Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/340

déterminer la valeur de ${\frac {\partial ^{n}u}{\partial \alpha ^{n}}}$ dans le cas de $\alpha =0\,;$ or on a

${\frac {\partial u}{\partial \alpha }}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \alpha }}=z{\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}},$

à cause de ${\frac {\partial x}{\partial \alpha }}=z{\frac {\partial x}{\partial t}},$ partant

$(a)\qquad \qquad \qquad \qquad {\frac {\partial u}{\partial x}}={\frac {\partial \int z{\cfrac {\partial u}{\partial x}}dx}{\partial t}}.$

En différenciant cette équation par rapport à $\alpha ,$ on aura

${\frac {\partial ^{2}u}{\partial \alpha ^{2}}}={\frac {\partial ^{2}\int z{\cfrac {\partial u}{\partial x}}dx}{\partial t\partial \alpha }}\,;$

or l’équation $(a)$ donne, en y changeant $u$ en $\int z{\frac {\partial u}{\partial x}}dx,$

${\frac {\partial \int z{\cfrac {\partial u}{\partial x}}\partial x}{\partial \alpha }}={\frac {\partial \int z^{2}{\cfrac {\partial u}{\partial x}}\partial x}{\partial t}},$

partant

${\frac {\partial ^{2}u}{\partial \alpha ^{2}}}={\frac {\partial ^{2}\int z^{2}{\cfrac {\partial u}{\partial x}}dx}{\partial t^{2}}}.$

En différenciant encore par rapport à $\alpha ,$ on aura

${\frac {\partial ^{3}u}{\partial \alpha ^{3}}}={\frac {\partial ^{3}\int z^{2}{\cfrac {\partial u}{\partial x}}dx}{\partial t^{2}\partial \alpha }},$

et, en changeant dans l’équation $(a)\ u$ en $\int z^{2}{\frac {\partial u}{\partial x}}\partial x,$ on aura

${\frac {\partial \int z^{2}{\cfrac {\partial u}{\partial x}}dx}{\partial \alpha }}={\frac {\partial \int z^{3}{\cfrac {\partial u}{\partial x}}dx}{\partial t}},$

partant

${\frac {\partial ^{3}u}{\partial \alpha ^{3}}}={\frac {\partial ^{3}\int z^{3}{\cfrac {\partial u}{\partial x}}dx}{\partial t^{3}}}\,;$