Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/35

égaler d’abord à zéro la variation de la valeur (51) de $r$, prise à la fois par rapport à ces deux variables, ce qui donnera, en supprimant les accens qui, dans cette rencontre, ne sont d’aucune utilité,

$3\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}-2{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}+\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\right\}\times$

$\left\{{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}\delta x+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\delta x+{\frac {\operatorname {d} S}{\operatorname {d} x}}\delta y\right){\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}+{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\delta y\right\}$

$-2\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}\right\}\left\{\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\right\}\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\delta x+{\frac {\operatorname {d} S}{\operatorname {d} y}}\delta y\right)$

$=\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}\right\}\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{3}}}\delta x+{\frac {\operatorname {d} S}{\operatorname {d} y}}\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\delta y-2{\frac {\operatorname {d} S}{\operatorname {d} x}}\delta x\right){\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} y}}\right.$

$\left.+{\frac {\operatorname {d} S}{\operatorname {d} x}}\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\delta x-2{\frac {\operatorname {d} S}{\operatorname {d} y}}\delta y\right){\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}}+\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{3}}}\delta y\right\}\,;$

mais l’équation (1) donne

${\frac {\operatorname {d} S}{\operatorname {d} x}}\delta x+{\frac {\operatorname {d} S}{\operatorname {d} y}}\delta y=0\,;$

ce qui fait d’abord disparaître la seconde partie du premier membre de la prccédente ; substituant dans l’équation restante la valeur de $\delta x$ ou de $\delta y$ tirée de cette dernière, l’autre variation en disparaîtra aussi, et l’on obtiendra, pour solution du problème,

$3\left\{{\frac {\operatorname {d} S}{\operatorname {d} y}}\left({\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)-{\frac {\operatorname {d} S}{\operatorname {d} x}}\left({\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)\right\}\times$

$\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}-2{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}+\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\right\}$

$=\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}\right\}\times$

$\left\{\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{3}{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{3}}}-3{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} y}}+3\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y^{2}}}-\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{3}{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{3}}}\right\}\,;\mathrm {(62)}$