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



values of these angles which will eliminate the variable factor 

 in the equation. 



Law II. All rays from an object point to points on a 

 spherical refracting surface equidistant from the line from the 

 object point normal to the surface focus at a common point. 



This law is the definition of a focus, the concentration of a 

 very large number of rays to a point. A focus, according to 

 this definition, is always a point. The line referred to in 

 the discussion of the first law is in fact a train of foci. There 

 should be separate words to designate these two conceptions. 

 Perhaps the line should be called the locus of the object point, 

 as will be done below. 



Laiv III. The location of the foci of rays through suc- 

 cessive zones depend on the relation of the angles o and oc or 

 e' and cl' . 



That which is called positive spherical aberration where the 

 focus of the outer zones is nearer the center of curvature than 

 of the rays of the inner zone is the condition where a increases 

 more rapidly than e. 



The above laws refer to refraction at a single surface. The 

 first law holds for all subsequent refractions through a centered 

 optical system for object points on the optical axis. It is not 

 true of the second or subsequent refractions of an object 

 point away from the optical axis though the locus of that 

 point may remain approximately linear. It is always more or 

 less curved. 



On the optical axis, while the locus remains linear and 

 fulfills the conditions of Law I and all foci conform to Law II, 

 they may fail to confonn to Law III because the successive 

 relative values of angles a and e contribute to the final 

 locations of the foci. The locus of an object point thus comes 

 to be a very complicated thing even in the simple case of an 

 axial object. Instead of attempting to study the subject ex- 

 haustively it is enough for practical purposes to determine 

 the limits which for an axial point requires the calculation of 

 at least two rays and for a point away from the axis at least 

 three or four rays. 



The usual discussion of the loci in oblique refraction illus- 

 trates the errors in not recognizing these laws of focus forma- 



