remarquable, qu’elle renferme les lois de la propagation du son, en y supposant
et en déterminant d’une manière particulière les constantes
Pour l’intégrer, on changera les variables
et
en d’autres
et
telles que l’on ait
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ac8b46c20a6c65f1c992bba2b48cf7a0cff590)
ce qui donne
![{\displaystyle \varpi =x\left(-{\frac {1}{2}}a+{\sqrt {{\frac {1}{4}}a^{2}-b}}\right)+y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45cf3a73cdf39a641cbd3ace9a648a136abf42c3)
et
![{\displaystyle \theta =x\left(-{\frac {1}{2}}a-{\sqrt {{\frac {1}{4}}a^{2}-b}}\right)+y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e092e9ac6c45d2087c85bcd9fab741f4da41e7)
d’où l’on tire
![{\displaystyle x={\frac {\varpi -\theta }{\sqrt {a^{2}-4b}}}\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/81b1889db8ff30a5ab62198782aa1052bdbf6f63)
et
![{\displaystyle \qquad y={\frac {\varpi \left(a+{\sqrt {a^{2}-4b}}\right)-\theta \left(a-{\sqrt {a^{2}-4b}}\right)}{2{\sqrt {a^{2}-4b}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffd3a2181e01de4ba83dc0a4a93e97dab2c1f7d)
on aura ensuite
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38644bac1410efaffcd9fdc4635592b388bd9cbe)