Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/64

peut ne pas avoir lieu ; en supposant dans ce cas

${\begin{aligned}\mathrm {K} =&{\frac {{\cfrac {\partial ^{2}\mathrm {B} '}{\partial \varpi \partial \theta }}+m{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n{\cfrac {\partial \mathrm {B} '}{\partial \theta }}+l\mathrm {B} '}{{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n\mathrm {B} '}},\\\mathrm {K} '=&{\frac {{\cfrac {\partial ^{2}\mathrm {B} }{\partial \varpi \partial \theta }}+m{\cfrac {\partial \mathrm {B} }{\partial \varpi }}+n{\cfrac {\partial \mathrm {B} }{\partial \theta }}+lb}{{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n\mathrm {B} '}},\\\mathrm {K} ''=&{\frac {{\cfrac {\partial \mathrm {B} }{\partial \varpi }}+n\mathrm {B} }{{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n\mathrm {B} '}},\\\mathrm {M} =&-{\frac {\mathrm {L} }{{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n\mathrm {B} '}},\\\mathrm {M} '=&-{\frac {\mathrm {L} '}{{\cfrac {\partial \mathrm {B} '}{\partial \varpi }}+n\mathrm {B} '}},\\\ldots &\ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura

$(\lambda ')\qquad \left\{{\begin{aligned}\int {\frac {\partial \mathrm {C} '}{\partial \theta }}\varphi (\varpi )d\varpi &+\mathrm {K} \int \mathrm {C} '\varphi (\varpi )d\varpi \\&+\mathrm {K} '\int \mathrm {C} \varphi (\varpi )d\varpi +\mathrm {K} ''\int {\frac {\partial \mathrm {C} }{\partial \theta }}\varphi (\varpi )d\varpi \\=\mathrm {M} \varphi '(\varpi )&+\mathrm {M} '\varphi (\varpi )+\mathrm {M} ''\varphi _{1}(\varpi )+\ldots \,;\end{aligned}}\right.$

en différenciant par rapport à $\pi ,$ on aura

${\begin{aligned}{\frac {\partial \mathrm {C} '}{\partial \theta }}\varphi (\varpi )&+{\frac {\partial \mathrm {K} }{\partial \varpi }}\int \mathrm {C} '\varphi (\varpi )d\varpi +\mathrm {KC'} \varphi (\varpi )+{\frac {\partial \mathrm {K} '}{\partial \varpi }}\int \mathrm {C} \varphi (\varpi )d\varpi \\&+\mathrm {K'C} \varphi (\varpi )+{\frac {\partial \mathrm {K} ''}{\partial \varpi }}\int {\frac {\partial \mathrm {C} }{\partial \theta }}\varphi (\varpi )d\varpi +\mathrm {K} ''{\frac {\partial \mathrm {C} }{\partial \theta }}\varphi (\varpi )\\=\mathrm {M} \varphi ''&(\varpi )+\left({\frac {\partial \mathrm {M} }{\partial \varpi }}+\mathrm {M} '\right)\varphi '(\varpi )+\ldots .\end{aligned}}$

Si ${\frac {\partial \mathrm {K} }{\partial \varpi }}$ n’est pas nul, on aura une valeur de $\int \mathrm {C} '\varphi (\varpi )d\varpi ,$ qui, substi-