70 Dr Anderson on Curves generated hy 



will increase with P L, the radius of the small circle described 

 by the axis of the revolving mirror, and therefore will be as r 

 tan \ (3. The value of the greatest ordinate, however, when |3 

 is different from X, is modified by the relative values of /3 and 

 X, and must always increase, the more /3 exceeds X. By in- 

 creasing X, and diminishing jS, the generated curves are cur- 

 tailed in all their dimensions. The curves L G L', R'G^U, and 

 PG"C, exhibit three forms of the curve, when PL measures 

 the angle which, in each of the three cases, the direction of 

 the light makes with the axis of rotation ; and when P L, P M, 

 and P M'j measure the respective inclinations of the mirrors 

 to the plane of rotation: PM being, in the present figure, 

 equal to the half of P L, according to the principles of the pro- 

 jection. 



The positions of the various points in the curves formed by 

 two reflections, though determined in a similar manner, are 

 yet assigned with much more labour than is requisite in the 

 case of those obtained for a single reflection. This is partly 

 owing to the greater complexity, and more tedious nature, of 

 the operations themselves ; but chiefly to the circumstance, 

 that, in the case of a double reflection, the direction of the lumi- 

 nous ray, as well as the magnitude of the angles which it forms 

 with the axis of the mirrors, is constantly changing in the 

 course of their revolution. In some instances, too, it happens 

 that, with the same disposition of the mirrors, and the same 

 original direction of the light, two different angles of incidence 

 may each have a second reflection ; while in others, the re- 

 flected ray may not quit the mirrors till after three or more 

 reflections ; as when both X and /3 are small angles. 



In consequence of the circumstances which have been men- 

 tioned, the curves produced by the doubly reflected images are 

 exceedingly varied, both in appearance and magnitude : inso- 

 much that a special construction is necessary to determine their 

 peculiar figures, according to the different values of X and jB. 

 Before exhibiting some of the more remarkable forms which 

 they assume, we shall briefly describe the method of their con- 

 struction. Let A B C D, fig. 4, (Plate II.) represent the primi- 

 tive ; then drawing the two diameters A C and B D at right 

 angles to each other, let BD represent the line of common 



