Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/133

Longitude vraie du deuxième satellite.

${\begin{aligned}\varphi _{2}=&u_{2}-\left(114^{\circ }35'\right)n\varepsilon _{2}\sin \mathrm {V} _{2})\\&+{\frac {{\text{☾}}_{1}}{\mathbb {Z} \!^{\upsilon }}}\left[-387482'\sin(u_{1}-u_{2})-2727'\sin 2(u_{1}-u_{2})\right.\\&\qquad \qquad \qquad \left.-509'\sin 3(u_{1}-u_{2})-12'\sin 4(u_{1}-u_{2})\ldots \right]\\&+{\frac {{\text{☾}}_{3}}{\mathbb {Z} \!^{\upsilon }}}\left[10067'\sin(u_{3}-u_{2})-626246'\sin 2(u_{3}-u_{2})\right.\\&\qquad \qquad \qquad \left.-3717'\sin 3(u_{3}-u_{2})-825'\sin 4(u_{3}-u_{2})\ldots \right]\\&+{\frac {{\text{☾}}_{4}}{\mathbb {Z} \!^{\upsilon }}}\left[506'\sin(u_{4}-u_{2})-3123'\sin 2(u_{4}-u_{2})\right.\\&\qquad \qquad \qquad \left.-52'\sin 3(u_{4}-u_{2})-17'\sin 4(u_{4}-u_{2})\ldots \right]\\&-0{,}190\sin 2(v-u_{2}).\end{aligned}}$

Longitude vraie du troisième satellite.

${\begin{aligned}\varphi _{3}=&u_{3}-\left(114^{\circ }35'\right)n\varepsilon _{3}\sin \mathrm {V} _{3})\\&+{\frac {{\text{☾}}_{1}}{\mathbb {Z} \!^{\upsilon }}}\left[1306'\sin(u_{1}-u_{3})-38'\sin 2(u_{1}-u_{3})\right.\\&\qquad \qquad \qquad \left.-8'\sin 3(u_{1}-u_{3})-1'\sin 4(u_{1}-u_{3})\ldots \right]\\&+{\frac {{\text{☾}}_{2}}{\mathbb {Z} \!^{\upsilon }}}\left[-207375'\sin(u_{2}-u_{3})-2760'\sin 2(u_{2}-u_{3})\right.\\&\qquad \qquad \qquad \left.-559'\sin 3(u_{2}-u_{3})-142'\sin 4(u_{2}-u_{3})\ldots \right]\\&+{\frac {{\text{☾}}_{4}}{\mathbb {Z} \!^{\upsilon }}}\left[5703'\sin(u_{4}-u_{3})-14911'\sin 2(u_{4}-u_{3})\right.\\&\qquad \qquad \qquad \left.-1376'\sin 3(u_{4}-u_{3})-306'\sin 4(u_{4}-u_{3})\ldots \right]\\&-0{,}977\sin 2(v-u_{3}).\end{aligned}}$

Longitude vraie du quatrième satellite.

${\begin{aligned}\varphi _{4}=&u_{4}-\left(114^{\circ }35'\right)n\varepsilon _{4}\sin \mathrm {V} _{4})\\&+{\frac {{\text{☾}}_{1}}{\mathbb {Z} \!^{\upsilon }}}\left[768'\sin(u_{1}-u_{4})-0'\sin 2(u_{1}-u_{4})\right.\\&\qquad \qquad \qquad \left.-0'\sin 3(u_{1}-u_{4})-0'\sin 4(u_{1}-u_{4})\ldots \right]\\&+{\frac {{\text{☾}}_{2}}{\mathbb {Z} \!^{\upsilon }}}\left[1219'\sin(u_{2}-u_{4})-15'\sin 2(u_{2}-u_{4})\right.\\&\qquad \qquad \qquad \left.-3'\sin 3(u_{2}-u_{4})-0'\sin 4(u_{2}-u_{4})\ldots \right]\\&+{\frac {{\text{☾}}_{3}}{\mathbb {Z} \!^{\upsilon }}}\left[-1691'\sin(u_{3}-u_{4})-444'\sin 2(u_{3}-u_{4})\right.\\&\qquad \qquad \qquad \left.-180'\sin 3(u_{3}-u_{4})-59'\sin 4(u_{3}-u_{4})\ldots \right]\\&-4{,}208\sin 2(v-u_{4}).\end{aligned}}$