Page:Optics.djvu/107


 * When $Rq=RnsinRnq⁄sinRqn=Rn·sinSRE⁄sinSRq=rcosφ′^{2}·sinφ⁄sin(φ−φ′)$ is a right angle, or $φ$ a tangent to the surface, $QR$.

In this case we have only to draw $v=rcosφ′$ perpendicular to the refracted ray.
 * (3) When $Em$ is to be infinite, or the refracted rays parallel $v$;


 * (4) When $utanφ−tanφ′−rcosφtanφ′=0$ is a right angle, $∴ u=rcosφtanφ′⁄tanφ′−tanφ=rcosφ^{2}·sinφ′⁄sin(φ′−φ)$.

108. It has been shown that an infinite number of different surfaces may reflect rays proceeding from the same point, so as to produce the same caustic: the same thing is true of refracting surfaces, for the equation

will have for its integral an arbitrary constant, as well when $φ′$ the line represented by $v=0$ is drawn every where to one point (as in last Chapter,) as when $du+mdv=0$ is always a tangent to a certain curve, or in short whatever law it is guided by.

We may now easily see what will be the form of the caustic in particular instances.

109. Let the refracting body be a cylinder of glass terminated by a hemisphere. (Fig. 107.)

Let the incident rays be parallel to the axis. Taking $Rq$, we have $v$, so that if $Rq$ be taken equal to three times $m=3⁄2$, $F=mr⁄m−1=3r$ is the principal focus, and it will easily be seen that there must be a cusp at that point.

In the next place, take the extreme ray $AF$, which touches the surface at $AE$, making the angle of incidence a right angle: the angle of refraction is, of course, that