48 Mr. A. Whitwell oti Refraction 



It may be stated here that the equation represents not only 

 the locus of the intersections of the real refracted rays, but 

 also of the false refracted rays. These false ravs are equally 

 inclined to but on the opposite side of the normal to the real 

 rays. This arises from the fact that we have to square to 

 get rid of the radical sign in the denominator of the ex- 

 pression for d; for this reason it is immaterial, as far as the 

 -equation arrived at is concerned, whether we take the -|- or 

 — sign before the radical quantitv in the denominator of cL 



The curves represented by this equation, taking /a= .^ and 



r = '2 and various values of a, are shown plotted in fig. 4, the 

 light proceeding from left to right. When the radiant-point 

 is at infinitv, the focal line will be the straight Une I. at a 



distance l from the surface, or = from the axis of the 



cylinder. As the radiant-point moves up to the surface, the 

 focal line gradually bulges out to the right at its centre, and 

 its upper and lower ends bend towards and become asymptotic 

 to the axis as shown by curve II., for which « = 10.'^ IVhen 



the distance of the radiant-point from the surface = 



or when a= -, the centre of the focal line breaks, and 



its two ends become parallel to, but at an infinite distance 

 from, the axis of x. The curve III. has parabolic asym- 

 ptotes, the equation of which, referred to the radiant-point as 

 origin, is 



^^2 = 3-2 ■x'-f- 22-4 



LLV 



This point, a = - -, will be recognized as the principal 



focus for light proceeding from right to left. When the 

 radiant-point is inside this focus, the carve, lY., has two 

 branches and a pair of rectilineal asymptotes, the axis of the 

 cylinder remaining an asymptote. The branch on the left 

 is of course virtual. 



As the radiant-point moves to the right, the angle which 

 the rectilineal asymptote makes with the axis of x increases 

 from zero to a maximum, and then diminishes to zero. 

 When the radiant-point is on the surface there is no focal 

 line ; an incident cone of light produces a refracted cone, the 

 ratio of the sines of the semi-angles of the cones being =//,. 



I have not plotted the false focal lines in curves I.-IV. ; 

 they nil lie between the surface of the cylinder and the axis, 

 to which they are asymptotic. 



