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Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/97
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XXIII.
Cela posé, on aura
r
1
Δ
(
r
1
,
r
2
)
3
=
a
1
[
Γ
(
a
1
,
a
2
)
+
Γ
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
+
Γ
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
+
…
]
+
n
x
1
a
1
[
3
Π
(
a
1
,
a
2
)
+
Γ
(
a
1
,
a
2
)
+
[
3
Π
1
(
a
1
,
a
2
)
+
Γ
1
(
a
1
,
a
2
)
]
cos
(
φ
2
−
φ
1
)
+
[
3
Π
2
(
a
1
,
a
2
)
+
Γ
2
(
a
1
,
a
2
)
]
cos
2
(
φ
2
−
φ
1
)
+
…
]
+
3
n
x
2
a
1
[
Ψ
(
a
1
,
a
2
)
+
Ψ
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
+
Ψ
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
+
…
]
.
{\displaystyle {\begin{aligned}&{\frac {r_{1}}{\Delta (r_{1},r_{2})^{3}}}\\&=a_{1}\left[\Gamma (a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +nx_{1}a_{1}\left[3\Pi (a_{1},a_{2})+\Gamma (a_{1},a_{2})+\left[3\Pi _{1}(a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\right.\\&\qquad \qquad \qquad \qquad \left.+\left[3\Pi _{2}(a_{1},a_{2})+\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +3nx_{2}a_{1}\left[\Psi (a_{1},a_{2})+\Psi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Psi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right].\end{aligned}}}
On trouvera de la même manière
r
2
Δ
(
r
1
,
r
2
)
3
=
a
2
[
Γ
(
a
1
,
a
2
)
+
Γ
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
+
Γ
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
+
…
]
+
3
n
x
1
a
2
[
Π
(
a
1
,
a
2
)
+
Π
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
+
Π
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
+
…
]
+
n
x
2
a
2
[
3
Ψ
(
a
1
,
a
2
)
+
Γ
(
a
1
,
a
2
)
+
[
3
Ψ
1
(
a
1
,
a
2
)
+
Γ
1
(
a
1
,
a
2
)
]
cos
(
φ
2
−
φ
1
)
+
[
3
Ψ
2
(
a
1
,
a
2
)
+
Γ
2
(
a
1
,
a
2
)
]
cos
2
(
φ
2
−
φ
1
)
+
…
]
.
{\displaystyle {\begin{aligned}&{\frac {r_{2}}{\Delta (r_{1},r_{2})^{3}}}\\&=a_{2}\left[\Gamma (a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +3nx_{1}a_{2}\left[\Pi (a_{1},a_{2})+\Pi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Pi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +nx_{2}a_{2}\left[3\Psi (a_{1},a_{2})+\Gamma (a_{1},a_{2})+\left[3\Psi _{1}(a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\right.\\&\qquad \qquad \qquad \qquad \left.+\left[3\Psi _{2}(a_{1},a_{2})+\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right].\end{aligned}}}
On aura ensuite
1
r
2
2
(
1
+
p
2
2
)
3
2
=
1
a
2
2
(
1
−
2
n
x
2
+
…
)
.
{\displaystyle {\frac {1}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}={\frac {1}{a_{2}^{2}}}(1-2nx_{2}+\ldots ).}
Donc
−
r
2
Δ
(
r
1
,
r
2
)
3
+
1
r
2
2
(
1
+
p
2
2
)
3
2
{\displaystyle -{\frac {r_{2}}{\Delta (r_{1},r_{2})^{3}}}+{\frac {1}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}}
=
1
a
2
2
−
a
2
Γ
(
a
1
,
a
2
)
−
a
2
Γ
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
−
a
2
Γ
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
−
…
−
3
n
x
1
a
2
[
Π
(
a
1
,
a
2
)
+
Π
1
(
a
1
,
a
2
)
cos
(
φ
2
−
φ
1
)
+
Π
2
(
a
1
,
a
2
)
cos
2
(
φ
2
−
φ
1
)
+
…
]
−
n
x
2
[
2
a
2
2
+
3
a
2
Ψ
(
a
1
,
a
2
)
+
a
2
Γ
(
a
1
,
a
2
)
+
[
3
a
2
Ψ
1
(
a
1
,
a
2
)
+
a
2
Γ
1
(
a
1
,
a
2
)
]
cos
(
φ
2
−
φ
1
)
+
[
3
a
2
Ψ
2
(
a
1
,
a
2
)
+
a
2
Γ
2
(
a
1
,
a
2
)
]
cos
2
(
φ
2
−
φ
1
)
+
…
]
.
{\displaystyle {\begin{aligned}&={\frac {1}{a_{2}^{2}}}-a_{2}\Gamma (a_{1},a_{2})-a_{2}\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})-a_{2}\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})-\ldots \\&\quad -3nx_{1}a_{2}\left[\Pi (a_{1},a_{2})+\Pi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Pi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad -nx_{2}\left[{\frac {2}{a_{2}^{2}}}+3a_{2}\Psi (a_{1},a_{2})+a_{2}\Gamma (a_{1},a_{2})\right.\\&\qquad \qquad \qquad \qquad +\left[3a_{2}\Psi _{1}(a_{1},a_{2})+a_{2}\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\\&\qquad \qquad \qquad \qquad +{\biggl .}\left[3a_{2}\Psi _{2}(a_{1},a_{2})+a_{2}\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots {\biggr ]}.\end{aligned}}}