Philosophiae naturalis principia mathematica . PER ^QUATIONES INFINITAS. fAltarum Ommum ^adratura^R E G U L A III. Sm valor ipfius y, vel aUqms ejus Termtnusfttpracedentthusmagts compofttus, in Termmos Simplictores reducendus ejf;operando in Ltterts ad eundem Modum quo Arithmeticiin Numerts Dectmaltbus divtdunt ^ Radtces extrabuntyvel affeEias ^quationes folvunt j ©* ex ijlis Terminisquafitam Curva Superficiem . per pracedentes Regulasdeinceps elictes. Exempla Dividendo. Sit^^ —y; Curva nempe exiftente ut -^quatio ifta a Denominatore fuo liberetur, Divifionemfic inftituo. ^ + ^)r
Philosophiae naturalis principia mathematica . PER ^QUATIONES INFINITAS. fAltarum Ommum ^adratura^R E G U L A III. Sm valor ipfius y, vel aUqms ejus Termtnusfttpracedentthusmagts compofttus, in Termmos Simplictores reducendus ejf;operando in Ltterts ad eundem Modum quo Arithmeticiin Numerts Dectmaltbus divtdunt ^ Radtces extrabuntyvel affeEias ^quationes folvunt j ©* ex ijlis Terminisquafitam Curva Superficiem . per pracedentes Regulasdeinceps elictes. Exempla Dividendo. Sit^^ —y; Curva nempe exiftente ut -^quatio ifta a Denominatore fuo liberetur, Divifionemfic inftituo. ^ + ^)rf^ + o(-—jr+TT —77 &C. o _^ + o ?—T — *. o o+^ + aaXi 4«» o — + 0 — a/ixi17 4ltX*. A S &c. Et fic vice hujus^ —^,, nova proditjv rz ^ — ^ + ^—-^, & iftac infinite continuata; Adeoque (per Regulam SecunJam) A 3 Area ?^e D E A N A L Y S I Area quacfita ABDC sequalis erit ipfi ^—i? + Sr —-^*&c. infini- t£E etiam feriei, cujus tamen Termini pauci initiales funt in ufumquemvis fatis exadi, fi modo x fit aliquoties minor quam ^. Eodemmodo, fi fit^-^f^irjv, Dividendo prodibit y — I •„ .v^ + x^i ? x^ + X* &c. Unde (per Regulani Secundam) erit ABDC zzx— \x + i x —Vx^ + ^ x^ 8cc. Vel fi Terminus xxponaturindiviforeprimus, hocmodoxx + i)j prodibit A--^ x-^^ + x-^—x^-^Scc. pro valore ipfiusjy; Unde (per Regulam Secundam) erit BD* — — x~ + jx~^—fx-^ +jx-^ 8ic. Priori modo proce- de cum x elt fatis parva, pofteriori cufn fatis magna fupponitur. Denique fi -^—^zzy; Dividendo prodit 2,xi — IX + yx^ — i3Ar + 34^-^ &c. unde eritABDCzzwv^ — x^ +i x^- — l^v^&c. Exempla Radicem Extrahendo, Si fit v^^ — jv, Radic
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