Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/391

383

FRAGMENTS.

Or

$\mathrm {x^{2}+y^{2}+z^{2}=r^{2}} ,\qquad \xi ^{2}+\eta ^{2}+\zeta ^{2}=a^{2}+b^{2}+c^{2}=r^{2}\,;$

donc

$\Pi ={\frac {1}{\sqrt {\mathrm {r} ^{2}+r^{2}-2\mathrm {\left(x\xi +y\eta +z\zeta \right)} }}}={\frac {1}{\mathrm {r} }}+{\frac {\mathrm {x\xi +y\eta +z\zeta } -{\cfrac {r^{2}}{2}}}{\mathrm {r} ^{3}}}+{\frac {3}{2}}\mathrm {\frac {\left(x\xi +y\eta +z\zeta \right)^{2}}{r^{5}}} .$

Or

$\xi =a\xi '+b\xi ''+c\xi '''+\ldots \,;$

donc

$\mathrm {\left(x\xi +y\eta +z\zeta \right)} =a\mathrm {\left(x\xi '+y\eta '+z\zeta '\right)} +b\mathrm {\left(x\xi ''+y\eta ''+z\zeta ''\right)} +c\mathrm {\left(x\xi '''+y\eta '''+z\zeta '''\right)} .$

Soit

${\begin{alignedat}{4}&\mathrm {x} \xi '&+&\mathrm {y} \eta '&+&\mathrm {z} \zeta '&=&\lambda ,\\&\mathrm {x} \xi ''&+&\mathrm {y} \eta ''&+&\mathrm {z} \zeta ''&=&\mu ,\\&\mathrm {x} \xi '''&+&\mathrm {y} \eta '''&+&\mathrm {z} \zeta '''&=&\nu \,;\end{alignedat}}$

on aura

$\Pi ={\frac {1}{\mathrm {r} }}+{\frac {a\lambda +b\mu +c\nu }{\mathrm {r} ^{3}}}-{\frac {a^{2}+b^{2}+c^{2}}{2\mathrm {r} ^{3}}}+{\frac {3(a\lambda +b\mu +c\nu )^{2}}{2\mathrm {r} ^{5}}}$

et (en ne retenant que le dernier terme)

$\mathrm {V} ={\frac {3}{2\mathrm {r} ^{5}}}\mathrm {\left({\frac {B+C-A}{2}}\lambda ^{2}+{\frac {A+C-B}{2}}\mu ^{2}+{\frac {A+B-C}{2}}\nu ^{2}+2F\mu \nu +2G\lambda \nu +2H\lambda \mu \right)} .$

En n’ayant toujours égard qu’au dernier terme, on aura

${\begin{alignedat}{4}&b{\frac {\partial \Pi }{\partial c}}&-&c{\frac {\partial \Pi }{\partial b}}&=&{\frac {3(a\lambda +b\mu +c\nu )}{\mathrm {r} ^{5}}}(b\nu &-&c\mu ),\\&c{\frac {\partial \Pi }{\partial a}}&-&a{\frac {\partial \Pi }{\partial c}}&=&{\frac {3(a\lambda +b\mu +c\nu )}{\mathrm {r} ^{5}}}(c\lambda &-&a\nu ),\\&a{\frac {\partial \Pi }{\partial b}}&-&b{\frac {\partial \Pi }{\partial a}}&=&{\frac {3(a\lambda +b\mu +c\nu )}{\mathrm {r} ^{5}}}(a\mu &-&b\lambda ).\end{alignedat}}$

Donc ; multipliant par $\operatorname {D} \mathrm {m}$ et intégrant, on aura

_S$\left(b{\frac {\partial \Pi }{\partial c}}-c{\frac {\partial \Pi }{\partial b}}\right)\operatorname {D} \mathrm {m} ={\frac {3}{r^{5}}}\mathrm {\left[(C-B)\mu \nu +H\lambda \nu +F(\nu ^{2}-\mu ^{2})-G\mu \lambda \right]} ,$

_S$\left(c{\frac {\partial \Pi }{\partial a}}-a{\frac {\partial \Pi }{\partial c}}\right)\operatorname {D} \mathrm {m} ={\frac {3}{r^{5}}}\mathrm {\left[G(\lambda ^{2}-\nu ^{2})+F\lambda \mu +(A-C)\lambda \nu -H\mu \nu \right]} ,$

_S$\left(a{\frac {\partial \Pi }{\partial b}}-b{\frac {\partial \Pi }{\partial a}}\right)\operatorname {D} \mathrm {m} ={\frac {3}{r^{5}}}\mathrm {\left[(B-A)\lambda \mu +H(\mu ^{2}-\lambda ^{2})+G\mu \nu -F\lambda \nu \right]} .$