Images from Plane Mirrors in rapid rotation. 75 



It is worthy of remark, that the formula (Z), (F'), and (I'), 

 which relate to some of the most important properties con- 

 nected with the forms of the secondary curves, present a very 

 curious analogy: the first being, cos a=cot X cot /3; the 

 second, cos a = cot X cot 2 /3 ; and the third, cos a = cot X cot 

 3/3. 



But the inquiry, attractive as it is, from the beautiful ex- 

 amples which it affords of the application of trigonometrical 

 analysis to the solution of the various cases of an interesting 

 problem in catoptrics, must be brought to a close. If I have 

 already trespassed beyond the proper limits allowable for such 

 an investigation, I trust some apology for it will be found in 

 the singularity, as well as the novelty, of the results which 

 have been disclosed, connected though they be with a subject 

 of physical research, which, at first sight, seemed to promise 

 nothing beyond the mere gratification of the eye. 



I shall, therefore, merely add, that when three mirrors are 

 made to revolve round the same axis, with various inclina- 

 tions and different exposures to the direction of the light, the 

 curves generated by the primary and secondary reflections ex- 

 hibit still more complicated forms than those which I have 

 attempted to investigate ; but still combining, in every case, 

 variety, with the most perfect symmetry of figure. If the re- 

 fracted light of the prism be conjoined with the direct rays of 

 the sun, the most gorgeous and beautiful curves are obtained, 

 of which no just conception can be formed from the most la- 

 boured description. 



In fig. 5, (Plate III.) S v R X R'T represents the curve formed 

 by the doubly reflected images when X=50°, and /3=40°. In 

 fig. 6, (Plate III.) XRvSTR' represents the same curve, 

 when X=40°, and /3=50^. In both curves v exhibits a node. 



plies either that cot 3/3 = 0, or cot x = «. The former coincides with 

 the conclusion already deduced, that /S = 30° ; the latter intimates, that 

 "when X = 90°, there can be no secondary curve, whatever be the value 



of/3. 



