24 Professor Airy on' the Spherical Aberration 



circular, it will easily be seen that their lengths are equal. AH 

 the others are ellipses, with the exception of the one equidistant 

 from the two lines, which is a circle whose diameter is half the 

 length of either line. The image of a point, therefore, on the 

 retina, when viewed out of the center of the field of view, is ge- 

 nerally an ellipse, becoming sometimes a circle, and sometimes 

 a straight line. 



It may happen that the two lines of convergence intersect each 

 other, in which case, the rays tend, after refraction, to converge 

 to a point. Instead of the images in Fig. 6, we should then have 

 (upon moving the retina) the series in Fig. 7, a series of circles 

 and a point. If this condition then be satisfied, the image of a 

 point viewed out of the center of the field of view, is a circle, but 

 on pushing in the eye-piece it may be made a point, as it ought 

 to be. 



We shall now proceed with the mathematical investigations 

 suggested by these considerations. 



Prop. III. A pencil of rays, whose axis intersects the axis 

 of the lens, is incident on a lens : to find the distance (from the 

 lens) of the plane perpendicular to the axis of the lens, at which 

 the convergence of rays, in a perpendicular plane, takes place. 



Let L, Fig. 4, be the point from which the rays diverge : let LAF 

 be the axis of the incident pencil, and let it be refracted first in the di- 

 rection FD, and afterwards in the direction GM. Let H and K be the 

 centers of the first and second surfaces ; join LH, and produce it to meet 

 the first refracted ray in D: join KD, cutting the emergent ray in M. 

 Now, because LHD is perpendicular to the first surface, all rays in the 

 conical surface, generated by the revolution of LF about LH, will converge 

 to D, and, therefore, rays near each other in a perpendicular plane will, 

 after refraction, converge to D. The refracted rays will, therefore, be 

 in the surface of the cone generated by the revolution of GD about KD ; 



