Page:Mémoires de l’Académie des sciences, Tome 8.djvu/882

les valeurs de $v'$ et $w',$ on aura

${\begin{aligned}v'=&{\frac {d.}{dx}}\int _{0}^{\pi }\int _{0}^{2\pi }f_{1}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \\&+{\frac {d^{2}.}{dxdt}}\int _{0}^{\pi }\int _{0}^{2\pi }\operatorname {F} _{1}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta ,\\w'=&{\frac {d.}{dx}}\int _{0}^{\pi }\int _{0}^{2\pi }f_{2}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \\&+{\frac {d^{2}.}{dxdt}}\int _{0}^{\pi }\int _{0}^{2\pi }\operatorname {F} _{2}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \,;\end{aligned}}$

et pour satisfaire en même temps à l’équation (7), de la manière la plus générale, il faudra prendre

${\begin{aligned}u'=\varpi &-{\frac {d.}{dy}}\int _{0}^{\pi }\int _{0}^{2\pi }f_{1}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \\&-{\frac {d.}{dz}}\int _{0}^{\pi }\int _{0}^{2\pi }f_{2}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \\&-{\frac {d^{2}.}{dydt}}\int _{0}^{\pi }\int _{0}^{2\pi }\operatorname {F} _{1}\left(x+{\frac {at}{\sqrt {3}}}\cos .\alpha ,\ y+{\frac {at}{\sqrt {3}}}\sin .\alpha \sin .\beta ,\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \left.z+{\frac {at}{\sqrt {3}}}\sin .\alpha \cos .\beta \right)t\sin .\alpha d\alpha d\beta \end{aligned}}$