Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 12.djvu/527

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(2)\left\{{\begin{aligned}367^{\mathrm {m} }{,}468=&56{,}879696.1{,}022486.2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}+54{,}260856.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}\\+&3{,}333442.2(\mathrm {A'-B} ).1{,}022486{\frac {\mathrm {L} '}{r'^{3}}}+4{,}685805.2(\mathrm {A-B} ){\frac {\mathrm {L} }{r^{3}}},\end{aligned}}\right.

(3)\left\{{\begin{aligned}160^{\mathrm {m} }{,}850=&56{,}962913.0{,}977514.2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}-63{,}635484.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}\\+&3{,}267165.0{,}977514.2(\mathrm {A'-B} ).{\frac {\mathrm {L} '}{r'^{3}}}+4{,}667171.2(\mathrm {A-B} ){\frac {\mathrm {L} }{r^{3}}},\end{aligned}}\right.

(4)\left\{{\begin{aligned}203^{\mathrm {m} }{,}948=&63{,}497242.0{,}977514.2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}-54{,}301142.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}\\-&3{,}267165.0{,}977514.2(\mathrm {A'-B} ).{\frac {\mathrm {L} '}{r'^{3}}}-4{,}667171.2(\mathrm {A-B} ){\frac {\mathrm {L} }{r^{3}}}.\end{aligned}}\right.

Dans toutes ces équations, les valeurs ${\frac {\mathrm {L} '}{r'^{3}}}$ et ${\frac {\mathrm {L} }{r^{3}}}$ sont relatives aux moyennes distances de la Lune et du Soleil à la Terre.

Le système $+(1)+(2)$ des équations précédentes donne

(5)\qquad 778^{\mathrm {m} }{,}708=120{,}426277.1{,}022486.2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}+117{,}893323.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}\,;

le système des équations $+(3)+(4)$ donne

(6)\qquad 364^{\mathrm {m} }{,}798=120{,}460155.0{,}977514.2A'{\frac {\mathrm {L} '}{r'^{3}}}-117{,}936626.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}

De ces deux équations on tire

2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}=4^{\mathrm {m} }{,}74788,\qquad 2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}=1^{\mathrm {m} }{,}64658.

Le système des équations $+(1)-(2)$ donne

(7)\left\{{\begin{aligned}43^{\mathrm {m} }{,}891=&6{,}666885.1{,}022486.2\mathrm {A} '{\frac {\mathrm {L} '}{r'^{3}}}\qquad \quad \ \ +9{,}371611.2\mathrm {A} {\frac {\mathrm {L} }{r^{3}}}\\-&6{,}666885.1{,}022486.2\mathrm {{\frac {A'-B}{A'}}A'} {\frac {\mathrm {L} '}{r'^{3}}}-9{,}371611.2\mathrm {{\frac {A-B}{A}}A} {\frac {\mathrm {L} }{r^{3}}}\,;\end{aligned}}\right.