Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/78

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INTÉGRALES

{\frac {\operatorname {d} x}{\operatorname {d} t}},\quad {\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}},\quad {\frac {\operatorname {d} ^{3}x}{\operatorname {d} t^{3}}},\ldots {\frac {\operatorname {d} y}{\operatorname {d} t}},\quad {\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}},\quad {\frac {\operatorname {d} ^{3}y}{\operatorname {d} t^{3}}},\ldots ,

{\frac {\operatorname {d} z}{\operatorname {d} t}},\quad {\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}},\quad {\frac {\operatorname {d} ^{3}z}{\operatorname {d} t^{3}}},\ldots

que nous représenterons respectivement par

x',x'',x''',\ldots ,y',y'',y''',\ldots ,z',z'',z''',\ldots ,

Supposant ensuite que $x,y,z$ deviennent respectivement

x+iX,\qquad y+iY,\qquad z+iZ,

où $X,Y,Z$ sont des fonctions arbitraires de $t,$ et $i$ un nombre abstrait d’une petitesse illimitée, $\int V\operatorname {d} t$ deviendra alors

\int V\operatorname {d} t+{\frac {i}{1}}\int \left\{{\begin{aligned}&\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X+\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''+\ldots \\\\+&\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y+\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''+\ldots \\\\+&\left({\frac {\operatorname {d} V}{\operatorname {d} z}}\right)Z+\left({\frac {\operatorname {d} V}{\operatorname {d} z'}}\right)Z'+\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)Z''+\ldots \end{aligned}}\right\}\operatorname {d} t+\ldots (11)

ce qui donnera, pour la condition commune au maximum et au minimum,

0=\left\{{\begin{aligned}&\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X+\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''+\ldots \\\\+&\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y+\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''+\ldots \\\\+&\left({\frac {\operatorname {d} V}{\operatorname {d} z}}\right)Z+\left({\frac {\operatorname {d} V}{\operatorname {d} z'}}\right)Z'+\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)Z''+\ldots \end{aligned}}\right\}.\quad (12)