OPTICS. 



17 



C fig. 20.) be a concave mirror, of which 

 R ( ' E is the axis;, or the line, by a 

 motion round which the section M N 



Fig. 20. 



would generate a concave mirror. Let 

 C be the centre of its concave surface 

 MEN, and let parallel rays R A, R E, 

 R B fall upon it at the points A, E, B ; 

 these rays will be all reflected or made 

 to converge to a focus /, half way be- 

 tween C and E, so that the principal 

 focal distance E / is half the radius C E 

 of the concave surface. The ray R E 

 falling perpendicularly at E, will be re- 

 flected backwards in the same line E R, 



Fig. 21. 



and will consequently pass through/ 

 In order to find the direction of R A 

 after reflection, draw C A P, which will 

 be perpendicular to the spherical sur- 

 face at A ; then as R A C is the angle 

 of incidence, make C Af equal to it, 

 and A / will be the reflected ray ; in 

 like manner find B /, the reflected ray 

 for R B. Now, since R A and RE are 

 parallel, R A C is equal to A C/, that 

 is, C A/ is equal to AC/; conse- 

 quently C/ is equal to /A. But as 

 the point A approaches to E,/ A will 

 become equal to /E, and consequently 

 /Eto/C. 



By continuing all the lines in the 

 figure to the other side of the mirror, 

 the very same reasoning may be used 

 to prove, that when parallel rays R' A, 

 R' E, R' B fall upon a convex mirror 

 M A E B N, the reflected rays A r, E R', 

 B r will diverge as if they came from / 

 which is called their virtual focus, and 

 which is the principal focus of parallel 

 rays. 



* Reflexion of diverging rays by con-- 

 cave and convex mirrors. Let M N 

 {fig. 21.) be a concave mirror, whose 

 axis is C E, and centre C, and let O be 



\ 



its principal focus or focus of parallel 

 rays, such as /was in fig. 20. Then if 

 rays RA, RE, RB, diverging from F, 

 fall upon it, they will be reflected to a 

 focus/ between O and C, so that ROis 

 to O C as O C is to O/; that is, the 

 distance / O is equal to half the ra- 

 dius of the mirror multiplied by itself, 

 and divided by the distance of the diver- 

 gent point 11 or F from the point O. 

 Hence by adding/ O to half the radius 

 O E, we obtain/ E, the conjugate focal 

 length of the mirror for rays proceeding 

 from F. The truth of this may be easily 

 proved by projecting the reflected rays, 

 and measuring the distances on a scale 

 of equal parts; but the following de- 



monstration of it is so simple, that we 

 shall lay it before the reader. Let A O 

 be the reflected ray, corresponding to 

 the incident ray D A, parallel to the 

 axis C E ; then, since D A C is equal to 

 CAO, and since RAG is equal to 

 C A/, the remainder D A R is equal to 

 the remainder O A/ But in the trian- 

 gles AR O, A/O, the angle A O/ is 

 common, and A R O equal to D A R, 

 which is equal to A/O ; hence the trian- 

 gles are similar, and R O is to O A, as 

 O A is to O/; but O A is equal to O C, 

 consequently, R O is to O C, as O C 

 is to O/. 



From this rule we conclude, and it may 

 be clearly proved by projecting the inci- 



