Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/112

106

CALCUL

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

Il n’y a plus, dans cette formule, dont la loi est très-élégante, que des fonctions de quantités polynômiales à développer ; et elles se succèdent par colonnes dans l’ordre suivant :

$(68)\left\{{\begin{aligned}&a_{1}^{n}\qquad ,\operatorname {D} .a_{1}^{n-1},\qquad {\tfrac {1}{2}}\operatorname {D} ^{2}.a_{1}^{n-2}\quad ,\ldots {\tfrac {\operatorname {D} ^{n-2}.a_{1}^{2}}{1.2\ldots (n-2)}},\ldots {\tfrac {\operatorname {D} ^{n-1}.a_{1}}{1.2\ldots (n-1)}},\\&a_{1}^{n-1}b_{1},\operatorname {D} .\left(a_{1}^{n-2}b_{1}\right),{\tfrac {1}{2}}\operatorname {D} ^{2}.\left(a_{1}^{n-3}b_{1}\right),\ldots {\tfrac {\operatorname {D} ^{n-2}\left(a_{1}b_{1}\right)}{1.2\ldots (n-2)}},\ldots {\tfrac {\operatorname {D} ^{n-1}.b_{1}}{1.2\ldots (n-1)}},\\&a_{1}^{n-2}b_{1}^{2},\operatorname {D} .\left(a_{1}^{n-3}b_{1}^{2}\right),{\tfrac {1}{2}}\operatorname {D} ^{2}.\left(a_{1}^{n-4}b_{1}^{2}\right),\ldots {\tfrac {\operatorname {D} ^{n-2}.b_{1}}{1.2\ldots (n-2)}}\\&a_{1}^{n-2}b_{1}^{3},\operatorname {D} .\left(a_{1}^{n-4}b_{1}^{3}\right),{\tfrac {1}{2}}\operatorname {D} ^{2}.\left(a_{1}^{n-5}b_{1}^{3}\right),\ldots \\&\ldots \ldots \ldots \ldots \ldots \\&a_{1}^{2}b_{1}^{n-2},\operatorname {D} .\left(a_{1}b_{1}^{n-2}\right),{\tfrac {1}{2}}\operatorname {D} ^{2}.b_{1}^{n-2},\\&a_{1}b_{1}^{n-1},\operatorname {D} .b_{1}^{n-1}\\&b_{1}^{n},\\\end{aligned}}\right.$