilon}{C'E'}, \ldots $$ and in the second we have, $$ \frac{B}{B'} - \frac{D}{D'} = \frac{\delta}{B'D'} $$ $$ \frac{D}{D'} - \frac{F}{F'} = \frac{\zeta}{D'F'}, \ldots $$ Now, if the numbers $\gamma, \delta, \varepsilon, \ldots$ were all equal to unity, we might prove, as in Art. 12, that between any two consecutive fractions of either of the preceding series, there could never be found any other fraction, whose denominator would be less than the denominators of those two fractions; but it will not be the same, when the numbers $\gamma, \delta, \varepsilon, \ldots$ are greater than unity; for, in that case, we may insert between the fractions in question as many intermediate fractions as there are units in the numbers $\gamma - 1, \delta - 1, \varepsilon - 1, \ldots$ and for this purpose we shall only have to substitute, successively, in the values of $c$ and $c'$, (Art. 10), the numbers, $1, 2, 3, \ldots \gamma$, instead of $\gamma$; and, in the values of $D$ and $D'$, the numbers $1, 2, 3, \ldots \delta$, instead of $\delta$, and so on. 17. Suppose, for example, that $\gamma = 4$, we have $c = 4B + A$ and $c' = 4B' + A'$, and we may insert between the fractions $\frac{A}{A'}$ and $\frac{C}{C'}$, three intermediate fractions, which will be