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Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/55
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a
′
N
(
1
)
=
e
′
768
{\displaystyle a'{\rm {N}}^{(1)}={\frac {e'}{768}}}
×
{
−
(
20267
e
′
2
+
248960
e
2
)
α
b
1
2
(
2
)
−
(
7223
e
′
2
+
8144
e
2
)
α
d
b
1
2
(
2
)
d
α
+
(
1094
e
′
2
+
3692
e
2
)
α
2
d
2
b
1
2
(
2
)
d
α
2
+
(
482
e
′
2
+
1436
e
2
)
α
3
d
3
b
1
2
(
2
)
d
α
3
+
(
41
e
′
2
+
140
e
2
)
α
4
d
4
b
1
2
(
2
)
d
α
4
+
(
e
′
2
+
4
e
2
)
α
5
d
5
b
1
2
(
2
)
d
α
5
}
{\displaystyle \times \left\{{\begin{aligned}-&\left(20267e'^{2}+248960e^{2}\right)\alpha b_{\frac {1}{2}}^{(2)}-\left(7223e'^{2}+8144e^{2}\right)\alpha {\frac {db_{\frac {1}{2}}^{(2)}}{d\alpha }}\\\\+&\left(1094e'^{2}+3692e^{2}\right)\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{2}}}+\left(482e'^{2}+1436e^{2}\right)\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{3}}}\\\\+&\left(41e'^{2}+140e^{2}\right)\alpha ^{4}{\frac {d^{4}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{4}}}+\left(e'^{2}+4e^{2}\right)\alpha ^{5}{\frac {d^{5}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{5}}}\end{aligned}}\right\}}
−
e
′
3
γ
2
384
{
590
α
(
b
3
2
(
1
)
+
b
3
2
(
3
)
)
+
255
α
2
(
d
b
3
2
(
1
)
d
α
+
d
b
3
2
(
3
)
d
α
)
+
30
α
3
(
d
2
b
3
2
(
1
)
d
α
2
+
d
2
b
3
2
(
3
)
d
α
2
)
+
α
4
(
d
3
b
3
2
(
1
)
d
α
3
+
d
3
b
3
2
(
3
)
d
α
3
)
}
{\displaystyle -{\frac {e'^{3}\gamma ^{2}}{384}}\left\{{\begin{aligned}&590\alpha \left(b_{\frac {3}{2}}^{(1)}+b_{\frac {3}{2}}^{(3)}\right)+255\alpha ^{2}\left({\frac {db_{\frac {3}{2}}^{(1)}}{d\alpha }}+{\frac {db_{\frac {3}{2}}^{(3)}}{d\alpha }}\right)\\\\&+30\alpha ^{3}\left({\frac {d^{2}b_{\frac {3}{2}}^{(1)}}{d\alpha ^{2}}}+{\frac {d^{2}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{2}}}\right)+\alpha ^{4}\left({\frac {d^{3}b_{\frac {3}{2}}^{(1)}}{d\alpha ^{3}}}+{\frac {d^{3}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{3}}}\right)\end{aligned}}\right\}}
a
′
N
(
2
)
=
e
′
e
768
{\displaystyle a'{\rm {N}}^{(2)}={\frac {e'e}{768}}}
×
{
−
(
109392
e
′
2
+
53064
e
2
)
α
b
1
2
(
3
)
−
(
42368
e
′
2
+
42368
e
2
)
α
d
b
1
2
(
3
)
d
α
+
(
1064
e
′
2
+
2088
e
2
)
α
2
d
2
b
1
2
(
3
)
d
α
2
+
(
1572
e
′
2
+
1710
e
2
)
α
3
d
3
b
1
2
(
3
)
d
α
3
+
(
152
e
′
2
+
192
e
2
)
α
4
d
4
b
1
2
(
3
)
d
α
4
+
(
4
e
′
2
+
6
e
2
)
α
5
d
5
b
1
2
(
3
)
d
α
5
}
{\displaystyle \times \left\{{\begin{aligned}&-\left(109392e'^{2}+53064e^{2}\right)\alpha b_{\frac {1}{2}}^{(3)}-\left(42368e'^{2}+42368e^{2}\right)\alpha {\frac {db_{\frac {1}{2}}^{(3)}}{d\alpha }}\\\\&+\left(1064e'^{2}+2088e^{2}\right)\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{2}}}+\left(1572e'^{2}+1710e^{2}\right)\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{3}}}\\\\&+\left(152e'^{2}+192e^{2}\right)\alpha ^{4}{\frac {d^{4}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{4}}}+\left(4e'^{2}+6e^{2}\right)\alpha ^{5}{\frac {d^{5}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{5}}}\end{aligned}}\right\}}
