Page:Lagrange - Œuvres (1867) vol. 1.djvu/640

9^o ${\frac {d^{2}v_{4}}{dt^{2}}}=yz{\frac {d^{2}y}{dt^{2}}}+z{\frac {dy^{2}}{dt^{2}}}+y{\frac {dydz}{dt^{2}}}\,;$

donc $yz\times (\mathrm {L} )+z\times (\mathrm {N} )+y\times (\mathrm {P} )$ donnera

(11)

$\left\{{\begin{aligned}{\frac {d^{2}v_{4}}{dt^{2}}}&+\mathrm {C} y+\mathrm {A} z+fu+4\mathrm {F} u_{1}+{\frac {g}{2}}v\\&+3\mathrm {G} v_{1}+{\frac {3\mathrm {H} }{2}}v_{2}+(h-\mathrm {G} )v_{6}+2\mathrm {H} v_{7}=0.\end{aligned}}\right.$

10^o ${\frac {d^{2}v_{5}}{dt^{2}}}=z^{2}{\frac {d^{2}y}{dt^{2}}}+2z{\frac {dydz}{dt^{2}}}\,;$

donc $z^{2}\times (\mathrm {L} )+2z\times (\mathrm {P} )$ donnera

(12)

${\frac {d^{2}v_{5}}{dt^{2}}}+2\mathrm {C} z+2fu_{1}+3\mathrm {F} u_{2}+gv_{1}+3\mathrm {G} v_{2}+2\mathrm {H} v_{3}+2(h-\mathrm {G} )v_{7}=0.$

11^o ${\frac {d^{2}v_{6}}{dt^{2}}}=\left(yz+\int zdy\right){\frac {d^{2}y}{dt^{2}}}+2z{\frac {dy^{2}}{dt^{2}}}+y{\frac {dydz}{dt^{2}}}\,;$

donc $\left(yz+\int zdy\right)\times (\mathrm {L} )+2z\times (\mathrm {N} )+y\times (\mathrm {P} )$ donnera

(13)

$\left\{{\begin{aligned}{\frac {d^{2}v_{6}}{dt^{2}}}&+\mathrm {C} y+2\mathrm {A} z+fu_{1}+6\mathrm {F} u+\mathrm {F} u_{3}\\&+{\frac {g}{2}}v+4\mathrm {G} v_{1}+{\frac {3\mathrm {H} }{2}}v_{2}+hv_{6}+5\mathrm {H} v_{7}=0.\end{aligned}}\right.$

12^o ${\frac {d^{2}v_{7}}{dt^{2}}}=z^{2}{\frac {d^{2}y}{dt^{2}}}+{\frac {d^{2}z}{dt^{2}}}\int zdy+3z{\frac {dydz}{dt^{2}}}\,;$

donc $z^{2}\times (\mathrm {L} )+(\mathrm {M} )\times \int zdy+3z\times (\mathrm {P} )$ donnera

(14)

$\left\{{\begin{aligned}{\frac {d^{2}v_{7}}{dt^{2}}}&+3\mathrm {C} z+3fu_{1}+4\mathrm {F} u_{2}+fu_{3}\\&+{\frac {3g}{2}}v_{1}+4\mathrm {G} v_{2}+{\frac {5\mathrm {H} }{2}}v_{3}+gv_{6}+(4h-3\mathrm {G} )=0.\end{aligned}}\right.$

Ayant ainsi autant d’équations que de variables, l’intégration qui doit donner la valeur de $y$ et de $z$ est facile par la méthode du n^o 26 ; de sorte