Images from Plane Mirrors in rapid rotation, 69 



the plane of rotation ; and the two diameters A C and B D being 

 at right angles to each other, the point P will be the projected 

 pole of the primitive. Make PL equal to the measure of the 

 angle which the direction of the light makes with the axis of ro- 

 tation ; then, whatever be the inclination of the mirrors, all the 

 planes of reflection must be great circles passing through L. 

 Find L', a point diametrically opposite to L, and these circles 

 must also pass through L'. Hence, if K K' bisect L L' at right 

 angles, it must contain (if produced indefinitely) the centres 

 of projection of all the great circles in which the reflections 

 take place. Moreover, if the great circle B X D be described 

 about L as a pole, it must pass through the poles of all these 

 great circles. If the inclination of the mirror to the plane of 

 rotation be equal to the angle which the direction of the light 

 makes with the axis of rotation, then will \1 L be equal to the 

 axis of the curve generated by the successive reflections ; and 

 a small circle described about P, with the radius P L, will give 

 L ^t, X /a', the path described by the extremity of the axis of 

 the mirror in the course of each revolution. At the end of 

 the first quadrant of rotation, L will arrive at ^ ; and, there- 

 fore, if a great circle be described through L, /i, and L', the 

 arch L /a will measure the angle of incidence ; and, conse- 

 quently, if ih G' be made equal to L /*, by the principles of the 

 stereographic projection, the point G' will be a point in the 

 curve, formed by the reflected images. In the same manner 

 may any number of points be determined ; and a line traced 

 through them will give the curve L G Ylg. 



Let the radius of the primitive, P D, be denoted by r ; and 



DL = -4^ = ^^ = ^^. ButL'L °^ 



COS P D L cos 1^ B P / cos ^ A cos D L L 



= ; : — ; — = -; — . Hence L'L, the principal axis, is in- 



cos ^ A sin ^ A sin A ^ r i: ^ 



versely as the sine of the angle which the direction of the light 



makes with the axis of rotation. 



With regard to the inclination of the mirror to the plane of 



rotation, three cases present themselves : For that inclination, 



which we have represented by /3, may be equal to X ; greater 



than X ; or less than X. Where it is equal to X, as supposed in 



the above construction, it is evident that the greatest ordinate 



