Page:Lagrange - Œuvres (1867) vol. 1.djvu/607

Donc, si l’on fait ces substitutions, et qu’on fasse, pour abréger,

${\begin{aligned}{\frac {2\alpha ^{2}k^{2}t^{2}}{2^{6}}}\Delta ^{6}\mathrm {Y} ^{(s)}-{\frac {2\alpha \beta k^{2}t^{2}}{2^{8}}}\Delta ^{8}\mathrm {Y} ^{(s)}+\ldots =&\mathrm {P} ^{(s)},\\-{\frac {\alpha kt}{2^{2}}}\Delta ^{2}\mathrm {Y} ^{(s)}+{\frac {1}{2}}(\alpha +2\beta ){\frac {kt}{2^{4}}}\Delta ^{4}\mathrm {Y} ^{(s)}-{\frac {1}{2}}\left({\frac {3\alpha }{4}}+\beta +2\gamma \right){\frac {kt}{2^{6}}}\Delta ^{6}\mathrm {Y} ^{(s)}+\ldots =&\mathrm {Q} ^{(s)},\end{aligned}}$

et de plus

${\begin{aligned}\mathrm {P} '\sin '{\frac {m\pi }{n+1}}+\mathrm {P} ''\sin {\frac {2m\pi }{n+1}}+\mathrm {P} '''\sin {\frac {3m\pi }{n+1}}+\ldots +\mathrm {P} ^{(n)}\sin {\frac {nm\pi }{n+1}}=&\mathrm {P} _{m},\\\mathrm {Q} '\sin '{\frac {m\pi }{n+1}}+\mathrm {Q} ''\sin {\frac {2m\pi }{n+1}}+\mathrm {Q} '''\sin {\frac {3m\pi }{n+1}}+\ldots +\mathrm {Q} ^{(n)}\sin {\frac {nm\pi }{n+1}}=&\mathrm {Q} _{m},\end{aligned}}$

on aura

${\frac {2\sin {\cfrac {ms\pi }{n+1}}}{n+1}}\mathrm {Y} _{m}\cos \left[2tk\sin {\frac {m\pi }{2(n+1)}}\right]$

${\begin{aligned}=&{\frac {2\sin {\cfrac {ms\pi }{n+1}}}{n+1}}(\mathrm {Y} _{m}+\mathrm {P} _{m})\cos {\frac {mkt}{n+1}}\pi +{\frac {2\sin {\cfrac {ms\pi }{n+1}}}{n+1}}\mathrm {Q} _{m}\sin {\frac {m\pi }{n+1}}\sin {\frac {mkt}{n+1}}\pi \\=&{\frac {\mathrm {Y} _{m}+\mathrm {P} _{m}}{n+1}}\left[\sin {\frac {m(s+kt)}{n+1}}\pi +\sin {\frac {m(s-kt)}{n+1}}\pi \right]\\&+{\frac {\mathrm {Q} _{m}}{2(n+1)}}\left[\sin {\frac {m(s+1+kt)}{n+1}}\pi -\sin {\frac {m(s-1+kt)}{n+1}}\pi \right.\\&\qquad \qquad \qquad -\left.\sin {\frac {m(s+1-kt)}{n+1}}\pi +\sin {\frac {m(s-1-kt)}{n+1}}\pi \right].\end{aligned}}$

Donc, si l’on fait $m$ successivement égal à $1,2,3,\ldots ,n,$ et qu’on suppose en général

${\begin{aligned}\varphi (x)=&2{\frac {\mathrm {Y_{1}+P_{1}} }{n+1}}\sin x\pi +2{\frac {\mathrm {Y_{2}+P_{2}} }{n+1}}\sin 2x\pi +2{\frac {\mathrm {Y_{3}+P_{3}} }{n+1}}\sin 3x\pi +\ldots \\&+2{\frac {\mathrm {Y} _{n}+\mathrm {P} _{n}}{n+1}}\sin nx\pi ,\\\\\psi (x)=&2{\frac {\mathrm {Q} _{1}}{n+1}}\sin x\pi +2{\frac {\mathrm {Q} _{2}}{n+1}}\sin 2x\pi +2{\frac {\mathrm {Q} _{3}}{n+1}}\sin 3x\pi +\ldots \\&+2{\frac {\mathrm {Q} _{n}}{n+1}}\sin nx\pi ,\end{aligned}}$