Introductio in analysin infinitorum

NUMERORUM. 271 325. Quod si autem hoc productum CAP. XVI. $(1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)(1+x^6)$ &c., evolvatur, sequens prodibit Series $1+x+x^2+2x^3+2x^4+3x^5+4x^6+5x^7+6x^8+8x^9+10x^{10}+$ &c., in qua quisque coëfficiens indicat, quot variis modis Exponens ipsius $x$ per additionem numerorum inæqualium oriri possit. Sic numerus 9 octo variis modis per additionem ex numeris inæqualibus formari potest. $9=9$ $9=6+2+1$ $9=8+1$ $9=5+4$ $9=7+2$ $9=5+3+1$ $9=6+3$ $9=4+3+2$ 326. Ut comparationem inter has formas instituamus, fit $P = (1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ &c., & $Q = (1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)(1+x^6)$ &c., erit $PQ=(1-x^2)(1-x^4)(1-x^6)(1-x^8)(1-x^{10})(1-x^{12})$ &c., qui Factores cum omnes in P contineantur, dividatur P per PQ, erit $\frac{1}{Q}=(1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)$ &c., ideoque $Q = \frac{1}{(1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9) \text{ &c.}},$ quæ fractio si evolvatur, prodibit Series, in qua quisque coëfficiens indicabit, quot variis modis Exponens ipsius $x$, per additionem ex numeris imparibus produci possit. Cum igitur hæc expressio æqualis sit illi, quam in §. præcedente contemplati sumus, sequitur hinc istud theorema. Quot