les équations différentielles
donneront les suivantes
![{\displaystyle (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} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1191710280278a813cb78503844ea51cd3bb873f)
Les valeurs de
et
renferment six arbitraires introduites par les intégrations. Considérons trois quelconques de ces arbitraires
et
on aura les trois équations suivantes :
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/234c87024f0ea10cea56dc066c9720f37d29b864)
On tirera de ces équations la valeur de
et, si l’on fait
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900c636153b796c6cfd809c4775e39c061f56de4)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06a7667f37bb359b1f3c2a4494bd5817cad6c497)
![{\displaystyle -{\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a263e4151f6c7356ecacaea9fccdc15418263b0)
on aura
![{\displaystyle {\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17ab92a97f9ca51135766216a9814cc28e72d491)
Si l’on fait pareillement
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4331bab13b96024c9e3d18867d30f19401cd2cd4)