NIPHER ON CERTAIN PROBLEMS IN REFRACTION. 33 I 



This discussion also furnishes a simple construction of the inci- 

 dent and refracted rays for any radiant point in the more refract- 

 ing medium. Supposing the line representing the bounding 

 surface to be horizontal, draw a vertical axis through the radiant 

 point, d' representing its distance below the surface. Select any 

 point in the axis above the surface whose distance from the sur- 

 face is d '. From this point draw a line parallel to the surface, 



making its length AB equal to "^ . This line will subtend 



V «^ — I 



the critical angle, being d" tan c. From the free end of this line 

 drop a vertical line to the surface, determining thus a point in the 



. . d' -ird" 



surface whose distance from the vertical axis is a := -> - - • 



Vn^ — I 



On the vertical axis, lay oft' from the surface a distance 3 := - • 

 On a and d as semi-axes construct an ellipse. Vertical lines, 

 drawn from any point in the line AB to the ellipse, will deter- 

 mine the direction of any incident and its corresponding refracted 

 ray. 



4°. 7o find the apparent form of a horizontal plane in a less re- 

 fracting medium, as viewed from a point in a more refracting medium y 

 the refracting surface being supposed horizontal. 



Let d" = the constant distance between the plane and the 

 refracting surface ; 

 d' = distance from the eye to the surface ; 

 d and L = coordinates of any point in the apparent surface r 



then sin i = — . sin r . 



n 



cos / j^, ,„ i_sm' 

 . d" = nd" 



and as 



we have finally 



cos r 

 Z2 



V I_«2 



L'^J^Kd' ^dY 



which is an equation of the fourth degree and of the same general 

 form as the equation of the former refraction conchoid in the first 

 problem. It cannot be referred to the conchoid of Nicomedes 



