Page:Optics.djvu/107

83

Rq=RnsinRnq/sinRqn=Rn·sinSRE/sinSRq=rcosφ′²·sinφ/sin(φ−φ′).

When φ is a right angle, or QR a tangent to the surface, v=rcosφ′.

In this case we have only to draw Em perpendicular to the refracted ray.

(3) When v is to be infinite, or the refracted rays parallel utanφ−tanφ′−rcosφtanφ′=0;

∴ u=rcosφtanφ′/tanφ′−tanφ=rcosφ²·sinφ′/sin(φ′−φ).

(4) When φ′ is a right angle, v=0.

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

du+mdv=0,

will have for its integral an arbitrary constant, as well when Rq the line represented by v is drawn every where to one point (as in last Chapter,) as when Rq 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.)

1. Let the incident rays be parallel to the axis. Taking m=3/2, we have F=mr/m−1=3r, so that if AF be taken equal to three times AE, F 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 QC, which touches the surface at C, making the angle of incidence a right angle: the angle of refraction is, of course, that