Images from Plane Mirrors in rapid rotation, 67 



each other, 1 + 2 cos 2 jS and 1 — 2 cos 2 /3 become each equal 

 to unity. And 



Cos i = cos X sin A V^ + cos A V^ = (cos ac sin A + cos a) V^ (V) 

 If we have also X = 45°, this expression becomes 



Cos i = J cos ot 4* ^ = cos* ^ « (XJ) 



It may be remarked, that this result coincides with the for- 

 mula (E) for the angle of incidence, in the case of a single re- 

 flection, when the inclination of the mirror, and the direction 

 of the luminous ray with respect to the axis, are both 45°. In 

 consequence, however, of the variable direction of the light, 

 after it has been reflected from the first mirror, the images 

 produced by the double reflections, do not at all coincide with 

 those produced by the single reflections, and the curves are 

 therefore totally different in form. 



In the formula (V), if X = o we have cos i = ^i, or i = 45** 



Hence the curves produced by the doubly reflected rays are, 

 in that case, conic sections. 



In the formula (U), if a = 90°, we have cos i = cos^ 45°=^, 

 or«=60°. 



Lastly, if it be assumed that in the formula (S) we have 

 /3=60°5 the mirrors being, in that case, also inclined to each 

 other, at an angle of 60°, we have 1 + 2 cos 2 /3 = 1 + 2 cos 120° 

 = 1 — 2 sin 30°=(?, and 1 — 2 cos 2 /3=2. Hence cos« = cos X, 

 or i=zX. 



Therefore, in every position of the revolving mirrors, the 

 reflected ray, after it has suffered two reflections, always makes 

 with the axis of the second mirror the same angle that the in- 

 cident light originally makes with the axis of rotation. This 

 must be admitted to be a very curious result, and, though con- 

 firmed by experiment, might not have been discovered by ob- 

 servation, without the aid of analysis. It need scarcely be re- 

 marked, that when X=o, we have also i=o, so that the doubly 

 reflected ray always coincides with the axis of the mirror from 

 which it is last reflected. Hence the curves are, in this case, 

 all conic sections. 



The various formulae which have been deduced for the 

 angle of incidence, in the case of two reflections, as well as in 

 that of one, are obviously of the general form y = M a: + N; 



