Mr. J. Bridge on the Oblique Aberration of Lenses. 351 

 In a similar manner for the secondary focus, 



Also, by substituting for /u, and /a' in the expression for refraction 

 at one spherical surface, their values 



y—b y-b yZ + z* 





u u 2ru 



and 



y-v + v—v . y*+z* _ y-v + %. f+s?_y—b_ f±f near[ 



v v 2rv v r 2rv u 2rv 



and —1 for m, we find 



v u L J \r uJ «JrV y T \r uJ u J 



for the point of intersection of a reflected ray with a plane per- 

 pendicular to the axis through the focus of rays proceeding to 

 the projection on the axis of the focus of incident rays. 



By limiting the expressions to the case considered in Potter's 

 ' Optics/ part 2. p. 1 10, &c, they will be found to give easily 

 the same results as those in pp. 117, 118, 119 of the same work. 



To find the plane perpendicular to the axis of the lens on 

 which the aberration of a given ray of an oblique pencil is a 

 minimum, that is, the distance between the points of intersec- 

 tion by it and by the central ray. 



The equations of the ray, taking the origin at distance v from 

 the lens, are (putting //■ for fi v [x for the central ray) 



Z = vX + ?; 

 and for the central ray, 



Y =/a X + ?? 



Z =v X + f ; 



the distance of the points of intersection of these rays with a 

 plane perpendicular to the axis at distance X from the origin is 



D= •(Y=Y )*+(Z-Zo) s 

 and 

 D 9 = O-a0 2 + (v-v ) 2 ]X 2 + 2(^ . ^=^o+"^vF?o)X 



+0-%) 2 +(£-g 2 )- 



Tliis is a minimum when 



x __ _ (/* ~ ft>) JV — 9o) + ( v - v ) (?— &) . 

 V-^+(v-v )S 



