Introductio in analysin infinitorum

NUMERORUM. 263 $Z = 1 + Pz + Qz^2 + Rz^3 + Sz^4 + Tz^5 + \&c..$ CAP. XVI. Perspicuum autem est, si loco $z$ ponatur $xz$, prodire $\frac{1}{(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)\&c.} = (1-xz)Z$. Facta ergo in Serie evoluta eadem mutatione, fiet $(1-xz)Z = 1 + Pxz + Qx^2z^2 + Rx^3z^3 + Sx^4z^4 + \&c..$ Multiplicetur ergo superior Series pariter per $(1-xz)$, eritque $(1-xz)Z = 1 + Pz + Qz^2 + Rz^3 + Sz^4 + \&c.$ $-xz - Pxz^2 - Qxz^3 - Rxz^4 - \&c..$ Comparatione ergo instituta orietur $P = \frac{x}{1-x}$; $Q = \frac{Px}{1-x^2}$; $R = \frac{Qx}{1-x^3}$; $S = \frac{Rx}{1-x^4}$ \&c., unde pro $P, Q, R, S [TRUNCATED]