Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/72

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on aura, en n’ayant égard qu’à ces termes et observant que l’on a

0={\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}},

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 {m'^{2}mt}{(5n'-2n){\sqrt {a}}}}\left\{{\begin{aligned}&{\frac {\partial {\rm {P'}}}{\partial e}}\left(e{\frac {\partial ^{2}{\rm {P}}}{\partial e^{2}}}+e'{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial e'}}+\gamma {\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}\right)\\\\-&{\frac {\partial {\rm {P}}}{\partial e}}\left(e{\frac {\partial ^{2}{\rm {P'}}}{\partial e^{2}}}+e'{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial e'}}+\gamma {\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)\\\\+&{\frac {\partial {\rm {P'}}}{\partial \gamma }}\left(e{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}+e'{\frac {\partial ^{2}{\rm {P}}}{\partial e'\partial \gamma }}+\gamma {\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}\right)\\\\-&{\frac {\partial {\rm {P}}}{\partial \gamma }}\left(e{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}+e'{\frac {\partial ^{2}{\rm {P'}}}{\partial e'\partial \gamma }}+\gamma {\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}}\right)\end{aligned}}\right\}.

${\frac {\rm {P}}{\partial e}}$ et ${\frac {\rm {P'}}{\partial e}}$ sont homogènes en $e,e'$ et $\gamma$ , de la seconde dimension, ce qui donne

{\begin{aligned}e{\frac {\partial ^{2}{\rm {P}}}{\partial e^{2}}}+e'{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial e'}}+\gamma {\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}&=2{\frac {\partial {\rm {P}}}{\partial e}},\\\\e{\frac {\partial ^{2}{\rm {P'}}}{\partial e^{2}}}+e'{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial e'}}+\gamma {\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}&=2{\frac {\partial {\rm {P'}}}{\partial e}}\,;\end{aligned}}

de plus, ${\frac {\partial {\rm {P}}}{\partial \gamma }}$ et ${\frac {\partial {\rm {P'}}}{\partial \gamma }}$ sont homogènes en $e,e'$ et $\gamma$ , de la seconde dimension, ce qui donne

{\begin{aligned}+&e{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}+e'{\frac {\partial ^{2}{\rm {P}}}{\partial e'\partial \gamma }}+\gamma {\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}=2{\frac {\partial {\rm {P}}}{\partial \gamma }}\\\\+&e{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}+e'{\frac {\partial ^{2}{\rm {P'}}}{\partial e'\partial \gamma }}+\gamma {\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}}=2{\frac {\partial {\rm {P'}}}{\partial \gamma }}\,;\end{aligned}}

on a donc encore, en n’ayant égard qu’à ces termes,

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 enfin le terme de $\delta e,$

{\frac {mm't}{(5n'-2n){\sqrt {aa'}}}}\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)\,;