Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 13.djvu/51

Formules du mouvement de Saturne.

L’expression de la longitude vraie $\mathrm {V} ^{\text{v}},$ de Saturne dans son orbite, comptée de l’équinoxe moyen, est

${\begin{aligned}\mathrm {V} ^{\text{v}}=\varphi ^{\text{v}}&+\ t.154''{,}03+(71663''{,}37-t.3''{,}9673)\sin \left(\varphi ^{\text{v}}-\omega ^{\text{v}}\right)\\&+(2520''{,}02-t.0''{,}2793)\sin 2\left(\varphi ^{\text{v}}-\omega ^{\text{v}}\right)\\&+(\ \ 122''{,}87-t.0''{,}0204)\sin 3\left(\varphi ^{\text{v}}-\omega ^{\text{v}}\right)\\&+(\quad \ \ 6''{,}85-t,0''{,}0015)\sin 4\left(\varphi ^{\text{v}}-\omega ^{\text{v}}\right)\\&+\ \ \quad \qquad \qquad \qquad 0''{,}41\sin 5\left(\varphi ^{\text{v}}-\omega ^{\text{v}}\right)\end{aligned}}$

$+\left\{{\begin{aligned}&89''{,}40\sin \left(\ \ \varphi ^{\text{ıv}}-\ \ \varphi ^{\text{v}}+86^{\circ }{,}73\right)\\-&92''{,}33\sin \left(2\varphi ^{\text{ıv}}-2\varphi ^{\text{v}}-\ \ 6^{\circ }{,}34\right)\\-&20''{,}27\sin \left(3\varphi ^{\text{ıv}}-3\varphi ^{\text{v}}\right)\\-&\ \ 6''{,}07\sin \left(4\varphi ^{\text{ıv}}-4\varphi ^{\text{v}}\right)\\-&\ \ 2''{,}15\sin \left(5\varphi ^{\text{ıv}}-5\varphi ^{\text{v}}\right)\\-&\ \ 0''{,}84\sin \left(6\varphi ^{\text{ıv}}-6\varphi ^{\text{v}}\right)\\-&\ \ 0''{,}36\sin \left(7\varphi ^{\text{ıv}}-7\varphi ^{\text{v}}\right)\end{aligned}}\right.$

${\begin{aligned}&-(1303''{,}78+t.0''{,}0682)\sin \left(\ \ \varphi ^{\text{ıv}}-2\varphi ^{\text{v}}-16^{\circ }{,}47\ \ +t.41''{,}67\right)\\&-(2066''{,}92-t.0''{,}0477)\sin \left(2\varphi ^{\text{ıv}}-4\varphi ^{\text{v}}+62^{\circ }{,}425+t.151''{,}77\right)\\&-(\ \ 149''{,}05-t.0''{,}0011)\sin \left(3\varphi ^{\text{v}}\ -\ \varphi ^{\text{ıv}}+86^{\circ }{,}495-t.106''{,}64\right)\\&-(\quad 75''{,}84-t.0''{,}0136)\sin \left(2\varphi ^{\text{ıv}}-3\varphi ^{\text{v}}+16^{\circ }{,}45\ \ -t.\ \ 38''{,}23\right)\\&+34''{,}81\sin \left(\ \ \varphi ^{\text{ıv}}+95^{\circ }{,}11\right).\\&-46''{,}08\sin \left(4\varphi ^{\text{ıv}}-9\varphi ^{\text{v}}+57^{\circ }{,}555\right)\\&+15''{,}12\sin \left(3\varphi ^{\text{ıv}}-4\varphi ^{\text{v}}-69^{\circ }{,}76\ \ \right)\\&+\ \ 9''{,}28\sin \left(2\varphi ^{\text{ıv}}-\ \ \varphi ^{\text{v}}+35^{\circ }{,}23\ \ \right)\\&+\ \ 9''{,}06\sin \left(3\varphi ^{\text{ıv}}-5\varphi ^{\text{v}}+63^{\circ }{,}50\ \ \right)\\&+\ \ 4''{,}38\sin \left(4\varphi ^{\text{ıv}}-3\varphi ^{\text{v}}-69^{\circ }{,}98\ \ \right)\end{aligned}}$

$+\left\{{\begin{aligned}&-28''{,}54\sin \left(\ \ \varphi ^{\text{v}}-\ \ \varphi ^{\text{vı}}\right)\\&+44''{,}60\sin \left(2\varphi ^{\text{v}}-2\varphi ^{\text{vı}}\right)\\&+\ \ 5''{,}91\sin \left(3\varphi ^{\text{v}}-3\varphi ^{\text{vı}}-76^{\circ }{,}06\right)\\&+\ \ 0''{,}97\sin \left(4\varphi ^{\text{v}}-4\varphi ^{\text{vı}}\right)\\&+\ \ 0''{,}28\sin \left(5\varphi ^{\text{v}}-5\varphi ^{\text{vı}}\right)\end{aligned}}\right.$