80
(Fig. 103.) be two incident rays inclined to each other at an infinitely small angle, DK, EL, the refracted rays. Let EF, EG be perpendicular to AD, DK. Then PF=Ep, QG=Eq; therefore DF=d·DP, DG=d·DQ. Now it was proved in the former proposition that DF=m·DG, and DP, DQ are evidently integrals of these differentials beginning together from nothing.
To return then to our problem,
| m= | PDDQ, | |
| but m= | DLDK=FBFK | =PH−PD−FBFQ−QD; |
| ∴ m= | PH−FBFQ | =CE+EG−BR−FRCE−FS |
| =CE+BG−FRCE−FS | ||
| =CE+(m−1)·CE−FRCE−FS; | ||
| ∴ m | =FRFS, |
and therefore the ray ADF is refracted to B.
CHAP. XI.
CAUSTICS PRODUCED BY REFRACTION.
106. These caustics are exactly analogous to those before treated of, being formed by the successive intersections of refracted rays, as those were by reflected ones.
Prop. Required the caustic to a plane refracting surface.
Let QR (Fig. 104.) represent an incident ray, qRS the refracted one; AM, MP are the rectangular co-ordinates of a point P on that line; PN is parallel to AM.
