Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/245

${\begin{aligned}\left(1-\mu ^{2}\right)i^{2}{\rm {P}}+2ni\mu {\sqrt {1-\mu ^{2}}}.{\rm {K}}&={\frac {\partial {\rm {F}}'}{\partial \varpi }},\\\left(1-\mu ^{2}\right)i^{2}{\rm {Q}}+2ni\mu {\sqrt {1-\mu ^{2}}}.{\rm {H}}&={\frac {\partial {\rm {G}}'}{\partial \varpi }},\end{aligned}}$

${\begin{aligned}F&={\frac {\partial .\gamma {\rm {H}}{\sqrt {1-\mu ^{2}}}}{\partial \mu }}-{\frac {\partial .\gamma {\rm {P}}}{\partial \varpi }},\\G&={\frac {\partial .\gamma {\rm {K}}{\sqrt {1-\mu ^{2}}}}{\partial \mu }}-{\frac {\partial .\gamma {\rm {Q}}}{\partial \varpi }}.\end{aligned}}$

Il est facile d’en conclure

${\begin{aligned}{\rm {H}}&={\frac {-{\frac {\partial F'}{\partial \mu }}{\sqrt {1-\mu ^{2}}}-{\frac {{\frac {2n}{i}}\mu {\frac {\partial {\rm {G}}'}{\partial \varpi }}}{\sqrt {1-\mu ^{2}}}}}{i^{2}-4n^{2}\mu ^{2}}}\\\\{\rm {K}}&={\frac {-{\frac {\partial G'}{\partial \mu }}{\sqrt {1-\mu ^{2}}}-{\frac {{\frac {2n}{i}}\mu {\frac {\partial {\rm {F}}'}{\partial \varpi }}}{\sqrt {1-\mu ^{2}}}}}{i^{2}-4n^{2}\mu ^{2}}}\\\\{\rm {P}}&={\frac {{\frac {\partial F'}{\partial \varpi }}-{\frac {2n}{i}}\mu \left(1-\mu ^{2}\right){\frac {\partial {\rm {G}}'}{\partial \mu }}}{\left(1-\mu ^{2}\right)\left(i^{2}-4n^{2}\mu ^{2}\right)}}\\\\{\rm {Q}}&={\frac {{\frac {\partial G'}{\partial \varpi }}+{\frac {2n}{i}}\mu \left(1-\mu ^{2}\right){\frac {\partial {\rm {F}}'}{\partial \mu }}}{\left(1-\mu ^{2}\right)\left(i^{2}-4n^{2}\mu ^{2}\right)}}\end{aligned}}$

ce qui donne

${\rm {F}}=-{\frac {\gamma \left(1-\mu ^{2}\right){\frac {\partial ^{2}{\rm {F}}'}{\partial \mu ^{2}}}}{i^{2}-4n^{2}\mu ^{2}}}-{\frac {\gamma {\frac {\partial ^{2}{\rm {F}}'}{\partial \varpi ^{2}}}}{\left(1-\mu ^{2}\right)\left(i^{2}-4n^{2}\mu ^{2}\right)}}-{\frac {\partial .{\frac {\gamma \left(1-\mu ^{2}\right)}{i^{2}-4n^{2}\mu ^{2}}}}{\partial \mu }}{\frac {\partial {\rm {F}}'}{\partial \mu }}$

$-{\frac {{\frac {\partial \gamma }{\partial \varpi }}{\frac {\partial {\rm {F}}'}{\partial \varpi }}}{\left(1-\mu ^{2}\right)\left(i^{2}-4n^{2}\mu ^{2}\right)}}+{\frac {{\frac {2n}{i}}\mu {\frac {\partial \gamma }{\partial \varpi }}}{i^{2}-4n^{2}\mu ^{2}}}{\frac {\partial {\rm {G}}'}{\partial \mu }}-{\frac {2n}{i}}{\frac {\partial .{\frac {\gamma u}{i^{2}-4n^{2}\mu ^{2}}}}{\partial \mu }}{\frac {\partial {\rm {G}}'}{\partial \varpi }}.$

En changeant dans cette équation ${\rm {F}}$ en ${\rm {G,\ F'}}$ en ${\rm {G'}}$ et réciproquement, et en changeant le signe des termes multipliés par ${\frac {2n}{i}},$ on aura une nouvelle équation entre ${\rm {G,G',F',}}$ qui, combinée avec celle-ci, déterminera ${\rm {F}}$ et ${\rm {G.}}$