192 CALIFORNIA ACADEMY OF SCIENCES [Peoc. 4th Ser. 



a and b throug-h a doublet objective is studied. One of these 

 points is twice as far from the optical axis as the other. 

 Parallel rays from these two points to the lens are selected 

 to show at the same time distortion and oblique aberrations. 

 The graphic calculation of the two rays is given to the right 

 above and all the oblique axes are drawn, the successive foci 

 being represented by letters with subnumbers. Each image 

 plane is also drawn and on the longer the length of the shorter 

 is laid off by short curves. The distance between these curves 

 is the amount of the distortion which is not serious till at 

 the third refraction. Because of the lateral aberrations, the 

 axial displacement is quite large at the second refraction and 

 very serious at the third. 



A simple inspection will show that both axial displacement 

 and distortion will be most greatly improved by a slight 

 increase in the length of the radius of the third surface, and 

 that this would have a very much more profound effect than 

 a change in the curvature of the second surface, and that if 

 these two are changed in the proper degrees the aberration 

 could be greatly improved without changing the magnification 

 of the lens as a whole. 



The great advantage of this system of lens calculation is 

 the facility with which the manner of correcting aberrations 

 can be located by inspection of the drawings. 



The graphic calculation, which is preliminary to the oblique 

 axis calculation in these examples, is described below. 



The method of calculation is illustrated in Figure 6, which 

 shows the calculation of the axial spherical aberration at the 

 principal focus of a Coddington lens. A is the center of 

 curvature of the first surface, B of the second. C is the 

 Graph Center, D and E are so located that CD/CE=n/n'. 



The method of calculation of a ray parallel with the optical 

 axis is as follows : 



1. Draw DL' parallel with J'A 3. Draw L'M' parallel with K'B 



2. Draw J'K' parallel with CL' 4. Draw K'G parallel with CM' 



The point G is the principal focus for the zone J' of the lens. 



Exactly the same method applies to the calculation of 

 paraxial rays except that straight lines perpendicular with 

 the axis replace all the cur\'^es. The steps in the process are 



