Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 1.djvu/201

on aura

$P'={\frac {\partial Q}{\partial \theta }}.$

Les valeurs de $r,v$ et $\theta$ renferment six arbitraires introduites par les intégrations. Considérons trois quelconques de ces arbitraires, que nous désignerons par $a,b,c$ ; l’équation $\mathrm {M} '={\frac {\partial Q}{\partial r}}$ donnera les trois suivantes

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

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

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

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

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

on aura

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

Si l’on fait pareillement

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