COLOURS OF THICK PLATES. [165] 



The formulae (27), (28), and (29) shew that X is equal to the semi-sum of {• and % lt and 

 for the same reason Fis equal to the semi-sum of rj and ft. Hence the geometrical construc- 

 tion given in Art. 11 for finding the centre of the system in the case of a plane mirror applies 

 equally to a curved mirror, even when the curvatures of the two surfaces are different. Since 

 the retardation vanishes for the image itself, it follows that the achromatic line is a circle 

 having the two points of intersection above mentioned for opposite extremities of a diameter. 



20. It follows from the expressions for X and Y, or from the geometrical construction to 

 which they lead, that if the eye be not in the line joining the luminous point and its image, 

 whenever it crosses either of two planes drawn perpendicular to the axis, and passing, one 

 through the luminous point, and the other through its image, the centre of curvature of the 

 bands moves off to an infinite distance, and the bands become straight, and then bend round 

 the other way. 



When the eye coincides with the luminous point, /, g, h become equal to a, b, c, respec- 

 tively, and R vanishes independently of x and y. The same takes place when the eye 

 coincides with the image, since in this case 



/ _ a g _ b 1 1_2 

 h c h c h c p 



Hence, when the eye crosses either of the planes above mentioned, remaining in the line 

 joining the luminous point and its image, instead of bands which become straight and then 

 change curvature, we have rings which disappear by moving off to infinity, and then appear 

 again. 



I have verified these conclusions by experiment, substituting when necessary a virtual 

 image of the eye for the eye itself, in the manner explained in Art. 15. The experiments 

 embraced the following cases, in the description of which will be used to denote the centre of 

 curvature of the mirror, F its principal focus, L the luminous point, and L 3 its image. 



Concave mirror: L beyond O. Eye (l) beyond L; (2) passing L; (3) between L 

 and L 3 ; (4) passing L 3 \ (5) between L 3 and the mirror. — Concave mirror: L between and 

 F. Eye (1) beyond L 3 ; (2) passing L 3 ; (3) between L 3 and L; (4) passing/,; (5) between 

 L and the mirror. — Concave mirror: L between F and the mirror. Eye (l) beyond L; 

 (2) passing L ; (3) between L and the mirror. — Convex mirror. Eye (l) beyond L ; 

 (2) passing L ; (3) between L and the mirror. 



The mirrors employed were formed, as usual, with surfaces of equal curvature. When the 

 observation was made directly, there was no difficulty in determining at which side of the line 

 joining the luminous point and its image the eye lay, and consequently in deciding whether 

 the direction of curvature agreed with theory or not. When the observation was made 

 by reflexion in a plate of glass, the eye was placed so that its virtual image fell in the line LL 3 

 by moving the head till the image of the luminous point was seen in the centre of the system of 

 rings. The radii of the two surfaces of the mirror being the same, or only differing by a small 

 quantity comparable with the thickness of the glass, the surfaces may be regarded as forming a 

 pair of concentric spheres ; and therefore, everything being symmetrical with respect to the line 



