Page:Joseph Louis de Lagrange - Œuvres, Tome 14.djvu/160

in $an$ transit, hic valor

$-\int {\frac {ds\,v\,\delta y}{2u^{\frac {3}{2}}}}+\int {\frac {dy\,\delta \,dy}{ds{\sqrt {u}}}}$

quod reducitur ad

$\int \left(-d{\frac {dy}{ds{\sqrt {u}}}}-{\frac {v\,ds}{2u^{\frac {3}{2}}}}\right)dy+{\frac {dy}{ds{\sqrt {u}}}}\delta y.$

Verum quia ex hypothesi citissimi appulsus integrale $\int {\frac {ds}{\sqrt {u}}}$ pro curva $\mathrm {AQN}$ accipi debet per totam $\mathrm {AM} ,$ loco, quod pro $an$ accipiendum est

courbe brachistochrone

tantum per $\mathrm {A} m,$ hinc sit ut mutata curva $\mathrm {AQN}$ in $an$ hoc integrale decrescat suo elemento quod respondet elemento $\mathrm {M} m=dx,$ axis $\mathrm {AM} \,;$ igitur verum ipsius $\int {\frac {ds}{\sqrt {u}}}$ differentiale hoc casu fiet

$\int \left(-d{\frac {dy}{ds{\sqrt {u}}}}-{\frac {v\,ds}{2u^{\frac {3}{2}}}}\right)dy+{\frac {dy}{ds{\sqrt {u}}}}\delta y-{\frac {ds}{\sqrt {u}}}$

ponendo in his duobus postremis terminis a pro $x\,;$ quod adeo nihilo æquatum dabit primò pro æquatione ad curvam quæsitam,

$-d{\frac {dy}{ds{\sqrt {u}}}}-{\frac {v\,ds}{2u^{\frac {3}{2}}}}=0$

(ut nempe nullum ex indeterminatis $\delta y$ in ipsa ingredi possit) deinde præbebit etiam, pro puncto intersectionis $\mathrm {N}$ quod respondet abscis-