Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/71

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-{\frac {3m'\left(5m{\sqrt {a}}+2m'{\sqrt {a'}}\right)}{a{\sqrt {a'}}(5n'-2n)^{2}}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}nt\left({\rm {P}}{\frac {\partial {\rm {P'}}}{\partial \gamma }}-{\rm {P'}}{\frac {\partial {\rm {P}}}{\partial \gamma }}\right).

On aura, en n’ayant égard qu’à ces termes,

e\delta e.m{\sqrt {a}}+e'\delta e'.m'{\sqrt {a'}}+{\frac {mm'{\sqrt {aa'}}}{m{\sqrt {a}}+m'{\sqrt {a'}}}}\gamma \delta \gamma

=-{\frac {3mm'\left(5m{\sqrt {a}}+2m'{\sqrt {a'}}\right)}{(5n'-2n)^{2}{\sqrt {aa'}}}}nt\left\{{\begin{aligned}&{\rm {P}}\left(e{\frac {\partial {\rm {P'}}}{\partial e}}+e'{\frac {\partial {\rm {P'}}}{\partial e'}}+\gamma {\frac {\partial {\rm {P'}}}{\partial \gamma }}\right)\\\\-&{\rm {P'}}\left(e{\frac {\partial {\rm {P}}}{\partial e}}+e'{\frac {\partial {\rm {P}}}{\partial e'}}+\gamma {\frac {\partial {\rm {P}}}{\partial \gamma }}\right)\end{aligned}}\right\}\,;

or ${\rm {P}}$ et ${\rm {P'}}$ étant des fonctions homogènes en $e,e'$ et $\gamma ,$ de la troisième dimension, on a

{\begin{aligned}&e{\frac {\partial {\rm {P}}}{\partial e}}+e'{\frac {\partial {\rm {P}}}{\partial e'}}+\gamma {\frac {\partial {\rm {P}}}{\partial \gamma }}=3{\rm {P,}}\\\\&e{\frac {\partial {\rm {P'}}}{\partial e}}+e'{\frac {\partial {\rm {P'}}}{\partial e'}}+\gamma {\frac {\partial {\rm {P'}}}{\partial \gamma }}=3{\rm {P'\,;}}\end{aligned}}

on a donc

0=e\delta e.m{\sqrt {a}}+e'\delta e'.m'{\sqrt {a'}}+{\frac {mm'{\sqrt {aa'}}}{m{\sqrt {a}}+m'{\sqrt {a'}}}}\gamma \delta \gamma .

Considérons ensuite le terme de $\delta e,$

{\frac {m'^{2}t}{a(5n'-2n)}}\left({\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e^{2}}}-{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e^{2}}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)\,;

le terme de $\delta e',$

{\frac {mm't}{{\sqrt {aa'}}(5n'-2n)}}\left({\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial e'}}-{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial e'}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e'\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e'\partial \gamma }}\right)\,;

enfin le terme de $\delta \gamma ,$

{\frac {m'^{2}t}{a(5n'-2n)}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}\left({\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)\,;