326 TRANS. ST. LOUIS ACAD. SCIENCE. 



The two triangles having a common base in the water surface 

 (Fig. i), their vertices being at the bottom and the apparent bot- 

 tom, give the relation 



, tan r . ^ i cos i ,,, , . 



d^ r a' =1 a" . . . . (2) 



tan I n cos r 



From equations (i) and (2) we have 

 ^" I I _sin2 i 



d = 



" ii 



But 



(3) 



and this value in (3) gives 



d^ I n\d>-\-d)^ 



which readily takes the form 



[^^]=,d>^dy[^ 



J • . (4) 



(5> 



d^ 

 The equation of the conchoid of Nicomedes is 



where b is the polar distance and a is the modulus of the curve. 

 Hence (4) is the equation of a conchoid, which would be 

 obtained by constructing the conchoid of Nicomedes on an ex- 

 tensible surface, and then stretching the surface in the direction 



n 

 of the X axis so that a unit of length becomes , , , which tor 



*= V — I 



water and air is 1.5, 2- These two curves have a common axis^ 

 to which they are tangent at an infinite distance, and they also 

 touch each other at the point whose coordidates are Z z= o ^ 



n 

 This furnishes a simple way of constructing the apparent bot- 

 tom of a sheet of water, of uniform depth d'\ as seen from a point 

 whose elevation above the surface is d' . With the point of the 

 eye as a pole, and with the axis in the water surface, draw the 



