1Aſymptoton ejus in V,fuerit VGreciproce ut ipſius ZXvel DN
dignitas aliqua DNn,cujus index eſt numerus n: & quæratur
Medii denſitas, qua Projectile progrediatur in hac curva.
dignitas aliqua DNn,cujus index eſt numerus n: & quæratur
Medii denſitas, qua Projectile progrediatur in hac curva.
Pro BN, BD, NXſcribantur A, O, C reſpective, ſitque VZ
ad XZvel DNut dad e,& VGæqualis (bb/DNn), & erit DNæqua
lis A-O, VG=(bb/—nA-O), VZ=d/e—A-O, & GDſeu NX-VZ
-VGæqualis C-d/eA+d/eO-(bb/—nA-O). Reſolvatur terminus ille
(bb/—nA-O) in ſeriem infinitam (bb/An)+(nbb/An+1)O+(nn+n/2An+2)bbO2+
(n3+3nn+2n/6An+3)bbO3 &c. ac fiet GDæqualis C-d/eA-(bb/An)+
d/eO-(nbb/An+1)O-(+nn+n/2An+2)bbO2-(+n3+3nn+2n/6An+3)bbO3 &c. Hu
jus ſeriei terminus ſecundus d/eO-(nbb/An+1)O uſurpandus eſt pro Qo,
tertius (nn+n/2An+2)bbO2 pro Ro2, quartus (n3+3nn+2n/6An+3)bbO3 pro
So3. Et inde Medii denſitas (S/R√1+QQ), in loco quovis G,fit
(n+2/3√A2+(dd/ee)A2-(2dnbb/eAn)A+(nnb4/A2n)), adeoque ſi in VZcapiatur VY
æqualis nXVG,denſitas illa eſt reciproce ut XY.Sunt enim A2
& (dd/ee)A3-(2dnbb/eAn)A+(nnb4/A2n) ipſarum XZ& ZYquadrata. Reſiſten
tia autem in eodem loco Gfit ad gravitatem ut 3S in (XY/A) ad 4RR,
id eſt, XYad (2nn+2n/n+2)VG.Et velocitas ibidem ea ipſa eſt qua
cum corpus projectum in Parabola pergeret, verticem G,diametrum
GD& latus rectum (1+QQ/R) ſeu (2XYquad./—nn+ninVG) habente. Q.E.I.
ad XZvel DNut dad e,& VGæqualis (bb/DNn), & erit DNæqua
lis A-O, VG=(bb/—nA-O), VZ=d/e—A-O, & GDſeu NX-VZ
-VGæqualis C-d/eA+d/eO-(bb/—nA-O). Reſolvatur terminus ille
(bb/—nA-O) in ſeriem infinitam (bb/An)+(nbb/An+1)O+(nn+n/2An+2)bbO2+
(n3+3nn+2n/6An+3)bbO3 &c. ac fiet GDæqualis C-d/eA-(bb/An)+
d/eO-(nbb/An+1)O-(+nn+n/2An+2)bbO2-(+n3+3nn+2n/6An+3)bbO3 &c. Hu
jus ſeriei terminus ſecundus d/eO-(nbb/An+1)O uſurpandus eſt pro Qo,
tertius (nn+n/2An+2)bbO2 pro Ro2, quartus (n3+3nn+2n/6An+3)bbO3 pro
So3. Et inde Medii denſitas (S/R√1+QQ), in loco quovis G,fit
(n+2/3√A2+(dd/ee)A2-(2dnbb/eAn)A+(nnb4/A2n)), adeoque ſi in VZcapiatur VY
æqualis nXVG,denſitas illa eſt reciproce ut XY.Sunt enim A2
& (dd/ee)A3-(2dnbb/eAn)A+(nnb4/A2n) ipſarum XZ& ZYquadrata. Reſiſten
tia autem in eodem loco Gfit ad gravitatem ut 3S in (XY/A) ad 4RR,
id eſt, XYad (2nn+2n/n+2)VG.Et velocitas ibidem ea ipſa eſt qua
cum corpus projectum in Parabola pergeret, verticem G,diametrum
GD& latus rectum (1+QQ/R) ſeu (2XYquad./—nn+ninVG) habente. Q.E.I.