332 TRANS. ST. LOUIS ACAD. SCIENCE. 



by projection along the x axis, as the projection factor J ^ 



is imaginary. The form of the curve is shown in Fig. 2. It of 

 course lies wholly within the critical angle, and the surface of revo- 

 tution will lie within the cone whose apex is at the eye, and having 

 an angle of twice the critical angle, to whichit is asymptotic. The 

 surface will be a frustum of the critical angle cone when the plane 

 is in the less refracting medium, but at the refracting surface. 

 The equation of the curve (13) then becomes 



^ = — ^' zt \/«'— I L . 

 This equation fails for that part of the plane within the cone- 

 A complete solution is however furnished by the polar equation, 

 which is 



/> = ^' sec/+ I I . . . (14) 



If ^ " = o, then for all values of / between 0° and the critical 

 angle (14) becomes 



fj = d' sec i -\- o . . . . . (15) 



If i = the critical angle, sin ?' becomes — and (14) becomes 



p = d' secz-\- — (16) 



If I > the critical angle, p becomes imaginary. When d" ^o 

 eq. (14) is therefore the equation of the broken line abed of 

 Fig. 2, which is the envelope of all curves for which d" > o. 



The results here reached are. of course, simple consequences of 

 well known relations, but it seems that the constructions involved 

 have not received the attention which they deserve. 



The additional fact that many lives are doubtless lost every year 

 by reason of these deceptive appearances is, perhaps, a sufficient 

 w^arrant for the publication of this discussion. 



