Page:Mémoires de l’Académie des sciences, Tome 4.djvu/601

on aura

${\frac {dv}{dt}}=\mathrm {\frac {K}{CD}} \left\{{\frac {d^{2}v}{dr^{2}}}\left[{\frac {d^{2}r}{dx^{2}}}+{\frac {d^{2}r}{dy^{2}}}+{\frac {d^{2}r}{dz^{2}}}\right]+{\frac {dv}{dr}}\left({\frac {d^{2}r}{dx^{2}}}+{\frac {d^{2}r}{dy^{2}}}+{\frac {d^{2}r}{dz^{2}}}\right)\right\}\quad (a)$

L’équation $x^{2}+y^{2}+z^{2}=r^{2}$ donnera

${\begin{alignedat}{3}x&=r{\frac {dr}{dx}}\qquad {\text{et}}\qquad 1&&=\left({\frac {dr}{dx}}\right)^{2}&&+r{\frac {d^{2}r}{dx^{2}}}\\y&=r{\frac {dr}{dy}}\qquad {\text{et}}\qquad 1&&=\left({\frac {dr}{dy}}\right)^{2}&&+r{\frac {d^{2}r}{dy^{2}}}\\z&=r{\frac {dr}{dz}}\qquad {\text{et}}\qquad 1&&=\left({\frac {dr}{dz}}\right)^{2}&&+r{\frac {d^{2}r}{dz^{2}}}.\end{alignedat}}$

Les trois équations du premier ordre donnent

$x^{2}+y^{2}+z^{2}=r^{2}\left[\left({\frac {dr}{dx}}\right)^{2}+\left({\frac {dr}{dy}}\right)^{2}+\left({\frac {dr}{dz}}\right)^{2}\right],$

ou

$1=\left({\frac {dr}{dx}}\right)^{2}+\left({\frac {dr}{dy}}\right)^{2}+\left({\frac {dr}{dz}}\right)^{2}.$

Les trois équations du second ordre donnent

$3=\left({\frac {dr}{dx}}\right)^{2}+\left({\frac {dr}{dy}}\right)^{2}+\left({\frac {dr}{dz}}\right)^{2}+r\left({\frac {d^{2}r}{dx^{2}}}+{\frac {d^{2}r}{dy^{2}}}+{\frac {d^{2}r}{dz^{2}}}\right)\,;$

et mettant pour $\left({\frac {dr}{dx}}\right)^{2}+\left({\frac {dr}{dy}}\right)^{2}+\left({\frac {dr}{dz}}\right)^{2}$ la valeur $1,$

${\frac {d^{2}r}{dx^{2}}}+{\frac {d^{2}r}{dy^{2}}}+{\frac {d^{2}r}{dz^{2}}}={\frac {2}{r}}.$

Faisant les substitutions convenables dans l’équation $(a),$ on aura l’équation

${\frac {dv}{dt}}=\mathrm {\frac {K}{CD}} \left({\frac {d^{2}v}{dr^{2}}}+{\frac {2}{r}}{\frac {dv}{dr}}\right),$

que l’on avait trouvée précédemment.