Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/209

$\left.{\begin{aligned}{\frac {\operatorname {d} p}{\operatorname {d} x}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} x}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} x_{1}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} x_{1}}}+{\frac {\operatorname {d} P}{\operatorname {d} x}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} x_{2}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} x_{2}}}+{\frac {\operatorname {d} P}{\operatorname {d} x_{1}}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \\{\frac {\operatorname {d} p}{\operatorname {d} x_{m-1}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} x_{m-1}}}+{\frac {\operatorname {d} P}{\operatorname {d} x_{m-2}}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} x_{m}}}&={\frac {\operatorname {d} P}{\operatorname {d} x_{m-1}}}.\end{aligned}}\right\}(1)\left.{\begin{aligned}{\frac {\operatorname {d} p}{\operatorname {d} y}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} y}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} y_{1}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} y_{1}}}+{\frac {\operatorname {d} P}{\operatorname {d} y}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} y_{2}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} y_{2}}}+{\frac {\operatorname {d} P}{\operatorname {d} y_{1}}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \\{\frac {\operatorname {d} p}{\operatorname {d} y_{n-1}}}&=\operatorname {d} {\frac {\operatorname {d} P}{\operatorname {d} y_{n-1}}}+{\frac {\operatorname {d} P}{\operatorname {d} y_{n-2}}},\\\\{\frac {\operatorname {d} p}{\operatorname {d} y_{n}}}&={\frac {\operatorname {d} P}{\operatorname {d} y_{n-1}}}.\end{aligned}}\right\}(2)$

Du premier de ces deux systèmes d’équations on déduit, par l’élination successive des différentielles de ${\frac {\operatorname {d} P}{\operatorname {d} x_{m-1}}},{\frac {\operatorname {d} P}{\operatorname {d} x_{m-2}}},\ldots {\frac {\operatorname {d} P}{\operatorname {d} x_{1}}},{\frac {\operatorname {d} P}{\operatorname {d} x}},$

$\left.{\begin{aligned}{\frac {\operatorname {d} P}{\operatorname {d} x_{m-1}}}&={\frac {\operatorname {d} p}{\operatorname {d} x_{m}}},\\\\{\frac {\operatorname {d} P}{\operatorname {d} x_{m-2}}}&={\frac {\operatorname {d} p}{\operatorname {d} x_{m-1}}}-\operatorname {d} {\frac {\operatorname {d} p}{\operatorname {d} x_{m}}},\\\\{\frac {\operatorname {d} P}{\operatorname {d} x_{m-3}}}&={\frac {\operatorname {d} p}{\operatorname {d} x_{m-2}}}-\operatorname {d} {\frac {\operatorname {d} p}{\operatorname {d} x_{m-1}}}+\operatorname {d} ^{2}{\frac {\operatorname {d} p}{\operatorname {d} x_{m}}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \\{\frac {\operatorname {d} P}{\operatorname {d} x}}&={\frac {\operatorname {d} p}{\operatorname {d} x_{1}}}-\operatorname {d} {\frac {\operatorname {d} p}{\operatorname {d} x_{2}}}+\operatorname {d} ^{2}{\frac {\operatorname {d} p}{\operatorname {d} x_{3}}}+\ldots \mp \operatorname {d} ^{m-1}{\frac {\operatorname {d} p}{\operatorname {d} x_{m}}},\\\\0&={\frac {\operatorname {d} p}{\operatorname {d} x}}-\operatorname {d} {\frac {\operatorname {d} p}{\operatorname {d} x_{1}}}-\operatorname {d} ^{2}{\frac {\operatorname {d} p}{\operatorname {d} x_{2}}}-\ldots \mp \operatorname {d} ^{m-1}{\frac {\operatorname {d} p}{\operatorname {d} x_{m-1}}}\pm \operatorname {d} ^{m}{\frac {\operatorname {d} p}{\operatorname {d} x_{m}}}\end{aligned}}\right\}(3)$