Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/257

 Les corrections sont expliquées en page de discussion

et par suite, les équations (17), (18), (19), (20) du n.^o  précédent, deviendront

$\operatorname {d} 'x={\frac {\operatorname {d} \phi }{\operatorname {d} x}},\quad \operatorname {d} 'y={\frac {\operatorname {d} \phi }{\operatorname {d} y}},\quad \operatorname {d} 'z={\frac {\operatorname {d} \phi }{\operatorname {d} z}}\,;\qquad$(21)

$\nu -\int {\frac {\delta p}{\Delta }}={\frac {\operatorname {d} \phi }{\operatorname {d} t}}+{\frac {1}{2}}\left\{\left({\frac {\operatorname {d} \phi }{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} y}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} z}}\right)^{2}\right\},\qquad$(22)

$0={\frac {\operatorname {d} \Delta }{\operatorname {d} t}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} x}}{\frac {\operatorname {d} \phi }{\operatorname {d} x}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} y}}{\frac {\operatorname {d} \phi }{\operatorname {d} y}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} z}}{\frac {\operatorname {d} \phi }{\operatorname {d} z}}+\Delta \left({\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} z^{2}}}\right)$ (23)

$0={\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} z^{2}}},\qquad$(24)

on peut observer de plus que le quarré de la vitesse absolue du fluide est

$\left({\frac {\operatorname {d} \phi }{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} y}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} z}}\right)^{2}.$

15. Lorsque les mouvemens du fluide sont très-petits, on peut négliger les termes affectés des puissances, et produits de puissances des vitesses $\operatorname {d} 'x,\operatorname {d} 'y,\operatorname {d} 'z.$ Dans ce cas les équations (3) deviennent

${\begin{aligned}{\frac {\operatorname {d} \nu }{\operatorname {d} x}}-{\frac {1}{\Delta }}{\frac {\operatorname {d} p}{\operatorname {d} x}}&={\frac {\operatorname {d} 'x}{\operatorname {d} t}},\\\\{\frac {\operatorname {d} \nu }{\operatorname {d} y}}-{\frac {1}{\Delta }}{\frac {\operatorname {d} p}{\operatorname {d} y}}&={\frac {\operatorname {d} 'y}{\operatorname {d} t}},\\\\{\frac {\operatorname {d} \nu }{\operatorname {d} z}}-{\frac {1}{\Delta }}{\frac {\operatorname {d} p}{\operatorname {d} z}}&={\frac {\operatorname {d} 'z}{\operatorname {d} t}}\,;\end{aligned}}$

desquelles nous tirerons