Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/93

${\frac {{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}-{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}{1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}}-c=U=0,$

d’où

$\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right)={\frac {\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}{\left(1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}},\quad \left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right)=-{\frac {\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}{\left(1+{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}}.$

Or, en désignant par $m$ l’angle de la droite $V=0$ avec l’axe des $x,$ on a en général,

$\operatorname {Tang} .m={\frac {\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right){\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}+\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right){\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}}{\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha }}\right)+\left({\frac {\operatorname {d} U}{\operatorname {d} \alpha '}}\right)}}\,;$

on aura donc, dans le cas présent

$\operatorname {Tang} .m={\frac {\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}.{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}-\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}.{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}}{\left[1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}-\left[1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}\right]{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}}$

$={\frac {{\frac {1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}{\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}-{\frac {1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}{\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}}{{\frac {1+\left({\frac {\operatorname {d} \beta }{\operatorname {d} \alpha }}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta }{\operatorname {d} \alpha ^{2}}}}-{\frac {1+\left({\frac {\operatorname {d} \beta '}{\operatorname {d} \alpha '}}\right)^{2}}{\frac {\operatorname {d} ^{2}\beta '}{\operatorname {d} \alpha '^{2}}}}}}.$

Cela posé, soient $\mathrm {P,P'}$ les points $(\alpha ,\beta ),(\alpha ',\beta ')\,;$ soient $\mathrm {C,C'}$ les centres de courbure de ces deux points, dont nous supposerons les coordonnées respectives $(t,u),(t',u')\,;$ soient enfin $\mathrm {Q,Q'}$ les milieux des droites $\mathrm {PC'}$ et $\mathrm {P'C} \,;$ les coordonnées de ces deux points seront respectivement