72 Br Anders on on Curves (generated hy 



the supplement of twice the inclination of the mirrors, or what 

 is the same thing, twice the complement of the inclination of 

 the mirrors, 



That is, D= 180° — 2 A=2 (90° — A). 



Since the inclination of the mirrors is expressed by 180°— 2 ft 

 we obtain, by substitution, 



D=4/3 — 180° (X) 



Hence, when /3= 45°, we have I) = o; so that the doubly-re- 

 flected ray suffers no deviation ; and therefore passes through 

 the luminous point, or is reflected directly backward. The 

 deviation of the ray, after two reflections, being, according to 

 the formula (X), altogether independent of the direction of 

 the light (providing it move in a plane cutting at right angles 

 the line of common section of the mirrors), it follows that, 

 when the mirrors are made to revolve around that line of com- 

 mon section, the direction of the ray, after two reflections, 

 must remain unchanged. This simple and beautiful property 

 affords an easy method of determining the point where the se- 

 condary curves intersect the principal axis. 



The condition that the angle of incidence, after two reflec- 

 tions, shall be equal to the angle of incidence after one reflec- 

 tion, may be derived from the formula (R). Two cases, how- 

 ever, present themselves, both of which lead to interesting re- 

 sults. In one case, the angle of incidence, after two reflec- 

 tions, may be equal to the angle of incidence for one reflection 

 belonging to the mirror from which the second reflection took 

 place ; and, in the other, it may be equal to the angle of inci- 

 dence from which the first reflection took place. In the for- 

 mer, we have cos e = cos T; and, in the latter, cos e = cos I. 

 We shall investigate these cases in their order : 



According to the first supposition, we have, by the formula 

 (R) 



Cos t=co8 r=i COS I' — 2 COS 2 /3 cos I. 

 Hence, — 2 cos 2 /3 . cos 1=8 



This equation holds true whether cos 2 iS=o; or cos 1 = 0. 

 From the first of these results, 



Cos 2 ^ = cos 90", or /3=45° . . . . . (Y) 

 From the second we obtain Cos r=«, 



Or, — cos « sin A sin ^-f cos A cos /3=o. 



