Images from Plane Mirrors in, rapid rotation. 63 



Also, in the spherical triangle P L N', we have — 

 Cos I' = cos « sin A sin (3' + cos A cos /8', . . . (B) 



Cos a being positive in the one expression and negative in 

 the other, according to the supposed value of a. 



When a=o, we obtain from the first of these equations — 

 Cos I = sin A sin /3 + cos A cos /3 = cos (A ^ /3), . (C) 



Hence I:±:X — /3, the minimum angle of incidence for the 

 first mirror, when M' coincides with M. 



Again, when a = cr=180^, that is, when M' arrives at »t, we 

 have — 



Cos I = —sin A sin /3 + cos A cos /3 = cos (a -f- /3), . (D) 



Heftce, I = X + /3 being the maximum angle of incidence for 

 the same mirror. 



The general formula (A), which expresses the value of the 

 angle of incidence for the first mirror, for every value of the 

 angle of rotation, admits of a great variety of forms, some of 

 which become extremely simple, when particular values are 

 assigned to a, as well as to \ and j8. Thus, if x=/3=45°, we 

 have, in the first and fourth quadrants of rotation — 



CosI = ^cos« + i = i±|^^ = co82^«, . . . (E) 



Moreover, in the second and third quadrants of rotation — 



Co8l=-^co8« + -i=^^=|^ = sin«i«, . . . . (F) 



Hence, cos I + cos F = 1, when a = ^ = 45". 



K/, o/\o 1 _ 3 4-cos<« l+cos^i« ,_,^ 



A = ^ = 30, we have cos I = — -!-- = o (G) 



A J T «/^o T 1 -i- 3 COS* ,__. 



And if A = /8 = 60°, COS I = J (H) 



Moreover, when X and jS are complemental angles — 



Cos I = COS « COS /3 sin /3 H- COS /3 sin /3 =r (1 -f- cos «) cos /8 sin /3 = 



=: COS 'J « sin 2 /8, or cos 2^ « sin 2 A, (I) 



When 



/8 = 15° and A = 75°, or a = 15° and ^ = 75° cos I = J cos «i «, (K) 



When |S=/3', that is, when the two mirrors have the same 

 inclination to the plane of rotation, we obtain, by the formulsB 



(A) and (B)— 



