52 Mr. A. Whitwell on Refraction 



When G= — = 2'68, the principal focus for rays 



V'/A^'^ — 1 



going from right to left, the curve, XYIII., has parabolic 



asymptotes. 



When a lies between and r, or between 



and —r, the curve has three branches and a })air of recti- 

 lineal asymptotes. Curve XIX. is for a = 2*4 ; the false 

 focal line is not drawn, and the branch on the left is virtual. 

 When a lies Ijetween r and —r, we cannot have an angle 



TT 



of incidence of ^ . 



When a lies between — r and — ^y-, ~.: we o^et a curve 



\/fX^ — i ® 



like XIX., but reversed about the axis of y. As the 

 radiant-point moACS to the right, the focal line gradually 

 approaches and finally coincides with the line XV. (See 

 curve XX., which is for « = — 10.) 



When the light passes from a denser to a rarer medium we 

 cannot obtain the equation of the locus of the intersection of 

 symmetrical refracted rays by putting fju less than unity in 



equation (4), because for an angle of incidence of — the rays 



would be totally reflected. The maximum angle of incidence 

 will be the critical angle; we shall, therefore, find the equa- 

 tion of the locus for symmetrical rays having an angle of 

 incidence the sine of which =|. When the triangle oag is 

 folded down into the yertical plane on the line or/, we shall 

 get the construction shown in fig. 3, in which 6 is the angle 

 of incidence, ^ the angle of refraction. We haye 



or 



sin 



6 = 



= IM 



;/3- 

 d 



e 



-a ; 



7'sin0=: \/d^ + lr 



sin a ; 



d = 



1 

 cos 



/S' 





r \/«^ + A2 



(5) 





\/Y- 



-^2 yja'^ + li'-fjC'r 



^^ + /.V 



x = a—d cos yjr, 

 and putting li= — we get finally 



The locus represented by this equation is shown plotted in 

 fig. 7, r being =2 and yu,= |. 



