Introductio in analysin infinitorum

CONTINUIS. 309 CAP. XVIII. $x = \frac{A}{1+\frac{Bz}{A-Bz+\frac{ACz}{B-Cz+\frac{BDz}{C-Dz+\&c.}}}}$ 372. Quo autem hoc negotium generalius absolvamus, ponamus esse $x = \frac{A}{L} - \frac{By}{Mz} + \frac{Cy^2}{Nz^2} + \frac{Dy^3}{Oz^3} - \frac{Ey^4}{Pz^4} + \&c.$, fietque, comparatione instituta, $\mathfrak{A} = \frac{Ab}{L}; \mathfrak{B} = \frac{BLbcy}{AMz-BLy}; \gamma = \frac{ACM^2cdyz}{(AMz-BLy)(BNz-CMy)};$ $\delta = \frac{BDN^2deyz}{(BNz-CMy)(COz-DNy)}; \&c.,$ statuantur valores $b, c, d, \&c.$, sequenti modo $b = L;$ $c = AMz - BLy;$ $d = BNz - CMy;$ $e = COz - DNy;$ $f = DPz - EOy;$ $\&c.$ erit $\alpha = A$ $\beta = BLLy$ $\gamma = ACM^2yz$ $\delta = BDN^2yz$ $\epsilon = CEO^2yz$ $\&c.$ unde Series proposita per sequentem fractionem continuam exprimetur $x = \frac{A}{L+\frac{BLLy}{AMz-BLy+\frac{ACMMyz}{BNz-CMy+\frac{BDNNyz}{COz-DNy+\&c.}}}}$ 373. Habeat denique Series proposita hujusmodi formam $x = \frac{A}{L} - \frac{ABy}{LMz} + \frac{ABC y^2}{LMN z^2} - \frac{ABCD y^3}{LMNO z^3} + \&c.,$ atque sequentes valores prodibunt Qq 3