Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/345

Si l’on considère deux variables $x$ et $x_{1},$ on aura d’abord

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

partant

${\frac {\partial ^{2}u}{\partial \alpha \partial \alpha _{1}}}=z{\frac {\partial ^{2}u}{\partial \alpha _{1}\partial t}}+{\frac {\partial u}{\partial t}}{\frac {\partial z}{\partial \alpha _{1}}}\,;$

or on a

${\frac {\partial u}{\partial \alpha _{1}}}=z_{1}{\frac {\partial u}{\partial t_{1}}},$

et, en changeant $u$ en $z$ dans cette équation, on a

${\frac {\partial z}{\partial \alpha _{1}}}=z_{1}{\frac {\partial z}{\partial t_{1}}}\,;$

donc

${\frac {\partial ^{2}u}{\partial \alpha \partial \alpha _{1}}}=z{\frac {\partial .z_{1}{\cfrac {\partial u}{\partial t_{1}}}}{\partial t}}+z_{1}{\frac {\partial u}{\partial t}}{\frac {\partial z}{\partial t_{1}}},$

ou

${\frac {\partial ^{2}u}{\partial \alpha \partial \alpha _{1}}}=zz_{1}{\frac {\partial ^{2}u}{\partial t\partial t_{1}}}+z{\frac {\partial u}{\partial t_{1}}}{\frac {\partial z_{1}}{\partial t}}+z_{1}{\frac {\partial u}{\partial t}}{\frac {\partial z}{\partial t_{1}}}.$

En changeant $z$ en $z'',\ z_{1}$ en $z_{1}^{n_{1}},$ et nommant $\scriptstyle \mathrm {U} \displaystyle ,\mathrm {Z,Z_{1}}$ ce que deviennent $u,z,z_{1},$ lorsqu’on y substitue $\varphi (t),\varphi _{1}(t_{1})$ au lieu de $x,x_{1},$ on aura

$q_{n,n_{1}}={\frac {\partial ^{n+n_{1}-2}\left(\mathrm {Z} ^{n}\mathrm {Z} _{1}^{n_{1}}{\frac {\partial ^{2}\scriptstyle \mathrm {U} \displaystyle }{\partial t\partial t_{1}}}+n\mathrm {Z} ^{n-1}\mathrm {Z} _{1}^{n_{1}}{\frac {\partial \scriptstyle \mathrm {U} \displaystyle }{\partial t}}{\frac {\partial \mathrm {Z} }{\partial t_{1}}}+n_{1}\mathrm {Z} ^{n}\mathrm {Z} _{1}^{n_{1}-1}{\frac {\partial \scriptstyle \mathrm {U} \displaystyle }{\partial t_{1}}}{\frac {\partial \mathrm {Z} _{1}}{\partial t}}\right)}{1.2.3\ldots n\partial t^{n-1}.1.2.3\ldots n_{1}\partial t_{1}^{n_{1}-1}}}.$

Si l’on considère trois variables $x,x_{1},x_{2},$ l’équation

${\frac {\partial ^{2}u}{\partial \alpha \partial \alpha _{1}}}=zz_{1}{\frac {\partial ^{2}u}{\partial t\partial t_{1}}}+z{\frac {\partial u}{\partial t_{1}}}{\frac {\partial z_{1}}{\partial t}}+z_{1}{\frac {\partial u}{\partial t}}{\frac {\partial z}{\partial t_{1}}}$

donnera, en la différenciant par rapport à $\alpha _{2},$

${\frac {\partial ^{3}u}{\partial \alpha \partial \alpha _{1}\partial \alpha _{2}}}=zz_{1}{\frac {\partial ^{3}u}{\partial t\partial t_{1}\partial \alpha _{2}}}+{\frac {\partial ^{2}u}{\partial t\partial t_{1}}}\left(z{\frac {\partial z_{1}}{\partial \alpha _{2}}}+z_{1}{\frac {\partial z}{\partial \alpha _{2}}}\right)+\ldots \,;$

or on a

${\frac {\partial u}{\partial \alpha _{2}}}=z_{2}{\frac {\partial u}{\partial t_{2}}},\qquad {\frac {\partial z}{\partial \alpha _{2}}}=z_{2}{\frac {\partial z}{\partial t_{2}}},\qquad {\frac {\partial z_{1}}{\partial \alpha _{2}}}=z_{2}{\frac {\partial z_{1}}{\partial t_{2}}},$