18 



OPTICS. 



dent and reflected rays, that when one 

 of the conjugate foci R approaches to C, 

 the other focus /also approaches to C ; 

 and when F coincides with C,/also co- 

 incides with it ; so that it follows, that 

 when rays diverge from the centre of a 

 sphere or a spherical surface, they are 

 all reflected back again to the same 

 point from which they diverged. When 

 R passes C towards O,/ will then pass 

 beyond C, and move farther off as R 

 approaches to O. When F coincides 

 with O, f will be infinitely distant, or 

 the reflected rays will be parallel. 

 When R passes O towards E, the re- 

 flected rays will diverge like AD', and 

 will have their virtual focus about f 

 behind the mirror ; and as R approaches 

 E,/' will also approach E. 



If we continue the lines C A, C E, 

 CB behind the mirror in fig. 21, and 

 suppose MEN the surface of a convex 

 mirror, upon which rays R' A, R' E, and 

 R' B fall, diverging from R', then it may 

 be proved, by the very same reasoning, 

 that they wall be reflected in the direc- 

 tions A r, E R, B r in lines which di- 

 verge from a virtual focus f", whose 

 distance from O or E is found by the 

 rule above given for concave mirrors. 

 As R' recedes from the mirror, /" will 

 approach to O, with which it will coin- 

 cide when R' is infinitely distant, and 

 the rays become parallel. When R' 

 approaches to E, /" also approaches 

 toE. 



Reflexion of converging rays by 

 concave and convex mirrors. It is ob- 

 vious, from fig. 21, that all rays, such 

 as D' A, which fall converging upon the 

 concave mirror M N, will be reflected 

 to a focus/ ' between O and E, and this 

 focus will approach to E, as the point 

 of convergence/' approaches to E. It 

 may be shown by the same reasoning as 

 for diverging rays, that /' O is to O C, 

 as O C is tot)/",/" being now between 

 O and E. 



When converging rays r A, r B (fig .21 .) 

 fall upon a convex mirror M N, as if they 

 proceeded to some point /" between O 

 and E, they will be reflected to R' whose 

 distance from O or E is found by 

 the very same reasoning which we have 

 given for diverging rays. From this it 

 follows, and it may be proved also by 

 projecting the rays, that when they con- 

 verge to any point between O and C, 

 they will be reflected, as if they diverged 

 from R beyond C. When they con- 

 verge to C they will be reflected in the 

 same direction as if they came from 



C ; and if they converge to a point be- 

 yond C, they will be reflected, diverging 

 as if they proceeded from some point 

 between C and O. When they con- 

 verge to O, they will be reflected, in pa- 

 rallel lines, or their focus will be infi- 

 nitely distant ; and if they converge to 

 a point /" between O and E, they will 

 be reflected to a real focus at R, which 

 will approach to E, as/" approaches to 

 E, according to the law already given. 



CHAPTER VI. CATOPTRICS continued. 



Formation of Images by Plane, Con- 

 cave, and Convex Mirrors Reflect- 

 ing Telescopes Reflecting Micro- 

 scopes. 



THE principle of the formation of images 

 by mirrors is exactly the same as by 

 lenses, and the place of the image may 

 be determined from the place of the ob- 

 ject, and the radius of the mirror, by 

 finding the foci or points of convergence 

 of the rays, from the rules in the preced- 

 ing chapter. The application of these 

 rules we shall now exemplify. 



Formation of images by plane mirrors. 

 Let R S (fig. 22.) be the surface of a 



plane mirror, and M N any object placed 

 before it, and let the eye of the observer 

 be placed any where before the mirror, 

 as at F G. Of all the rays which pro- 

 ceed in every direction from the points 

 M, N of the object, and are reflected from 

 the mirror, those which enter the eye are 

 few in number and must be reflected 

 from portions A B, C D of the mirror, 

 so situated with respect to the eve and 

 the object, that the angles of incidence 

 of the rays which fall on these portions 

 must be equal to the angles of reflexion 

 of those which enter the eye between F 

 and G. The ray M A, for example, will 

 be reflected in the direction A F, and 

 the ray M B in the direction B G ; in 



