Introductio in analysin infinitorum

IN RADICIBUS ÆQUATION. INDAGAND. 279 fit, satis est perspicuum. Ex Factoribus enim denominatoris $1 - \alpha z - \beta z^2 - \gamma z^3 - \delta z^4 - \&c.$, cognitis facile assignantur radices æquationis hujus $1 - \alpha z - \beta z^2 - \gamma z^3 - \delta z^4 - \&c. = 0$, ita ut, si Factor fuerit $1 - pz$, hujus æquationis radix una futura fit $z = \frac{1}{p}$. Cum igitur ex Serie recurrente reperiatur maximus numerus $p$, indidem obtinebitur minima radix æquationis $1 - \alpha z - \beta z^2 - \gamma z^3 - \&c. = 0$. Vel, si ponatur $z = \frac{1}{x}$ ut prodeat hæc æquatio $x^m - \alpha x^{m-1} - \beta x^{m-2} - \gamma x^{m-3} - \&c. = 0$, ejusdem methodi ope eruitur maxima hujus æquationis radix $x = p$. 338. Si igitur proponatur æquatio hæc $x^m - \alpha x^{m-1} - \beta x^{m-2} - \gamma x^{m-3} - \&c. = 0$, quæ omnes radices habeat reales & inter se inæquales, harum radicum maxima sequenti modo reperietur. Formetur ex coëfficientibus hujus æquationis fractio $\frac{a+bz+cz^2+dz^3+\&c.}{1-\alpha z - \beta z^2 - \gamma z^3 - \delta z^4 - \&c.}$ Hincque formetur Series recurrens, assumendo pro arbitrio numeratorem, seu, quod eodem redit, assumendo pro libitu terminos initiales; quæ sit $A+Bz+Cz^2+Dz^3+ \dots + Pz^n + Qz^{n+1}$ dabitque fractio $\frac{Q}{P}$ valorem radicis maximæ $x$ pro æquatione proposita, eo propius, quo major fuerit numerus $n$. EXEM-