Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/381

et, comme

$1+\mathrm {A} t+\mathrm {B} t^{2}+\mathrm {C} t^{3}+\ldots =\theta =\left(1-{\frac {t}{t'}}\right)\left(1-{\frac {t}{t''}}\right)\left(1-{\frac {t}{t'''}}\right)\ldots ,$

on aura, en différentiant et faisant successivement $t=t',\ t=t'',$

${\begin{aligned}\mathrm {A} +2\mathrm {B} t'\ +3\mathrm {C} t'^{2}\ +\ldots =&-{\frac {1}{t'}}\left(1-{\frac {t'}{t''}}\right)\left(1-{\frac {t'}{t'''}}\right)\left(1-{\frac {t'}{t^{\scriptscriptstyle {\text{IV}}}}}\right)\ldots ,\\\mathrm {A} +2\mathrm {B} t''+3\mathrm {C} t''^{2}+\ldots =&-{\frac {1}{t''}}\left(1-{\frac {t''}{t'}}\right)\left(1-{\frac {t''}{t'''}}\right)\left(1-{\frac {t''}{t^{\scriptscriptstyle {\text{IV}}}}}\right)\ldots ,\end{aligned}}$

où l’on voit que dans le cas de $t'=t''$ ces deux quantités seront nulles.

Supposons pour un moment que $t''=t'+\omega ,$ $\omega$ étant une quantité infiniment petite, on aura, en négligeant les infiniment petits du second ordre,

${\begin{aligned}\mathrm {A} +2\mathrm {B} t'\ +3\mathrm {C} t'^{2}\ +\ldots =&-{\frac {\omega }{t'^{2}}}\left(1-{\frac {t'}{t'''}}\right)\left(1-{\frac {t'}{t^{\scriptscriptstyle {\text{IV}}}}}\right)\ldots ,\\\mathrm {A} +2\mathrm {B} t''+3\mathrm {C} t''^{2}+\ldots =&+{\frac {\omega }{t'^{2}}}\left(1-{\frac {t'}{t'''}}\right)\left(1-{\frac {t'}{t^{\scriptscriptstyle {\text{IV}}}}}\right)\ldots .\end{aligned}}$

Donc faisant, pour abréger,

$\Pi ={\frac {1}{t'^{2}}}\left(1-{\frac {t'}{t'''}}\right)\left(1-{\frac {t'}{t^{\scriptscriptstyle {\text{IV}}}}}\right)\ldots ,$

on aura

$y'={\frac {{\dfrac {\mathrm {P} }{t'}}+{\dfrac {\mathrm {Q} }{t'^{2}}}+{\dfrac {\mathrm {R} }{t'^{3}}}+\ldots }{\omega \Pi }},\quad y''={\frac {{\dfrac {\mathrm {P} }{t''}}+{\dfrac {\mathrm {Q} }{t''^{2}}}+{\dfrac {\mathrm {R} }{t''^{3}}}+\ldots }{\omega \Pi }}\,;$

mais

${\begin{aligned}{\frac {\mathrm {P} }{t''}}+{\frac {\mathrm {Q} }{t''^{2}}}+{\frac {\mathrm {R} }{t''^{3}}}+\ldots =&{\frac {\mathrm {P} }{t'}}+{\frac {\mathrm {Q} }{t'^{2}}}+{\frac {\mathrm {R} }{t'^{3}}}+\ldots +\omega {\frac {d\left({\dfrac {\mathrm {P} }{t'}}+{\dfrac {\mathrm {Q} }{t'^{2}}}+{\dfrac {\mathrm {R} }{t'^{3}}}+\ldots \right)}{dt'}}\\=&{\frac {\mathrm {P} }{t'}}+{\frac {\mathrm {Q} }{t'^{2}}}+{\frac {\mathrm {R} }{t'^{3}}}+\ldots -\omega \left({\frac {\mathrm {P} }{t'^{2}}}+{\frac {2\mathrm {Q} }{t'^{3}}}+{\frac {3\mathrm {R} }{t'^{4}}}+\ldots \right)\,;\end{aligned}}$

donc on aura

$y''=-y'+{\frac {{\dfrac {\mathrm {P} }{t'^{2}}}+{\dfrac {2\mathrm {Q} }{t'^{3}}}+{\dfrac {3\mathrm {R} }{t'^{4}}}+\ldots }{\Pi }},$

et de là

$y'+y''={\frac {{\dfrac {\mathrm {P} }{t'^{2}}}+{\dfrac {2\mathrm {Q} }{t'^{3}}}+{\dfrac {3\mathrm {R} }{t'^{4}}}+\ldots }{\Pi }}.$