Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/363

et des inclinaisons, les formules dont il s’agit se réduisent à celles-ci

${\begin{aligned}d\mathrm {M} =&-r^{2}{\frac {d\Omega }{dr}}\cos qdq+{\frac {d\Omega }{dq}}(2r\sin qdq-\cos qdr),\\d\mathrm {N} =&r^{2}{\frac {d\Omega }{dr}}\sin qdq+{\frac {d\Omega }{dq}}(2r\cos qdq+\sin qdr).\end{aligned}}$

Faisant donc dans ces formules les substitutions du n^o 4 ci-dessus, et changeant $\mathrm {M,N}$ en $x,y,$ on aura des équations de cette forme

$dx=\left(\mathrm {X} \sin {\overline {p}}-\mathrm {Y} \cos {\overline {p}}\right)d{\overline {p}},\quad dy=\left(\mathrm {X} \cos {\overline {p}}+\mathrm {Y} \sin {\overline {p}}\right)d{\overline {p}},$

dans lesquelles

${\begin{aligned}\mathrm {X} =&2{\overline {r}}{\frac {d\Omega }{d{\overline {p}}}}+\left(-2{\overline {r}}{\frac {d\Omega }{d{\overline {p}}}}+2{\overline {r}}^{2}{\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {r}}}}\right)\left(x\sin {\overline {p}}+y\cos {\overline {p}}\right)\\&+2{\overline {r}}{\overline {r}}'\ {\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {r}}'}}\left(x'\ \sin {\overline {p}}'\ +y'\ \cos {\overline {p}}'\right)\\&+2{\overline {r}}{\overline {r}}''{\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {r}}''}}\left(x''\sin {\overline {p}}''+y''\cos {\overline {p}}''\right)+\ldots \\&+\left(2{\overline {r}}^{2}{\frac {d\Omega }{d{\overline {r}}}}+4{\overline {r}}{\frac {d^{2}\Omega }{d{\overline {p}}^{2}}}\right)\left(x\cos {\overline {p}}-y\sin {\overline {p}}\right)\\&+4{\overline {r}}{\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {p}}'\,}}\left(x'\,\cos {\overline {p}}'\,-y'\ \sin {\overline {p}}'\ \right)\\&+4{\overline {r}}{\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {p}}''}}\left(x''\cos {\overline {p}}''-y''\sin {\overline {p}}''\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\mathrm {Y} =&{\overline {r}}^{2}{\frac {d\Omega }{d{\overline {r}}}}+{\overline {r}}^{3}{\frac {d^{2}\Omega }{d{\overline {r^{2}}}}}\left(x\sin {\overline {p}}+y\cos {\overline {p}}\right)\\&+{\overline {r}}^{2}{\overline {r}}'\ {\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {r}}'}}\left(x'\ \sin {\overline {p}}'\ +y'\ \cos {\overline {p}}'\right)\\&+{\overline {r}}^{2}{\overline {r}}''{\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {r}}''}}\left(x''\sin {\overline {p}}''+y''\cos {\overline {p}}''\right)+\ldots \\&+\left(-3{\overline {r}}{\frac {d\Omega }{d{\overline {p}}}}+2{\overline {r}}^{2}{\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {p}}}}\right)\left(x\cos {\overline {p}}-y\sin {\overline {p}}\right)\\&+2{\overline {r}}^{2}{\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {p}}'}}\left(x'\,\cos {\overline {p}}'\ -y'\ \sin {\overline {p}}'\ \right)\\&+2{\overline {r}}^{2}{\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {p}}''}}\left(x''\cos {\overline {p}}''-y''\sin {\overline {p}}''\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\end{aligned}}$