DE AFQVATIONIOJ^S INFINITIS. 77 



§. 9. Dictae autem feries algebraicae fiue con- 

 tinuae (iue interpolatae non lolae funt quae alte- 

 ram reddunt recurrentem: nam poftunt etiam eflfe 

 recurrentes qualescunque , haeque vt de algebrai- 

 cis oftenfum eft, cum continuae, tum nullioni- 

 bus interruptae. Sit v. gr. (A) 2.3.5 9.17.33.65. 

 129. etc. cuius feriei terminus generalis eft i x ~ l 

 H- 1 x liue 2 X ~~ 'H-i, quaeque adeo recurrens eft, 

 indice 3,-2 praedita:. fit feries (B) 2.7.25.90. 

 325. 1 175. 4.250. 15375. etc. quae itidem recur- 

 rens indicem habet 5 , — 5. Sin autem fuerit (A) 

 2.0.3.0.5.0.9.0.17.0.33. etc. oritur (B) 2.4. 11, 

 28. 73, 1 S9..491. 1 274.3308. etc. quae eft feries 

 exponentialis quarti ordinis, cui index eft 2, g r 

 -3,-2. 



1 §. 10. Tota denique res in hoc confiftit theo- 

 remate generali, vt fi primus feriei (A) terminus 

 intelligatur multiplicatus per X) fecnndus per x 2 

 tertius per x 3 et fic in infinitum, ficque habea- 

 tur ax-\-bxx-\-cx li -\-dx*-\-ex < ' -\-fx 6 etc. refpicien- 

 dum llt , an rnec vltima feries inflniti in firm- 

 mam finitam algebraice expreflam reduci poffit 

 nec ne? i\ prius indicium eft r feriem (B) recur- 

 rentem fore; tunc autem fumma • aequalis- ftatuen- 

 da eft vnitati , pofteaque aequationi concilian.ia.' 



eft haec forma 1 zzzolx-\- %xx--\~yx y -j- fix* 



-\-(px l , quo ficto erit index feriei (B) ceu re- 



currentis a, S, y, $ (J) pertinens ad or- 



dinem /. Si vero expofita feries ax-\-bxx-\-':x^ 



