Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/244

${\begin{aligned}{\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}t'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}u'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}}v'&={\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {t'^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}{\frac {t'u'}{1.2}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {u'^{2}}{1.2}}+\ldots \\\\&\qquad +2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {v't'}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {u'v'}{1.2}}\\\\&\qquad \qquad +{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {v'^{2}}{1.2}}\end{aligned}}$

ce qui donnera, en substituant dans l’équation (54) du plan tangent en $(t',u',v')$,

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

${\begin{aligned}&={\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {t'^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}{\frac {t'u'}{1.2}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {u'^{2}}{1.2}}+\ldots \,;\ \mathrm {(57)} \\\\&\qquad +2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {v't'}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {u'v'}{1.2}}\\\\&\qquad \qquad +{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {v'^{2}}{1.2}}\end{aligned}}$

Ce plan coupera le plan tangent à l’origine des $t,u,v$ suivant une droite qui sera donnée par l’ensemble de cette dernière équation et de l’équation

${\frac {\operatorname {d} S}{\operatorname {d} x}}t+{\frac {\operatorname {d} S}{\operatorname {d} y}}u+{\frac {\operatorname {d} S}{\operatorname {d} z}}v=0\,;\qquad$(58)

on pourra d’ailleurs, dans la recherche de cette droite, substituer à l’une ou à l’autre équation toute combinaison qu’on en voudra faire, de manière à n’en pas élever le degré. On pourra, par exemple, substituer à l’équation (57) sa différence avec l’équation (58), et dire, en conséquence, que l’intersection du plan