at a Cylindrical Surface, 51 



about the axis of the cylinder they will represent the focal 

 lines produced by lioht passing from left to right and falling 

 on the concave surface of the cylinder if the corresponding 

 values of a be also reversed in sign. 



We shall now find the equation of the locus of the inter- 

 section of two symmetrical rays which have the greatest 



angle of incidence possible, viz. ^ . Suppose the triangle 



oag, fig. 1, to be turned down into the plane of the paper 

 on the line og, as shown in fig. 2. We know that 



, - TT 



oc= s/a'-^ + h'^ ; oa= \/d'^r^ + /i^ ; ^= 2" ' 

 and that sin 9= -. 



We also have r sin (f) = d sin a5, 





r r,/a^-^/i^ 



Taking the radiant-point as origin, a) = a + d cos yjr, or 



ra 



,v = a 4- 



and putting A= ^, tl^e equation reduces to 



y 



~aHfJ^l U-«)- J 



(^) 



3 

 Puttino- ^= - , r = 2, the locus represented by this equation 



is shown plotted in fig. 6. The general nature of the curves 

 is the same as that of the curves for small apertures shown 

 hj fig. 4. As before, we can get a relation between a and h 

 for which the corresponding value of d is infinite. 



In this case d=-JO when a^ + Ji^ = -^-^; this relation 

 represents a circle of radius 



When a = yo the focal line is a straight line, XY., at a 

 distance — or 1*78 from the axis. As the radiant- 



point moves to the right, the curve bulges out at the centre 

 and bends towards the axis at each end, as shown by curves 

 XVI. and XYIL, for which a=\0 and a = 4. 



E 2 



