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Fourier - Théorie analytique de la chaleur, 1822.djvu/257
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225
CHAPITRE III.
sont exprimées ainsi,
1
2
a
=
A
−
B
(
π
2
2.3
−
1
1
2
)
+
C
(
π
4
2.3.4.5
−
1
1
2
⋅
π
2
2.3
+
1
1
4
)
−
D
(
π
6
2.3.4.5.6.7
−
1
1
2
⋅
π
4
2.3.4.5
+
1
1
4
⋅
π
2
2.3
−
1
1
6
)
+
E
(
π
8
2.3.4.5.6.7.8.9
−
1
1
2
⋅
π
6
2.3.4.5.6.7
+
1
1
4
⋅
π
4
2.3.4.5
−
1
1
6
⋅
π
2
2.3
+
1
1
8
)
−
e
t
c
.
−
1
2
2
b
=
A
−
B
(
π
2
2.3
−
1
2
2
)
+
C
(
π
4
2.3.4.5
−
1
2
2
⋅
π
2
2.3
+
1
2
4
)
−
D
(
π
6
2.3.4.5.6.7
−
1
2
2
⋅
π
4
2.3.4.5
+
1
2
4
⋅
π
2
2.3
−
1
2
6
)
+
E
(
π
8
2.3.4.5.6.7.8.9
−
1
2
2
⋅
π
6
2.3.4.5.6.7
+
1
2
4
⋅
π
4
2.3.4.5
−
1
2
6
⋅
π
2
2.3
+
1
2
8
)
−
e
t
c
.
1
2
3
c
=
A
−
B
(
π
2
2.3
−
1
3
2
)
+
C
(
π
4
2.3.4.5
−
1
3
2
⋅
π
2
2.3
+
1
3
4
)
−
D
(
π
6
2.3.4.5.6.7
−
1
3
2
⋅
π
4
2.3.4.5
+
1
3
4
⋅
π
2
2.3
−
1
3
6
)
+
E
(
π
8
2.3.4.5.6.7.8.9
−
1
3
2
⋅
π
6
2.3.4.5.6.7
+
1
3
4
⋅
π
4
2.3.4.5
−
1
3
6
⋅
π
2
2.3
+
1
3
8
)
−
e
t
c
.
−
1
2
4
d
=
A
−
B
(
π
2
2.3
−
1
4
2
)
+
C
(
π
4
2.3.4.5
−
1
4
2
⋅
π
2
2.3
+
1
4
4
)
−
D
(
π
6
2.3.4.5.6.7
−
1
4
2
⋅
π
4
2.3.4.5
+
1
4
4
⋅
π
2
2.3
−
1
4
6
)
+
E
(
π
8
2.3.4.5.6.7.8.9
−
1
4
2
⋅
π
6
2.3.4.5.6.7
+
1
4
4
⋅
π
4
2.3.4.5
−
1
4
6
⋅
π
2
2.3
+
1
4
8
)
−
e
t
c
.
e
t
c
.
{\displaystyle {\begin{aligned}{\frac {1}{2}}\;\;a&=\mathrm {A} -\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{1^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{1^{2}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{1^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{1^{2}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{1^{4}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}-{\frac {1}{1^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{1^{2}}}\!\cdot \!{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{1^{4}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{1^{6}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{1^{8}}}\right)\\&-\mathrm {etc.} \\-{\frac {1}{2}}2b&=\mathrm {A} -\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{2^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{2^{2}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{2^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{2^{2}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{2^{4}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}-{\frac {1}{2^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{2^{2}}}\!\cdot \!{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{2^{4}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{2^{6}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{2^{8}}}\right)\\&-\mathrm {etc.} \\{\frac {1}{2}}3c&=\mathrm {A} -\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{3^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{3^{2}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{3^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{3^{2}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{3^{4}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}-{\frac {1}{3^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{3^{2}}}\!\cdot \!{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{3^{4}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{3^{6}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{3^{8}}}\right)\\&-\mathrm {etc.} \\-{\frac {1}{2}}4d&=\mathrm {A} -\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{4^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{4^{2}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{4^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{4^{2}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{4^{4}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}-{\frac {1}{4^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{4^{2}}}\!\cdot \!{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{4^{4}}}\!\cdot \!{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{4^{6}}}\!\cdot \!{\frac {\pi ^{2}}{2.3}}+{\frac {1}{4^{8}}}\right)\\&-\mathrm {etc.} \\&\qquad \qquad \qquad \mathrm {etc.} \end{aligned}}}
