![{\displaystyle {\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}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/843e032ec157badc2d8fbdd3da3dd839ece93c07)
mais (6, 6’)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1e59e85b220637281879166bdd711c8f640b9d)
donc, en substituant, et remplaçant
et
par
et ![{\displaystyle \operatorname {Sin} .^{2}(R,d')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f84cadf4f1ac2a4b16a832bbb47f4ecef4f6f8)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f78d81927b6b16e8fa2fa48b561d1ec3faedba83)
mais on a
![{\displaystyle {\frac {\operatorname {Sin} .^{2}(R,d)}{\lambda ^{2}}}={\frac {\operatorname {Sin} .^{2}(R,d')}{\lambda '^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8603cf8c645ea04cc1ddde4f8f9dce212682c67b)
donc finalement
![{\displaystyle {\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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80145353e221669024297c649df4c4aa00471538)
Ainsi étant donné le rapport de
à
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