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



represents only the behavior of a very narrow bundle of rays, 

 so narrow that it cannot be studied experimentally because of 

 defraction phenomena. This begging of the question does 

 not avail, however, since there is no room for argument in 

 this case for the reason that the laws here formulated are the 

 direct consequences of the fundamental laws of refraction 

 upon which the whole superstructure of Geometrical Optics 

 rests, and apply with equal force to all refractions including 

 those for narrow bundles. 



Again in the case of curved surfaces the construction 

 usually presented (See, for instance, Southall's "Geometrical 

 Optics," pp. 49-50, fig, 15a and 15b) one can readily conceive 

 what the nature of the surface must be to produce the conoid 

 of Sturm. It would not be difficult to calculate nor would it 

 be impossible to grind such a lens, though it would not by any 

 means be a spherical lens. 



As applied to a spherical lens it could be shown by calcula- 

 tions that every detail of the construction shown in these 

 figures fails to conform to the laws above enunciated in the 

 same definite way explained above in the case of refraction at 

 a plane surface. The writer has calculated the rays and con- 

 structed models and verified them in every particular experi- 

 mentally, and has proven beyond controversy that the trans- 

 formation of a beam of light while passing through a locus 

 behaves as the laws indicated and not at all according to 

 Sturm's theory. 



Here again the assumption is made that the construction 

 illustrated in these figures applies only to very narrow bundles, 

 an assumption as will be shown below necessary to give the 

 theory any standing at all because all the observable facts 

 contradict it and it has no more basis in sound theory than in 

 the previous case. 



The final consequence of any focus is that all rays concerned 

 are completely reversed in relation to each other and in case 

 lateral aberrations and a section of a beam of light beyond a 

 locus assumes a different and characteristic shape. 



A very simple and effective way of showing how the re- 

 versal is accomplished when a beam of light passes through 

 an oblique locus is by pasting strips of paper on a common 

 reading glass, leaving four equal windows, thus conforming 



