NIPHER ON CERTAIN PROBLEMS IN REFRACTION. 325 



On Certain Problems in Refraction. 

 Bj Francis E. Nipher. 

 While observing, as I had often done before, the dished ap- 

 pearance of a flat-bottomed vessel as seen through a vs^ater surface, 

 it occurred to me that I had never seen a discussion of the shape 

 of such an apparent bottom, nor have I since been able to find 

 such discussion, 



r. To find the shape of the apparent bottom of xvaier when the real 

 botto7n is a horizontal plane. 



Let d' = the altitude of the eye above the surface ; 

 d" = the real depth of the w^ater ; 

 d = the apparent depth of the v^rater at a distance ; 

 Z = abscissa of the point at d. 

 The origin of the coordinates Z and d is in the surface imme- 

 diately under the eye. 



The angle of incidence has been taken in the medium contain- 

 ing the eye, and hence the point occupied by the eye may also be 

 considered as a radiant point. 

 By the law of refraction 



sin / ( s 



sm r 



1 



EXPLANATION OF CUT. 

 ss = water surface. I t' i' = apparent bottom for Problem II. 



cc = conchoid of Nicomedes. 1 b' b' = bottom for Problem I. 



bb = corresponding refraction conchoid. I &" b" = bottom for Problem II. 



