at a ( \iflifulrical Suvfdce. 49 



LbV 



When a lies betwetMi r and , the real focal line lies 



between the surface and axis of the cylinder, and the false 

 focalline has two branches and a pair of rectilineal asymptotes. 



When (f = -' — - the real line is shown by curye Y.. and the 



false line has parabolic asymptotes, the equation of which is 



A graphical method of drawing the asymptotes may Ije 

 obtained by considering equation (1), 





I£ /A-(rt — ?')-+ {/JL-—l)h'^ = a^, d is infinite, and this relation 

 between a and h giyes us the circle shown in fig. 4. 



To obtain the asymptotes, draw an ordinate to the circle 

 through the radiant-point, and project horizontally to a point 

 on the axis of the cylinder. The asymptote will pass through 

 this point on the axis, and through the radiant-point. When 

 a = the focal line will be the axis itself, and this is the only 

 case in which the focal Hne will be a mathematical straight 

 line whether the aperture be large or small. 



When a becomes negative the curve is still asymptotic to 

 the axis (see curve VI., for which a = — 10), and as a 

 increases the curve gradually moves to the right and 

 approaches the line I., which it reaches when a is inlinite. 



The set of curves shown in fig. 5 are for the case in which 

 light proceeds from a denser to a rarer medium, and are 



obtained by putting i^= ^ in equation (2). 



When the radiant-point is at infinity the focal line is a 



T 



straio'ht line, YII., at a distance from the surface, or 



^ /JL — 1 



— — from the axis : it is virtual. As the radiant-point 

 yu.— 1 



moves to the right, the curve becomes of the form shown by 



VIIL, for which « = !(). In this curve, when the vertical 



aperture Ji = + , , total reflexion occurs, and the focal 



line cuts the surface ; the continuation of the curve inside 

 the cylinder is the false focal line. 



As a diminishes, the curve becomes smaller (curve IX. is 

 for a^Q)), and finally diminishes to a point when a = r \ in 



Pliil. Mag, S. 6. A^ol. 6. No. 31. July 1903. E 



