83
SECONDE PARTIE. — SECTION VII.
aura
![{\displaystyle {\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} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9259edaf482d04ca5c9ef29a1904be29073b195b)
Or
étant fonctions de
on a
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ded04529a355846229fc1a03365a21a19fd60b39)
Substituant ces valeurs et ordonnant les termes par rapport aux variations
on aura
![{\displaystyle {\frac {\partial \Omega }{\partial a}}dt=[a,b]db+[a,c]dc+[a,h]dh+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa468f6218dfabc1f8bc2d0a7ced676cb3dc777)
où les symboles
sont exprimés par ces formules
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72eb27ec73257dcdb823ede84b509782d0a9db8d)
On aura de même, à cause de ![{\displaystyle [b,a]=-[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c9de8071f3f8548c0cb99798395b537bb9b776)
![{\displaystyle {\frac {\partial \Omega }{\partial b}}dt=-[a,b]da+[b,c]dc+[b,h]dh+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bacaec695fbe67805d60c9ba2bdd998f1c5fb5bf)
![{\displaystyle [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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00c5b1767e3e7261ef21bc4565310c9f24c04e3)
et ainsi de suite, en changeant simplement les quantités
entre elles, prises deux à deux, et en observant que l’on a, en général,
![{\displaystyle [b,a]=-[a,b],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0661ea138bf4a57fa27596471e41785ff5d69c2e)
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-