Page:Fourier - Théorie analytique de la chaleur, 1822.djvu/353

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CHAPITRE IV.

272.

Il faut maintenant substituer les valeurs connues des coëfficients $\mathrm {A_{1}B_{1}} \;\mathrm {A_{2}B_{2}} \;\mathrm {A_{3}B_{3}} \dots$ etc., dans les équations $(\mu ),$ art. 266, et l’on trouvera les valeurs suivantes :

${\begin{aligned}\alpha _{1}&=\mathrm {N} _{0}\qquad \qquad \quad +\qquad \qquad \mathrm {N} _{1}\varepsilon ^{t\,\mathrm {sin.V} .q_{1}}\qquad \qquad +\qquad \qquad \mathrm {N} _{2}\varepsilon ^{t\,\mathrm {sin.V} .q_{2}}\quad \qquad \qquad \qquad +\mathrm {etc.} \\\alpha _{2}&=\mathrm {N} _{0}+(\mathrm {M} _{1}\sin .q_{1}+\mathrm {N} _{1}\cos .q_{1})\varepsilon ^{t\,\mathrm {sin.V} .q_{1}}\quad +(\mathrm {M} _{2}\sin .q_{2}+\mathrm {N} _{2}\cos .q_{2})\varepsilon ^{t\,\mathrm {sin.V} .q_{2}}\quad \qquad \qquad +\mathrm {etc.} \\\alpha _{3}&=\mathrm {N} _{0}+(\mathrm {M} _{1}\sin .2q_{1}+\mathrm {N} _{1}\cos .2q_{1})\varepsilon ^{t\,\mathrm {sin.V} .q_{1}}+(\mathrm {M} _{2}\sin .2q_{2}+\mathrm {N} _{2}\cos .2q_{2})\varepsilon ^{t\,\mathrm {sin.V} .q_{2}}\quad \qquad \quad +\mathrm {etc.} \\\vdots &\\\alpha _{j}&=\mathrm {N} _{0}+(\mathrm {M} _{1}\sin .{\overline {j-1}}\,q_{1}+\mathrm {N} _{1}\cos .{\overline {j-1}}\,q_{1})\varepsilon ^{t\,\mathrm {sin.V} .q_{1}}+(\mathrm {M} _{2}\sin .{\overline {j-1}}\,q_{2}+\mathrm {N} _{2}\cos .{\overline {j-1}}\,q_{2})\varepsilon ^{t\,\mathrm {sin.V} .q_{2}}\\\vdots &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\mathrm {etc.} \\\alpha _{n}&=\mathrm {N} _{0}+(\mathrm {M} _{1}\sin .{\overline {n-1}}\,q_{1}\!+\!\mathrm {N} _{1}\cos .{\overline {n-1}}\,q_{1})\varepsilon ^{t\,\mathrm {sin.V} .q_{1}}+(\mathrm {M} _{2}\sin .{\overline {n-1}}\,q_{2}\!+\!\mathrm {N} _{2}\cos .{\overline {n-1}}\,q_{2})\varepsilon ^{t\,\mathrm {sin.V} .q_{2}}\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\mathrm {etc.} \\\end{aligned}}$

dans ces équations

${\begin{array}{c}\varepsilon =e^{-2{\frac {k}{m}}},\qquad q_{1}=1.{\frac {2\pi }{n}},\qquad q_{2}=2.{\frac {2\pi }{n}},\qquad q_{3}=3.{\frac {2\pi }{n}},\qquad \mathrm {etc.} \\{\begin{array}{lll}\mathrm {N} _{0}={\frac {1}{n}}\mathrm {S} \,a_{i}&\\\mathrm {N} _{1}={\frac {2}{n}}\mathrm {S} \,a_{i}\cos .{\overline {i-1}}\,q_{1}&&\mathrm {M} _{1}={\frac {2}{n}}\mathrm {S} \,a_{i}\sin .{\overline {i-1}}\,q_{1}\\\mathrm {N} _{2}={\frac {2}{n}}\mathrm {S} \,a_{i}\cos .{\overline {i-1}}\,q_{2}&&\mathrm {M} _{2}={\frac {2}{n}}\mathrm {S} \,a_{i}\sin .{\overline {i-1}}\,q_{2}\\\mathrm {N} _{3}={\frac {2}{n}}\mathrm {S} \,a_{i}\cos .{\overline {i-1}}\,q_{3}&&\mathrm {M} _{3}={\frac {2}{n}}\mathrm {S} \,a_{i}\sin .{\overline {i-1}}\,q_{3}\\\qquad \mathrm {etc.} &&\qquad \mathrm {etc.} \end{array}}\\\end{array}}$