C& Dr Anderson on Curves generated by 



Let the great circle A B C D fig. 2, (Plate I.) representing 

 the plane of rotation, be the primitive ; then P being the stereo- 

 graphic projection of its pole, let there be described around that 

 point the two small circles M-W mm' and N N' ^^ w/, at the dis- 

 tances P M and P N, the measures of the inclination of the 

 two mirrors to the plane of rotation. These small circles will 

 evidently represent the varying positions of the poles, or the 

 extremities of the axes of the mirrors, in their revolutions 

 round the axis of rotation. Let A C represent the plane of 

 reflection common to both mirrors, when the angle of reflec- 

 tion is in a maximum state for the one, and in a minimum 

 state for the other ; or, to speak more correctly, let A C be 

 the line of common section of a plane passing through the 

 luminous point L, with the plane of rotation, and cutting at 

 right angles the planes of both mirrors. Then B D will be 

 the line of common section of the two mirrors, and P L will 

 measure the angle which the direction of the light makes, 

 with the axis of rotation, supposed to be constant. 



Since all the planes in which the angles of incidence take 

 place pass through L, and also through the axes of the mirrors, 

 they will be great circles of the sphere, and consequently, they 

 must all pass through L', a point taken diametrically opposite 

 to L, the former point being found according to the principles 

 of the stereographic projection. It is evident from the nature 

 of the construction, that LM = PL + PM, the maximum 

 angle of incidence, and Lw=PL — PM the minimum angle 

 of incidence, for the mirror whose inclination with the plane 

 of rotation is measured by P M. Also L N and L n measure 

 the corresponding angles for the other mirror. 



Let it be now supposed that the mirrors revolve through 

 the angular space measured by A R, so that the pole M arrives 

 at M', and the pole N at N' ; then the great circles L M' L' 

 and L N' L' being described through L, and the poles M' and 

 N', it is obvious that L M' will measure the angle of incidence 

 for the one mirror, and L N' that for the other. To obtain 

 analytical expressions for these quantities, let the angle of ro- 

 tation A PR=a ; PL=X; PM = i8; PN=/?'; LM'=I;and 

 L N'=r. Then, in the spherical triangle P L M', we have — 

 Cos I = — cos flc sin A sin /3 + cos A cos |3, ... (A) 



