10
PARALLÉLISME DES LIGNES
Cela posé, en élimment
entre les équations (1, 2), on a
![{\displaystyle u-y=\pm {\frac {k}{\sqrt {1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/230ca468c47d3e914efb216c53f0b1f6ca1c9c8b)
en conséquence de quoi la valeur de
devient
![{\displaystyle {\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}={\frac {\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}{1\mp {\frac {k{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}{\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14275b59ee815cefd9739a8037da2c0504a8c7fb)
donc, en vertu de l’équation (3),
![{\displaystyle {\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}={\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{{\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k}}={\frac {\left\{1+\left({\frac {\operatorname {d} u}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{{\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80c0271f078a84418e60dd9293cf3e40777fa435)
ou encore
![{\displaystyle {\frac {\left\{1+\left({\frac {\operatorname {d} u}{\operatorname {d} t}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}}={\frac {\left\{1+\left({\frac {\operatorname {d} y}{\operatorname {d} x}}\right)^{2}\right\}^{\frac {1}{2}}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} x^{2}}}}\mp k\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f57f2b50c06cda1f771d5f5a8606c4e23af212c)
mais, en représentant par
le rayon de courbure de la courbe dont les coordonnées sont
et par
celui de la courbe dont les coordonnées sont
et
on a