+
e
′
2
e
γ
2
128
{
595
α
(
b
3
2
(
2
)
+
b
3
2
(
4
)
)
+
245
α
2
(
d
b
3
2
(
2
)
d
α
+
d
b
3
2
(
4
)
d
α
)
+
29
α
3
(
d
2
b
3
2
(
2
)
d
α
2
+
d
2
b
3
2
(
4
)
d
α
2
)
+
α
4
(
d
3
b
3
2
(
2
)
d
α
3
+
d
3
b
3
2
(
4
)
d
α
3
)
}
{\displaystyle +{\frac {e'^{2}e\gamma ^{2}}{128}}\left\{{\begin{aligned}&595\alpha \left(b_{\frac {3}{2}}^{(2)}+b_{\frac {3}{2}}^{(4)}\right)+245\alpha ^{2}\left({\frac {db_{\frac {3}{2}}^{(2)}}{d\alpha }}+{\frac {db_{\frac {3}{2}}^{(4)}}{d\alpha }}\right)\\\\&+29\alpha ^{3}\left({\frac {d^{2}b_{\frac {3}{2}}^{(2)}}{d\alpha ^{2}}}+{\frac {d^{2}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{2}}}\right)+\alpha ^{4}\left({\frac {d^{3}b_{\frac {3}{2}}^{(2)}}{d\alpha ^{3}}}+{\frac {d^{3}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{3}}}\right)\end{aligned}}\right\}}
a
′
N
(
3
)
=
e
′
e
2
768
{\displaystyle a'{\rm {N}}^{(3)}={\frac {e'e^{2}}{768}}}
×
{
−
(
42912
e
2
+
199848
e
′
2
)
α
b
1
2
(
4
)
−
(
21728
e
2
+
82032
e
′
2
)
α
d
b
1
2
(
4
)
d
α
−
(
640
e
2
+
2970
e
′
2
)
α
2
d
2
b
1
2
(
4
)
d
α
2
+
(
864
e
2
+
1854
e
′
2
)
α
3
d
3
b
1
2
(
4
)
d
α
3
+
(
116
e
2
+
210
e
′
2
)
α
4
d
4
b
1
2
(
4
)
d
α
4
+
(
4
e
2
+
6
e
′
2
)
α
5
d
5
b
1
2
(
4
)
d
α
5
}
{\displaystyle \times \left\{{\begin{aligned}&-\left(42912e^{2}+199848e'^{2}\right)\alpha b_{\frac {1}{2}}^{(4)}-\left(21728e^{2}+82032e'^{2}\right)\alpha {\frac {db_{\frac {1}{2}}^{(4)}}{d\alpha }}\\\\&-\left(640e^{2}+2970e'^{2}\right)\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{2}}}+\left(864e^{2}+1854e'^{2}\right)\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{3}}}\\\\&+\left(116e^{2}+210e'^{2}\right)\alpha ^{4}{\frac {d^{4}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{4}}}+\left(4e^{2}+6e'^{2}\right)\alpha ^{5}{\frac {d^{5}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{5}}}\end{aligned}}\right\}}
−
e
′
e
2
γ
2
128
{
580
α
(
b
3
2
(
3
)
+
b
3
2
(
5
)
)
+
234
α
2
(
d
b
3
2
(
3
)
d
α
+
d
b
3
2
(
5
)
d
α
)
+
28
α
3
(
d
2
b
3
2
(
3
)
d
α
2
+
d
2
b
3
2
(
5
)
d
α
2
)
+
α
4
(
d
3
b
3
2
(
3
)
d
α
3
+
d
3
b
3
2
(
5
)
d
α
3
)
}
{\displaystyle -{\frac {e'e^{2}\gamma ^{2}}{128}}\left\{{\begin{aligned}&580\alpha \left(b_{\frac {3}{2}}^{(3)}+b_{\frac {3}{2}}^{(5)}\right)+234\alpha ^{2}\left({\frac {db_{\frac {3}{2}}^{(3)}}{d\alpha }}+{\frac {db_{\frac {3}{2}}^{(5)}}{d\alpha }}\right)\\\\&+28\alpha ^{3}\left({\frac {d^{2}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{2}}}+{\frac {d^{2}b_{\frac {3}{2}}^{(5)}}{d\alpha ^{2}}}\right)+\alpha ^{4}\left({\frac {d^{3}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{3}}}+{\frac {d^{3}b_{\frac {3}{2}}^{(5)}}{d\alpha ^{3}}}\right)\end{aligned}}\right\}}