• Images from Plane Mirrors in rapid rotation. 65 



N' ; in which position L M' is evidently the angle of incidence 

 for the one mirror, and L N' that for the other. But in the 

 triangle M'L N' we have M'N' = jS + jS' ; L M' = I ; and L N' = I' ; 

 and therefore, if the auxiliary angle sought be denoted by p, 

 we have 



Cos ozz Co8(/3 + /30 — cosIcosF .Q. 



^ Sin I sin r, ^^ 



Let M'E be made equal to L M', then M'E being the measure 

 of the angle of reflection, corresponding to the angle of inci- 

 dence I, it follows that P E will represent the direction of the 

 ray after it is reflected from the first mirror. Produce E P, in 

 the opposite direction, till it intersect the sphere, in a point 

 ' F diametrically opposite to E ; and F L will be equal to ^— 2 I. 

 Since E P is the direction in which the light, after reflection 

 from the first mirror, falls upon the second mirror, let a great 

 circle be described through F and N', and the arch F N' will 

 evidently measure the angle of incidence for the reflection 

 from the second mirror, after the ray has been reflected from 

 the first. To determine F N', which we shall denote by /, we 

 have in the triangle F L N', the sides F L and L N', respec- 

 tively equal to ^—2 I, and I, and the angle FLN'=:'T— p. 



TT ^ , , Cos i — cos V cos (v — 21) 



Hence Cos (^r — <p) = g. t/ ♦ / o ta 



^ ^ Sin r. sm (tt — -2 I) 



And Cos { ■=. cos (tt — (p) sin I', sin {v — 2 I) -f- (cos I', cos {tc — 2 1) 



Or, Cos %■=. — cos ^ sin I', sin 2 1 — - cos I", cos 2 I 



Substituting the value of cos <p in equation (Q), and supposing 



at the same time, that ^—^ 



^ , /Cos 2/3 — cos I. cos I'\ c.- T • / n TN T/ «x 



Cos i = I . T — . T. 1 Sin I. sua {'r^2 I) — cos T cos 2 1 



\ — sm I. sm I / 



This equation reduced assumes the simple form — 



Cos i = cos I' — 2 cos I. cos 2 /» (R) 



The same result is derived by computing E N', the supple- 

 ment of F N', in the triangle L E N', in which L E = 2I ; angle 

 ELN' = ^; LN = I; and E N= -jr— i ; or more simply, by 

 equating the values of cos F M'N' in the two triangles L M'N', 

 and F M'N'. 



The formula (R) may be reduced to a more independent 

 form by eliminating cos I and cos T, according to the values of 

 these quantities in the formulae (A) and (B). We thus obtain 



VOL. XL. NO. LXXIX.— JANUARY 1846. E 



