Multiple Reflexion. 491 



Returning to the general formula (6) we have for the 

 angle 6 between the rays r and r', cos# = rllr, i. e. 



J(l— cos 0)=r 1 2 +r 2 2 — 2ar 1 r 2 , 



where r 1 = rn,, r 2 =rn 2 are the projections of the incident 

 ray upon the mirror normals. On the other hand, since 

 ;r = r 1 n 1 +r 2 n 2 + (re)e and r is a unit vector, we have 



r : 2 + r 2 2 + 2ar x r 2 = 1 — (re) 2 , 



so that the last equation can be written 



i(l-cos5') = l-4ar 1 r 2 -(re) 2 . ... (9) 



This gives the angle 6 for any double mirror and for any r. 



From this general formula we see at once that there is no 

 such double mirror which would send back (parallel to its 

 own path) every incident ray, in short, that there are no 

 central double mirrors. In fact, cos#=— 1 would mean 

 Aai\r 2 + (re) 2 = 0, and this cannot be fulfilled for all directions 

 of r. 



Triple mirror. — The reflector in this case is n = i2 3 n 2 lli, 

 that is, the product of 12 3 = 1 — 2n 3 . n 3 into the operator (6) 

 of the preceding section. Writing, therefore, n^ — a 12 , etc., 

 we have, for any triple mirror (the order of reflexions being 

 1, 2, 3), 



12 = n i23 = l — 2[n 1 .n! + n 2 .n 2 + n 3 .n 3 ] 



+ 4[a ]2 n 2 . n 2 + a 23 n 3 . n 2 + « 31 n 3 . x^] — 8a 12 a 23 n 3 . n 2 . (10) 



Here again, the third and fourth terms being non- 

 symmetrical, an incident beam of parallel rays will give rise 

 to six reflected beams X^i 23 r, fl u . 2 r, etc., which will, in 

 general, not be parallel to one another. These reflected 

 beams become parallel to one another, i. e. 11 becomes 

 independent of the order of reflexions 1, 2, 3, when, and only 

 when, 



i. e. when the three component mirrors are perpendicular to 

 one another. In that case, n^ n 2 , n 3 being a triad of normal 

 unit vectors, the dyadic n^ . ii! -f n 2 . n 2 + n 3 . n 3 becomes an 

 idemfactor or 1, and therefore, 



H=-l ; r' = -r (11) 



That is to say, every incident ray is sent back parallel to 

 itself. The orthogonal triple mirror is a central mirror. 

 Returning to the general triple mirror, let r u r 2 , ?•■> be the 



2 L2 



