Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/282

 Les corrections sont expliquées en page de discussion

ce qui donnera, en intégrant,

$\left.{\begin{aligned}t+D&={\frac {p}{2k}}\operatorname {Log} .\left\{2k{\frac {x-a}{p}}+{\sqrt {4k^{2}\left({\frac {x-a}{p}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\alpha -4k^{2}{\frac {a^{2}}{p^{2}}}}}\right\},\\\\t+E&={\frac {q}{2k}}\operatorname {Log} .\left\{2k{\frac {y-b}{q}}+{\sqrt {4k^{2}\left({\frac {y-b}{q}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\beta -4k^{2}{\frac {b^{2}}{q^{2}}}}}\right\},\\\\t+F&={\frac {r}{2k}}\operatorname {Log} .\left\{2k{\frac {z-c}{r}}+{\sqrt {4k^{2}\left({\frac {z-c}{r}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\gamma -4k^{2}{\frac {c^{2}}{r^{2}}}}}\right\},\end{aligned}}\right\}$ (21)

$D,E,F$ étant trois nouvelles constantes arbitraires.

Pour les déterminer, fixons l’origine des temps au passage de la molécule par l’origine des coordonnées ; alors $x,y,z$ devront être nuls en même temps que $t\,;$ ce qui donnera

${\begin{aligned}D&={\frac {p}{2k}}\operatorname {Log} .\left(-2k{\frac {a}{p}}+V\operatorname {Cos} .\alpha \right),\\\\E&={\frac {q}{2k}}\operatorname {Log} .\left(-2k{\frac {b}{q}}+V\operatorname {Cos} .\beta \right),\\\\F&={\frac {r}{2k}}\operatorname {Log} .\left(-2k{\frac {c}{r}}+V\operatorname {Cos} .\gamma \right)\,;\end{aligned}}$

d’où, en retranchant,

$\left.{\begin{aligned}t&={\frac {p}{2k}}\operatorname {Log} .{\frac {2k{\frac {x-a}{p}}+{\sqrt {4k^{2}\left({\frac {x-a}{p}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\alpha -4k^{2}{\frac {a^{2}}{p^{2}}}}}}{V\operatorname {Cos} .\alpha -2k{\frac {a}{p}}}},\\\\t&={\frac {q}{2k}}\operatorname {Log} .{\frac {2k{\frac {y-b}{q}}+{\sqrt {4k^{2}\left({\frac {y-b}{q}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\beta -4k^{2}{\frac {b^{2}}{q^{2}}}}}}{V\operatorname {Cos} .\beta -2k{\frac {b}{q}}}},\\\\t&={\frac {r}{2k}}\operatorname {Log} .{\frac {2k{\frac {z-c}{r}}+{\sqrt {4k^{2}\left({\frac {z-c}{r}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\gamma -4k^{2}{\frac {c^{2}}{r^{2}}}}}}{V\operatorname {Cos} .\gamma -2k{\frac {c}{r}}}}\,;\end{aligned}}\right\}$ (22)