Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/241

${\sqrt {\operatorname {d} 'x^{2}+\operatorname {d} 'y^{2}+\operatorname {d} 'z^{2}}}$

et que les forces accélératrices correspondantes seront représentées par $\operatorname {d} '^{2}x+\operatorname {d} '^{2}y+\operatorname {d} '^{2}z.$

4. On peut observer, d’après ces conventions, que $\operatorname {d} 'u\operatorname {d} t$ n’est autre chose que la différentielle totale de $u,$ prise en regardant $x,y,z$ comme fonctions de $t,$ et que, par suite, on doit avoir identiquement

$\operatorname {d} 'u={\frac {\operatorname {d} u}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} u}{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} u}{\operatorname {d} z}}\operatorname {d} 'z.$

On trouvera aussi

${\begin{aligned}{\frac {\operatorname {d} u}{\operatorname {d} a}}&={\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {d} y}{\operatorname {d} a}}+{\frac {\operatorname {d} u}{\operatorname {d} z}}{\frac {\operatorname {d} z}{\operatorname {d} a}},\\\\{\frac {\operatorname {d} u}{\operatorname {d} b}}&={\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} b}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {d} y}{\operatorname {d} b}}+{\frac {\operatorname {d} u}{\operatorname {d} z}}{\frac {\operatorname {d} z}{\operatorname {d} b}},\\\\{\frac {\operatorname {d} u}{\operatorname {d} c}}&={\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} c}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {d} y}{\operatorname {d} c}}+{\frac {\operatorname {d} u}{\operatorname {d} z}}{\frac {\operatorname {d} z}{\operatorname {d} c}}\,;\end{aligned}}$

transformations dont nous ferons usage par la suite.

5. D’après les principes connus du calcul différentiel partiel, ou aura identiquement

${\begin{array}{rrr}{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} x}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} x\operatorname {d} t}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} y}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} y\operatorname {d} t}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} z}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} z\operatorname {d} t}},\\\\{\frac {\operatorname {d} ^{2}u}{\operatorname {d} x\operatorname {d} y}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} y\operatorname {d} x}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} x\operatorname {d} z}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} z\operatorname {d} x}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} y\operatorname {d} z}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} z\operatorname {d} y}},\end{array}}$

et encore