Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/391

373

SECONDE PARTIE. — SECTION VI.

$[1]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{1}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{1}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{1}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&2a_{1}b_{1}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+2a_{1}c_{1}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}+2b_{1}c_{1}{\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

$[2]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{2}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{2}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{2}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&2a_{2}b_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+2a_{2}c_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}+2b_{2}c_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

$[3]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{3}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{3}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{3}^{2}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&2a_{3}b_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+2a_{3}c_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}+2b_{3}c_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

;

$[1,2]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{1}a_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{1}b_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{1}c_{2}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&(a_{1}b_{2}+a_{2}b_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+(a_{1}c_{2}+a_{2}c_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}\\+&(b_{1}c_{2}+b_{2}c_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

$[1,3]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{1}a_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{1}b_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{1}c_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&(a_{1}b_{3}+a_{3}b_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+(a_{1}c_{3}+a_{3}c_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}\\+&(b_{1}c_{3}+b_{3}c_{1}){\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

$[2,3]=$_S$\mathrm {m} \left\{{\begin{aligned}&a_{2}a_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial a^{2}}}+b_{2}b_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial b^{2}}}+c_{2}c_{3}{\frac {\partial ^{2}\operatorname {F} }{\partial c^{2}}}\\+&(a_{2}b_{3}+a_{3}b_{2}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial b}}+(a_{2}c_{3}+a_{3}c_{2}){\frac {\partial ^{2}\operatorname {F} }{\partial a\partial c}}\\+&(b_{2}c_{3}+b_{3}c_{2}){\frac {\partial ^{2}\operatorname {F} }{\partial b\partial c}}\end{aligned}}\right\},$

 

3. Ayant ainsi les valeurs de $\mathrm {T}$ et $\mathrm {V}$ exprimées en fonctions des variables $\xi ,\,\psi ,\,\varphi ,\ldots ,$ indépendantes entre elles, on n’aura plus aucune équation de condition à employer ; et comme la quantité $\mathrm {T}$ ne contient