Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/125

107

PREMIÈRE PARTIE. — SECTION IV.

et qu’on observe que

${\begin{alignedat}{4}{\frac {\partial z_{_{'}}}{\partial x}}=&{\frac {\partial z'}{\partial y}},&{\frac {\partial z'}{\partial y}}=&{\frac {\partial z_{_{'}}}{\partial x}},&{\frac {\partial ^{2}z'}{\partial x^{2}}}=&{\frac {\partial z''}{\partial x}},&{\frac {\partial ^{2}z_{_{'}}}{\partial x^{2}}}=&{\frac {\partial z''}{\partial y}},\\{\frac {\partial ^{2}z'}{\partial x\partial y}}=&{\frac {\partial z'_{_{'}}}{\partial x}},\qquad &{\frac {\partial ^{2}z_{_{'}}}{\partial x\partial y}}=&{\frac {\partial z'_{_{'}}}{\partial y}},\qquad &{\frac {\partial ^{2}z'}{\partial y^{2}}}=&{\frac {\partial z_{_{''}}}{\partial x}},\qquad &\ldots ,\quad &\\\end{alignedat}}$

on aura plus simplement

${\begin{alignedat}{4}&\delta z'\ &=&{\frac {\partial \delta u}{\partial x}}&+&{\frac {\partial z'}{\partial x}}\delta x&+&{\frac {\partial z'}{\partial y}}\delta y,\\&\delta z_{_{'}}\ &=&{\frac {\partial \delta u}{\partial y}}&+&{\frac {\partial z'}{\partial x}}\delta x&+&{\frac {\partial z_{_{'}}}{\partial y}}\delta y,\\&\delta z''&=&{\frac {\partial ^{2}\delta u}{\partial x^{2}}}&+&{\frac {\partial z''}{\partial x}}\delta x&+&{\frac {\partial z''}{\partial y}}\delta y,\\&\delta z'_{_{'}}\ &=&{\frac {\partial ^{2}\delta u}{\partial x\partial y}}&+&{\frac {\partial z'_{_{'}}}{\partial x}}\delta x&+&{\frac {\partial z'_{_{'}}}{\partial y}}\delta y,\\&\delta z_{_{''}}&=&{\frac {\partial ^{2}\delta u}{\partial y^{2}}}&+&{\frac {\partial z_{_{''}}}{\partial x}}\delta x&+&{\frac {\partial z_{_{''}}}{\partial y}}\delta y,\\\end{alignedat}}$

 

Faisant ces substitutions dans l’expression de $\delta \mathrm {U} ,$ mettant

$\delta u+{\frac {\partial z}{\partial x}}\delta x+{\frac {\partial z}{\partial y}}\delta y$

à la place de $\delta z$ et ordonnant les termes par rapport à $\delta x,\delta y,\delta u,$ on aura

${\begin{aligned}\delta \mathrm {U} &=\left({\frac {\partial \mathrm {U} }{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z}}{\frac {\partial z}{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z'}}{\frac {\partial z'}{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z_{_{'}}}}{\frac {\partial z_{_{'}}}{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z''}}{\frac {\partial z''}{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z'_{_{'}}}}{\frac {\partial z'_{_{'}}}{\partial x}}+\ldots \right)\delta x\\&+\,\left({\frac {\partial \mathrm {U} }{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z}}{\frac {\partial z}{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z'}}{\frac {\partial z'}{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z_{_{'}}}}{\frac {\partial z_{_{'}}}{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z''}}{\frac {\partial z''}{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z'_{_{'}}}}{\frac {\partial z'_{_{'}}}{\partial y}}+\ldots \right)\delta y\\&+{\frac {\partial \mathrm {U} }{\partial z}}\delta u+{\frac {\partial \mathrm {U} }{\partial z'}}{\frac {\partial \delta u}{\partial x}}+{\frac {\partial \mathrm {U} }{\partial z_{_{'}}}}{\frac {\partial \delta u}{\partial y}}+{\frac {\partial \mathrm {U} }{\partial z''}}{\frac {\partial ^{2}\delta u}{\partial x^{2}}}+{\frac {\partial \mathrm {U} }{\partial z'_{_{'}}}}{\frac {\partial ^{2}\delta u}{\partial x\partial y}}+\ldots .\end{aligned}}$

Désignons par $\left({\frac {\partial \mathrm {U} }{\partial x}}\right),\left({\frac {\partial \mathrm {U} }{\partial y}}\right)$ les différences partielles de $\mathrm {U} ,$ relatives à $x$ et $y,$ en regardant $z$ comme fonction de ces deux variables ; il est