50
INTÉGRALES
afin donc que
soit maximum ou minimum, il faudra, suivant les principes connus, que le multiplicateur de
soit nul ; et alors le maximum ou le minimum aura lieu, suivant que le multiplicateur de
. sera constamment négatif ou constamment positif. La condition commune au maximum et au minimum sera donc exprimée par l’équation
![{\displaystyle \iint \left\{{\begin{array}{r}KZ+L{\frac {\operatorname {d} Z}{\operatorname {d} x}}+N\ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+Q\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{3}}}+\ldots \\\\+M{\frac {\operatorname {d} Z}{\operatorname {d} y}}+O{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x\operatorname {d} y}}+R{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\+P\ \ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+S{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\+T\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}}+\ldots \\\\+\ldots \end{array}}\right\}\operatorname {d} x\operatorname {d} y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a3a7be22950cd94c34b82ae4b477c4cdeb51ed)
ou plus simplement, en différentiant,
![{\displaystyle \left.{\begin{array}{r}0=KZ+L{\frac {\operatorname {d} Z}{\operatorname {d} x}}+N\ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+Q\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{3}}}+\ldots \\\\+M{\frac {\operatorname {d} Z}{\operatorname {d} y}}+O{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x\operatorname {d} y}}+R{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\+P\ \ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+S{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\+T\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}}+\ldots \\\\+\ldots \end{array}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24a38a38ae9935f575a3d52a579eb6fbd9c27aa0)
(XXV)
45. Cela posé, en ayant successivement égard à la variabilité de
et à celle de
la formule
donne