Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/316

308

MÉCANIQUE ANALYTIQUE.

laquelle, devant avoir lieu lorsqu’on fait $z=\alpha ,$ se réduira à cette forme plus simple

${\begin{aligned}\varphi ''&-{\frac {\partial }{\partial x}}\left(\alpha {\frac {\partial \varphi '}{\partial x}}\right)-{\frac {\partial }{\partial y}}\left(\alpha {\frac {\partial \varphi '}{\partial y}}\right)-{\frac {1}{2}}{\frac {\partial }{\partial x}}\left(\alpha ^{2}{\frac {\partial \varphi ''}{\partial x}}\right)-{\frac {1}{2}}{\frac {\partial }{\partial y}}\left(\alpha ^{2}{\frac {\partial \varphi ''}{\partial y}}\right)\\&+{\frac {1}{2.3}}{\frac {\partial }{\partial x}}\left[\alpha ^{3}\left({\frac {\partial ^{3}\varphi '}{\partial x^{3}}}+{\frac {\partial ^{3}\varphi '}{\partial x\partial y^{2}}}\right)\right]+{\frac {1}{2.3}}{\frac {\partial }{\partial y}}\left[\alpha ^{3}\left({\frac {\partial ^{3}\varphi '}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi '}{\partial y^{3}}}\right)\right]+\ldots =0\,;\end{aligned}}$

et il faudra que cette équation soit vraie dans toute l’étendue des parois données.

27. Enfin l’équation de la surface extérieure et libre du fluide, étant $\lambda =0,$ sera de la forme

$\lambda '+z\lambda ''+z^{2}\lambda '''+z^{3}\lambda ^{\scriptscriptstyle {\text{IV}}}+\ldots =0\,;$

et l’équation de condition pour que les mêmes particules demeurent à la surface (art. 24) sera

${\begin{aligned}{\frac {\partial \lambda '}{\partial t}}&+{\frac {\partial \varphi '}{\partial x}}{\frac {\partial \lambda '}{\partial x}}+{\frac {\partial \varphi '}{\partial y}}{\frac {\partial \lambda '}{\partial y}}+\varphi ''\lambda ''\\&+z\ \left[{\frac {\partial \lambda ''}{\partial t}}+{\frac {\partial \varphi ''}{\partial x}}{\frac {\partial \lambda '}{\partial x}}+{\frac {\partial \varphi '}{\partial x}}{\frac {\partial \lambda ''}{\partial x}}+{\frac {\partial \varphi ''}{\partial y}}{\frac {\partial \lambda '}{\partial y}}+{\frac {\partial \varphi '}{\partial y}}{\frac {\partial \lambda ''}{\partial y}}\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+2\varphi ''\lambda ''-\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}\right)\lambda ''\right]\\\\&+z^{2}\left[{\frac {\partial \lambda '''}{\partial t}}+{\frac {\partial \varphi ''}{\partial x}}{\frac {\partial \lambda ''}{\partial x}}+{\frac {\partial \varphi '}{\partial x}}{\frac {\partial \lambda '''}{\partial x}}+{\frac {\partial \varphi ''}{\partial y}}{\frac {\partial \lambda ''}{\partial y}}+{\frac {\partial \varphi '}{\partial y}}{\frac {\partial \lambda '''}{\partial y}}\right.\\&\qquad \qquad \quad -{\frac {1}{2}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{3}}}+{\frac {\partial ^{3}\varphi '}{\partial x\partial y^{2}}}\right){\frac {\partial \lambda '}{\partial x}}-{\frac {1}{2}}\left({\frac {\partial ^{3}\varphi '}{\partial x^{2}\partial y}}+{\frac {\partial ^{3}\varphi '}{\partial y^{3}}}\right){\frac {\partial \lambda '}{\partial y}}\\&\qquad \qquad \quad \ \left.-2\left({\frac {\partial ^{2}\varphi '}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi '}{\partial y^{2}}}\right)\lambda '''-{\frac {1}{2}}\left({\frac {\partial ^{2}\varphi ''}{\partial x^{2}}}+{\frac {\partial ^{2}\varphi ''}{\partial y^{2}}}\right)\lambda ''+3\varphi ''\lambda ^{\scriptscriptstyle {\text{IV}}}\right]+\ldots =0.\end{aligned}}$

Chassant $z$ de ces deux équations, on en aura une qui devra subsister d’elle-même pour tous les points de la surface extérieure.