at a Cylindyieal Surface. 53 



The equation to the circle which gives the asymptotes is 



a^ + A-^^. 

 When rt=x> the focal line is the straight line XXI. at a 

 distance ~jr^^^., or '2-^S from the axis. As the radiant- 



V 1 — IJb^ 



point moves to the right the focal line also moves to the 

 right and becomes asymptotic to the axis, as shown by 

 curve XXII. for which a= 10. The branch of the curve XXII. 

 on the right is the focal line for light falling on the concave 

 surface. The real and false focal lines coincide since the 

 refracted ray is at right angles to the normal at the point 

 of incidence. The parts of the curves inside the cylinder have 

 no real existence^ they are the loci of intersection supposing 

 that total reflexion did not occur. As the radiant-point 

 approaches the surface the real portion of the focal line becomes 

 shorter and dwindles to zero when the radiant-point reaches 



the surface. Curve XXIII. is for a= -^ ^ - or 1*78, 



V 1— //,- 



which is the point where the asymptote circle cuts the axis 

 of X ; the branch on the left, of which only a minute portion 

 is real, is the focal line for light falling on the convex surface 

 of the cylinder, and the branch on the right, which has para- 

 bolic asymptotes, is the focal line for light falling on the 

 concave surface of the cylinder. 



Curve XXIV. is for a=I'33. The branch on the right 

 and the left-hand part of the branch on the left above the 

 axis of X is the focal line for light falling on the concave part 

 of the cylinder, whilst the remainder of the left-hand branch 

 is the focal line for light falling on the convex portion of the 

 cylinder. When a = the axis is the focal line, when a is 

 negative the curves are got by reversing the corresponding- 

 curves for positive values of a. Thus curve XXY., which is 

 curve XXII. reversed, is for a= —10. 



As the radiant-point moves off to the right the focal line 

 gradually approaches and ultimately coincides with the line 

 XXI. from which we started. 



We have now plotted the focal lines for maximum and 

 minimum horizontal aperture in all possible cases. A curve 

 from fig. 4 or fig. 5 and the corresponding curve from fig. 6 

 or fig. 7, e, g. curves IV. and XVII., will define the focal 

 area, that is to say, all the light from the corresponding 

 radiant-point will after refraction pass through the area 

 between these curves. If we suppose ourselves at the radiant- 

 point and facing the cylinder half thb light will be bent from 



