I 1 30 APPENDICES AND TABLES 



of either a presbyopic or myopic person to a normal focus. Make p the 

 abnormal, and^/ the normal focus ; then/ will be the focus of the spectacles 

 required. 



In both cases p is a negative quantity, because it is on the same side 

 of the lens as p' ; it is usual to make p f 10 or 12 inches. 



Achromatism 



Let /LI be the refractive index of a mean ray (D line nearly) for a 

 certain material, p. v that for a blue ray, and /u r that for a red ray ; the dis- 



persive power of the material is /ig ~j ir ; this is usually written ^-, or -or. 



PjT * f* ~ 1 



The formula for achromatism is 



dp 1 ftp' 1. 



/i-r? jT^r/' 



that is, ^ + 5' = ......... (1) 



The foci of the two lenses are therefore directly as their dispersive 

 powers, and the focus of one will be negative. 



An achromatic effect, which is not achromatism in the strict mean- 

 ing of the term, can be obtained with two lenses of the same kind of glass 

 by making d the distance between the lenses : 



If p is large,/ in the denominator may be neglected ; this will make 

 d half the sum of the foci, which is the formula for both the Huyghenian 

 and Kamsden eyepieces ; but when p = /, d is the sum of the foci. 



Formulce relating to Spherical Mirrors 



Let p - one focus, p' = its conjugate, / = principal focus, and r 

 = radius of curvature ; then in concave mirrors 



- 



- = - ..... 



P+P P / 



To find# interchange^? and^?'. 



If o is the size of an object, and i the size of its image, and v the dis- 

 tance of the image from the principal focal point, then 



In convex mirrors prefix a negative sign, thus: r= 2/, and so on 

 with the other formulae. 



The formulae for mirrors may be derived from those of lenses by sub- 

 stituting - 1 for p; thusr= -2/(vii). 



Let y = the semi-aperture ; then the spherical aberration 



/= ~\ ' y j ..... (v) or (viii). 



A mirror to be aplanatic for parallel rays must have a parabolic curve. 



A mirror to reflect rays diverging from a point p, so that they may 

 converge aplanatically to another point p', must be elliptical, having p 

 and p' for its foci. 



