Page:Joseph Louis de Lagrange - Œuvres, Tome 4.djvu/332

21. Prenons l’Exemple du n^o 8, où l’on a trouvé

$\xi =a,\quad \xi '=mx,\quad \xi ''=-(m-1){\frac {x}{2}},\quad \xi '''={\frac {m+1}{3}}{\frac {x}{2}},$

$\xi ^{\scriptscriptstyle {\text{IV}}}=-{\frac {m-2}{3}}{\frac {x}{2}},\ldots \,;$

on trouvera les formules suivantes, où je suppose, pour plus de simplicité, $a=1,$

${\begin{alignedat}{2}&\mathrm {P} &=&0,\\&\mathrm {P} '&=&1,\\&\mathrm {P} ''&=&1,\\&\mathrm {P} '''&=&1-{\frac {(m-1)}{2}}x,\\&\mathrm {P} ^{\scriptscriptstyle {\text{IV}}}&=&1-{\frac {(m-2)}{3}}x,\\&\mathrm {P} ^{\scriptscriptstyle {\text{V}}}&=&1-{\frac {2(m-2)}{4}}x+{\frac {(m-2)(m-1)}{4.3}}x^{2},\\&\mathrm {P} ^{\scriptscriptstyle {\text{VI}}}&=&1-{\frac {2(m-3)}{5}}x+{\frac {(m-3)(m-2)}{5.4}}x^{2},\\&\mathrm {P} ^{\scriptscriptstyle {\text{VII}}}&=&1-{\frac {3(m-3)}{6}}x+{\frac {3(m-3)(m-2)}{6.5}}x^{2}-{\frac {(m-3)(m-2)(m-1)}{6.5.4}}x^{3},\\&\mathrm {P} ^{\scriptscriptstyle {\text{VIII}}}&=&1-{\frac {3(m-4)}{7}}x+{\frac {3(m-4)(m-3)}{7.6}}x^{2}-{\frac {(m-4)(m-3)(m-2)}{7.6.5}}x^{3},\\&\mathrm {P} ^{\scriptscriptstyle {\text{IX}}}&=&1-{\frac {4(m-4)}{8}}x+{\frac {6(m-4)(m-3)}{8.7}}x^{2}-{\frac {4(m-4)(m-3)(m-2)}{8.7.6}}x^{3},\\&&&+{\frac {(m-4)(m-3)(m-2)(m-1)}{8.7.6.5}}x^{4},\\&\ldots &&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\\\\&\mathrm {Q} &=&1,\\&\mathrm {Q} '&=&1,\\&\mathrm {Q} ''&=&1+mx,\\&\mathrm {Q} '''&=&1+{\frac {(m+1)}{2}}x,\\&\mathrm {Q} ^{\scriptscriptstyle {\text{IV}}}&=&1+{\frac {2(m+1)}{3}}x+{\frac {(m+1)m}{3.2}}x^{2},\end{alignedat}}$