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MÉCANIQUE ANALYTIQUE.
ce qui fournira trois équations qui serviront à déterminer les valeurs de
en
ce sera la même chose si l’on substitue dans la seconde expression de
les valeurs de
en
tirées des expressions de ces dernières quantités ; alors la comparaison des termes affectés de
donnera immédiatement les valeurs de
Or, par les règles ordinaires de l’élimination, on a
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8a7ff36267c0c727f09837598efd8b34c4588e)
en supposant
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/491e30ed21b298c6425025caf959816fbcd4dd4e)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35942e8a194fea729a4cad10cc60551e91dd075e)
Faisant donc ces substitutions dans l’expression
![{\displaystyle {\frac {\partial \lambda }{\partial a}}da+{\frac {\partial \lambda }{\partial b}}db+{\frac {\partial \lambda }{\partial c}}dc,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d73365e6f3f4525f44a62628fb03328c9ffb1c01)
et comparant ensuite avec l’expression identique
![{\displaystyle {\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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62713d9c7d76dc1f344f5fd47581dfea668ccb10)