Curvature Method of Teaching Optics. 473 



the resultant R must evidently be in kilodioptres, and the 



. . „ . . ,, 2000 h 2000, , n 



curvature in dioptres is then 5 ,., = — s— h tor small curva- 



c- + lr i- 



tnres. It is then pointed out that if c- = 2000 or c = 44'7 mm. 

 the sagitta or sag of the curve denotes directly the curvature 

 in dioptres, and Prof. Thompson's u dioptrie w spherometer 

 and other forms are explained. At the same time attention 

 is called to the facts, first that the curvatures of all circular 

 curves having the same chord are proportional to their 

 sagittas, and secondly, that this is only true for chords of 

 small lengths compared with the radius. This assists the 

 explanation of spherical aberration later. 



The properties of spherical mirrors follow immediately. 

 From the fact that lio-ht striking the surface travels back 

 the same distance as it would have gone forward in the 

 absence of the mirror, it follows immediately that the curva- 

 ture of the mirror is the mean between the curvatures of the 



V— u 



incident and reflected wave-fronts. Hence R = — » — or 



V — U = 2R, where V and U are the convergences of the 

 incident and reflected light and 2R = F is shown to be the 

 convergence of the mirror. By considering the image of an 

 object formed by the mirror when the latter is stopped down 

 to a pinhole at the vertex, we at once find the magnification 



v U 



m = - = ^ , and the whole of the properties of mirrors are 



deduced from the two equations 



V-U = 2R = F 



v U 

 m = - = =~ . 

 a v 



The error of considering the sagitta proportional to curva- 

 ture for large apertures is again pointed out, with reference 

 to parabolic mirrors, caustics, &c. 



Refraction and Dispersion. — It is at this point that the value 

 of the wave method begins to show itself most strikingly. 

 Refraction at a simple plane surface may be shown by 

 Huvghens' method. The writer has found it of the greatest 

 ance to students to illustrate the wave-fronts by parallel 

 ones of men marching towards a river which they can only 

 ford at some fraction of their marching speed. The swinging 

 round of the line is easily realized ; and if at the same time 

 the idea is introduced of the lines being made up of men of 

 different heights, with uniforms ranging from violet to red 



Phil. Moo. S. 6. VoL 9. No. 52. April 1905. 2 I 



