P E F 



q represent the first incident ray and g, q ...... the rays after the 1st, 2df, 



3d, &c. reflexions, and let the ^'s. between them be denoted by (q g,'} 

 (g q) &c. and we shall have 



(g g} = (g g) = ... ...... (q ?) = 2 / 



13 35 _! n+ l 



, and (g g} = 2 n ;, provided m be an odd number. 



1 2n+m 



Also (g g) = 2 Q 2i 



( 9) = 2 - 4 t 



(g g) =2 $ (2 m 2) t 

 i 2m 



II. Reflexion at spherical surfaces. 



1. Rays meeting in a point being incident on a spherical reflecting 1 sur- 

 face j to determine the directions of the reflected rays. 



Let r =" radius of the surface, g and g' distance of the foci of inci- 

 dent and reflected rays from the centre of the reflector, then when the 

 incident rays are nearly coincident with the axis, 



If q is infinite, or the rays parallel, 



1 2 r 



or g 1 = -. 

 q r 2 



This is technically called the principal focal distance' of the reflector, 

 and if we call it/, we have /= ^ 3 and ,'. by substituting in the first 

 equation, 



1-1 4.1 



r ~ 9 + f 



These formulae may be obtained in another form, which is often more 

 convenient, thus : 

 Let A and A' = distance of the foci of incident and reflected rays from 



1 12 



the surface of the mirror, then + = ; r being negative if the 



mirror be convex. 

 If A is infinite, or the rays parallel, 

 1 2 r 



230 



