Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/315

307

SECONDE PARTIE. — SECTION XI.

Ayant trouvé l’expression de $\varphi ,$ on aura par la différenciation celles de $p,\,q,\,r,$ comme il suit :

${\begin{aligned}p={\frac {\partial \varphi }{\partial x}}={\frac {\partial \varphi '}{\partial x}}&+z{\frac {\partial \varphi ''}{\partial x}}-{\frac {z^{2}}{2}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{3}}}+{\frac {\partial ^{3}\varphi '}{\partial x\partial y^{2}}}\right)-{\frac {z^{3}}{2.3}}\left({\frac {\partial ^{3}\varphi ''}{\partial x^{3}}}+{\frac {\partial ^{3}\varphi ''}{\partial x\partial y^{2}}}\right)+\ldots ,\\q={\frac {\partial \varphi }{\partial y}}={\frac {\partial \varphi '}{\partial y}}&+z{\frac {\partial \varphi ''}{\partial y}}-{\frac {z^{2}}{2}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi '}{\partial y^{3}}}\right)-{\frac {z^{3}}{2.3}}\left({\frac {\partial ^{3}\varphi ''}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi ''}{\partial y^{3}}}\right)+\ldots ,\\r={\frac {\partial \varphi }{\partial z}}=\ \varphi ''\ \ &-z\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}\right)-{\frac {z^{2}}{2}}\left({\frac {\partial ^{2}\varphi ''}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi ''}{\partial y^{2}}}\right)\\&+{\frac {z^{3}}{2.3}}\left({\frac {\partial ^{4}\varphi '}{\partial x^{4}}}+2{\frac {\partial ^{4}\varphi '}{\partial x^{2}\partial y^{2}}}+{\frac {\partial ^{4}\varphi '}{\partial y^{4}}}\right)+\ldots .\end{aligned}}$

Et substituant ces valeurs dans l’expression de $\lambda$ de l’article 23, elle deviendra de cette forme

$\lambda =\lambda '+z\lambda ''+z^{2}\lambda '''+z^{3}\lambda ^{\scriptscriptstyle {\text{IV}}}+\ldots ,$

dans laquelle

${\begin{aligned}\lambda '\ \,=&-g(x\cos \xi +y\cos \eta )+{\frac {\partial \varphi '}{\partial t}}+{\frac {1}{2}}\left({\frac {\partial \varphi '}{\partial x}}\right)^{2}+{\frac {1}{2}}\left({\frac {\partial \varphi '}{\partial y}}\right)^{2}+{\frac {1}{2}}\varphi ''^{2},\\\lambda ''\ =&-g\cos \zeta +{\frac {\partial \varphi ''}{\partial t}}+{\frac {\partial \varphi '}{\partial x}}{\frac {\partial \varphi ''}{\partial x}}+{\frac {\partial \varphi '}{\partial y}}{\frac {\partial \varphi ''}{\partial y}}-\varphi ''\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}\right),\\\lambda '''=&-{\frac {1}{2}}\left({\frac {\partial ^{3}\varphi '}{\partial t\partial x^{2}}}+{\frac {\partial ^{3}\varphi '}{\partial t\partial y^{2}}}\right)+{\frac {1}{2}}\left({\frac {\partial \varphi ''}{\partial x}}\right)^{2}-{\frac {1}{2}}{\frac {\partial \varphi '}{\partial x}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{3}}}+{\frac {\partial ^{3}\varphi '}{\partial x\partial y^{2}}}\right)+{\frac {1}{2}}\left({\frac {\partial \varphi ''}{\partial y}}\right)^{2}\\&-{\frac {1}{2}}{\frac {\partial \varphi '}{\partial y}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi '}{\partial y^{3}}}\right)+{\frac {1}{2}}\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}\right)^{2}-{\frac {1}{2}}\varphi ''\left({\frac {\partial ^{2}\varphi ''}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi ''}{\partial y^{2}}}\right),\end{aligned}}$

et ainsi de suite.

26. Maintenant, si $z=\alpha$ est l’équation des parois, $\alpha$ étant une fonction fort petite de $x$ et $y$ sans $z,$ l’équation de condition pour que les mêmes particules soient toujours contiguës à ces parois (art. 24) deviendra, par les substitutions précédentes,

${\begin{aligned}0=\varphi ''&-{\frac {\partial \varphi '}{\partial x}}{\frac {\partial \alpha }{\partial x}}-{\frac {\partial \varphi '}{\partial y}}{\frac {\partial \alpha }{\partial y}}-z\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}+{\frac {\partial \varphi ''}{\partial x}}{\frac {\partial \alpha }{\partial x}}+{\frac {\partial \varphi ''}{\partial y}}{\frac {\partial \alpha }{\partial y}}\right)\\&-{\frac {1}{2}}z^{2}\left[{\frac {\partial ^{2}\varphi ''}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi ''}{\partial y^{2}}}-\left({\frac {\partial ^{3}\varphi }{\partial x^{3}}}+{\frac {\partial ^{3}\varphi '}{\partial x\partial y^{2}}}\right){\frac {\partial a}{\partial x}}-\left({\frac {\partial ^{3}\varphi '}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi '}{\partial y^{3}}}\right){\frac {\partial a}{\partial y}}\right]\end{aligned}}$

+

,