Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/250

${\frac {1+{\frac {\lambda k\operatorname {Cos} .(R,d)}{R}}-{\frac {r\operatorname {Cos} .^{2}(R,d)}{d}}}{\lambda ^{2}}}={\frac {1+{\frac {\lambda 'k\operatorname {Cos} .(R,d')}{R}}-{\frac {r'\operatorname {Cos} .^{2}(R,d')}{d'}}}{\lambda '^{2}}}\,;$

mais (6, 6’)

${\begin{array}{rll}{\frac {r\operatorname {Cos} .^{2}(R,d)}{d}}&={\frac {(d-\lambda k)\operatorname {Cos} .^{2}(R,d)}{d}}&=\operatorname {Cos} .^{2}(R,d)-{\frac {\lambda k\operatorname {Cos} .^{2}(R,d)}{d}},\\\\{\frac {r'\operatorname {Cos} .^{2}(R,d')}{d'}}&={\frac {(d'-\lambda 'k)\operatorname {Cos} .^{2}(R,d')}{d'}}&=\operatorname {Cos} .^{2}(R,d')-{\frac {\lambda 'k\operatorname {Cos} .^{2}(R,d')}{d'}}\,;\end{array}}$

donc, en substituant, et remplaçant $1-\operatorname {Cos} .^{2}(R,d)$ et $1-\operatorname {Cos} .^{2}(R,d')$ par $\operatorname {Sin} .^{2}(R,d)$ et $\operatorname {Sin} .^{2}(R,d')$

${\begin{aligned}&{\frac {\operatorname {Sin} .^{2}(R,d)}{\lambda ^{2}}}+{\frac {k}{\lambda }}\left\{{\frac {\operatorname {Cos} .(R,d)}{R}}+{\frac {\operatorname {Cos} .^{2}(R,d)}{d}}\right\}\\\\=&{\frac {\operatorname {Sin} .^{2}(R,d')}{\lambda '^{2}}}+{\frac {k}{\lambda '}}\left\{{\frac {\operatorname {Cos} .(R,d')}{R}}+{\frac {\operatorname {Cos} .^{2}(R,d')}{d'}}\right\},\end{aligned}}$

mais on a

${\frac {\operatorname {Sin} .^{2}(R,d)}{\lambda ^{2}}}={\frac {\operatorname {Sin} .^{2}(R,d')}{\lambda '^{2}}}$

donc finalement

${\frac {1}{\lambda }}\left\{{\frac {\operatorname {Cos} .(R,d)}{R}}+{\frac {\operatorname {Cos} .^{2}(R,d)}{d}}\right\}={\frac {1}{\lambda '}}\left\{{\frac {\operatorname {Cos} .(R,d')}{R}}+{\frac {\operatorname {Cos} .^{2}(R,d')}{d'}}\right\}.$

Ainsi étant donné le rapport de $\lambda$ à $\lambda '$ qui est celui du sinus d’incidence au sinus de réfraction, la caustique des rayons incidens et la courbe séparatrice, et par suite le rayon de courbure