Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 12.djvu/149

les équations différentielles $(b)$ donneront les suivantes

$(c)\qquad \qquad \qquad {\frac {\partial \mathrm {V} '}{\partial r}}=\mathrm {EM} ,\qquad {\frac {\partial \mathrm {V} '}{\partial v}}=\mathrm {EN} ,\qquad {\frac {\partial \mathrm {V} '}{\partial \varpi }}=\mathrm {EP} .$

Les valeurs de $r,v$ et $\varpi$ renferment six arbitraires introduites par les intégrations. Considérons trois quelconques de ces arbitraires $a,b$ et $c\,;$ on aura les trois équations suivantes :

${\begin{aligned}{\frac {\partial ^{2}\mathrm {V} '}{\partial r^{2}}}{\frac {\partial r}{\partial a}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial v}}{\frac {\partial v}{\partial a}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial \varpi }}{\frac {\partial \varpi }{\partial a}}=&\mathrm {E} {\frac {\partial \mathrm {M} }{\partial a}},\\{\frac {\partial ^{2}\mathrm {V} '}{\partial r^{2}}}{\frac {\partial r}{\partial b}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial v}}{\frac {\partial v}{\partial b}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial \varpi }}{\frac {\partial \varpi }{\partial b}}=&\mathrm {E} {\frac {\partial \mathrm {M} }{\partial b}},\\{\frac {\partial ^{2}\mathrm {V} '}{\partial r^{2}}}{\frac {\partial r}{\partial c}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial v}}{\frac {\partial v}{\partial c}}+{\frac {\partial ^{2}\mathrm {V} '}{\partial r\partial \varpi }}{\frac {\partial \varpi }{\partial c}}=&\mathrm {E} {\frac {\partial \mathrm {M} }{\partial c}}.\end{aligned}}$

On tirera de ces équations la valeur de ${\frac {\partial ^{2}\mathrm {V} '}{\partial r^{2}}}$ et, si l’on fait

${\begin{aligned}m=&{\frac {\partial v}{\partial b}}{\frac {\partial \varpi }{\partial c}}-{\frac {\partial v}{\partial c}}{\frac {\partial \varpi }{\partial b}},\\n\ =&{\frac {\partial v}{\partial c}}{\frac {\partial \varpi }{\partial a}}-{\frac {\partial v}{\partial a}}{\frac {\partial \varpi }{\partial c}},\\p\ =&{\frac {\partial v}{\partial a}}{\frac {\partial \varpi }{\partial b}}-{\frac {\partial v}{\partial b}}{\frac {\partial \varpi }{\partial a}},\end{aligned}}$

${\text{ϐ}}={\frac {\partial r}{\partial a}}{\frac {\partial v}{\partial b}}{\frac {\partial \varpi }{\partial c}}-{\frac {\partial r}{\partial a}}{\frac {\partial v}{\partial c}}{\frac {\partial \varpi }{\partial b}}+{\frac {\partial r}{\partial b}}{\frac {\partial v}{\partial c}}{\frac {\partial \varpi }{\partial a}}$

$-{\frac {\partial r}{\partial b}}{\frac {\partial v}{\partial a}}{\frac {\partial \varpi }{\partial c}}+{\frac {\partial r}{\partial c}}{\frac {\partial v}{\partial a}}{\frac {\partial \varpi }{\partial b}}-{\frac {\partial r}{\partial c}}{\frac {\partial v}{\partial b}}{\frac {\partial \varpi }{\partial a}},$

on aura

${\frac {1}{\mathrm {E} }}{\frac {\partial ^{2}\mathrm {V} '}{\partial r^{2}}}={\frac {m{\cfrac {\partial \mathrm {M} }{\partial a}}+n{\cfrac {\partial \mathrm {M} }{\partial b}}+p{\cfrac {\partial \mathrm {M} }{\partial c}}}{\text{ϐ}}}.$

Si l’on fait pareillement

${\begin{aligned}m'=&{\frac {\partial r}{\partial c}}{\frac {\partial \varpi }{\partial b}}-{\frac {\partial r}{\partial b}}{\frac {\partial \varpi }{\partial c}},\\n'\ =&{\frac {\partial r}{\partial a}}{\frac {\partial \varpi }{\partial c}}-{\frac {\partial r}{\partial c}}{\frac {\partial \varpi }{\partial a}},\\p'\ =&{\frac {\partial r}{\partial b}}{\frac {\partial \varpi }{\partial a}}-{\frac {\partial r}{\partial a}}{\frac {\partial \varpi }{\partial b}},\end{aligned}}$