Page:Fourier - Théorie analytique de la chaleur, 1822.djvu/257

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CHAPITRE III.

sont exprimées ainsi,

${\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}}$