36 Professor Airy on the Sphei'ical Aberration 



4th. If the breadth of the pencil, supposed circular, be X, the rays 



will meet the plane perpendicular to the axis of the lens at the distance 



U 3'- \ ^'* 



Z, in an ellipse, of which the axes are ^ » • "J .> '^"^ ^ 2 ' ~2 ' ^"'^ 



when V=Q or U=\, the ellipse becomes a circle : its diameter is 



Z' n ' 2 ' 



5 th. The rays will meet the plane at the distance Z — p in 

 an ellipse whose axes are 



If p=U.^—, or if p = Y ^— , the ellipse becomes a straight line. 



\ B'' 

 the ellipse becomes a circle whose diameter = -^ F . — . 

 ^ Z 2 



If V=o, the ellipse becomes a circle, whatever be the value 



of p ; the diameter is 



Z\n 2 ^/ 



If at the same time p='-i- .^, the diameter is o, or the circle 

 becomes a point, and a distinct image is formed. 



6th. If V-\- ^ =0, the image formed on the plane at the 

 distance Z will be circular. Its diameter will be 



^r^ or ^ ^ ^' 

 Z*^- 2' °' Z-2n- 2- 



