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Joseph Louis de Lagrange - Œuvres, Tome 7.djvu/545
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C
=
−
4
a
sin
2
φ
2
sin
α
,
C
1
=
−
4
a
sin
2
φ
2
sin
(
α
+
φ
)
,
C
2
=
−
4
a
sin
2
φ
2
sin
(
α
+
2
φ
)
,
…
;
2
D
=
−
8
a
sin
3
φ
2
cos
[
α
−
(
1
+
1
2
)
φ
]
,
1
D
=
−
8
a
sin
3
φ
2
cos
(
α
−
φ
)
,
D
1
=
−
8
a
sin
3
φ
2
cos
(
α
+
φ
2
)
,
D
2
=
−
8
a
sin
3
φ
2
cos
[
α
+
(
1
+
1
2
)
φ
]
,
…
;
1
E
=
16
a
sin
4
φ
2
sin
(
α
−
φ
)
,
E
=
16
a
sin
4
φ
2
sin
α
,
E
1
=
16
a
sin
4
φ
2
sin
(
α
+
φ
)
,
…
.
{\displaystyle {\begin{aligned}\mathrm {C} \ \ =&-4a\sin ^{2}{\frac {\varphi }{2}}\sin \alpha ,\\\mathrm {C} _{1}=&-4a\sin ^{2}{\frac {\varphi }{2}}\sin(\alpha +\varphi ),\\\mathrm {C} _{2}=&-4a\sin ^{2}{\frac {\varphi }{2}}\sin(\alpha +2\varphi ),\quad \ldots \,;\\\\{}_{2}\!\mathrm {D} \ \ =&-8a\sin ^{3}{\frac {\varphi }{2}}\cos \left[\alpha -\left(1+{\frac {1}{2}}\right)\varphi \right],\\{}_{1}\!\mathrm {D} \ \ =&-8a\sin ^{3}{\frac {\varphi }{2}}\cos(\alpha -\varphi ),\\\mathrm {D} _{1}=&-8a\sin ^{3}{\frac {\varphi }{2}}\cos \left(\alpha +{\frac {\varphi }{2}}\right),\\\mathrm {D} _{2}=&-8a\sin ^{3}{\frac {\varphi }{2}}\cos \left[\alpha +\left(1+{\frac {1}{2}}\right)\varphi \right],\quad \ldots \,;\\\\{}_{1}\!\mathrm {E} \ \ =&16a\sin ^{4}{\frac {\varphi }{2}}\sin(\alpha -\varphi ),\\\mathrm {E} \ \ =&16a\sin ^{4}{\frac {\varphi }{2}}\sin \alpha ,\\\mathrm {E} _{1}=&16a\sin ^{4}{\frac {\varphi }{2}}\sin \left(\alpha +\varphi \right),\quad \ldots .\end{aligned}}}
De sorte qu’on aura
A
=
a
sin
α
,
B
=
2
a
sin
φ
2
cos
φ
2
cos
α
,
C
=
−
4
a
sin
2
φ
2
sin
α
,
D
=
−
8
a
sin
3
φ
2
cos
φ
2
cos
α
,
E
=
16
a
sin
4
φ
2
sin
α
,
F
=
32
a
sin
5
φ
2
cos
φ
2
cos
α
,
.
…
…
…
…
…
…
…
…
.
{\displaystyle {\begin{aligned}\mathrm {A} =&a\sin \alpha ,\\\mathrm {B} =&2a\sin {\frac {\varphi }{2}}\cos {\frac {\varphi }{2}}\cos \alpha ,\\\mathrm {C} =&-4a\sin ^{2}{\frac {\varphi }{2}}\sin \alpha ,\\\mathrm {D} =&-8a\sin ^{3}{\frac {\varphi }{2}}\cos {\frac {\varphi }{2}}\cos \alpha ,\\\mathrm {E} =&16a\sin ^{4}{\frac {\varphi }{2}}\sin \alpha ,\\\mathrm {F} =&32a\sin ^{5}{\frac {\varphi }{2}}\cos {\frac {\varphi }{2}}\cos \alpha ,\\.\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}}