Elements of Algebra

500 ADDITIONS. CHAP. II. $\mu = \frac{aq''-p''}{p'-aq'} + \frac{p-aq}{p'-aq'}$ $\mu' = \frac{aq'''-p'''}{p''-aq''} + \frac{p'-aq'}{p''-aq''}$ $\mu'' = \frac{aq^{iv}-p^{iv}}{p'''-aq'''} + \frac{p''-aq''}{p'''-aq'''}$ $\mu''' = \frac{aq^v-p^v}{p^{iv}-aq^{iv}} + \frac{p'''-aq'''}{p^{iv}-aq^{iv}}$, &c. Now, as by hypothesis the quantities $p-aq$, and $p'-aq'$, are of different signs; and farther, as $p'-aq'$ (abstracting from the signs) must be greater than $p-aq$, it follows that $\frac{p-aq}{p'-aq'}$ will be a negative quantity, and less than unity. Therefore, in order that $\mu$ may be an integer, positive number (as it must), it is evident, that $\frac{aq''-p''}{p'-aq'}$ must be a positive quantity greater than unity; and it is obvious, at the same time, that $\mu$ can only be the integer number, that is immediately less than $\frac{aq''-p''}{p'-aq'}$; that is to say, contained between the limits $\frac{aq''-p''}{p'-aq'}$, and $\frac{aq''-p''}{p'-aq'} - 1$; for since $-\frac{p-aq}{p'-aq'} > 0$, and $< 1$, we shall have $\mu < \frac{aq''-p''}{p'-aq'}$, and $> \frac{aq''-p''}{p'-aq'} - 1$. Also, since we have seen, that $\frac{aq''-p''}{p'-aq'}$ must be a positive quantity greater than unity, it follows that $\frac{p'-aq'}{p''-aq''}$ will be a negative quantity less than unity, (I say less than unity, abstracting from the sign). Wherefore, in order that $\mu'$ may be an integer, positive number, $\frac{aq'''-p'''}{p''-aq''}$ must be a positive quantity greater than unity, and consequently the number $\mu'$ can only be the integer number, which will be immediately below the quantity $\frac{aq'''-p'''}{p''-aq''}$. In the same manner, and from the consideration, that $\mu''$