Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/368

c’est-à-dire pour $\xi$ et $\psi ,$ des expressions de cette forme

$\xi =\Xi \sin {\overline {p}}-\Psi \cos {\overline {p}},\quad \psi =\Xi \cos {\overline {p}}+\Psi \sin {\overline {p}},$

dans lesquelles

${\begin{aligned}\Xi =&\mathrm {T} '\ {\Bigl [}\left[\left[{\overline {r}},{\overline {r}}'\ \right]\right.\left[\left[{\overline {r}},{\overline {r}}'\,\ \right]\right._{1}\cos \left({\overline {p}}-{\overline {p}}'\ \right)+\left[\left[{\overline {r}},{\overline {r}}'\ \right]_{2}\right.\cos 2\left({\overline {p}}-{\overline {p}}'\right)+\ldots {\Bigr ]}\\+&\mathrm {T} ''{\Bigl [}\left[\left[{\overline {r}},{\overline {r}}''\,\right]\right.\left[\left[{\overline {r}},{\overline {r}}''\,\right]\right._{1}\cos \left({\overline {p}}-{\overline {p}}''\right)+\left[\left[{\overline {r}},{\overline {r}}''\right]_{2}\right.\cos 2\left({\overline {p}}-{\overline {p}}''\right)+\ldots {\Bigr ]}\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\psi =&\mathrm {T} '\,\ {\Bigl [}\left.\left[{\overline {r}},{\overline {r}}'\,\ \right]\right]_{1}\sin \left({\overline {p}}-{\overline {p}}'\ \ \right)+\left.\left[{\overline {r}},{\overline {r}}'\ \right]_{2}\right]\sin 2\left({\overline {p}}-{\overline {p}}'\ \right)+\ldots {\Bigr ]}\\+&\mathrm {T} ''\ {\Bigl [}\left.\left[{\overline {r}},{\overline {r}}''\,\right]\right]_{1}\sin \left({\overline {p}}-{\overline {p}}''\ \right)+\left.\left[{\overline {r}},{\overline {r}}''\ \right]_{2}\right]\sin 2\left({\overline {p}}-{\overline {p}}''\right)+\ldots {\Bigr ]}\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\end{aligned}}$

en supposant

${\begin{aligned}&\left[\left[{\overline {r}},{\overline {r}}'\right]_{1}\right.={\overline {r}}^{2}{\frac {\left[{\overline {r}},{\overline {r}}'\right]}{dr}},\\&\left[\left[{\overline {r}},{\overline {r}}'\right]_{1}\right.={\frac {{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{1}}{d\,{\overline {r}}}}+2n'{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{1}-{\frac {{\overline {r}}^{2}}{{\overline {r}}'^{2}}}(1+2n')}{1-n'^{2}}},\\&\left[\left[{\overline {r}},{\overline {r}}'\right]_{2}\right.={\frac {{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{2}}{d\,{\overline {r}}}}+2.4n'{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{2}}{1-4n'^{2}}},\\&\left[\left[{\overline {r}},{\overline {r}}'\right]_{3}\right.={\frac {{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{3}}{d\,{\overline {r}}}}+2.9n'{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{3}}{1-9n'^{2}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\&\left.\left[{\overline {r}},{\overline {r}}'\right]_{1}\right]={\frac {n'{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{1}}{d\,{\overline {r}}}}+2{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{1}-{\frac {{\overline {r}}^{2}}{{\overline {r}}'^{2}}}(n'+2)}{1-n'^{2}}},\\&\left.\left[{\overline {r}},{\overline {r}}'\right]_{2}\right]=2{\frac {n'{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{2}}{d\,{\overline {r}}}}+2{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{2}}{1-4n'^{2}}},\\&\left.\left[{\overline {r}},{\overline {r}}'\right]_{3}\right]=3{\frac {n'{\overline {r}}^{2}{\cfrac {d\left[{\overline {r}},{\overline {r}}'\right]_{3}}{d\,{\overline {r}}}}+2{\overline {r}}\left[{\overline {r}},{\overline {r}}'\right]_{3}}{1-9n'^{2}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

et ainsi des autres fonctions semblables.