Page:Lagrange - Œuvres (1867) vol. 1.djvu/263

Soit supposé

${\frac {d\varphi (x)}{dx}}=\varphi '(x),\qquad {\frac {d\varphi '(x)}{dx}}=\varphi ''(x),\ldots ,$

$\int \varphi (x)dx={\grave {\,}}\!\varphi (x),\qquad \int {\grave {\,}}\!\varphi (x)dx={\grave {\,}}\!{\grave {\,}}\!\varphi (x),\ldots ,$

et ainsi pour la fonction $\psi ,$ on aura

${\begin{aligned}\int x\varphi \left(x\pm t{\sqrt {c}}\right)dx=&x\int \varphi \left(x\pm t{\sqrt {c}}\right)dx-\int dx\int \varphi \left(x\pm t{\sqrt {c}}\right)dx\\=&x{\grave {\,}}\!\varphi \left(x\pm t{\sqrt {c}}\right)-{\grave {\,}}\!{\grave {\,}}\!\varphi \left(x\pm t{\sqrt {c}}\right)\,;\end{aligned}}$

traitant de la même manière les autres formules intégrales qui composent les valeurs de $\int \mathrm {P} xdx$ et de $\int \mathrm {Q} xdx,$ on aura, après toutes les substitutions,

${\begin{aligned}z&={\frac {{\grave {\,}}\!\varphi \left(x+t{\sqrt {c}}\right)}{2x}}-{\frac {{\grave {\,}}\!{\grave {\,}}\!\varphi \left(x+t{\sqrt {c}}\right)}{2x^{2}}}\\&+{\frac {{\grave {\,}}\!{\grave {\,}}\!\psi \left(x+t{\sqrt {c}}\right)}{2x{\sqrt {c}}}}-{\frac {{\grave {\,}}\!{\grave {\,}}\!{\grave {\,}}\!\psi \left(x+t{\sqrt {c}}\right)}{2x^{2}{\sqrt {c}}}}\\&+{\frac {{\grave {\,}}\!\varphi \left(x-t{\sqrt {c}}\right)}{2x}}-{\frac {{\grave {\,}}\!{\grave {\,}}\!\varphi \left(x-t{\sqrt {c}}\right)}{2x^{2}}}\\&-{\frac {{\grave {\,}}\!{\grave {\,}}\!\psi \left(x-t{\sqrt {c}}\right)}{2x{\sqrt {c}}}}+{\frac {{\grave {\,}}\!{\grave {\,}}\!{\grave {\,}}\!\psi \left(x-t{\sqrt {c}}\right)}{2x^{2}{\sqrt {c}}}}\\u&={\frac {{\grave {\,}}\!\psi \left(x+t{\sqrt {c}}\right)}{2x}}-{\frac {{\grave {\,}}\!{\grave {\,}}\!\psi \left(x+t{\sqrt {c}}\right)}{2x^{2}}}\\&+{\sqrt {c}}{\frac {\varphi \left(x+t{\sqrt {c}}\right)}{2x}}-{\sqrt {c}}{\frac {{\grave {\,}}\!\varphi \left(x+t{\sqrt {c}}\right)}{2x^{2}}}\\&+{\frac {{\grave {\,}}\!\psi \left(x-t{\sqrt {c}}\right)}{2x}}-{\frac {{\grave {\,}}\!{\grave {\,}}\!\psi \left(x-t{\sqrt {c}}\right)}{2x^{2}}}\\&-{\sqrt {c}}{\frac {\varphi \left(x-t{\sqrt {c}}\right)}{2x}}+{\sqrt {c}}{\frac {{\grave {\,}}\!\varphi \left(x-t{\sqrt {c}}\right)}{2x^{2}}}.\end{aligned}}$

21. On peut simplifier ces expressions de la manière suivante. Au lieu de ${\grave {\,}}\!{\grave {\,}}\!\varphi \left(x+t{\sqrt {c}}\right)+{\frac {{\grave {\,}}\!{\grave {\,}}\!{\grave {\,}}\!\psi \left(x+t{\sqrt {c}}\right)}{\sqrt {c}}}$ je pose simplement $\Delta \left(x+t{\sqrt {c}}\right),$ et au lieu de ${\grave {\,}}\!{\grave {\,}}\!\varphi \left(x-t{\sqrt {c}}\right)-{\frac {{\grave {\,}}\!{\grave {\,}}\!{\grave {\,}}\!\psi \left(x-t{\sqrt {c}}\right)}{\sqrt {c}}}$ je substitue de même la seule