Formules du mouvement de Jupiter.
Soient
![{\displaystyle {\begin{aligned}n^{\text{ıv}}t+\varepsilon ^{\text{ıv}}=&\quad 4^{\circ }{,}17858+t.33^{\circ }{,}721121,\\n^{\text{v}}\ t+\varepsilon ^{\text{v}}\ =&257^{\circ }{,}07259+t.13{,}579357,\\n^{\text{vı}}t+\varepsilon ^{\text{vı}}=&353^{\circ }{,}96753+t.\ \ 4^{\circ }{,}760710.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edbc061ac1d1ca6f7ee63188433062e18f413e2e)
Ces trois quantités sont les longitudes moyennes de Jupiter, de Saturne et d’Uranus, comptées de l’équinoxe fixe de 1750.
Soient encore
![{\displaystyle {\begin{aligned}\omega ^{\text{ıv}}=&\ \ 11^{\circ }{,}47940+t.\ \ 20''{,}527446+t^{2}.0''{,}00061759,\\\omega ^{\text{v}}\ =&\ \ 97^{\circ }{,}96370+t.\ \ 59''{,}736418+t^{2}.0''{,}00049627,\\\theta ^{\text{ıv}}\,=&108^{\circ }{,}82267+t.105''{,}832375,\\\theta ^{\text{v}}\ \,=&123^{\circ }{,}88341+t.\ \ 94''{,}677500,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af14755f3520de48660b3f339c407b1ceaf09b0)
précession annuelle des équinoxes ![{\displaystyle =154''{,}63\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2aa46348409f493dcec3519c5d2b887dfb1325f)
![{\displaystyle \varphi ^{\text{ıv}}=n^{\text{ıv}}t+\varepsilon ^{\text{ıv}}+\left(3739''{,}05-t.0''{,}11224+t^{2}.0''{,}0001078\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87a27959ba4a9d64f1692b6ab99c9dc278ada24e)
![{\displaystyle {\begin{aligned}\times \sin &\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0093\right.\\&\ \ \ \quad \qquad \qquad -\left.t.242''{,}25+t^{2}.0''{,}03789\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3a14785127beb7605f8dc073052814eefa77fb)
![{\displaystyle {\begin{aligned}-40''{,}86\sin 2&\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0093\right.\\&\ \ \ \quad \qquad \qquad -\left.t.242''{,}25+t^{2}.0''{,}03789\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a389fa5b426d4c25c47d891ab7773bc87934c2)
![{\displaystyle \varphi ^{\text{v}}=n^{\text{v}}t+\varepsilon ^{\text{v}}-\left(9111''{,}406-t.0''{,}2738+t^{2}.0''{,}0002534\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbbfb7861916d5892b9b42107d0b64922d13c42)
![{\displaystyle {\begin{aligned}\times \sin &\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0510\right.\\&\ \ \ \quad \qquad \qquad -\left.t.237''{,}84+t^{2}.0''{,}03635\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ecda40cee2e9461c5da9616e2712fc8e2aef4c9)
![{\displaystyle {\begin{aligned}&+(94''{,}72-t.0''{,}0053)\sin 2\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0510\right.\\&\qquad \qquad \qquad \qquad -\left.t.237''{,}84+t^{2}.0''{,}03635\right).\\&+95''{,}76\sin \left(3n^{\text{vı}}t-n^{\text{v}}t+3\varepsilon ^{\text{vı}}-\varepsilon ^{\text{v}}-95^{\circ }{,}0779\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8eaf17fda99c64a60ed3c8ad13c47c2c48f924)
La longitude vraie
de Jupiter dans son orbite, et comptée de l’équinoxe moyen, sera
![{\displaystyle {\begin{aligned}\mathrm {V} ^{\text{ıv}}=\varphi ^{\text{ıv}}&+\ t.154''{,}63+(61208''{,}23+t.1''{,}9446)\sin \left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(1838''{,}54+t.0''{,}1168)\sin 2\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(\quad 76''{,}57+t.0''{,}0072)\sin 3\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(\quad \ \ 3''{,}65+t.0''{,}0005)\sin 4\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+\ \ \quad \qquad \qquad \qquad 0''{,}19\sin 5\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c382b8281661a3ac8100db0d859a5f51b53e8f1b)