Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/255

 Les corrections sont expliquées en page de discussion

${\begin{array}{rll}{\frac {\operatorname {d} z}{\operatorname {d} t}}&={\frac {\operatorname {d} z}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} t}}+{\frac {\operatorname {d} z}{\operatorname {d} y}}{\frac {\operatorname {d} y}{\operatorname {d} t}}&=p{\frac {\operatorname {d} x}{\operatorname {d} t}}+q{\frac {\operatorname {d} y}{\operatorname {d} t}},\\\\{\frac {\operatorname {d} z}{\operatorname {d} u}}&={\frac {\operatorname {d} z}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} u}}+{\frac {\operatorname {d} z}{\operatorname {d} y}}{\frac {\operatorname {d} y}{\operatorname {d} u}}&=p{\frac {\operatorname {d} x}{\operatorname {d} u}}+q{\frac {\operatorname {d} y}{\operatorname {d} u}},\\\\{\frac {\operatorname {d} z'}{\operatorname {d} t}}&={\frac {\operatorname {d} z'}{\operatorname {d} x'}}{\frac {\operatorname {d} x'}{\operatorname {d} t}}+{\frac {\operatorname {d} z'}{\operatorname {d} y'}}{\frac {\operatorname {d} y'}{\operatorname {d} t}}&=p'{\frac {\operatorname {d} x'}{\operatorname {d} t}}+q'{\frac {\operatorname {d} y'}{\operatorname {d} t}},\\\\{\frac {\operatorname {d} z'}{\operatorname {d} u}}&={\frac {\operatorname {d} z'}{\operatorname {d} x'}}{\frac {\operatorname {d} x'}{\operatorname {d} u}}+{\frac {\operatorname {d} z'}{\operatorname {d} y'}}{\frac {\operatorname {d} y'}{\operatorname {d} u}}&=p'{\frac {\operatorname {d} x'}{\operatorname {d} u}}+q'{\frac {\operatorname {d} y'}{\operatorname {d} u}},\end{array}}$

en prenant donc, sous ce point de vue, les deux différentielles partielles de l’équation (1), par rapport à $t$ et $u$, on trouvera

${\begin{aligned}&{\frac {(t-x)\left(1-{\frac {\operatorname {d} x}{\operatorname {d} t}}\right)-(u-y){\frac {\operatorname {d} y}{\operatorname {d} t}}+(v-z)\left({\frac {\operatorname {d} v}{\operatorname {d} t}}-p{\frac {\operatorname {d} x}{\operatorname {d} t}}-q{\frac {\operatorname {d} y}{\operatorname {d} t}}\right)}{\lambda ^{2}}}\\\\=&{\frac {(t-x')\left(1-{\frac {\operatorname {d} x'}{\operatorname {d} t}}\right)-(u-y'){\frac {\operatorname {d} y'}{\operatorname {d} t}}+(v-z')\left({\frac {\operatorname {d} v}{\operatorname {d} t}}-p'{\frac {\operatorname {d} x'}{\operatorname {d} t}}-q'{\frac {\operatorname {d} y'}{\operatorname {d} t}}\right)}{\lambda '^{2}}},\\\\&{\frac {-(t-x){\frac {\operatorname {d} x}{\operatorname {d} u}}+(u-y)\left(1-{\frac {\operatorname {d} y}{\operatorname {d} u}}\right)+(v-z)\left({\frac {\operatorname {d} v}{\operatorname {d} u}}-p{\frac {\operatorname {d} x}{\operatorname {d} u}}-q{\frac {\operatorname {d} y}{\operatorname {d} u}}\right)}{\lambda ^{2}}}\\\\=&{\frac {-(t-x'){\frac {\operatorname {d} x'}{\operatorname {d} u}}+(u-y')\left(1-{\frac {\operatorname {d} y'}{\operatorname {d} u}}\right)+(v-z')\left({\frac {\operatorname {d} v}{\operatorname {d} u}}-p'{\frac {\operatorname {d} x'}{\operatorname {d} u}}-q'{\frac {\operatorname {d} y'}{\operatorname {d} u}}\right)}{\lambda '^{2}}}\,;\end{aligned}}$

ou encore

${\frac {(t-x)+(v-z){\frac {\operatorname {d} v}{\operatorname {d} t}}-\left[(t-x)+p(v-z)\right]{\frac {\operatorname {d} x}{\operatorname {d} t}}-\left[(u-y)+q(v-z)\right]{\frac {\operatorname {d} y}{\operatorname {d} t}}}{\lambda ^{2}}}$