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PARALLÉLISME DES LIGNES
![{\displaystyle {\frac {\operatorname {d} \Sigma }{\operatorname {d} t}}={\frac {\operatorname {d} \Sigma }{\operatorname {d} x}}.{\frac {\operatorname {d} x}{\operatorname {d} u}}+{\frac {\operatorname {d} \Sigma }{\operatorname {d} y}}.{\frac {\operatorname {d} y}{\operatorname {d} u}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7826e12fc7e5183a16fcd747e5303fd572687b7f)
d’où
![{\displaystyle {\frac {\operatorname {d} .{\frac {\operatorname {d} \Sigma }{\operatorname {d} t}}}{\operatorname {d} x}}={\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} x^{2}}}.{\frac {\operatorname {d} x}{\operatorname {d} u}}+{\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} x\operatorname {d} y}}.{\frac {\operatorname {d} y}{\operatorname {d} u}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e454dd70f1c1b90d9d0fa87ee92f8e1f05d86d)
![{\displaystyle {\frac {\operatorname {d} .{\frac {\operatorname {d} \Sigma }{\operatorname {d} t}}}{\operatorname {d} y}}={\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} x\operatorname {d} y}}.{\frac {\operatorname {d} x}{\operatorname {d} u}}+{\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} y^{2}}}.{\frac {\operatorname {d} y}{\operatorname {d} u}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9b8497734fa2eee41dcc662b023f7ae71f55f0)
ce qui donnera, en substituant
![{\displaystyle {\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} t\operatorname {d} u}}={\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}={\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} x^{2}}}\left({\frac {\operatorname {d} x}{\operatorname {d} u}}\right)^{2}+{\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} y^{2}}}\left({\frac {\operatorname {d} y}{\operatorname {d} u}}\right)^{2}+2{\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} x\operatorname {d} y}}.{\frac {\operatorname {d} x}{\operatorname {d} u}}.{\frac {\operatorname {d} y}{\operatorname {d} u}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058014420e63b9584513c6605b516bcb789544ab)
Cette équation différentielle contient implicitement l’expression cherchée de
en
On pourra la résoudre dans des cas particuliers.
13.
Ayant trouvé les expressions des élémens
![{\displaystyle {\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}},\qquad {\frac {\operatorname {d} ^{2}\Sigma }{\operatorname {d} t\operatorname {d} u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a89126b2a7eadcf6e8f371a5235640b3850b43f)
en
on aura aussi celle de l’élément du volume compris entre les deux surfaces parallèles ; car cet élément peut être considéré comme une pyramide tronquée, dont nos deux surfaces élémentaires sont les deux bases parallèles et dont la hauteur est
Il serait intéressant d’appliquer ces formules aux surfaces développables.