Elements of Algebra

CHAP. II. ADDITIONS. 515 Whence, because $\mu^{iv}$, $\mu^v$, &c. are positive numbers, it is evident that all the quantities $p^v - bq^v$, $p^{vi} - bq^{vi}$, &c. to infinity, will have the same signs as the quantities $p''' - bq'''$, and $p^{iv} - bq^{iv}$; consequently, all the terms $p'''$, $p^{iv}$, $p^v$, &c. to infinity, will alternately have the signs *plus* and *minus*. From the preceding equations, we shall now have $\mu^{iv} = \frac{p^v - bq^v}{p^{iv} - bq^{iv}} - \frac{p''' - bq'''}{p^{iv} - bq^{iv}}$ $\mu^v = \frac{p^{vi} - bq^{vi}}{p^v - bq^v} - \frac{p^{iv} - bq^{iv}}{p^v - bq^v}$ $\mu^{vi} = \frac{p^{vii} - bq^{vii}}{p^{vi} - bq^{vi}} - \frac{p^v - bq^v}{p^{vi} - bq^{vi}}$ &c. where the quantities, $\frac{p''' - bq'''}{p^{iv} - bq^{iv}}$, $\frac{p^{iv} - bq^{iv}}{p^v - bq^v}$, &c. will be all positive. Wherefore, since the numbers $\mu^{iv}$, $\mu^v$, $\mu^{vi}$, &c. must be all positive integers, by hypothesis, the quantity $\frac{p^v - bq^v}{p^{iv} - bq^{iv}}$ must be positive, and $> 1$; so also must the quantities $\frac{p^{vi} - bq^{vi}}{p^v - bq^v}$, $\frac{p^{vii} - bq^{vii}}{p^{vi} - bq^{vi}}$, &c.; wherefore the quantities $\frac{p''' - bq'''}{p^{iv} - bq^{iv}}$, $\frac{p^{iv} - bq^{iv}}{p^v - bq^v}$, &c. will be positive, and less than unity; so that the numbers $\mu^v$, $\mu^{vi}$, &c. can only be the integer numbers, which are immediately less than the values of $\frac{p^{vi} - bq^{vi}}{p^v - bq^v}$, $\frac{p^{vii} - bq^{vii}}{p^{vi} - bq^{vi}}$, &c. As to the number $\mu^{iv}$, it will also be equal to the integer number, which is immediately less than the value of $\frac{p^v - bq^v}{p^{iv} - bq^{iv}}$, whenever we have $\frac{p''' - bq'''}{p^{iv} - bq^{iv}} < 1$. Thus, we shall have $\mu^{iv} < \frac{p^v - bq^v}{p^{iv} - bq^{iv}}$ if $\frac{p''' - bq'''}{p^{iv} - bq^{iv}} < 1$. $\mu^v < \frac{p^{vi} - bq^{vi}}{p^v - bq^v}$, $\mu^{vi} < \frac{p^{vii} - bq^{vii}}{p^{vi} - bq^{vi}}$, &c. L L 2