Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/219

${\begin{array}{rr}0=M-{\frac {\operatorname {d} M}{\operatorname {d} a}}.{\frac {\alpha }{1}}+{\frac {\operatorname {d} ^{2}M}{\operatorname {d} a^{2}}}.{\frac {\alpha ^{2}}{1.2}}-\ldots &0=N-{\frac {\operatorname {d} N}{\operatorname {d} a}}.{\frac {\alpha }{1}}+{\frac {\operatorname {d} ^{2}N}{\operatorname {d} a^{2}}}.{\frac {\alpha ^{2}}{1.2}}-\ldots \\\\-{\frac {\operatorname {d} M}{\operatorname {d} b}}.{\frac {\beta }{1}}+2{\frac {\operatorname {d} ^{2}M}{\operatorname {d} a\operatorname {d} b}}.{\frac {\alpha \beta }{1.2}}-\ldots &-{\frac {\operatorname {d} N}{\operatorname {d} b}}.{\frac {\beta }{1}}+2{\frac {\operatorname {d} ^{2}N}{\operatorname {d} a\operatorname {d} b}}.{\frac {\alpha \beta }{1.2}}-\ldots \\\\+{\frac {\operatorname {d} ^{2}M}{\operatorname {d} b^{2}}}.{\frac {\beta ^{2}}{1.2}}-\ldots &+{\frac {\operatorname {d} ^{2}N}{\operatorname {d} b^{2}}}.{\frac {\beta ^{2}}{1.2}}-\ldots \end{array}}$

mais puisque, par l’hypothèse, $\alpha$ et $\beta$ sont de forts petits nombres, nous pourrons, sans beaucoup d’erreur, écrire simplement

${\frac {\operatorname {d} M}{\operatorname {d} a}}\alpha +{\frac {\operatorname {d} M}{\operatorname {d} b}}\beta =M,\qquad {\frac {\operatorname {d} N}{\operatorname {d} a}}\alpha +{\frac {\operatorname {d} N}{\operatorname {d} b}}\beta =N,$

d’où nous conclurons

${\begin{aligned}&\alpha ={\frac {M{\frac {\operatorname {d} N}{\operatorname {d} b}}-N{\frac {\operatorname {d} M}{\operatorname {d} a}}}{{\frac {\operatorname {d} M}{\operatorname {d} a}}{\frac {\operatorname {d} N}{\operatorname {d} b}}-{\frac {\operatorname {d} N}{\operatorname {d} a}}{\frac {\operatorname {d} M}{\operatorname {d} b}}}},\\\\&\beta ={\frac {N{\frac {\operatorname {d} M}{\operatorname {d} b}}-M{\frac {\operatorname {d} N}{\operatorname {d} a}}}{{\frac {\operatorname {d} M}{\operatorname {d} a}}{\frac {\operatorname {d} N}{\operatorname {d} b}}-{\frac {\operatorname {d} N}{\operatorname {d} a}}{\frac {\operatorname {d} M}{\operatorname {d} b}}}}\end{aligned}}$

ces valeurs ne seront qu’approchées ; mais il n’en résultera pas moins, pour $x$ et $y$, les valeurs

$a-{\frac {M{\frac {\operatorname {d} N}{\operatorname {d} b}}-N{\frac {\operatorname {d} M}{\operatorname {d} a}}}{{\frac {\operatorname {d} M}{\operatorname {d} a}}{\frac {\operatorname {d} N}{\operatorname {d} b}}-{\frac {\operatorname {d} N}{\operatorname {d} a}}{\frac {\operatorname {d} M}{\operatorname {d} b}}}},\qquad b-{\frac {N{\frac {\operatorname {d} M}{\operatorname {d} b}}-M{\frac {\operatorname {d} N}{\operatorname {d} a}}}{{\frac {\operatorname {d} M}{\operatorname {d} a}}{\frac {\operatorname {d} N}{\operatorname {d} b}}-{\frac {\operatorname {d} N}{\operatorname {d} a}}{\frac {\operatorname {d} M}{\operatorname {d} b}}}},$

généralement plus approchées que ne l’étaient les premières $a$ et