3*4 



FOCAL LINES OF A REFRACTED PENCIL. 



3. These relations of the quantities A, B, C may be found by the ful- 

 lowing construction : 



Let IO be the incident and OR the refracted ray, and let ON be the 

 normal to the surface; NOI=e NOR = 6 r 



Find the points A, B, C, whose co-ordinates are 



for A, 

 for B, 



x=A 

 z = 0, 

 x=<7 



cos* 0,- cos' 0. 

 cot0 1 -cot0, ' 



',-0080. 

 C0t0,-C0t0,' 



z = A 



z = < 



cos* 0. cot 0, - cos* 0. cot 0. 

 cot 0, cot 0, 



cos 0, cot 0, cos 2 cot 0, 



(12); 



cot 0j cot a 



The positions of these points depend only on the form of the surface and 

 on the directions of the axes of the incident and refracted pencil. The point 

 B is absolutely fixed, being the centre of curvature of a normal section of the 

 surface perpendicular to the plane of refraction. 



Let OA t , OB lt 0(7, be the values of A u B v (7, for the incident ray. To 

 find the corresponding quantities for the refracted ray, draw the straight lines 

 A A u BB lt (7(7, intersecting the refracted ray in A v B u C t . OA OB V and 0(7, 

 are the required values of A v B v C r 



When any of these quantities becomes infinite, the line must be drawn 

 in a given direction. For A t , B u or (7, infinite, it must be parallel to the 

 incident ray. For A v B v or (7, infinite, it must be parallel to the refracted 

 ray. For B infinite, it must be parallel to the normal ; for A infinite, it must 

 make with the normal an angle whose tangent is 



9,-cos*0 1 



cos 



cos' t cot 6 l cos 8 0, cot 0, ' 



