Accueil
Au hasard
Se connecter
Configuration
Faire un don
À propos de Wikisource
Avertissements
Rechercher
Page
:
Lagrange - Œuvres (1867) vol. 1.djvu/333
Langue
Suivre
Modifier
Le texte de cette page a été
corrigé
et est conforme au fac-similé.
1
4
L
(
X
,
Y
+
q
t
,
Z
−
r
t
)
+
1
4
L
(
X
,
Y
−
q
t
,
Z
+
r
t
)
+
1
4
L
(
X
,
Y
+
q
t
,
Z
−
r
t
)
+
1
4
L
(
X
,
Y
−
q
t
,
Z
−
r
t
)
{\displaystyle {\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} -qt,\mathrm {Z} +rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} -qt,\mathrm {Z} -rt)}}
=
L
+
1
2
t
2
(
q
2
d
2
L
d
Y
2
+
r
2
d
2
L
d
Z
2
)
{\displaystyle =\mathrm {L} +{\frac {1}{2}}t^{2}\left(q^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} ^{2}}}+r^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Z} ^{2}}}\right)}
+
1
2.3.4
t
4
(
q
4
d
4
L
d
Y
4
+
6
q
2
r
2
d
4
L
d
Y
2
d
Z
2
+
r
4
d
4
L
d
Z
4
)
,
{\displaystyle +{\frac {1}{2.3.4}}t^{4}\left(q^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{4}}}+6q^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{2}d\mathrm {Z} ^{2}}}+r^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} ^{4}}}\right),}
1
8
L
(
X
+
p
t
,
Y
+
q
t
,
Z
+
r
t
)
+
1
8
L
(
X
+
p
t
,
Y
−
q
t
,
Z
+
r
t
)
+
1
8
L
(
X
−
p
t
,
Y
−
q
t
,
Z
+
r
t
)
+
1
8
L
(
X
−
p
t
,
Y
−
q
t
,
Z
−
r
t
)
−
1
8
L
(
X
+
p
t
,
Y
−
q
t
,
Z
+
r
t
)
−
1
8
L
(
X
+
p
t
,
Y
−
q
t
,
Z
−
r
t
)
−
1
8
L
(
X
−
p
t
,
Y
+
q
t
,
Z
+
r
t
)
−
1
8
L
(
X
−
p
t
,
Y
+
q
t
,
Z
−
r
t
)
=
1
2
2
p
q
t
2
d
2
L
d
X
d
Y
+
1
2.3.4
t
4
(
4
p
3
q
d
4
L
d
X
3
d
Y
+
4
p
q
3
d
4
L
d
X
d
Y
3
{\displaystyle {\begin{aligned}&\quad \ {\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}\\=&{\frac {1}{2}}2pqt^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} }}+{\frac {1}{2.3.4}}t^{4}\left(4p^{3}q{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Y} }}+4pq^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{3}}}\right.\\\end{aligned}}}
+
12
p
q
r
2
d
4
L
d
X
d
Y
d
Z
2
)
,
{\displaystyle +\left.12pqr^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}\right),}
1
8
L
(
X
+
p
t
,
Y
+
q
t
,
Z
+
r
t
)
+
1
8
L
(
X
+
p
t
,
Y
−
q
t
,
Z
+
r
t
)
+
1
8
L
(
X
−
p
t
,
Y
+
q
t
,
Z
−
r
t
)
+
1
8
L
(
X
−
p
t
,
Y
−
q
t
,
Z
−
r
t
)
−
1
8
L
(
X
+
p
t
,
Y
+
q
t
,
Z
−
r
t
)
−
1
8
L
(
X
+
p
t
,
Y
−
q
t
,
Z
−
r
t
)
−
1
8
L
(
X
−
p
t
,
Y
+
q
t
,
Z
+
r
t
)
−
1
8
L
(
X
−
p
t
,
Y
−
q
t
,
Z
+
r
t
)
=
1
2
2
p
r
t
2
d
2
L
d
X
d
Z
+
1
2.3.4
t
4
(
4
p
3
r
d
4
L
d
X
3
d
Z
+
4
p
r
3
d
4
L
d
X
d
Z
3
{\displaystyle {\begin{aligned}&\quad \ {\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\=&{\frac {1}{2}}2prt^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} }}+{\frac {1}{2.3.4}}t^{4}\left(4p^{3}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Z} }}+4pr^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} ^{3}}}\right.\\\end{aligned}}}
+
12
p
q
2
r
d
4
L
d
X
d
Y
2
d
Z
)
.
{\displaystyle +\left.12pq^{2}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{2}d\mathrm {Z} }}\right).}
On formera de pareilles formules à l’égard des expressions
M
(
X
+
p
t
,
Y
+
q
t
,
Z
+
r
t
)
et
N
(
X
+
p
t
,
Y
+
q
t
,
Z
+
r
t
)
.
{\displaystyle \mathrm {M} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}\quad {\text{et}}\quad \mathrm {N} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}.}