NIPHER — ON CERTAIN PROBLEMS IN REFRACTION. 327 



superior branch of a conchoid of Nicomedes, making the nodulus 



— . Through any point on this conchoid whose distance from 

 n 



the y axis is L draw a horizontal line, and determine the point 



on this line whose abscissa is =^ L (or 1.J12 -^)' This is a 



Jn'^ — I 

 point on the apparent bottom, and is the apparent position of the 

 point lying directly below upon the real bottom. This determines 

 the direction of any incident and the corresponding refracted ray. 



An inspection of the diagram (Fig. 1) led to the discovery of 

 an interesting property of the refraction conchoid. Calling the 

 tangent to this curve at the point just under the eye the principal 

 tangent, this property may be stated as follows. 



From any point of incidence in the surface, the distance to the 

 apparent bottom, measured along the incident ray produced, is equal to 

 the distance to the pri?icipal tangent tneasured along the refracted ray. 



Calling these distances s and 5'^, their values are 



_ d ,_ d" 



cos i ?i cos r 



hence 



s dn cos r 



s' d" cos / 



A reference to eq. (2) shows that the only condition which can 

 satisfy this equation is 



5 = 5'. 



If s" ^= the distance of the observed point on the real bottom 

 from the point of incidence, then, as is well known, 



s" zzz ns ; 

 which means that light will travel over the distance s" in the less 

 refracting medium, in the same time required for it to traverse a 

 distance s in the more refracting medium. 



2°. To find the shape of the bottom of a pool of water., if it appears 

 plane as seen from any point above the surface. 



Let d" =z the apparent depth which is constant; 

 d' z= the height of the eye above the surface ; 

 D and Z = the coordinates of any point on the bot- 

 tom, the origin being in the surface below the eye. 

 Then, similarly to eq. (2), we have 



