Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/331

${\begin{aligned}{\frac {\operatorname {d} V'}{\operatorname {d} x}}&={\frac {\operatorname {d} V}{\operatorname {d} x}}+{\frac {\operatorname {d} y}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y}}+{\frac {\operatorname {d} y_{1}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{1}}}+{\frac {\operatorname {d} y_{2}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}+\ldots +{\frac {\operatorname {d} y_{n}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{n}}},\\\\{\frac {\operatorname {d} V'}{\operatorname {d} x_{1}}}&={\frac {\operatorname {d} V}{\operatorname {d} x_{1}}}+{\frac {\operatorname {d} y_{1}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{1}}}+{\frac {\operatorname {d} y_{2}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}+\ldots +{\frac {\operatorname {d} y_{n}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{n}}},\\\\{\frac {\operatorname {d} V'}{\operatorname {d} x_{2}}}&={\frac {\operatorname {d} V}{\operatorname {d} x_{2}}}+{\frac {\operatorname {d} y_{2}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}+{\frac {\operatorname {d} y_{3}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{3}}}+\ldots +{\frac {\operatorname {d} y_{n}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{n}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\{\frac {\operatorname {d} V'}{\operatorname {d} x_{n}}}&={\frac {\operatorname {d} V}{\operatorname {d} x_{n}}}+{\frac {\operatorname {d} y_{n}}{\operatorname {d} x_{n}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{n}}}\,;\end{aligned}}$

ce qui donnera, en substituant,

$X'={\frac {\operatorname {d} V}{\operatorname {d} x}}+{\frac {\operatorname {d} y}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y}}+{\frac {\operatorname {d} y_{1}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{1}}}+{\frac {\operatorname {d} y_{2}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}+\ldots {\frac {\operatorname {d} y_{i}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}+\ldots$

${\begin{aligned}&-{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x_{1}}}\right)}{\operatorname {d} t}}-{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{1}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{1}}}\right)}{\operatorname {d} t}}-{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{2}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}\right)}{\operatorname {d} t}}-\ldots -{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t}}-\ldots \\\\&+{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} V}{\operatorname {d} x_{2}}}\right)}{\operatorname {d} t^{2}}}\qquad \qquad +{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} y_{2}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{2}}}\right)}{\operatorname {d} t^{2}}}+\ldots +{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{2}}}+\ldots \\\\&-{\frac {\operatorname {d} ^{3}\left({\frac {\operatorname {d} V}{\operatorname {d} x_{3}}}\right)}{\operatorname {d} t^{3}}}\qquad \qquad \qquad -\ldots -{\frac {\operatorname {d} ^{3}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{3}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{3}}}-\ldots \\&+\ldots \ldots \ldots \ldots \qquad \qquad +\ldots \ldots \ldots \ldots \ldots \ldots \\&+{\frac {\operatorname {d} ^{i+1}\left({\frac {\operatorname {d} V}{\operatorname {d} x_{i+1}}}\right)}{\operatorname {d} t^{i+1}}}\end{aligned}}$