images from Plane Mirrors in rapid rotation. 73 



Hence, Cos «=,cot a cot /3 . (Z) 



When a =0, this expression becomes TanA=±cot|3 . . {Af) 

 It follows, therefore, that when X and j3 are complemental 

 angles, the primary and secondary images coalesce on the same 

 point of the 9-xis, both reflections being from the same mirror. 

 But when X and /3 are complemental angles, the general value 

 of cos 2, given by the formula (R), is, after the suitable reduc- 

 tions, 



Cos {±i(cos «^ « — 2 cos 2 ^ sin 2^ «) sin 2 A . . (B') 



When jS=45°, this expression becomes 



Cos i = cos ^^ cc sin 2 A, 



Or, since a=:45°, cos i=cos'* ^ » (0') 



The other case, connected with the secondary reflections, re- 

 lates to cos e=cos I. In fig. 1, this implies that M''E is equal 

 to Ny. According to this supposition, cos «=cos I ; and the 

 formula (R) becomes 



Cos i = cos I = cos r — 2 cos 2 /3 cos I. 



After the admissible reductions, this expression becomes, 



_ cos 2 /2 . cos A cos yg \ .-p.,v 



^^^'*- (14.cos2^)smAsin^l • ' • ' ' ' K^ ) 



^ ^ cos 2 /3 . cot A cot /S 



Ur, Cos a = 



1 + cos 2 /d 



By farther reduction, we obtain the more simple and elegant 

 expression, 



Cos ec =.cot A . cot 2 /3 (E') 



When cos a = l, or when a=o, we have 



Tan A = cot 2 /3 (F') 



Hence the primary image from the first mirror coalesces 

 with the secondary image from the other mirror, on the same 

 point of the axis, when X is the complement of 2 j3. 



When the point, where the primary and secondary images 

 thus concur, coincides with the point where by the formula 

 (X) the ray of deviation also intersects the axis, the curves 

 formed by the secondary images have no node. But a node is 

 formed when X exceeds the complement of 2 jS ; and in all 

 cases its distance from the pole of the primitive is equal to the 

 tangent of half the angle, which the direction of the light 



