Page:Lagrange - Œuvres (1867) vol. 1.djvu/695

${\begin{aligned}&+[-2i+2i\mu -i\nu \left[2h^{2}+i(20h\mathrm {f-8g} )\right]-3i^{2}\sigma (\mathrm {g} -2h\mathrm {f} )]{\frac {dydz}{dt^{2}}}\\&+\left({\frac {3}{2}}i^{2}\mu h^{2}-6i^{2}\nu h^{4}-9i^{2}\pi h^{2}\right)z^{3}\\&+\left(-10i^{2}h^{2}+7i^{2}\mu h^{2}-20i^{2}\nu h^{4}-5i^{2}\rho h^{2}+4i^{2}\sigma h^{4}\right)y^{2}z\\&+\left(2i^{2}-2i^{2}\mu +16i^{2}\nu h^{2}+4i^{2}\rho -5i^{2}\sigma h^{2}\right)y{\frac {dydz}{dt^{2}}}\\&+\mathrm {n} \left(-i\Psi -i\mu z\Phi -i\nu {\frac {d\Phi dz}{dt^{2}}}+2i\nu h^{2}z\Phi \right).\end{aligned}}$

Je mets donc ces valeurs de $\mathrm {y,z} ,{\frac {d^{2}\mathrm {y} }{dt^{2}}}$ et ${\frac {d^{2}\mathrm {z} }{dt^{2}}}$ dans les équations

${\frac {d^{2}\mathrm {y} }{dt^{2}}}+\mathrm {K^{2}y+n} {\mathfrak {Y}}=0,\qquad {\frac {d^{2}\mathrm {z} }{dt^{2}}}+\mathrm {L^{2}y} +i\mathrm {n} {\mathfrak {Z}}=0,$

et ensuite j’égale à zéro les termes homogènes, ce qui me donne les équations suivantes :

$-h^{2}-i\left(6h\mathrm {f} -2\mathrm {g} +3i\mathrm {f} ^{2}+\mathrm {n} {\mathfrak {P}}_{4}\right)-6\beta \left(\mathrm {g} -2h\mathrm {f} -i\mathrm {f} ^{2}+{\frac {2}{3}}i\mathrm {B} +\mathrm {n} {\mathfrak {A}}_{2}\right)$

$+8i^{2}\gamma \mathrm {B} -6i^{2}\delta \mathrm {A} -2i^{2}\varepsilon \mathrm {B} +2i^{2}\eta h^{2}\mathrm {B+K^{2}} =0,$

${\begin{aligned}&-\left(g-2hf-if^{2}+n{\mathfrak {A}}_{2}\right)-2i\beta \mathrm {A} -2i\gamma \mathrm {B} +2i^{2}\nu (g-2hf)\mathrm {B} +\alpha \mathrm {K} ^{2}=0,\\&3h^{2}+3i(4h\mathrm {f} -\mathrm {g} )-4\beta \left[h^{2}+i(6h\mathrm {f} -2\mathrm {g} )\right]-15i\delta (\mathrm {g} -2h\mathrm {f} )+\beta \mathrm {K} ^{2}=0,\\&{\frac {3}{2}}\left(h^{2}+i\mathrm {g} \right)-2i\beta (\mathrm {g} -2h\mathrm {f} )-4\gamma \left(h^{2}+2ih\mathrm {f} \right)-i\varepsilon (\mathrm {g} -2h\mathrm {f} )\\&\qquad \qquad \qquad \qquad \qquad \qquad +4i\eta h^{2}(\mathrm {g} -2h\mathrm {f} )+\gamma \mathrm {K} ^{2}=0,\\&-6h^{2}+10\beta h^{2}-9\delta h^{2}+\delta \mathrm {K} ^{2}=0,\\&-3h^{2}+\beta h^{2}+16\gamma h^{2}-5\varepsilon h^{2}+4\eta h^{4}+\varepsilon \mathrm {K} ^{2}=0,\\&-4\beta -4\gamma +4\varepsilon -5\eta h^{2}+\eta \mathrm {K} ^{2}=0,\\&-\Phi -2i\beta \Phi y-4i\beta \int \Phi dy-4i^{2}\gamma {\mathfrak {B}}_{5}\int (\cos \mathrm {H} t)z'dz+{\mathfrak {Y}}=0,\\&-h^{2}-i\left(2h\mathrm {f} +i\mathrm {f} ^{2}+\mathrm {n} {\mathfrak {A}}_{4}\right)-i\mu \left(\mathrm {g} -2h\mathrm {f} -i\mathrm {f} ^{2}+\mathrm {n} {\mathfrak {A}}_{3}\right)\\&\quad +2i\nu \left(\mathrm {g} -2h\mathrm {f} -i\mathrm {f} ^{2}+\mathrm {n} {\mathfrak {A}}_{2}\right)\left(h^{2}+2ih\mathrm {f} \right)\\&-3i^{2}\nu h^{2}(2\mathrm {A+B} )-6i^{2}\pi \mathrm {B} -2i^{2}\rho \mathrm {A} +2i^{2}\sigma h^{2}\mathrm {A+L^{2}} =0,\end{aligned}}$