Page:Lagrange - Œuvres (1867) vol. 1.djvu/480

Substituant ces valeurs dans l’équation (L) ci-dessus, et faisant sortir hors du signe $\mathrm {S}$ les quantités $d\mathrm {X} ,d\mathrm {Y} ,d\mathrm {Z} ,\delta \mathrm {X} ,\delta \mathrm {Y} ,\delta \mathrm {Z} ,d\mathrm {P} ,d\mathrm {Q} ,d\mathrm {R} ,$$\delta \mathrm {P} ,\delta \mathrm {Q} ,\delta \mathrm {R}$ qui sont les mêmes pour chaque point du corps, enfin ordonnant les termes par rapport à $\delta \mathrm {X} ,\delta \mathrm {Y} ,\delta \mathrm {Z} ,\delta \mathrm {P} ,\delta \mathrm {Q} ,\delta \mathrm {R} ,$ on aura une équation de la forme suivante :

(S)

$\int \left([\mathrm {X} ]\delta \mathrm {X} +[\mathrm {Y} ]\delta \mathrm {Y} +[\mathrm {Z} ]\delta \mathrm {Z} +[\mathrm {P} ]\delta \mathrm {P} +[\mathrm {Q} ]\delta \mathrm {Q} +[\mathrm {R} ]\delta \mathrm {R} \right)=0,$

dans laquelle

${\begin{aligned}&[\mathrm {X} ]=\mathrm {M} d{\frac {d\mathrm {X} }{dt}}+\mathrm {S} rdm\left(d{\frac {d\mathrm {Q} }{dt}}+{\frac {d\mathrm {P} d\mathrm {R} }{dt}}\right)+\mathrm {S} qdm\left(d{\frac {d\mathrm {R} }{dt}}-{\frac {d\mathrm {P} d\mathrm {Q} }{dt}}\right)\\&\qquad -\mathrm {S} pdm{\frac {d\mathrm {Q} ^{2}+d\mathrm {R} ^{2}}{dt}}+\mathrm {S} \Pi dm\,dt,\\&[\mathrm {Y} ]=\mathrm {M} d{\frac {d\mathrm {Y} }{dt}}+\mathrm {S} rdm\left(d{\frac {d\mathrm {P} }{dt}}+{\frac {d\mathrm {Q} d\mathrm {R} }{dt}}\right)+\mathrm {S} qdm{\frac {d\mathrm {P} ^{2}+d\mathrm {R} ^{2}}{dt}}\\&\qquad -\mathrm {S} pdm\left(d{\frac {d\mathrm {R} }{dt}}+{\frac {d\mathrm {P} d\mathrm {Q} }{dt}}\right)+\mathrm {S} \varpi dm\,dt,\\&[\mathrm {Z} ]=\mathrm {M} d{\frac {d\mathrm {Z} }{dt}}-\mathrm {S} rdm{\frac {d\mathrm {P} ^{2}+d\mathrm {Q} ^{2}}{dt}}-\mathrm {S} qdm\left(d{\frac {d\mathrm {P} }{dt}}+{\frac {d\mathrm {P} d\mathrm {R} }{dt}}\right)\\&\qquad -\mathrm {S} pdm\left(d{\frac {d\mathrm {Q} }{dt}}+{\frac {d\mathrm {P} d\mathrm {R} }{dt}}\right)+\mathrm {S} \Psi dm\,dt.\\&[\mathrm {P} ]=\mathrm {S} rdm\,d{\frac {d\mathrm {Y} }{dt}}-\mathrm {S} qdm\,d{\frac {d\mathrm {Z} }{dt}}+\left(\mathrm {S} r^{2}dm+\mathrm {S} q^{2}dm\right)d{\frac {d\mathrm {P} }{dt}}\\&+\mathrm {S} pq\,dm\,d{\frac {d\mathrm {Q} }{dt}}-\mathrm {S} pr\,dm\,d{\frac {d\mathrm {R} }{dt}}+\mathrm {S} qr\,dm\,{\frac {d\mathrm {Q} ^{2}-d\mathrm {R} ^{2}}{dt}}-\mathrm {S} pr\,dm{\frac {d\mathrm {P} d\mathrm {Q} }{dt}}\\&-\mathrm {S} pq\,dm{\frac {d\mathrm {P} d\mathrm {R} }{dt}}+\left(\mathrm {S} q^{2}dm-\mathrm {S} r^{2}dm\right){\frac {d\mathrm {Q} d\mathrm {R} }{dt}}+\mathrm {S} \varpi rdm\,dt-\mathrm {S} \Psi qdm\,dt.\\&[\mathrm {Q} ]=\mathrm {S} rdm\,d{\frac {d\mathrm {X} }{dt}}-\mathrm {S} qdm\,d{\frac {d\mathrm {Z} }{dt}}+\left(\mathrm {S} r^{2}dm+\mathrm {S} p^{2}dm\right)d{\frac {d\mathrm {Q} }{dt}}\\&+\mathrm {S} pq\,dm\,d{\frac {d\mathrm {P} }{dt}}+\mathrm {S} qr\,dm\,d{\frac {d\mathrm {R} }{dt}}+\mathrm {S} pr\,dm\,{\frac {d\mathrm {P} ^{2}-d\mathrm {R} ^{2}}{dt}}-\mathrm {S} qr\,dm{\frac {d\mathrm {P} d\mathrm {Q} }{dt}}\\&+\mathrm {S} pq\,dm{\frac {d\mathrm {Q} d\mathrm {R} }{dt}}+\left(\mathrm {S} r^{2}dm-\mathrm {S} p^{2}dm\right){\frac {d\mathrm {P} d\mathrm {R} }{dt}}+\mathrm {S} \Pi rdm\,dt-\mathrm {S} \Psi pdm\,dt.\end{aligned}}$