178 CALIFORNIA ACADEMY OF SCIENCES [Proc. 4th Ser. 



Coma is also due to a radial shifting of images, but is dis- 

 tinguished by the fact that one edge of the image is sharp and 

 distinct instead of fading off in both directions as is true of 

 astigmatism. Like distortion positive and negative coma, 

 representing under and over correction, are strikingly different 

 in appearance. 



In the calculation of aberrations, a series of rather compli- 

 cated formulae have been developed which are supposed to be 

 measures of these aberrations, and in some cases faulty defini- 

 tions of the aberration have sprung from these mathematical 

 formulae. Among these is a prevailing conception of the 

 nature of astigmatism, not conforming to anything that can 

 be verified experimentally, as has been pointed out by the 

 writer in another place ("Science", vol. XLVII, pp. 459-460). 



These effects shown in the figure are the things visible to 

 the eye that the systems of calculation have been devised to 

 measure in order that lenses can be so designed as to elimi- 

 nate them. 



As has been already intimated, it is not usual to find these 

 aberrations present singly as here drawn, but they are all 

 distinct enough so that they can be distinguished even when 

 simultaneously present. Thus the distortion of an image 

 would not prevent the recognition of any of the other aberra- 

 tions that might be present. 



When curvature is involved, a manipulation of the fine ad- 

 justment might be necessary to make the distinction, and 

 between coma and astigmatism the distinction might require 

 changes in the method of illumination. 



11. Laws of Focus Formation 



The study of the behavior of rays of light from a point 

 source when refracted at a spherical surface presents no special 

 difficulties, but the laws of focus formation under these condi- 

 tions do not seem to have been formulated, though they lie 

 at the foundation of all practical optics. The attempt made 

 below to formulate these laws presents nothing new and 

 nothing not thoroughly accepted by physicists. 



The need of the definite statement of these laws will be 

 evident when it is appreciated that the prevailing theory of 



