Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/82

76

CALCUL

 Les corrections sont expliquées en page de discussion

où chaque dérivée de $\phi a$ et de $\psi b$ doit être développée comme les équations (10) ; c’est-à-dire, d’après la règle du n.^o 8. Les formules précédentes contiennent donc, au fond, tout ce qu’il faut pour la solution complète de la question ; mais, pour ne rien laisser à désirer, nous allons en déduire les moyens d’exécuter immédiatement le développement complet de ces formules.

14. En exécutant les dérivations indiquées, au moyen de la règle 8, effectuant les multiplications, et ordonnant le tout par rapport aux exposans des dérivées, on obtient

$(\mathrm {18} )\left\{{\begin{aligned}&A=\phi a.\psi b\\&A_{1}=\phi a.\operatorname {D} \psi b.b_{1}\\&\quad +\operatorname {D} \phi a.\psi b.a_{1}\\&A_{2}=\phi a.\operatorname {D} \psi b.b_{2}+\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.b_{1}^{2}\\&\quad +\operatorname {D} \phi a.\psi b.a_{2}+\operatorname {D} \phi a.\operatorname {D} \psi b.a_{1}b_{1}\\&\quad \qquad \qquad \qquad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\psi b.a_{1}^{2}\\&A_{3}=\phi a.\operatorname {D} \psi .b_{3}+\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}.\psi b.2b_{1}b_{2}+\phi a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b.b_{1}^{3}\\&\quad +\operatorname {D} \phi a.\psi b.a_{3}+\operatorname {D} \phi a.\operatorname {D} \psi b\left(a_{1}b_{2}+a_{2}b_{1}\right)\operatorname {D} \phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.a_{1}b_{2}^{2}\\&\quad \qquad \qquad \qquad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\psi b.2a_{1}a_{2}+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\operatorname {D} \psi b.a_{1}^{2}b_{1}\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\psi b.a_{1}^{3}\\&A_{4}=\phi a.\operatorname {D} \psi .b_{4}+\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}.\psi b\left(2b_{1}b_{3}+b_{2}^{2}\right)+\phi a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b.3b_{1}^{2}b_{2}\\&\quad +\operatorname {D} \phi a.\psi b.a_{4}+\operatorname {D} \phi a.\operatorname {D} \psi b\left(a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1}\right)\\&\quad \qquad \qquad \qquad \qquad \qquad +\operatorname {D} \phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.\left(2a_{1}b_{1}b_{2}+a_{2}b_{1}^{2}\right)\\&\quad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\psi b\left(2a_{1}a_{3}+a_{2}^{2}\right)+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\operatorname {D} \psi b.\left(a_{1}^{2}b_{2}+2a_{1}a_{2}b_{1}\right)\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\psi b.3a_{1}^{2}a_{2}\\&\quad +\phi a.{\tfrac {1}{24}}\operatorname {D} ^{4}\psi b.b_{1}^{4}\\&\quad +\operatorname {D} \phi a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b.a_{1}b_{1}^{3}\\&\quad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b.a_{1}^{2}b_{1}^{2}\\&\quad +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\operatorname {D} \psi b.a_{1}^{3}b_{1}\\&\quad +{\tfrac {1}{24}}\operatorname {D} ^{4}\phi a.\psi b.a_{1}^{4}\\&A_{5}=\phi a.\operatorname {D} \psi .b_{5}+\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}.\psi b\left(2b_{1}b_{4}+2b_{2}b_{3}\right)\\&\quad \qquad \qquad \qquad +\phi a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b\left(3b_{1}^{2}b_{3}+3b_{1}b_{2}^{2}\right)\\&\quad +\operatorname {D} \phi a.\psi b.a_{5}+\operatorname {D} \phi a.\operatorname {D} \psi b\left(a_{1}b_{4}+a_{2}b_{3}+a_{3}b_{2}+a_{4}b_{1}\right)\\&\quad \qquad \qquad +\operatorname {D} \phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b\left(2a^{2}b_{1}b_{3}+a_{1}b_{1}+2a_{2}b_{1}b_{2}+a_{3}b_{1}\right)\\&\quad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\psi b\left(2a_{1}a_{4}+2a_{2}a_{3}\right)\\&\quad \qquad +{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.\operatorname {D} \psi b.\left(a_{1}^{2}b_{3}+2a_{1}a_{2}b_{2}+2a_{1}a_{3}b_{1}+a_{2}^{2}b_{1}\right)\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\psi b\left(3a_{1}^{2}a_{3}+3a_{1}a_{2}^{2}\right)\\&+\phi a.{\tfrac {1}{24}}\operatorname {D} ^{4}\psi b.4b_{1}^{3}b_{2}\ \qquad \qquad \qquad \qquad +\phi a.{\tfrac {1}{120}}\operatorname {D} ^{5}\psi b.b^{5}\\&+\operatorname {D} \phi a.{\tfrac {1}{6}}\operatorname {D} ^{4}\psi b\left(3a_{1}b_{1}^{2}b_{2}+a_{2}b_{1}^{3}\right)\,\qquad \ \ +\operatorname {D} \phi a.{\tfrac {1}{24}}\operatorname {D} ^{4}\psi b.a_{1}b_{1}^{4}\\&+{\tfrac {1}{2}}\operatorname {D} ^{2}\phi a.{\tfrac {1}{2}}\operatorname {D} ^{2}\psi b\left(2a_{1}^{2}b_{1}b_{2}+2a_{1}a_{2}b_{1}^{2}\right)+{\tfrac {1}{2}}\operatorname {D} ^{2}a.{\tfrac {1}{6}}\operatorname {D} ^{3}\psi b.a_{1}^{2}b_{1}^{3}\\&+{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.\operatorname {D} \psi b\left(a_{1}^{3}b_{2}+3a_{1}^{2}a_{2}b_{1}\right)\qquad \ \ +{\tfrac {1}{6}}\operatorname {D} ^{3}\phi a.{\tfrac {1}{6}}\operatorname {D} ^{2}\psi b.a_{1}^{3}b_{1}^{2}\\&+{\tfrac {1}{24}}\operatorname {D} ^{4}\phi a.\psi b.4a_{1}^{3}a_{2}\qquad \qquad \qquad \qquad +{\tfrac {1}{24}}\operatorname {D} ^{4}\phi a.\operatorname {D} \psi b.a_{1}^{4}b_{1}\\&+{\tfrac {1}{120}}\operatorname {D} ^{5}\phi a.\psi b.a_{1}^{5}\end{aligned}}\right.$