Page:Lagrange - Œuvres (1867) vol. 1.djvu/510

on trouvera de même

${\begin{aligned}\operatorname {d} \beta =&{\frac {d\mathrm {P} }{dt}}d\mathrm {X} +{\frac {d\mathrm {Q} }{dt}}d\mathrm {Y} +{\frac {d\mathrm {R} }{dt}}d\mathrm {Z} ,\\\operatorname {d} \gamma =&{\frac {d\mathrm {S} }{dt}}d\mathrm {X} +{\frac {d\mathrm {T} }{dt}}d\mathrm {Y} +{\frac {d\mathrm {U} }{dt}}d\mathrm {Z} \,;\end{aligned}}$

substituant, au lieu de $d\mathrm {X} ,d\mathrm {Y} ,d\mathrm {Z} ,$ les valeurs trouvées ci-devant, il viendra

${\begin{aligned}\operatorname {d} \alpha =&{\frac {(\mathrm {QU-RT} ){\cfrac {d\mathrm {L} }{dt}}+(\mathrm {RS-PU} ){\cfrac {d\mathrm {M} }{dt}}+(\mathrm {PT-QS} ){\cfrac {d\mathrm {N} }{dt}}}{\mathrm {K} }}\operatorname {d} x\\+&{\frac {(\mathrm {NT-MU} ){\cfrac {d\mathrm {L} }{dt}}+(\mathrm {LU-NS} ){\cfrac {d\mathrm {M} }{dt}}+(\mathrm {MS-LT} ){\cfrac {d\mathrm {N} }{dt}}}{\mathrm {K} }}\operatorname {d} y\\+&{\frac {(\mathrm {MR-NQ} ){\cfrac {d\mathrm {L} }{dt}}+(\mathrm {NP-LR} ){\cfrac {d\mathrm {M} }{dt}}+(\mathrm {LQ-MP} ){\cfrac {d\mathrm {N} }{dt}}}{\mathrm {K} }}\operatorname {d} z,\\\operatorname {d} \beta =&{\frac {(\mathrm {QU-RT} ){\cfrac {d\mathrm {P} }{dt}}+(\mathrm {RS-PU} ){\cfrac {d\mathrm {Q} }{dt}}+(\mathrm {PT-QS} ){\cfrac {d\mathrm {R} }{dt}}}{\mathrm {K} }}\operatorname {d} x\\+&{\frac {(\mathrm {NT-MU} ){\cfrac {d\mathrm {P} }{dt}}+(\mathrm {LU-NS} ){\cfrac {d\mathrm {Q} }{dt}}+(\mathrm {MS-LT} ){\cfrac {d\mathrm {R} }{dt}}}{\mathrm {K} }}\operatorname {d} y\\+&{\frac {(\mathrm {MR-NQ} ){\cfrac {d\mathrm {P} }{dt}}+(\mathrm {NP-LR} ){\cfrac {d\mathrm {Q} }{dt}}+(\mathrm {LQ-MP} ){\cfrac {d\mathrm {R} }{dt}}}{\mathrm {K} }}\operatorname {d} z,\\\operatorname {d} \gamma =&{\frac {(\mathrm {QU-RT} ){\cfrac {d\mathrm {S} }{dt}}+(\mathrm {RS-PU} ){\cfrac {d\mathrm {T} }{dt}}+(\mathrm {PT-QS} ){\cfrac {d\mathrm {U} }{dt}}}{\mathrm {K} }}\operatorname {d} x\\+&{\frac {(\mathrm {NT-MU} ){\cfrac {d\mathrm {S} }{dt}}+(\mathrm {LU-NS} ){\cfrac {d\mathrm {T} }{dt}}+(\mathrm {MS-LT} ){\cfrac {d\mathrm {U} }{dt}}}{\mathrm {K} }}\operatorname {d} y\\+&{\frac {(\mathrm {MR-NQ} ){\cfrac {d\mathrm {S} }{dt}}+(\mathrm {NP-LR} ){\cfrac {d\mathrm {T} }{dt}}+(\mathrm {LQ-MP} ){\cfrac {d\mathrm {U} }{dt}}}{\mathrm {K} }}\operatorname {d} z.\end{aligned}}$

Donc prenant, dans l’expression de $\operatorname {d} \alpha ,$ le coefficient de $\operatorname {d} x,$ dans celle de $\operatorname {d} \beta$ le coefficient de $\operatorname {d} y,$ et dans celle de $\operatorname {d} \gamma$ le coefficient de $\operatorname {d} z,$