On aura ainsi les quatre équations suivantes :
![{\displaystyle 411^{\mathrm {m} }{,}359=63{,}546581.1{,}022486.2\mathrm {A} '.{\frac {\mathrm {L'} }{r^{'3}}}+63{,}632467.2\mathrm {A} .{\frac {\mathrm {L} }{r^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05f1617f8e3bbdb6ae81d30f0276101b78208973)
![{\displaystyle -3{,}333442.2(\mathrm {A'-B} ).1{,}022486.{\frac {\mathrm {L'} }{r^{'3}}}-4{,}685805.2(\mathrm {A-B} ).{\frac {\mathrm {L} }{r^{3}}};\quad (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd179af47c8afa4c3bf7505f504df8c5e39630b)
![{\displaystyle 367^{\mathrm {m} }{,}468=56{,}879696.1{,}022486.2\mathrm {A} '.{\frac {\mathrm {L'} }{r^{'3}}}+54{,}260856.2A.{\frac {\mathrm {L} }{r^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e4a9e28e3ac977262d66da1ac825aa52ed0f23)
![{\displaystyle +3{,}333442.2(\mathrm {A'-B} ).1{,}022486.{\frac {\mathrm {L'} }{r^{'3}}}+4{,}685805.2(\mathrm {A-B} ).{\frac {\mathrm {L} }{r^{3}}};\quad (2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/973f1f67181d9c472968955e8cf2a6a312c394fb)
![{\displaystyle {\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(A'-B).{\frac {\mathrm {L'} }{r^{'3}}};\\+&4{,}667171.2(\mathrm {A-B} ).{\frac {\mathrm {L} }{r^{3}}};\quad (3)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63090d6bc51df209071de512e67be5e6718e39e)
![{\displaystyle 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd0e5947a32af0e2d30994f587805ace10c8a4a)
![{\displaystyle -3{,}267165.0{,}977514.2(\mathrm {A'-B} ).{\frac {\mathrm {L'} }{r^{'3}}}-4{,}667171.2(\mathrm {A-B} ).{\frac {\mathrm {L} }{r^{3}}};\quad (4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24862154dad579846441b43e7635f119dd81b21e)
Dans toutes ces équations, les valeurs
et
sont relatives aux moyennes distances de la lune et du soleil à la terre.
Le système
des équations précédentes, donne
![{\displaystyle 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}}};\quad (5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f55cf717148addb81978e3f8fece642c7203510)
Le système des équations
donne
![{\displaystyle 364^{\mathrm {m} }{,}798=120{,}460155.0{,}977514.2\mathrm {A} '.{\frac {\mathrm {L'} }{r^{'3}}}-117{,}936626.2\mathrm {A} .{\frac {\mathrm {L} }{r^{3}}};\quad (6)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/028e33a3940aa6890b815c7ecc36d37c35120fcb)
De ces deux équations, on tire
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094de2a5f2a99004ca83736999868bc11e874684)