Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/286

278

MÉCANIQUE ANALYTIQUE.

ce qui fournira trois équations qui serviront à déterminer les valeurs de ${\frac {\operatorname {D} \lambda }{\operatorname {D} x}},\,{\frac {\operatorname {D} \lambda }{\operatorname {D} y}},\,{\frac {\operatorname {D} \lambda }{\operatorname {D} z}}$ en ${\frac {\partial \lambda }{\partial a}},\,{\frac {\partial \lambda }{\partial b}},\,{\frac {\partial \lambda }{\partial c}}\,;$ ce sera la même chose si l’on substitue dans la seconde expression de $\operatorname {D} \lambda$ les valeurs de $da,\,db,\,dc$ en $\operatorname {D} x,\,\operatorname {D} y,\,\operatorname {D} z$ tirées des expressions de ces dernières quantités ; alors la comparaison des termes affectés de $\operatorname {D} x,\operatorname {D} y,\operatorname {D} z$ donnera immédiatement les valeurs de ${\frac {\operatorname {D} \lambda }{\operatorname {D} x}},\ldots .$

Or, par les règles ordinaires de l’élimination, on a

${\begin{aligned}da=&{\frac {\alpha \operatorname {D} x+\alpha '\operatorname {D} y+\alpha ''\operatorname {D} z}{\theta }},\\db=&{\frac {\beta \operatorname {D} x+\beta '\operatorname {D} y+\beta ''\operatorname {D} z}{\theta }},\\dc=&{\frac {\gamma \operatorname {D} x+\gamma '\operatorname {D} y+\gamma ''\operatorname {D} z}{\theta }},\end{aligned}}$

en supposant

${\begin{alignedat}{3}\alpha \ \,=&{\frac {\partial y}{\partial b}}{\frac {\partial z}{\partial c}}-{\frac {\partial y}{\partial c}}{\frac {\partial z}{\partial b}},\quad &\beta \ \,=&{\frac {\partial y}{\partial c}}{\frac {\partial z}{\partial a}}-{\frac {\partial y}{\partial a}}{\frac {\partial z}{\partial c}},\quad &\gamma \ \,=&{\frac {\partial y}{\partial a}}{\frac {\partial z}{\partial b}}-{\frac {\partial y}{\partial b}}{\frac {\partial z}{\partial a}},\\\alpha '\ =&{\frac {\partial x}{\partial c}}{\frac {\partial z}{\partial b}}-{\frac {\partial x}{\partial b}}{\frac {\partial z}{\partial c}},&\beta '\ =&{\frac {\partial x}{\partial a}}{\frac {\partial z}{\partial c}}-{\frac {\partial x}{\partial c}}{\frac {\partial z}{\partial a}},&\gamma '\ =&{\frac {\partial x}{\partial b}}{\frac {\partial z}{\partial a}}-{\frac {\partial x}{\partial a}}{\frac {\partial z}{\partial b}},\\\alpha ''=&{\frac {\partial x}{\partial b}}{\frac {\partial y}{\partial c}}-{\frac {\partial x}{\partial c}}{\frac {\partial y}{\partial b}},&\beta ''=&{\frac {\partial x}{\partial c}}{\frac {\partial y}{\partial a}}-{\frac {\partial x}{\partial a}}{\frac {\partial y}{\partial c}},&\gamma ''=&{\frac {\partial x}{\partial a}}{\frac {\partial y}{\partial b}}-{\frac {\partial x}{\partial b}}{\frac {\partial y}{\partial a}},\end{alignedat}}$

${\begin{aligned}\theta =&+{\frac {\partial x}{\partial a}}{\frac {\partial y}{\partial b}}{\frac {\partial z}{\partial c}}-{\frac {\partial x}{\partial b}}{\frac {\partial y}{\partial a}}{\frac {\partial z}{\partial c}}\\&+{\frac {\partial x}{\partial b}}{\frac {\partial y}{\partial c}}{\frac {\partial z}{\partial a}}-{\frac {\partial x}{\partial c}}{\frac {\partial y}{\partial b}}{\frac {\partial z}{\partial a}}\\&+{\frac {\partial x}{\partial c}}{\frac {\partial y}{\partial a}}{\frac {\partial z}{\partial b}}-{\frac {\partial x}{\partial a}}{\frac {\partial y}{\partial c}}{\frac {\partial z}{\partial b}}.\end{aligned}}$

Faisant donc ces substitutions dans l’expression

${\frac {\partial \lambda }{\partial a}}da+{\frac {\partial \lambda }{\partial b}}db+{\frac {\partial \lambda }{\partial c}}dc,$

et comparant ensuite avec l’expression identique

${\frac {\operatorname {D} \lambda }{\operatorname {D} x}}\operatorname {D} x+{\frac {\operatorname {D} \lambda }{\operatorname {D} y}}\operatorname {D} y+{\frac {\operatorname {D} \lambda }{\operatorname {D} z}}\operatorname {D} z,$