Introductio in analysin infinitorum

NUMERORUM. 275 five addendo sive subtrahendo: num vero omnes Potestates prodeant, singulæque semel, sic exploro. Sit CAP. XVI. $P = \&c. + cx^{-3} + bx^{-2} + ax^{-1} + 1 + \alpha x^1 + \beta x^2 + \gamma x^3 + \delta x^4 + \epsilon x^5 + \&c.$ , manifestum vero est, si $x^3$ loco $x$ scribatur, tum prodire $\frac{P}{x^{-1}+1+x^1} = bx^{-6} + ax^{-3} + 1 + \alpha x^3 + \beta x^6 + \gamma x^9 + \&c.$ . Hinc igitur reperitur $P = \&c.$ $+ ax^{-4} + ax^{-3} + ax^{-2} + x^{-1} + 1 + x + \alpha x^2 + \alpha x^3 + \alpha x^4 + \beta x^5 + \beta x^6 + \beta x^7 + \&c.$ , quæ expressio cum assumta comparata dabit $\alpha = 1; \beta = \alpha; \gamma = \alpha; \delta = \alpha; \epsilon = \beta; \zeta = \beta; \&c., \&$ $a = 1, b = a, c = a, d = a, e = b, \&c.$ . Hinc itaque erit $P = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + \&c.$ $+ x^{-1} + x^{-2} + x^{-3} + x^{-4} + x^{-5} + x^{-6} + x^{-7} + \&c.$ , unde patet omnes ipsius $x$ Potestates, tam affirmativas quam negativas, hic occurrere, atque adeo omnes numeros ex terminis progressionis geometricæ triplæ, vel addendo vel subtrahendo, formari posse; & unumquemque numerum unico tantum modo. Mm 2 a. CAPUT