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Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/253
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1
o
Pour les mouvements annuels des nœuds par rapport à l’écliptique vraie
d
Ω
d
t
=
−
(
0
,
1
)
−
(
0
,
2
)
−
(
0
,
3
)
−
(
0
,
4
)
−
(
0
,
5
)
−
(
3
,
0
)
+
[
(
0
,
1
)
−
(
3
,
1
)
]
θ
′
cos
(
ω
′
−
ω
)
θ
+
[
(
0
,
2
)
−
(
3
,
2
)
]
θ
″
cos
(
ω
″
−
ω
)
θ
+
[
(
0
,
4
)
−
(
3
,
4
)
]
θ
IV
cos
(
ω
IV
−
ω
)
θ
+
[
(
0
,
5
)
−
(
3
,
5
)
]
θ
V
cos
(
ω
V
−
ω
)
θ
,
d
Ω
′
d
t
=
−
(
1
,
0
)
−
(
1
,
2
)
−
(
1
,
3
)
−
(
1
,
4
)
−
(
1
,
5
)
−
(
3
,
1
)
+
[
(
1
,
0
)
−
(
3
,
0
)
]
θ
cos
(
ω
−
ω
′
)
θ
′
+
[
(
1
,
2
)
−
(
3
,
2
)
]
θ
″
cos
(
ω
″
−
ω
′
)
θ
′
+
[
(
1
,
4
)
−
(
3
,
4
)
]
θ
IV
cos
(
ω
IV
−
ω
′
)
θ
′
+
[
(
1
,
5
)
−
(
3
,
5
)
]
θ
V
cos
(
ω
V
−
ω
′
)
θ
′
,
d
Ω
″
d
t
=
−
(
2
,
0
)
−
(
2
,
1
)
−
(
2
,
3
)
−
(
2
,
4
)
−
(
2
,
5
)
−
(
3
,
2
)
+
[
(
2
,
0
)
−
(
3
,
0
)
]
θ
cos
(
ω
−
ω
″
)
θ
″
+
[
(
2
,
1
)
−
(
3
,
1
)
]
θ
′
cos
(
ω
′
−
ω
″
)
θ
″
+
[
(
2
,
4
)
−
(
3
,
4
)
]
θ
IV
cos
(
ω
IV
−
ω
″
)
θ
″
+
[
(
2
,
5
)
−
(
3
,
5
)
]
θ
V
cos
(
ω
V
−
ω
″
)
θ
″
,
d
Ω
IV
d
t
=
−
(
4
,
0
)
−
(
4
,
1
)
−
(
4
,
2
)
−
(
4
,
3
)
−
(
4
,
5
)
−
(
3
,
4
)
+
[
(
4
,
0
)
−
(
3
,
0
)
]
θ
cos
(
ω
−
ω
IV
)
θ
IV
+
[
(
4
,
1
)
−
(
3
,
1
)
]
θ
′
cos
(
ω
′
−
ω
IV
)
θ
IV
+
[
(
4
,
2
)
−
(
3
,
2
)
]
θ
″
cos
(
ω
″
−
ω
IV
)
θ
IV
+
[
(
4
,
5
)
−
(
3
,
5
)
]
θ
V
cos
(
ω
V
−
ω
IV
)
θ
IV
,
d
Ω
V
d
t
=
−
(
5
,
0
)
−
(
5
,
1
)
−
(
5
,
2
)
−
(
5
,
3
)
−
(
5
,
4
)
−
(
3
,
5
)
+
[
(
5
,
0
)
−
(
3
,
0
)
]
θ
cos
(
ω
−
ω
V
)
θ
V
+
[
(
5
,
1
)
−
(
3
,
1
)
]
θ
′
cos
(
ω
′
−
ω
V
)
θ
V
+
[
(
5
,
2
)
−
(
3
,
2
)
]
θ
″
cos
(
ω
″
−
ω
V
)
θ
V
+
[
(
5
,
4
)
−
(
3
,
4
)
]
θ
IV
cos
(
ω
IV
−
ω
V
)
θ
V
.
{\displaystyle {\begin{aligned}&{\frac {d\Omega }{dt}}\ \ \ =-(0,1)-(0,2)-(0,3)-(0,4)-(0,5)-(3,0)\\&\qquad +{\bigl [}(0,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega )}{\theta }}+{\bigl [}(0,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega )}{\theta }}\\&\qquad +{\bigl [}(0,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega )}{\theta }}+{\bigl [}(0,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega )}{\theta }},\\\\&{\frac {d\Omega '}{dt}}\ \ =-(1,0)-(1,2)-(1,3)-(1,4)-(1,5)-(3,1)\\&\qquad +{\bigl [}(1,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ')}{\theta '}}+{\bigl [}(1,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ')}{\theta '}}\\&\qquad +{\bigl [}(1,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega ')}{\theta '}}+{\bigl [}(1,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega ')}{\theta '}},\\\\&{\frac {d\Omega ''}{dt}}\ \ =-(2,0)-(2,1)-(2,3)-(2,4)-(2,5)-(3,2)\\&\qquad +{\bigl [}(2,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega '')}{\theta ''}}+{\bigl [}(2,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega '')}{\theta ''}}\\&\qquad +{\bigl [}(2,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega '')}{\theta ''}}+{\bigl [}(2,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega '')}{\theta ''}},\\\\&{\frac {d\Omega ^{\scriptscriptstyle {\text{IV}}}}{dt}}=-(4,0)-(4,1)-(4,2)-(4,3)-(4,5)-(3,4)\\&\qquad +{\bigl [}(4,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}+{\bigl [}(4,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}\\&\qquad +{\bigl [}(4,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}+{\bigl [}(4,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}},\\\\&{\frac {d\Omega ^{\scriptscriptstyle {\text{V}}}}{dt}}\ =-(5,0)-(5,1)-(5,2)-(5,3)-(5,4)-(3,5)\\&\qquad +{\bigl [}(5,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}+{\bigl [}(5,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}\\&\qquad +{\bigl [}(5,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}+{\bigl [}(5,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}.\end{aligned}}}