194 CALIFORNIA ACADEMY OF SCIENCES [Proc. 4th Sek. 



glass, respectively, and the graphs, three series of which are 

 shown, are drawn parallel with three radii of the lens from 

 the three points of refraction being studied. 



The object whose images are studied is the letter A stand- 

 ing edgewise to the lens and at some distance above the 

 optical axis. The "chief image" is the one produced by the 

 central portion of the lens and the incident pencil is indicated 

 by lines from the principal points of the image to the middle 

 point of the lens. 



Lines from the graph center to the air curve parallel with 

 three lines of the incident pencil, but which are omitted in 

 the drawing, locate the air ends of the chief graphs. Similar 

 lines, also not shown, from the graph center to the glass ends 

 of these graphs give the directions of the corresponding rays 

 of the refracted pencil. 



In precisely the same manner the pencils to the extreme 

 edges of the lens which produce images showing extreme 

 aberrations are indicated by lines, and likewise parallels to 

 the rays of these pencils from the graph center to the air 

 curve locate the positions of the other graphs shown. Like- 

 wise also the directions of the rays of the refracted pencil are 

 indicated by lines connecting the glass ends of these graphs 

 with the graph center. None of these lines from the graph 

 center are drawn nor need they be drawn when making 

 graphic calculations since the graphs themselves are the only 

 record that need be made of the process of calculation. 



The "oblique axis" method of calculation is as stated above 

 based on the fundamental fact of refraction at spherical sur- 

 faces that the focus conjugate with any image point must 

 lie on a line through that point and the center of curvature 

 of the lens. Thus all points on the optical axis have their 

 conjugate foci also on the optical axis. In the same manner 

 all conjugate foci away from the optical axis lie on oblique 

 axes. The oblique axes of the principal points of the object 

 A are shown in the figure. These axes are drawn away from 

 the lens because the object A is so located that virtual images 

 will be formed in that direction. 



Knowing the direction of each ray of the refracted chief 

 pencil as explained above, it is only necessary to find the 

 intersection of a line from the vertex of the lens, which is 



