Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/132

Rayon vecteur du troisième satellite.

${\begin{aligned}r_{3}=&14{,}38(1+n\varepsilon _{3}\cos \mathrm {V} _{3})\\&+{\frac {{\text{☾}}_{1}}{\mathbb {Z} \!^{\upsilon }}}\left[5{,}88\cos(u_{1}-u_{3})+0{,}19\cos 2(u_{1}-u_{3})\right.\\&\qquad \qquad \qquad \left.+0{,}03\cos 3(u_{1}-u_{3})+0{,}00\cos 4(u_{2}-u_{1})\ldots \right]\\&+{\frac {{\text{☾}}_{2}}{\mathbb {Z} \!^{\upsilon }}}\left[452{,}98\cos(u_{2}-u_{3})+9{,}13\cos 2(u_{2}-u_{3})\right.\\&\qquad \qquad \qquad \left.+2{,}16\cos 3(u_{2}-u_{3})+0{,}59\cos 4(u_{2}-u_{3})\ldots \right]\\&+{\frac {{\text{☾}}_{4}}{\mathbb {Z} \!^{\upsilon }}}\left[7{,}46\cos(u_{4}-u_{3})-33{,}62\cos 2(u_{4}-u_{3})\right.\\&\qquad \qquad \qquad \left.-3{,}95\cos 3(u_{4}-u_{3})+0{,}95\cos 4(u_{4}-u_{3})\ldots \right]\\&-0{,}0000392\cos 2(v-u_{3}).\end{aligned}}$

Rayon vecteur du quatrième satellite.

${\begin{aligned}r_{4}=&25{,}30(1+n\varepsilon _{4}\cos \mathrm {V} _{4})\\&+{\frac {{\text{☾}}_{1}}{\mathbb {Z} \!^{\upsilon }}}\left[5{,}66\cos(u_{1}-u_{4})+0{,}01\cos 2(u_{1}-u_{4})\right.\\&\qquad \qquad \qquad \left.+0{,}00\cos 3(u_{1}-u_{4})+0{,}00\cos 4(u_{1}-u_{4})\ldots \right]\\&+{\frac {{\text{☾}}_{2}}{\mathbb {Z} \!^{\upsilon }}}\left[8{,}18\cos(u_{2}-u_{4})+0{,}17\cos 2(u_{2}-u_{4})\right.\\&\qquad \qquad \qquad \left.+0{,}07\cos 3(u_{2}-u_{4})+0{,}00\cos 4(u_{3}-u_{2})\ldots \right]\\&+{\frac {{\text{☾}}_{3}}{\mathbb {Z} \!^{\upsilon }}}\left[31{,}55\cos(u_{3}-u_{4})+1{,}62\cos 2(u_{3}-u_{4})\right.\\&\qquad \qquad \qquad \left.+1{,}35\cos 3(u_{3}-u_{4})+0{,}43\cos 4(u_{3}-u_{4})\ldots \right]\\&-0{,}0003754\cos 2(v-u_{4}).\end{aligned}}$

LI.

Longitude vraie du premier satellite.

${\begin{aligned}\varphi _{1}=&u_{1}-\left(114^{\circ }35'\right)n\varepsilon _{1}\sin \mathrm {V} _{1})\\&+{\frac {{\text{☾}}_{2}}{\mathbb {Z} \!^{\upsilon }}}\left[9500'\sin(u_{2}-u_{1})-1227214'\sin 2(u_{2}-u_{1})\right.\\&\qquad \qquad \qquad \left.-3526'\sin 3(u_{2}-u_{1})-705'\sin 4(u_{2}-u_{1})\ldots \right]\\&+{\frac {{\text{☾}}_{3}}{\mathbb {Z} \!^{\upsilon }}}\left[1158'\sin(u_{3}-u_{1})-1007'\sin 2(u_{3}-u_{1})\right.\\&\qquad \qquad \qquad \left.-101'\sin 3(u_{3}-u_{1})-18'\sin 4(u_{3}-u_{1})\ldots \right]\\&+{\frac {{\text{☾}}_{4}}{\mathbb {Z} \!^{\upsilon }}}\left[188'\sin(u_{4}-u_{1})-81'\sin 2(u_{4}-u_{1})\right.\\&\qquad \qquad \qquad \left.-3'\sin 3(u_{4}-u_{1})-0'\sin 4(u_{4}-u_{1})\ldots \right]\\&-0{,}047\sin 2(v-u_{1}).\end{aligned}}$