Elements of Algebra

516 ADDITIONS. CHAP. II. the sign $\angle$ placed after the numbers $\mu'''$, $\mu^{\text{iv}}$, $\mu^{\text{v}}$, $\ldots$ denoting as before, the integer numbers which are immediately under the quantities which follow that same sign. Now, by reductions similar to those of Art. 33, it is easy to transform the quantities $\frac{p^{\text{v}}-bq^{\text{v}}}{p^{\text{iv}}-bq^{\text{iv}}}$, $\frac{p^{\text{vi}}-bq^{\text{vi}}}{p^{\text{v}}-bq^{\text{v}}}$, $\ldots$ into these, $\frac{Q^{\text{v}}+\frac{1}{2}\sqrt{E}}{p^{\text{iv}}}$, $\frac{Q^{\text{vi}}-\frac{1}{2}\sqrt{E}}{p^{\text{v}}}$, $\ldots$. Farther, the condition of $\frac{p''' - bq'''}{p^{\text{iv}} - bq^{\text{iv}}} \angle 1$ may be reduced to this, $\frac{-p'''}{p^{\text{iv}}} \angle \frac{aq''' - p'''}{p^{\text{iv}} - aq^{\text{iv}}}$; which, because $\frac{aq''' - p'''}{p^{\text{iv}} - aq^{\text{iv}}} > 1$, will certainly take place, when $\frac{-p'''}{p^{\text{iv}}} = \text{or} \angle 1$; wherefore we shall have $\mu^{\text{iv}} \angle \frac{Q^{\text{v}}+\frac{1}{2}\sqrt{E}}{p^{\text{iv}}}$, if $\frac{-p'''}{p^{\text{iv}}} = \text{or} \angle 1$. $\mu^{\text{v}} \angle \frac{Q^{\text{vi}}-\frac{1}{2}\sqrt{E}}{p^{\text{v}}}$, $\mu^{\text{vi}} \angle \frac{Q^{\text{vii}}+\frac{1}{2}\sqrt{E}}{p^{\text{vi}}}$, $\ldots$. Combining now these formulæ with those of Art. 33, which contain the law of the series $p'$, $p''$, $p'''$, $\ldots$ and $q'$, $q''$, $q'''$, $\ldots$ we shall easily see, that, if two corresponding terms of these two series be supposed to be given, the rank of which is higher than 3, we may go back to the preceding terms, as far as $p^{\text{iv}}$ and $q^{\text{iv}}$, and even to the terms $p'''$ and $q'''$, if the condition of $\frac{-p'''}{p^{\text{iv}}} = \text{or} \angle 1$ takes place; so that all these terms will be absolutely determined by those which we have supposed to be given. For example, knowing $p^{\text{vi}}$, and $q^{\text{vi}}$, we shall immediately know $p^{\text{v}}$ from the equation $Q^{\text{vi}} - p^{\text{v}}p^{\text{vi}} = \frac{1}{2}E$; then, having $q^{\text{vi}}$ and $p^{\text{v}}$, we shall find the value of $\mu^{\text{v}}$; by means of which we shall next find the value of $q^{\text{v}}$ from the equation $q^{\text{vi}} = \mu^{\text{v}}p^{\text{v}} + q^{\text{v}}$. Now, the equation $q^{\text{v}} - p^{\text{iv}}p^{\text{v}} = \frac{1}{2}E$, will give $p^{\text{iv}}$; and if we previously know, that $\frac{-p'''}{p^{\text{iv}}}$ must be $= \text{or} \angle 1$, we shall find $\mu^{\text{iv}}$; after which, we shall have $q^{\text{iv}}$ from