Introductio in analysin infinitorum

48 DE EXPLICATIONE FUNCTIONUM LIB. I. Cum igitur terminus primus $A$ fit determinatus $=\frac{a}{\alpha}$ ex eo sequentes litterae $B$, $C$, $D$, &c. definiuntur eodem modo, quo ex divisione sunt orti. Ceterum ex inspectione perspicitur in serie infinita pro $\frac{a}{\alpha+\zeta z}$ inventa potestatis $z^n$ coëfficientem fore $=\pm \frac{a\zeta^n}{\alpha^{n+1}}$, ubi signum $+$ locum habet si $n$ sit numerus par, signum $-$ autem si $n$ sit numerus impar: seu coëfficiens erit $=\frac{a}{\alpha}(-\frac{\zeta}{\alpha})^n$. 61. Simili modo ope divisionis continuatae haec Functio fracta $\frac{a+bz}{\alpha+\zeta z+\gamma zz}$ in seriem infinitam converti potest. Cum autem divisio sit taediosa, neque tam facile naturam seriei infinitae ostendat, commodius erit seriem quaesitam fingere, atque modo ante tradito determinare. Sit igitur $\frac{a+bz}{\alpha+\zeta z+\gamma zz} = A+Bz+Cz^2+Dz^3+Ez^4+\text{&c.}}$ multiplicetur utrinque per $\alpha+\zeta z+\gamma zz$, atque fiet $a+bz = \alpha A + \alpha B z + \alpha C z^2 + \alpha D z^3 + \alpha E z^4 + \text{&c.}$ $\qquad\qquad +\zeta A z + \zeta B z^2 + \zeta C z^3 + \zeta D z^4 + \text{&c.}$ $\qquad\qquad\qquad\qquad +\gamma A z^2 + \gamma B z^3 + \gamma C z^4 + \text{&c.}$ Hinc erit $\alpha A = a$; $\alpha B + \zeta A = b$; unde reperitur $A=\frac{a}{\alpha}$ & $B=\frac{b}{\alpha}-\frac{a\zeta}{\alpha\alpha}$; reliquae vero litterae ex sequentibus aequationibus determinabuntur: $\alpha C + \zeta B + \gamma A = 0$ hinc ergo ex binis quibusque coëfficientibus contiguis sequens reperitur. Sic si duo coëfficientes contigui fuerint $P$, $Q$ & sequens $R$, erit $\alpha R + \zeta Q + \gamma P = 0$ seu $R = \frac{-\zeta Q - \gamma P}{\alpha}$. $\alpha D + \zeta C + \gamma B = 0$ $\alpha E + \zeta D + \gamma C = 0$ $\alpha F + \zeta E + \gamma D = 0$ &c. Cum igitur duae litterae primae $A$ & $B$ jam sint inventae sequentes $C$, $D$, $E$, $F$ &c. omnes successive ex iis invenientur,