Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 7.djvu/298

grandes, alors on a, à très peu près,

$\mathrm {M} =-{\frac {\cfrac {\partial ^{2}\mathrm {Y} }{\partial x^{2}}}{2\mathrm {Y} }},\qquad 2\mathrm {N} =-{\frac {\cfrac {\partial ^{2}\mathrm {Y} }{\partial x\partial x'}}{\mathrm {Y} }},\qquad \mathrm {P} =-{\frac {\cfrac {\partial ^{2}\mathrm {Y} }{\partial x'^{2}}}{2\mathrm {Y} }},$

${\frac {\partial ^{2}\mathrm {Y} }{\partial x^{2}}},{\frac {\partial ^{2}\mathrm {Y} }{\partial x\partial x'}},{\frac {\partial ^{2}\mathrm {Y} }{\partial x'^{2}}}$ étant ce que deviennent ${\frac {\partial ^{2}y}{\partial x^{2}}},{\frac {\partial ^{2}y}{\partial x\partial x'}}$ et ${\frac {\partial ^{2}y}{\partial x'^{2}}}$ lorsqu’on y change $x$ et $x'$ en $a$ et $a'\,;$ l’intégrale $\iint ydxdx'$ devient ainsi, à fort peu près,

${\frac {2\pi \mathrm {Y} ^{2}}{\sqrt {{\cfrac {\partial ^{2}\mathrm {Y} }{\partial x^{2}}}{\cfrac {\partial ^{2}\mathrm {Y} }{\partial x'^{2}}}-\left({\cfrac {\partial ^{2}\mathrm {Y} }{\partial x\partial x'}}\right)^{2}}}}.$