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



Measured the same way as the Comatic ray. This ray deter- 

 mines astigmatism. 



The focus of the Zone and Paraxial rays is determined by 

 the intersection with the optical axis. The Distortion ray does 

 not require the determination of a focal point. The three 

 oblique rays require, for the three values of the X coordinates, 

 the determination of the oblique focus of each, which can be 

 most easily accomplished by the new" oblique axis method 

 described below. There is no true focus in the strict sense 

 of the word after a number of refractions, but if each suc- 

 cessive focus were considered the source of a radiant pencil 

 of light then all would be true foci. The theoretical foci 

 secured by assuming this character of each focus lie within 

 the loci of the point and this is the best if not the only method 

 available for determining the position of an element of the 

 locus of a point away from the optical axis after numerous 

 refractions. 



The oblique axis method is based on the first law of focus 

 formation, that all possible foci lie on a line from the object 

 normal to the refracting surface. Having calculated the path 

 of a ray in any of the usual ways, the determination of the 

 focus after the first refraction is accomplished according to 

 this method by locating the normal on which the focus must 

 lie and finding the intersection this normal makes with the 

 refracted ray which is accomplished by running a line from 

 the object point towards or through the center of curvature. 

 After thus locating the focus on the ray path after the first 

 refraction, this point is considered as an object and the focus 

 conjugate with it after the second refraction is determined in 

 the same manner. 



The accompanying figure (4) illustrates the calculation of 

 the foci of a ray in both directions through a doublet, assuming 

 the object to be first on one side of the lens and then on the 

 other at the two points marked Focus 0. The graphic calcula- 

 tion of these rays is shown above. 



The foci and 1 are on Axis 1 normal to the first surface, 

 foci 1 and 2 are on axis 2 normal to the second surface and 

 foci 2 and 3 are on axis 3 normal to the third surface. 



The application of this method to the study of aberrations 

 is seen in figure 5 where the successive foci of two points 



