492 Dr. L. Silberstein on 



direction cosines of the incident ray, i. e. the projections o£ r 

 upon n 1? n 2 , n 3 . Then the direction cosines r/, r 2 ', r s ' of the 

 reflected ray r' will be, by (10), 



n^nCi-^rO, J 



r 1 '=r a (l-2r 1 + 4« u r J ), V. . (12) 



r 3 ! = r 3 (1 — 2r 3 + 4a 23 r 2 + 4a 31 ri - 8a 12 a 2 3^i ) • ) 



These scalar formulae can at once be used for numerical 

 calculation. 



The angle 6= (r, r')' is given by cos <9 = rOr, i. e. by (10), 



^-±_ = Tl 2 -t- r 2 2 4- r 8 2 — 2 [a 12 ?v 2 + a 2i r 2 r B + a zl r z r{\ + 4a 12 a 23 r 3 r 1 . 



On the other hand, we have, by squaring 

 r^iiii + rsiis + rgiis*, 

 r 2 = 1 = r x 2 + r 2 2 + r 3 2 + 2\_a 12 r x r 2 + a 23 ?Vs + ag^^] , 



and therefore, for any incident ray, whose order of reflexions 

 is 123, 



i — POS (/ 



_ = ri 2 + r 2 2 + r 3 2 + 2a 12 a 23 r 3 r v . . (13) 



From this general formula, which enables us to calculate 

 at once the angle 6 for any incident ray, we can see also 

 that the orthogonal mirror is the only possible central 'mirror. 

 In fact, the right-hand member of (13) becomes equal to 1 

 (i. e. cos0=— 1) for any direction of r, when, and on] y 

 when, n 1? n 2 , n 3 are normal to one another. 



If the three mirrors constitute a regular pyramid, i. e. if 



«i2 = «23 = «3i= cos co, say, .... (14; 

 then the reflector (10) becomes 



11 = 1 — 2[n 1 .n 1 + n 2 .n 2 4-n 3 . n 3 ] 



+ 4a [n 2 . iix + n 3 . n 2 + n 3 . tl{\ — 8« 2 n 3 . n x , (10 a) 

 and the formula (13) for the angle = (r, r'), 



^| OS - 6, =r 1 2 + r 2 2 + r 3 2 4 2cos 2 ft)7V3, . (13 a) 



for any incident rav, the order of succession of the reflexions 

 being 123, for (10 a), and either 123 or 321 for (13 a). 

 If the order is 231 (or 132) or 312 (or 213), we have only 

 to write, in the last term of (13 a), r 2 r x and r z r 2 respectively. 



* We assume, of course, that xii, n 2 , n d are not coplanar, i. e. that the 

 three reflecting planes constitute a pyramid, not a prism. 



