Introductio in analysin infinitorum

PER SUBSTITUTIONEM. 43 $= \mu q$; & $q/p = \frac{m-n}{\mu}$, atque $r/p = m - \frac{am+an}{\mu}$. Hinc CAP. III. fit $p = \mu$; $q = m-n$; & $r = \mu m - am + an$; atque hæc æquatio resultabit: $Az^m + By^\mu z^n + Cy^\nu z^{\mu m - \frac{\nu(m-n)}{\mu}} + \&c. = ay^\alpha + by^\zeta z \frac{(\alpha-\zeta)(m-n)}{\mu} + cy^\gamma z \frac{(\alpha-\gamma)(m-n)}{\mu} + \&c.$ quæ ita resolvetur ut fit: $z = (\frac{ax^\alpha+bx^\beta+cx^\gamma+\&c.}{A+Bx^\mu+Cx^\nu+\&c.})^{\frac{\mu}{\mu m - am + an}}$ & $y = x(\frac{ax^\alpha+bx^\beta+cx^\gamma+\&c.}{A+Bx^\mu+Cx^\nu+\&c.})^{\frac{m-n}{\mu m - am + an}}$ 54. Si $y$ ita pendeat a $z$ ut fit $ayy + byz + czz + dy + ez = 0$, sequenti modo tam $y$ quam $z$ rationaliter per novam variabilem $x$ exprimetur. Ponatur $y = xz$, erit divisione per $z$ facta: $axxz + bxz + cz + dx + e = 0$, ex qua reperitur $z = \frac{-dx-e}{axx+bx+c}$, & $y = \frac{-dxx-ex}{axx+bx+c}$. At vero ad formam propositam reduci potest hæc æquatio inter $y$ & $z$: $ayy + byz + czz + dy + ez + f = 0$ diminuendo vel augendo utramque variabilem certa quadam quantitate constante, unde & hæc æquatio per novam variabilem $x$ rationaliter explicari potest. 55. Si $y$ ita pendeat a $z$, ut fit $ay^3+by^2z+cyz^2+dz^3+eyy+fyz+gzz=0$; sequenti modo tam $y$ quam $z$ rationaliter per novam variabilem $x$ exprimi poterit. Ponatur $y=xz$, & facta substitutione tota æquatio per $zz$ dividi poterit: prodibit autem $ax^3z+bxxz+cxz+dz+exx+fx+g=0$. Unde oritur $z = \frac{-exx-fx-g}{ax^3+bxx+cx+d}$. ex quo erit $y = \frac{-ex^3-fxx-gx}{ax^3+bxx+cx+d}$. F 2 Ex