Page:Optics.djvu/106

 Then we have these equations:

and the following, derived by differentiation,

Dividing the latter of these by the former, and putting $sinφ=m·sinφ′$ for $du+mdv=0$ and $q^{2}=u^{2}+r^{2}+2urcosφ$ for $t^{2}=v^{2}+r^{2}−2vrcosφ′$, we obtain

Particular cases.
 * (1) When $dφ⁄mdφ′=cosφ′⁄cosφ$ is infinite, or the incident rays are parallel

This is easily constructed:

Draw $0=(u+rcosφ)du−ursinφdφ$ (Fig. 106.) perpendicular to $0=(v−rcosφ′)dv+vrsinφ′dφ′$, $u$ to $=u+rcosφ⁄rsinφdu⁄dφ$; $v$ parallel to $=v−rcosφ′⁄rsinφ′·−mdv⁄mdφ′$, determines the point $1$.

It is easy to see that by this construction we have