488 Dr. L. Silberstein on 



In fact, squaring (A) and remembering that n^ — 1, we have 

 r 2 _ r 2 + 4( nir )2 _ 4( nir )2 _ ^ 



identically. Thus, r being a unit vector, so is also r x . 

 Notice in passing that 



X2 1 f2 1 = {2 1 2 = l — 4n x .n 1 + 4(n 1 .n 1 ) 2 = l — 4n x . ni + 4% . n 1? 



i. e. 12i 2 = l, which is an obvious property. 



Let, now, the ray r l5 reflected from the first mirror, 

 impinge upon a second mirror whose normal is n 2 , again a 

 unit vector. Then the ray r 2 , reflected from the second 

 mirror, will be 



r 2 =fl 2 ri=£V i ir, 



where X2 2 = 1 — 2n 2 . n 2 ; and since f2 2 applied to the vector 

 X2 : r gives the same result as 12 2 ^i applied to r (associative 

 property), no separating signs are needed. Thus the reflector 

 of the double mirror 1, 2 is simply X2 = f2 2 f2j. Similarly, 

 if r 2 impinges upon a third mirror, the reflected ray is 

 r 3 ~X2r, where ^s^H^^ and so on. 



Thus the reflector of a multiple mirror consisting of any 

 number k of plane mirrors, taken in the prescribed succession 

 1,2,3, ...k, is 



f2=12 K ...f2 3 I2 2 n 1 =ni2 l , . . . . (1) 

 where l=K 



O i =l-2n l .n i , (2) 



iii being the unit normal of the zth mirror *. If r is the 

 incident and r' the finally reflected ray, we have 



r' = I2r, (la) 



and, by what has been said before, 



/ 2 = r 2 (3) 



If, therefore, r is a unit vector, so is r', and if be the 

 angle between the incident and the finally reflected ray, we 

 have simply 



cos0=r!2r (4) 



* Notice that, for a given multiple mirror, i2 will, in general, be 

 different according- to the order of succession of the component reflexions, 

 that is, of the suffixes *. In short, the "products" £2 3 Q 2 &i, etc., are 

 associative, but, in general, not commutative. They become so only for 

 certain multiple mirrors, i. e. for some particular arrangements of the 

 component mirrors. If a finite beam of parallel rays r impinges upon, 

 say, a triple mirror, some pencils may be reflected as {2 3 J2 2 &ir, others as 

 £l 3 QiQ 2j and so on. Thus the several reflected pencils will, in general, 

 diverge from one another. 



