Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/354

${\begin{aligned}{\frac {d\Omega }{dq}}=&{\frac {d\Omega }{d{\overline {p}}}}+{\overline {r}}{\frac {d^{2}\Omega }{d{\overline {r}}d{\overline {p}}}}\left(\mathrm {M} \sin {\overline {p}}+\mathrm {N} \cos {\overline {p}}\right)\\&+{\overline {r}}'{\frac {d^{2}\Omega }{d{\overline {r}}'d{\overline {p}}}}\left(\mathrm {M} '\sin {\overline {p}}'+\mathrm {N} '\cos {\overline {p}}'\right)+\ldots \\&+2{\frac {d^{2}\Omega }{d{\overline {p}}^{2}}}\left(\mathrm {M} \cos {\overline {p}}-\mathrm {N} \sin {\overline {p}}\right)\\&+2{\frac {d^{2}\Omega }{d{\overline {p}}d{\overline {p}}'}}\left(\mathrm {M} '\cos {\overline {p}}'-\mathrm {N} '\sin {\overline {p}}'\right)+\ldots ,\\\Omega =&\mathrm {T} '\ \left[{\frac {{\overline {r}}\cos \left({\overline {p}}-{\overline {p}}'\ \right)}{{\overline {r}}'^{2}\ }}-{\frac {1}{\sqrt {{\overline {r}}^{2}-2{\overline {r}}{\overline {r}}'\ \cos \left({\overline {p}}-{\overline {p}}'\ \right)+{\overline {r}}'^{2}}}}\right]\\+&\mathrm {T} ''\left[{\frac {{\overline {r}}\cos \left({\overline {p}}-{\overline {p}}''\right)}{{\overline {r}}''^{2}}}-{\frac {1}{\sqrt {{\overline {r}}^{2}-2{\overline {r}}{\overline {r}}''\cos \left({\overline {p}}-{\overline {p}}''\right)+{\overline {r}}''^{2}}}}\right]\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Négligeant donc dans ces substitutions les secondes dimensions des quantités $\mathrm {M,N,M',N',\ldots } ,$ et changeant les lettres $\mathrm {M,N}$ en $x,y,$ ainsi que nous l’avons fait dans le n^o 50 de l’Ouvrage cité, on aura pour la variation de $\Sigma$ cette formule

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

dans laquelle il ne s’agira plus que de substituer la valeur précédente de $\Omega .$