328 TRANS. ST. LOUIS ACAD. SCIENCE. 



D^nd''^"^: , (6) 



COS I 



which readily takes the form 



Z>2r= n^-d' 



~^ I .... (7) 



sin^/J 



Substituting for sin^/ its value in terms of Z, d" and a?', and 

 reducing, the equation readily takes the form 



Z2 Z>2 



n'^{d''-\-d'Y n^d' 

 n^— I 



(8) 



which is the equation of an hyperbola the semi-axes of which are 



n{d"-\-d>) 

 \/«^ — I 

 The centre of this hyperbola is at our origin of coordinates. The 

 bottom will, of course, be generated by revolving the curve about 

 the vertical line passing through the eye. If the eye is at an infi- 

 nite height above the surface, the hyperboloid would become a 

 plane, the distance of which below the surface would be 

 D' = nd". 



If n= I the bottom would also be a plane whose depth below 

 the (imagined) surface would be d". This is shown in the ex- 

 pression for eccentricity, which is 



The conditions which make e ^ 00 , and which would reduce 

 the hyperboloid of revolution to a plane, are d' =:oo ; d" = o ; 

 or n=z I . 



The hyperbola would become equilateral if d'=d{^~n^—\—i), 

 and, in order to comply with the previously assumed conditions, 

 d' and d" must both be positive, or As/n^ _ i ^ i , and hence 

 n^ \^2. This condition cannot, therefore, be supplied physi- 

 cally in case of water. The hyperbola would, however, become 

 equilateral for glass (« ^ f ) if </' r=: o. , j g d". 



With any medium having a refractive index of «, the mini- 

 mum eccentricity physically possible is (eq. 9) 



