Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/206

l’action du Soleil et de la Lune. L’équation (1) du numéro précédent devient, en y faisant $\cos \theta =\mu$ et $\gamma =l,$

${\frac {y}{l}}={\frac {\partial \left(u{\sqrt {1-\mu ^{2}}}\right)}{\partial \mu }}-{\frac {\partial v}{\partial \varpi }}\,;$

l’équation (2) devient

$d\theta {\frac {\partial ^{2}u}{\partial t^{2}}}+d\varpi \left(1-\mu ^{2}\right){\frac {\partial ^{2}v}{\partial t^{2}}}=$

$-g{\frac {\partial y}{\partial \mu }}{\frac {\partial \mu }{\partial \theta }}d\theta -g{\frac {\partial y}{\partial \varpi }}d\varpi +{\frac {\partial {\rm {V'}}}{\partial \mu }}{\frac {\partial \mu }{\partial \theta }}d\theta +{\frac {\partial {\rm {V'}}}{\partial \varpi }}d\varpi \,;$

or on a ${\frac {\partial \mu }{\partial \theta }}=-{\sqrt {1-\mu ^{2}}}\,;$ on aura donc, en comparant les coefficients de $d\theta$ et de $d\varpi$,

${\begin{aligned}{\frac {\partial ^{2}u}{\partial t^{2}}}&=g{\frac {\partial y}{\partial \mu }}{\sqrt {1-\mu ^{2}}}-{\frac {\partial {\rm {V'}}}{\partial \mu }}{\sqrt {1-\mu ^{2}}},\\\\{\frac {\partial ^{2}v}{\partial t^{2}}}&=-{\frac {g{\frac {\partial y}{\partial \varpi }}}{1-\mu ^{2}}}+{\frac {\frac {\partial {\rm {V'}}}{\partial \varpi }}{1-\mu ^{2}}}\,;\end{aligned}}$

partant,

${\frac {\partial .{\frac {\partial ^{2}u}{\partial t^{2}}}{\sqrt {1-\mu ^{2}}}}{\partial \mu }}=g{\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial y}{\partial \mu }}}{\partial \mu }}-{\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial {\rm {V'}}}{\partial \mu }}}{\partial \mu }},$

${\frac {\partial {\frac {\partial ^{2}v}{\partial t^{2}}}}{\partial \varpi }}=-g{\frac {\frac {\partial ^{2}y}{\partial \varpi ^{2}}}{1-\mu ^{2}}}+{\frac {\frac {\partial ^{2}{\rm {V'}}}{\partial \varpi ^{2}}}{1-\mu ^{2}}}.$

L’expression précédente de ${\frac {y}{l}}$ donne

${\frac {\partial ^{2}y}{\partial t^{2}}}=l{\frac {\partial .{\frac {\partial ^{2}u}{\partial t^{2}}}{\sqrt {1-\mu ^{2}}}}{\partial \mu }}-l{\frac {\partial {\frac {\partial ^{2}v}{\partial t^{2}}}}{\partial \varpi }}\,;$

on aura donc

(3) ${\frac {\partial ^{2}y}{\partial t^{2}}}=lg{\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial y}{\partial \mu }}}{\partial \mu }}+{\frac {lg{\frac {\partial ^{2}y}{\partial \varpi ^{2}}}}{1-\mu ^{2}}}-l{\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial {\rm {V'}}}{\partial \mu }}}{\partial \mu }}-{\frac {l{\frac {\partial ^{2}{\rm {V'}}}{\partial \varpi ^{2}}}}{1-\mu ^{2}}}.$

Pour intégrer cette équation, nous ferons

$y={\rm {Y^{(0)}+Y^{(1)}+Y^{(2)}+Y^{(3)}+\ldots ,}}$