Page:Annales de mathématiques pures et appliquées, 1821-1822, Tome 12.djvu/14

10

PARALLÉLISME DES LIGNES

Cela posé, en élimment $t-x$ entre les équations (1, 2), on a

$u-y=\pm {\frac {k}{\sqrt {1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}}}}\,;$

en conséquence de quoi la valeur de ${\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}$ devient

${\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}={\frac {\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}{1\mp {\frac {k{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}{\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}}}}\,;$

donc, en vertu de l’équation (3),

${\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}={\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{{\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k}}={\frac {\left\{1+\left({\frac {\operatorname {d} u}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{{\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k}}\,;$

ou encore

${\frac {\left\{1+\left({\frac {\operatorname {d} u}{\operatorname {d} t}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}}={\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k\,;$

mais, en représentant par $r$ le rayon de courbure de la courbe dont les coordonnées sont $x,y,$ et par $\rho$ celui de la courbe dont les coordonnées sont $t$ et $u,$ on a