Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/110

104

CALCUL

(65)$(\phi a+\operatorname {D} \phi a.a_{1}x+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.a_{1}^{2}x^{2}+\ldots )(\psi b+\operatorname {D} \psi b.b_{1}x+{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.b_{1}^{2}x^{2}+\ldots )$

Ce produit étant développé, d’après l’équation (13), donne

(66)$\qquad \phi \left(a+a_{1}x\right)\times \psi \left(b+b_{1}x\right)=$

$A\qquad +A_{1}x\qquad \qquad +A_{2}x^{2}\qquad \qquad \qquad +A_{3}x^{3}$

${\begin{aligned}{\begin{aligned}&\left.{\begin{aligned}=\phi a\psi b+\phi a.\operatorname {D} \psi b.b_{1}&\\+\operatorname {D} \phi a.\psi b.a_{1}&\\\end{aligned}}\right\}x\\\\\\\\\\\end{aligned}}{\begin{aligned}\left.{\begin{aligned}&+\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.b_{1}^{2}\\&+\operatorname {D} \phi a.\operatorname {D} \psi b.a_{1}b_{1}\\&+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\psi b.a_{1}^{3}\\\end{aligned}}\right\}\\\\\\\end{aligned}}{\begin{aligned}{\begin{aligned}&x^{2}\end{aligned}}\\\\\\\end{aligned}}\left.{\begin{aligned}&+\phi a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b.b_{1}^{3}\\&+\operatorname {D} \phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.a_{1}b_{1}^{2}\\&+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\operatorname {D} \psi b.a_{1}^{2}b_{1}\\&+{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\psi b.a_{1}^{3}\\\end{aligned}}\right\}&x^{3}\end{aligned}}$

$+\ldots +\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots +A_{n}x^{n}+\ldots$

$\left.{\begin{aligned}&+\ldots +\phi a.{\frac {\operatorname {D} ^{n}\psi b}{1.2\ldots n}}.b_{1}^{n}\\&+\ldots +\operatorname {D} \phi a.{\frac {\operatorname {D} ^{n-1}\psi b}{1.2\ldots (n-1)}}.a_{1}b_{1}^{n-1}\\&+\ldots +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.{\frac {\operatorname {D} ^{n-2}\psi b}{1.2\ldots (n-2)}}.a_{1}^{2}b_{1}^{n-2}\\&+\ldots +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.{\frac {\operatorname {D} ^{n-3}\psi b}{1.2\ldots (n-3)}}.a_{1}^{3}b_{1}^{n-3}\\&+\ldots +\ldots \ldots \ldots \ldots \ldots \ldots \\&\quad \qquad +{\frac {\operatorname {D} ^{n-1}\phi a}{1.2\ldots (n-1)}}.\operatorname {D} \psi b.a_{1}^{n-1}b_{1}\\&\quad \qquad +{\frac {\operatorname {D} ^{n}\phi a}{1.2\ldots n}}.\psi b.a_{1}^{n}\\\end{aligned}}\right\}{\begin{aligned}&+\ldots \\\\&+\ldots \\\\&+\ldots \\\\&x^{n}\\\\&+\ldots \\\\&+\ldots \\\\&+\ldots \\\end{aligned}}$

Ici les colonnes qui forment les coefficient de $x$ ne sont autre chose