Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/245

tangent à l’origine des $t,u,v$, avec le plan tangent $t',u',v'$ est donnée par l’équation (58), combinée avec l’équation

$\left\{{\begin{aligned}&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'+\ldots \right)t\\\\+&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'+\ldots \right)u\\\\+&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'+\ldots \right)v\end{aligned}}\right\}$

${\begin{aligned}&={\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {t'^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}{\frac {t'u'}{1.2}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {u'^{2}}{1.2}}+\ldots \,;\ \mathrm {(59)} \\\\&\qquad +2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {v't'}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {u'v'}{1.2}}\\\\&\qquad \qquad +{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {v'^{2}}{1.2}}\end{aligned}}$

et, quant à la droite qui joindra les deux points de contact, elle sera donnéa par la double équation

${\frac {t}{t'}}={\frac {u}{u'}}={\frac {v}{v'}}.\qquad$(60)

Si présentement on suppose que le point $(t',u',v')$ est très voisin de l’origine des $t,u,v$, à raison de la petitesse de $t',u',v',$ on pourra, sans erreur sensible, supprimer, dans l’équation (59), tous les termes de plus d’une dimension, par rapport à ces coordonnées ; ce qui la changera en celle-ci :

$\left\{{\begin{aligned}&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'\right)t\\\\+&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'\right)u\\\\+&\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'\right)v\end{aligned}}\right\}=0\,;$

ou bien encore

$\left\{{\begin{aligned}&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}tt'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}(uv'+vu')\\\\+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}uu'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}(\tau '+tv')\\\\+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}vv'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}(tu'+ut')\end{aligned}}\right\}=0\,;\quad$(61)