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Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/276
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266
RECHERCHE
n
l
−
2
n
+
n
′
=
d
2
n
d
t
2
i
2
+
(
N
′
+
N
l
)
i
4
;
{\displaystyle n_{l}-2n+n'={\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}i^{2}+(N'+N_{l})i^{4}\,;}
m
′
−
m
l
=
2
d
m
d
t
i
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d
3
m
d
t
3
i
3
3
+
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M
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M
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)
i
4
,
n
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n
l
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2
d
n
d
t
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d
3
n
d
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3
i
3
3
+
(
N
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N
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)
i
4
,
Z
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Z
l
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d
Z
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d
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Z
d
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3
i
3
3
+
(
w
x
′
−
w
x
l
)
i
4
,
{\displaystyle {\begin{array}{rlll}m'-m_{l}&=2{\frac {\operatorname {d} m}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(M'-M_{l})i^{4},\\n'-n_{l}&=2{\frac {\operatorname {d} n}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(N'-N_{l})i^{4},\\Z'-Z_{l}&=2{\frac {\operatorname {d} Z}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(wx'-wx_{l})i^{4},\\\end{array}}}
m
−
m
l
=
d
m
d
t
i
1
−
d
2
m
d
t
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i
2
1.2
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d
3
m
d
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i
3
1.2.3
−
M
l
i
4
,
n
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n
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d
n
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1
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d
2
n
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d
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1.2.3
−
N
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4
,
m
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m
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d
m
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d
2
m
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1.2.3
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M
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i
4
,
n
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−
n
=
d
n
d
t
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1
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d
2
n
d
t
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i
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1.2
+
d
3
n
d
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3
i
3
1.2.3
+
N
′
i
4
,
{\displaystyle {\begin{array}{rllll}m-m_{l}&={\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {i}{1}}&-{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&-M_{l}i^{4},\\n-n_{l}&={\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {i}{1}}&-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&-N_{l}i^{4},\\m'-m&={\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+M'i^{4},\\n'-n&={\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+N'i^{4},\\\end{array}}}
d’après quoi, en posant pour abréger
S
=
(
N
′
+
N
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)
d
m
d
t
−
(
M
′
+
M
l
)
d
n
d
t
−
1
6
(
d
2
m
d
t
2
d
3
n
d
t
3
−
d
2
n
d
t
2
d
3
m
d
t
3
)
{\displaystyle S=(N'+N_{l}){\frac {\operatorname {d} m}{\operatorname {d} t}}-(M'+M_{l}){\frac {\operatorname {d} n}{\operatorname {d} t}}-{\tfrac {1}{6}}\left({\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}\right)}
−
1
2
{
(
N
′
−
N
l
)
d
2
m
d
t
2
−
(
M
′
−
M
l
)
d
2
n
d
t
2
}
i
{\displaystyle -{\tfrac {1}{2}}\left\{(N'-N_{l}){\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}-(M'-M_{l}){\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}\right\}i}
+
1
6
{
(
N
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+
N
l
)
d
3
m
d
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3
−
(
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M
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3
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d
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3
}
i
2
−
(
M
l
N
′
−
M
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N
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)
i
3
;
{\displaystyle +{\tfrac {1}{6}}\left\{(N'+N_{l}){\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}-(M'+M_{l}){\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}\right\}i^{2}-(M_{l}N'-M'N_{l})i^{3}\,;}
F
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H
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d
m
d
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2
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(
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3
d
h
n
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n
d
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3
d
2
g
d
t
2
)
{\displaystyle F=2(H'+H_{l}){\frac {\operatorname {d} m}{\operatorname {d} t}}-2(G'+G_{l}){\frac {\operatorname {d} n}{\operatorname {d} t}}+{\tfrac {1}{3}}\left({\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {\operatorname {d} ^{h}n}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right)}
+
{
(
M
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M
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h
d
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2
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2
}
i
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3
{
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d
3
m
d
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−
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l
)
d
3
n
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}
i
2
{\displaystyle +\left\{(M'-M_{l}){\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-(N'-N_{l}){\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}i+{\tfrac {1}{3}}\left\{(H'+H_{l}){\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}-(G'+G_{l}){\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}\right\}i^{2}}
+
{
(
M
′
−
M
l
)
(
H
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+
H
l
)
−
(
N
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l
)
(
G
′
+
G
l
)
}
i
3
;
{\displaystyle +\left\{(M'-M_{l})(H'+H_{l})-(N'-N_{l})(G'+G_{l})\right\}i^{3}\,;}