Page:Lagrange - Œuvres (1867) vol. 1.djvu/639

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

(6)

$\left\{{\begin{aligned}&{\frac {d^{2}u_{3}}{dt^{2}}}+\mathrm {C} +fy+2\mathrm {F} z+{\frac {g}{2}}u+2\mathrm {G} u_{1}+{\frac {3\mathrm {H} }{2}}u_{2}+(h-\mathrm {G} )u_{3}\\&+i\left[{\frac {k}{3}}v+2\mathrm {K} v_{1}+{\frac {3\mathrm {L} }{2}}v_{2}+{\frac {4\mathrm {M} }{3}}v_{3}+(l-2\mathrm {K} )v_{4}+\left(m-{\frac {\mathrm {L} }{2}}\right)v_{5}\right]=0.\end{aligned}}\right.$

5^o ${\frac {d^{2}v}{dt^{2}}}=3y^{2}{\frac {d^{2}y}{dt^{2}}}+6y{\frac {dy^{2}}{dt^{2}}}\,;$

donc $3y^{2}\mathrm {(L)+6(N)}$ donnera, en rejetant les termes affectés de $i,$ à cause que la variable $v$ est déjà elle-même multipliée par $i^{2}$ dans les équations (1) et (2),

(7)

${\frac {d^{2}v}{dt^{2}}}+6\mathrm {A} y+15\mathrm {F} u+g\mathrm {G} v+3\mathrm {H} v_{1}+12\mathrm {H} v_{6}=0.$

6^o ${\frac {d^{2}v_{1}}{dt^{2}}}=2yz{\frac {d^{2}y}{dt^{2}}}+y^{2}{\frac {d^{2}z}{dt^{2}}}+2z{\frac {dy^{2}}{dt^{2}}}+4y{\frac {dydz}{dt^{2}}}\,;$

donc $2yz\times (\mathrm {L} )+y^{2}\times (\mathrm {M} )+2z\times (\mathrm {N} )+4y\times (\mathrm {P} )$ donnera, en négligeant par la même raison que ci-devant les termes de l’ordre de $i,$

(8)

$\left\{{\begin{aligned}{\frac {d^{2}v_{1}}{dt^{2}}}&+4\mathrm {C} y+2\mathrm {A} z+5fu+10\mathrm {F} u_{1}+3gv\\&+(8\mathrm {G} +h)v_{1}+4\mathrm {H} v_{2}+4(h-\mathrm {G} )v_{6}+4\mathrm {H} v_{7}=0.\end{aligned}}\right.$

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

donc $z^{2}\times (\mathrm {L} )+2yz\times (\mathrm {M} )+2y\times (\mathrm {O} )+4z\times (\mathrm {P} )$ donnera

(9)

$\left\{{\begin{aligned}{\frac {d^{2}v_{2}}{dt^{2}}}&+2\mathrm {B} y+4\mathrm {C} z+10fu_{1}+5\mathrm {F} u_{2}+8gv_{1}\\&+(5\mathrm {G} +4h)v_{2}+3\mathrm {H} v_{3}-4gv_{6}+4(h-\mathrm {G} )v_{7}=0.\end{aligned}}\right.$

8^o ${\frac {d^{2}v_{3}}{dt^{2}}}=3z^{2}{\frac {d^{2}z}{dt^{2}}}+6z{\frac {dz^{2}}{dt^{2}}}\,;$

donc $3z^{2}\times (\mathrm {M} )+6z\times (\mathrm {O} )$ donnera

(10)

${\frac {d^{2}v_{3}}{dt^{2}}}+6\mathrm {B} z+15fu_{2}+15gv_{2}+ghv_{3}-12gv_{7}=0.$