Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/54

remarquable, qu’elle renferme les lois de la propagation du son, en y supposant $\mathrm {T} =0,$ et en déterminant d’une manière particulière les constantes $a,b,c,\ldots .$

Pour l’intégrer, on changera les variables $x$ et $y$ en d’autres $\varpi$ et $\theta ,$ telles que l’on ait

${\begin{aligned}{\frac {\partial \varpi }{\partial x}}&={\frac {\partial \varpi }{\partial y}}\left(-{\frac {1}{2}}a+{\sqrt {{\frac {1}{4}}a^{2}-b}}\right),\\{\frac {\partial \theta }{\partial x}}&={\frac {\partial \theta }{\partial y}}\left(-{\frac {1}{2}}a-{\sqrt {{\frac {1}{4}}a^{2}-b}}\right),\end{aligned}}$

ce qui donne

$\varpi =x\left(-{\frac {1}{2}}a+{\sqrt {{\frac {1}{4}}a^{2}-b}}\right)+y$

et

$\theta =x\left(-{\frac {1}{2}}a-{\sqrt {{\frac {1}{4}}a^{2}-b}}\right)+y,$

d’où l’on tire

$x={\frac {\varpi -\theta }{\sqrt {a^{2}-4b}}}\qquad$et$\qquad y={\frac {\varpi \left(a+{\sqrt {a^{2}-4b}}\right)-\theta \left(a-{\sqrt {a^{2}-4b}}\right)}{2{\sqrt {a^{2}-4b}}}}$

on aura ensuite

${\begin{aligned}{\frac {\partial z}{\partial x}}=&{\frac {1}{2}}{\frac {\partial z}{\partial \varpi }}\left(-a+{\sqrt {a^{2}-4b}}\right)+{\frac {1}{2}}{\frac {\partial z}{\partial \theta }}\left(-a-{\sqrt {a^{2}-4b}}\right),\\{\frac {\partial ^{2}z}{\partial x^{2}}}=&{\frac {1}{4}}{\frac {\partial ^{2}z}{\partial \varpi ^{2}}}\left(2a^{2}-2a{\sqrt {a^{2}-4b}}-4b\right)\\&+2b{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}+{\frac {1}{4}}{\frac {\partial ^{2}z}{\partial \theta ^{2}}}\left(2a^{2}+2a{\sqrt {a^{2}-4b}}-4b\right),\\{\frac {\partial z}{\partial y}}=&{\frac {\partial z}{\partial \varpi }}+{\frac {\partial z}{\partial \theta }},\\{\frac {\partial ^{2}z}{\partial x\partial y}}=&{\frac {1}{2}}{\frac {\partial ^{2}z}{\partial \varpi ^{2}}}\left(-a+{\sqrt {a^{2}-4b}}\right)-a{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}\\&\qquad \qquad +{\frac {1}{2}}{\frac {\partial ^{2}z}{\partial \theta ^{2}}}\left(-a-{\sqrt {a^{2}-4b}}\right),\\{\frac {\partial ^{2}z}{\partial y^{2}}}=&{\frac {\partial ^{2}z}{\partial \varpi ^{2}}}+2{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}+{\frac {\partial ^{2}z}{\partial \theta ^{2}}}\,;\end{aligned}}$