![{\displaystyle {\frac {{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}-{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}{1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}}-c=U=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/611c91b2d28fec484c8dc202a87cc2d7913f07fd)
d’où
![{\displaystyle \left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right)={\frac {\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}{\left(1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}},\quad \left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right)=-{\frac {\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}{\left(1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/312f0bc8ddc3624be78afa7223bb762c6e4f071d)
Or, en désignant par
l’angle de la droite
avec l’axe des
on a en général,
![{\displaystyle \operatorname {Tang} .m={\frac {\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right){\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}+\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right){\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}{\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right)+\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right)}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326f16f24de2f23b12df0edb6c8e70af31e58bbf)
on aura donc, dans le cas présent
![{\displaystyle \operatorname {Tang} .m={\frac {\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}-\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}.{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}}{\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}-\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4129c7a2d9b1ae57c3adc28e3f522fca1fbb61c0)
![{\displaystyle ={\frac {{\frac {1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}-{\frac {1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}}{{\frac {1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}-{\frac {1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d106d439803e825b91b69d77ba668b5ef900a79)
Cela posé, soient
les points
soient
les centres de courbure de ces deux points, dont nous supposerons les coordonnées respectives
soient enfin
les milieux des droites
et
les coordonnées de ces deux points seront respectivement