Images from Plane Mirrors in rapid rotation. 71 



section of the mirrors, when a = o. Make P L = X ; and P M 

 =/3, with the latter as a radius, describe the small circle Mm 

 N n. Let the pole M revolve to w, and N to n, each having 

 moved through 90°. Find L', a point diametrically opposite 

 to L, and the projections of the various great circles which 

 represent the planes of reflection will all pass through L and 

 L'. Through m describe one of these circles F L tw L', and 

 L m being the measure of the angle of incidence in the sup- 

 posed position of the mirror, whose pole is m, if 9^^ E be made 

 equal to L m, by the rules of the stereographic projection, the 

 point E will be a point in the primary curve, or the curve ge- 

 nerated by single reflection. Since P E evidently represents 

 the direction of the light after reflection from the first mirror, 

 let P E be produced in the opposite direction till it meet the 

 surface of the sphere in a point F, diametrically opposite to 

 E ; and F P will represent the direction in which the light, 

 after being reflected from the first mirror, falls upon the se- 

 cond. Hence, if a great circle be described through F, w, and 

 E, the arch F n will measure the angle of incidence for the 

 new direction of the light. Consequently, if w II be made 

 equal to F n, the point R will be a point in the curve gene- 

 rated by the doubly- reflected images. If E' be another point 

 in the primary curve, then F', being a point diametrically op- 

 posite, F' n' will measure the angle of incidence at which the 

 ray, reflected from the first mirror, falls upon the second ; and, 

 therefore, if n' R' be made equal to F' n\ the point R' will be 

 another point in the secondary curve. The curve L R R'R", 

 traced through the several points, R, R', &c., gives the curve 

 formed by the doubly-reflected images. When /3=45°, as is 

 the case in the present figure, the secondary curves are of a 

 circular form, arranged round the pole of the primitive, all of 

 them passing through the point L, either really or virtually. 

 The property of these curves, intersecting the principal axis 

 L'X, in the luminous point L, when /3=45°, is immediately 

 connected with the following catoptrical theorem, which is 

 easily demonstrated, namely, that if two plain mii-rors be in- 

 clined to each other, at an angle represented by A, and the in- 

 cident ray be at right angles to their line of common section, 

 the deviation, D, of the ray^ after two reflections, is equal to 



