Multiple Reflexion. 493 



More especially, if the incident ray is equally inclined to 

 the three reflecting planes of the regular pyramid, we have 



1 ,0) 



r x = r 2 = r 3 = — -= cot - 



and (13 a) becomes 



3- cos 6 



V3 2' 



(l+^cos^cot 2 **. . . (13 6) 



In this case the angle 6 is independent of the order of 

 reflexion?, as was to be expected. The reflected beams, 

 although not parallel to one another, are equally inclined to, 

 and symmetrically disposed around, the direction of the 

 incident beam r. These reflected beams coincide in direction 

 when, and only when, o) = 90°, i. e. when the mirror becomes 

 an orthogonal and, therefore, a central mirror. 



Further discussion of the above formulse and the con- 

 struction of similar ones for quadruple and more complicated 

 mirrors are left to the reader. Here but two further 

 remarks on the general reflector 12 : — 



Reversal of the order of reflexions. — Let r be the incident 

 ray, r ; the finally reflected ray when the order of reflexions 

 is 123... /c, and s' the finally reflected ray when the order of 

 reflexions is a:... 321. Then, if f2 = n K ...n 3 n 2 f2 1 , as in (1), 



r' = Hr, s'=r0 3 



whence 



rr'=rnr, s'r = rnr, 



and therefore, for any multiple mirror, 



r'r = s'r, (14) 



while r's' = r(l 2 r. That is, the reflected rays r', s', although 

 not parallel to one another, are always equally inclined to the 

 incident ray r. The equality #i23 = #;52i 5 exhibited by (13 a) 

 is but a special instance of this general property. 



Images of given objects. — Hitherto we have considered r 

 and r / as determining the directions of the incident and the 

 reflected rays. In order to obtain the image of a given 

 point-object, let the end-point of the vector 1% drawn from a 

 fixed origin 0, determine the position of the object, and the 

 end-point of r', drawn from the same origin, the position of 

 the image. Then, in the case of a simple mirror, we have 

 again 



r'zriVKl-L^.n^r, 

 provided that is a point of the reflecting plane itself. 



