Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/255

$\nu -\int {\frac {\delta p}{\Delta }}={\frac {\operatorname {d} \phi }{\operatorname {d} t}}+{\frac {\operatorname {d} \phi }{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} \phi }{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} \phi }{\operatorname {d} z}}\operatorname {d} 'z-{\frac {1}{2}}\left(\operatorname {d} 'x^{2}+\operatorname {d} 'y^{2}+\operatorname {d} 'z^{2}\right)$ (16)

si donc l’on fait

$\operatorname {d} 'x={\frac {\operatorname {d} \phi }{\operatorname {d} x}}+P,\quad \operatorname {d} 'y={\frac {\operatorname {d} \phi }{\operatorname {d} y}}+Q,\quad \operatorname {d} 'z={\frac {\operatorname {d} \phi }{\operatorname {d} z}}+R,\qquad$(17)

et qu’on substitue ces valeurs dans l’équation (16), elle deviendra, réductions faites,

$\nu -\int {\frac {\delta p}{\Delta }}={\frac {\operatorname {d} \phi }{\operatorname {d} t}}+{\frac {1}{2}}\left\{\left({\frac {\operatorname {d} \phi }{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} y}}\right)^{2}+\left({\frac {\operatorname {d} \phi }{\operatorname {d} z}}\right)^{2}\right\}-{\frac {1}{2}}\left(P^{2}+Q^{2}+R^{2}\right).$ (18)

au moyen des mêmes substitutions, l’équation (5) devient

${\begin{array}{cr}0={\frac {\operatorname {d} \Delta }{\operatorname {d} t}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} x}}{\frac {\operatorname {d} \phi }{\operatorname {d} x}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} y}}{\frac {\operatorname {d} \phi }{\operatorname {d} y}}+{\frac {\operatorname {d} \Delta }{\operatorname {d} z}}{\frac {\operatorname {d} \phi }{\operatorname {d} z}}+\Delta \left({\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} z^{2}}}\right)&\\&(19)\\+P{\frac {\operatorname {d} \Delta }{\operatorname {d} x}}+Q{\frac {\operatorname {d} \Delta }{\operatorname {d} y}}+R{\frac {\operatorname {d} \Delta }{\operatorname {d} z}}+\Delta \left({\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {\operatorname {d} Q}{\operatorname {d} y}}+{\frac {\operatorname {d} R}{\operatorname {d} z}}\right)\,;&\\\end{array}}$

et l’équation (7), qui a lieu quand le fluide est incompressible, deviendra

$0={\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} ^{2}\phi }{\operatorname {d} z^{2}}}+{\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {\operatorname {d} Q}{\operatorname {d} y}}+{\frac {\operatorname {d} R}{\operatorname {d} z}}\,;\qquad$(20)

d’où l’on voit que les équations du mouvement du fluide se simplifieraient d’une manière notable, si l’on pouvait déterminer les fonctions $P,Q,R.$ On a pour cela l’équation

$P\delta x+Q\delta y+R\delta z=A\delta a+B\delta b+C\delta c,$

que l’on obtient en mettant dans inéquation (14) les valeurs de