Introductio in analysin infinitorum

LIB. I. 42 DE TRANSFORMATIONE FUNCTIONUM minationes $z = (\frac{ax^\alpha + bx^\beta + cx^\gamma + \&c.}{A + Bx^\mu + Cx^\nu + \&c.})^{p/v}$, atque $y = x (\frac{ax^\alpha + bx^\beta + cx^\gamma + \&c.}{A + Bx^\mu + Cx^\nu + \&c.})^{q/r}$; eritque $y^p = x^p z^q$; & hinc $x = yz^{-q/p}$. Cum igitur fit $z^{r/p} = \frac{ax^\alpha + bx^\beta + cx^\gamma + \&c.}{A + Bx^\mu + Cx^\nu + \&c.}$, si loco $x$ ejus valorem $yz^{-q/p}$ substituamus; prodibit ista æquatio $z^{r/p} = \frac{ay^\alpha z^{-\alpha q/p} + by^\beta z^{-\beta q/p} + cy^\gamma z^{-\gamma q/p} + \&c.}{A + By^\mu z^{-\mu q/p} + Cy^\nu z^{-\nu q/p} + \&c.}$; quæ reducitur ad hanc $Az^{r/p} + By^\mu z^{(r-\mu q)/p} + Cy^\nu z^{(r-\nu q)/q} + \&c. = ay^\alpha z^{-\alpha q/p} + by^\beta z^{-\beta q/p} + cy^\gamma z^{-\gamma q/p} + \&c.$ quæ multiplicata per $z^{\alpha q/p}$ transibit in hanc: $Az^{(\alpha q+r)/p} + By^\mu z^{(\alpha q-\mu q+r)/p} + Cy^\nu z^{(\alpha q-\nu q+r)/p} + \&c. = ay^\alpha + by^\beta z^{(\alpha q-\beta q)/p} + cy^\gamma z^{(\alpha q-\gamma q)/p} + \&c.$ Ponatur $\frac{\alpha q+r}{p} = m$ & $\frac{\alpha q-\beta q}{p} = n$: fiet $p = \alpha - \beta$; $q=n$, & $r = \alpha m - \beta m - \alpha n$: atque nascetur ista æquatio: $Az^m + By^\mu z^{m-\frac{\mu n}{\alpha-\beta}} + Cy^\nu z^{m-\frac{\nu n}{\alpha-\beta}} + \&c. = ay^\alpha + by^\beta z^n + cy^\gamma z^{\frac{(\alpha-\gamma)n}{\alpha-\beta}} + \&c.$ quæ propterea ita resolvetur ut fit: $z = (\frac{ax^\alpha + bx^\beta + cx^\gamma + \&c.}{A + Bx^\mu + Cx^\nu + \&c.})^{\frac{\alpha-\beta}{\alpha m - \beta m - \alpha n}}$ & $y = x(\frac{ax^\alpha + bx^\beta + cx^\gamma + \&c.}{A + Bx^\mu + Cx^\nu + \&c.})^{\frac{n}{\alpha m - \beta m - \alpha n}}$. Vel ponatur $\frac{\alpha q+r}{p} = m$, & $\frac{\alpha q - \mu q + r}{p} = n$, crit $m-n =$