Page:Mémoires de l’Académie des sciences, Tome 4.djvu/651

posent les coëfficients de $\sin .2x,\,\sin .3x,\,\sin .4x,\ldots .$ On emploiera pour cela les équations

${\begin{alignedat}{4}\varphi '0&+{\frac {\pi ^{2}}{2.3}}\varphi '''0&&+{\frac {\pi ^{4}}{2.3.4.5}}\varphi ^{\scriptscriptstyle {\text{V}}}0&&+{\frac {\pi ^{6}}{2.3.4.5.6.7}}\varphi ^{\scriptscriptstyle {\text{VII}}}0\ldots &&={\frac {1}{\pi }}\varphi \ \ \pi ,\\\varphi '''0&+{\frac {\pi ^{2}}{2.3}}\varphi ^{\scriptscriptstyle {\text{V}}}0&&+{\frac {\pi ^{4}}{2.3.4.5}}\varphi ^{\scriptscriptstyle {\text{VII}}}0&&+{\frac {\pi ^{6}}{2.3.4.5.6.7}}\varphi ^{\scriptscriptstyle {\text{IX}}}0\ldots &&={\frac {1}{\pi }}\varphi ''\pi ,\\\varphi ^{\scriptscriptstyle {\text{V}}}0&+{\frac {\pi ^{2}}{2.3}}\varphi ^{\scriptscriptstyle {\text{VII}}}0&&+{\frac {\pi ^{4}}{2.3.4.5}}\varphi ^{\scriptscriptstyle {\text{IX}}}0&&+{\frac {\pi ^{6}}{2.3.4.5.6.7}}\varphi ^{\scriptscriptstyle {\text{XI}}}0\ldots &&={\frac {1}{\pi }}\varphi ^{\scriptscriptstyle {\text{IV}}}\pi ,\\&{\text{etc}}.\end{alignedat}}$

Et au moyen de ces réductions, on donnera facilement à l’équation $(\mathrm {A} )$ la forme suivante :

${\begin{aligned}(\mathrm {B} )\quad {\frac {1}{2}}\pi \varphi x=\quad \;\;&\left(\varphi \pi -{\frac {1}{1^{2}}}\varphi ''\pi +{\frac {1}{1^{4}}}\varphi ^{\scriptscriptstyle {\text{IV}}}\pi -{\frac {1}{1^{6}}}\varphi ^{\scriptscriptstyle {\text{VI}}}\pi +\ldots \right)\sin .\ \ x\\-{\frac {1}{2}}&\left(\varphi \pi -{\frac {1}{2^{2}}}\varphi ''\pi +{\frac {1}{2^{4}}}\varphi ^{\scriptscriptstyle {\text{IV}}}\pi -{\frac {1}{2^{6}}}\varphi ^{\scriptscriptstyle {\text{VI}}}\pi +\ldots \right)\sin .2x\\+{\frac {1}{3}}&\left(\varphi \pi -{\frac {1}{3^{2}}}\varphi ''\pi +{\frac {1}{3^{4}}}\varphi ^{\scriptscriptstyle {\text{IV}}}\pi -{\frac {1}{3^{6}}}\varphi ^{\scriptscriptstyle {\text{VI}}}\pi +\ldots \right)\sin .3x\\-{\frac {1}{4}}&\left(\varphi \pi -{\frac {1}{4^{2}}}\varphi ''\pi +{\frac {1}{4^{4}}}\varphi ^{\scriptscriptstyle {\text{IV}}}\pi -{\frac {1}{4^{6}}}\varphi ^{\scriptscriptstyle {\text{VI}}}\pi +\ldots \right)\sin .4x.\\&+{\text{etc}}.\end{aligned}}$

ou celle-ci :

${\begin{aligned}(\mathrm {C} )\quad {\frac {1}{2}}\pi \varphi x=\varphi \;\pi &\left(\quad \;\sin .1x-{\frac {1}{2}}\ \ \sin .2x+{\frac {1}{3}}\ \ \sin .3x-{\frac {1}{4}}\ \ \sin .4x+\ldots \right)\\-\varphi ''\pi &\left({\frac {1}{1^{3}}}\sin .1x-{\frac {1}{2^{3}}}\sin .2x+{\frac {1}{3^{3}}}\sin .3x-{\frac {1}{4^{3}}}\sin .4x+\ldots \right)\\+\varphi ^{\scriptscriptstyle {\text{IV}}}\pi &\left({\frac {1}{1^{5}}}\sin .1x-{\frac {1}{2^{5}}}\sin .2x+{\frac {1}{3^{5}}}\sin .3x-{\frac {1}{4^{5}}}\sin .4x+\ldots \right)\\-\varphi ^{\scriptscriptstyle {\text{VI}}}\pi &\left({\frac {1}{1^{7}}}\sin .1x-{\frac {1}{2^{7}}}\sin .2x+{\frac {1}{3^{7}}}\sin .3x-{\frac {1}{4^{7}}}\sin .4x+\ldots \right)\\&+{\text{etc}}.\end{aligned}}$