Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/91

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SECONDE PARTIE. — SECTION VII.

aura

${\frac {\partial \Omega }{\partial a}}dt={\frac {\partial \mathrm {x} }{\partial a}}d\mathrm {x} '+{\frac {\partial \mathrm {y} }{\partial a}}d\mathrm {y} '+{\frac {\partial \mathrm {z} }{\partial a}}d\mathrm {z} '-{\frac {\partial \mathrm {x} '}{\partial a}}d\mathrm {x} -{\frac {\partial \mathrm {y} '}{\partial a}}d\mathrm {y} -{\frac {\partial \mathrm {z} '}{\partial a}}d\mathrm {z} .$

Or $\mathrm {x,\,y,\,z,\,x} ',\ldots$ étant fonctions de $a,\,b,\,c,\ldots ,$ on a

${\begin{aligned}\partial \mathrm {x} \ =&{\frac {\partial \mathrm {x} }{\partial a}}\ da+{\frac {\partial \mathrm {x} }{\partial b}}\ db+{\frac {\partial \mathrm {x} }{\partial c}}\ dc+\ldots ,\\\partial \mathrm {x} '=&{\frac {\partial \mathrm {x} '}{\partial a}}da+{\frac {\partial \mathrm {x} '}{\partial b}}db+{\frac {\partial \mathrm {x} '}{\partial c}}dc+\ldots ,\\\ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Substituant ces valeurs et ordonnant les termes par rapport aux variations $da,\,db,\,dc,\ldots ,$ on aura

${\frac {\partial \Omega }{\partial a}}dt=[a,b]db+[a,c]dc+[a,h]dh+\ldots ,$

où les symboles $[a,b],[a,c],\ldots$ sont exprimés par ces formules

${\begin{aligned}&[a,b]={\frac {\partial \mathrm {x} }{\partial a}}{\frac {\partial \mathrm {x} '}{\partial b}}+{\frac {\partial \mathrm {y} }{\partial a}}{\frac {\partial \mathrm {y} '}{\partial b}}+{\frac {\partial \mathrm {z} }{\partial a}}{\frac {\partial \mathrm {z} '}{\partial b}}-{\frac {\partial \mathrm {x} '}{\partial a}}{\frac {\partial \mathrm {x} }{\partial b}}-{\frac {\partial \mathrm {y} '}{\partial a}}{\frac {\partial \mathrm {y} }{\partial b}}-{\frac {\partial \mathrm {z} '}{\partial a}}{\frac {\partial \mathrm {z} }{\partial b}},\\&[a,c]={\frac {\partial \mathrm {x} }{\partial a}}{\frac {\partial \mathrm {x} '}{\partial c}}+{\frac {\partial \mathrm {y} }{\partial a}}{\frac {\partial \mathrm {y} '}{\partial c}}+{\frac {\partial \mathrm {z} }{\partial a}}{\frac {\partial \mathrm {z} '}{\partial c}}-{\frac {\partial \mathrm {x} '}{\partial a}}{\frac {\partial \mathrm {x} }{\partial c}}-{\frac {\partial \mathrm {y} '}{\partial a}}{\frac {\partial \mathrm {y} }{\partial c}}-{\frac {\partial \mathrm {z} '}{\partial a}}{\frac {\partial \mathrm {z} }{\partial c}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

On aura de même, à cause de $[b,a]=-[a,b]$

${\frac {\partial \Omega }{\partial b}}dt=-[a,b]da+[b,c]dc+[b,h]dh+\ldots ,$

$[b,c]={\frac {\partial \mathrm {x} }{\partial b}}{\frac {\partial \mathrm {x} '}{\partial c}}+{\frac {\partial \mathrm {y} }{\partial b}}{\frac {\partial \mathrm {y} '}{\partial c}}+{\frac {\partial \mathrm {z} }{\partial b}}{\frac {\partial \mathrm {z} '}{\partial c}}-{\frac {\partial \mathrm {x} '}{\partial b}}{\frac {\partial \mathrm {x} }{\partial c}}-{\frac {\partial \mathrm {y} '}{\partial b}}{\frac {\partial \mathrm {y} }{\partial c}}-{\frac {\partial \mathrm {z} '}{\partial b}}{\frac {\partial \mathrm {z} }{\partial c}},$

et ainsi de suite, en changeant simplement les quantités $a,\,b,\,c,\,h,\,i,\,k$ entre elles, prises deux à deux, et en observant que l’on a, en général,

$[b,a]=-[a,b],$

de sorte que la valeur des symboles ne fait que changer de signe par la permutation des deux quantités qu’ils contiennent.

Si l’on compare les valeurs de ces symboles marqués par des crochets carrés avec celles des symboles analogues marqués par des cro-