Pagina:Principia newton la.djvu/166

Exempl. 3. Sit linea AGK Hyperbola, Asymptoton habens NX plano horizontali AK perpendicularem; & quæratur Medii densitas quæ faciat ut Projectile moveatur in hac linea.

Sit MX Asymptotos altera, ordinatim applicatæ DG productæ occurrens in V, & ex natura Hyperbolæ, rectangulum XV in VG dabitur. Datur autem ratio DN ad VX, & propterea datur etiam rectangulum DN in VG. Sit illud bb; & completo parallelogrammo DNXZ, dicatur BNa, BDo, NXc, & ratio data VZ ad ZX vel DN ponatur esse ${\tfrac {m}{n}}$. Et erit DN æqualisa −o, VG æqualis ${\tfrac {bb}{a-o}}$, VZ æqualis ${\tfrac {m}{na-o}}$, & GD seu NX−VZ−VG æqualis $c-{\tfrac {m}{n}}a+{\tfrac {m}{n}}o-{\tfrac {bb}{a-o}}$. Resolvatur terminus ${\tfrac {bb}{a-o}}$ in seriem convergentem ${\tfrac {bb}{a}}+{\tfrac {bb}{aao}}+{\tfrac {bb}{a^{3}}}oo+{\tfrac {bb3}{a^{4}}}o^{3}$ etc. & fiet GD æqualis $c-{\tfrac {m}{n}}a-{\tfrac {bb}{a}}+{\tfrac {m}{no}}-{\tfrac {bb}{aao}}-{\tfrac {bb}{a^{3}}}o^{2}-{\tfrac {bb}{a^{4}}}o^{3}$ &c. Hujus seriei terminus secundus ${\tfrac {m}{n}}o={\tfrac {bb}{aa}}o$ usurpandus est pro Qo, tertius cum signo mutato ${\tfrac {bb}{a^{3}}}o^{2}$ pro $Ro^{2}$, & quartus cum signo etiam mutato ${\tfrac {bb}{a^{4}}}o^{3}$ pro $So^{3}$, eorumq; coefficientes ${\tfrac {m}{n}}-{\tfrac {bb}{aa}}$, ${\tfrac {bb}{a^{3}}}$ & ${\tfrac {bb}{a^{4}}}$ scribendæ sunt, in Regula superi ore, pro Q, R & S. Quo facto prodit medii densitas ut ${\tfrac {\tfrac {bb}{a^{4}}}{{\tfrac {bb}{a^{3}}}{\sqrt {1-{\tfrac {mm}{nn}}-2{\tfrac {mbb}{naa}}+{\tfrac {b^{4}}{a^{4}}}}}}}$ seu ${\tfrac {1}{\sqrt {aa+{\tfrac {mm}{nn}}aa-{\tfrac {2mbb}{n}}+{\tfrac {b^{4}}{aa}}}}}$

est, si in VZ sumatur VY æqualis VG, ut ${\tfrac {1}{XY}}$. Namq; aa & ${\tfrac {mm}{nn}}aa-2{\tfrac {mbb}{n}}+{\tfrac {b^{4}}{aa}}$ sunt ipsarum XZ & ZY quadrata. Resistentia autem invenitur in ratione ad Gravitatem quam habet XY ad YG, & velocitas ea est quacum corpus in Parabola pergeret verticem G diametrum DG & latus rectum YX quad. VG habente. Ponatur itaq; quod Medii densitates in locis singulis G sint reciproce ut distantiæ XY, quodq; resistentia in loco aliquo G sit ad gravitatem ut XY ad YG; & corpus de loco A justa cum velocitate emissum describet Hyperbolam illam AGK. Q. E. I.

Exempl. 4. Ponatur indefinite, quod linea AGK Hyperbola sit, centro X Asymptotis MX, NX, ea lege descripta, ut constructo rectangulo XZDN cujus latus ZD secet Hyperbolam in G & Asymptoton ejus in V, fuerit VG reciproce ut ipsius ZX vel DN dignitas aliqua NDn, cujus index est numerus n: & quæratur Medii densitas, qua Projectile progrediatur in hac curva. Pro DN, BD, NX scribantur A, O, C respective, sitq; VZ ad ZX vel DN ut d ad e, & VG æqualis ${\tfrac {bb}{DN^{n}}}$, & erit DN æqualis A−O, $VG={\tfrac {bb}{A-O^{n}}}$, $VZ={\tfrac {d}{e}}$ in A−O, & GD seu NX−VZ−VG æqualis $C-{\tfrac {d}{e}}A+{\tfrac {d}{e}}O-{\tfrac {bb}{A-O^{n}}}$. Resolvatur terminus ille ${\tfrac {bb}{A-O^{n}}}$ in seriam infinitam ${\tfrac {bb}{A^{n}}}+{\tfrac {nbbO}{A^{n+1}}}+{\tfrac {nn+n}{2A^{n+2}}}bbO^{2}+{\tfrac {n^{3}+3nn+2n}{6A^{n+3}}}bbO^{3}$ &c. ac fiet GD æqualis $C-{\tfrac {d}{e}}A-{\tfrac {bb}{A^{n}}}$ +${\tfrac {d}{e}}O-{\tfrac {nbb}{A^{n+1}}}O-{\tfrac {nn+n}{2A^{n+2}}}bbO^{2}$-${\tfrac {n^{3}+3nn+2n}{6A^{n+3}}}bbO^{3}$ &c. Hujus seriei terminus secundus ${\tfrac {d}{e}}O$-${\tfrac {nbb}{A^{n+1}}}O$ usurpandus est