1126 APPENDICES AND TABLES 



= 2(N.A.)<; N.A. = -i- (xxix)' 



If X = the number of waves per inch of light of a given colour, L the 

 limit of resolving power of any objective with an illuminating beam of 

 maximum obliquity is 



L = 2X(N.A.) (xxx) 



But with a solid .7 axial cone and white light the resolving limit is 

 equal to the N.A. multiplied by 70,000. When Gilford's screen is used, or 

 photography employed, the limit is raised to the N.A. multiplied by 80,000. 



The aberration for non-parallel rays. It is a little more troublesome 

 to find the aberration of rays other than parallel, but if the following 

 instructions are carefully attended to the problem merely becomes the 

 simplification of a vulgar fraction. Let P and P' be the distances of the 

 point and its image from the lens. First find a, by either (xxxi) or 

 (xxxii) : 



o f 2 f 

 a=---l (xxxi); = !-- (xxxii) 



Next find x by (xxxiii) or (xxxiv) : 



- 1 . . . (xxxiii) ; x 1 - 



r 



Then find co by (xxxv) : 



co = , -- s x~ + 4i(fjL + 1) a, x 4- (3/x + 2) (jfj. 1) a~ + > (xxxv) 



8/x(/x -I)/ 3 L/i 1 n 1J 



1111, v -> t 



p' = f~ T5~ + -( X ~ a )~~A~f2 + <y~ ' (XXXVI) 



The aberration 8 P'= -coP' 2 ?/ 2 . . . (xxxvii) 



To find the aberration of two lenses in contact. Let Q and Q' be the 

 object and its conjugate at the second lens,/' be the focus of the second 

 lens, and F the focus of the combination ; then P' = Q. 



Ill 111 



p /== 7~p' Q /= r /+ ~Q ' ' ' ^ Xlx ^ 



for the first lens, - = - - _ + co y- . . . (xxxvi) 



for the second lens, -_= +: + co' y- . . . (xxxvi) 

 Q ./ Q 



for both lenses, = _ + - - + (co + co') y~ . (xxxviii) 



^e J J 



Therefore, for 'ii lenses, n = 2 - - - + 2 co y" . (xxxviii) 



Q ./ 



The aberration 8 Q' = - 2 co Q' 2 y- and 8 F = - 2 co F-y- . (xxxix) 



Example : Two plano-convex lenses of equal foci have their convex 

 surfaces in contact (tig. 7) ; find the aberration for parallel rays. Then 



3 . / ,v 



f-r. 



the first lens r = 00 ; therefore x = -1 (xxxiii); P = oo ; therefore 



a = - 1 (xxxi) ; and co -- .., ............. (xxxv) 



' 



For the second lens r. ; therefore rc = l (xxxiv); -= - ; there- 



y / 

 1 Journal 11. M.S. 



