490 Dr. L. Silberstein on 



included angle, we have, by (1), (2), 



n = n 2 H 1 = l — 2[n 1 .n 1 +n 2 .n 2 ] +4a n 2 .n^ . (6) 



or, introducing the vector p = n x — 2<xn 2 , 



Il=l-2p.n 1 -2n 2 .n 2 (7) 



The meaning of this operational equation is seen at once 

 by remembering that r' = Or ; thus 



r-r' = 2(rn 1 )p + 2(rn 2 )n 2 , . . . (7 a) 



i. e. whatever the incident ray r, the vector r' — r is normal 

 to the common edge of the two mirrors. In other words, 

 the projections of r and r' upon the edge are equal to one 

 another. 



From (6) we see that, in general, £2 2 &i differs from fl l n 2 , 

 since the last term 4an a . iij is not symmetrical. Thus, a 

 beam of parallel rays r (broad enough to impinge upon both 

 mirrors) is split by the double mirror into two beams r', r" 

 oblique to one another, e. g. such that 



r'— r" = 4a[n 2 . iix — iix . n 2 ]r. 



In particular, if the double mirror is orthogonal, we have 

 a=n!n 2 = 0, and, independently of the order of reflexions, 



0==i2 2 Xi 1 =0 1 2 = 1 — 211! .ii! — 2n 2 .n 2 . 



But since nij.n 2 , we have 



l = n 1 .n 1 + n 2 .n 2 + e .e, 



where e is a unit vector along the common edge of the two 

 mirrors *. Therefore, for an orthogonal double mirror, 



Q=-[l-2e.e], (8) 



that is to say, the reflexion from such a double mirror is 

 equivalent to the reflexion from a simple mirror whose 

 normal is e, followed by a simple reversal (—1). This is 

 valid for any incident ray r. More especially, if the incident 

 ray is normal to the edge, or re = 0, we have r' = I2r= — r, 

 that is to say, the ray is sent back parallel to itself. The 

 latter property is familiar from ordinary geometrical con- 

 structions. 



* If i, j, k be any normal system of unit vectors, the dyadic 

 i.i-M J-T*k.k is equivalent to 1, or is, in Gibbs's nomenclature, an 

 idem/actor: [i.i+j . j-fk.k]r=r, for any r. Notice that, since e 

 appears only through the dyad e . e, the sense of e is, obviously, a matter 

 of indifference. 



