Multiple Reflexion, 48 9- 



In particular, when the arrangement of the component 

 mirrors is such that 



o=-i, 



then every incident ray will be sent back parallel to its own 

 path. {Such multiple mirrors are called central mirrors. 



By a well-known theorem of vector algebra, the general 

 reflector (1), being a pure versor, could always be expressed 

 by I2=.a.i + b J + c .k, where both a, b, c, and i, j, k are 

 some normal systems of unit vectors, both right-handed, 

 or both left-handed. No use, however, will be made here 

 of this fundamental property, since obviously the most 

 natural entities to represent the properties of any multiple 

 mirror are the normals n^ n 2 , etc. themselves. These appear 

 in fl as dyads, such as n^ . n x or n 2 .n!, etc., or as scalar 

 products n 1 2 = l, n 1 n 2 = cos (n l5 n 2 ), and so on. In certain 

 cases it may be advantageous to employ the unit edges of 

 consecutive mirrors, i. e. apart from the scalar factors 

 sin (n b n 2 ), etc., the vector products Yn^, and so on. 



The utility of* the above method of treatment, in which the 

 clumsy and often unmanageable formulae of spherical 

 trigonometry are replaced by the simple operator (1), needing 

 no drawings whatever^ will best be exhibited on a number of 

 examples. We shall begin with the simplest case of a double 

 mirror and then proceed to more complicated ones. In 

 each case the procedure will consist in simply " multiplying " 

 out the dyads n . n contained in the component reflectors. 

 And in doing so we have only to remember that juxtaposed 

 vectors, not separated by dots, are fused into ordinary scalar 

 products. Thus, n 2 . n 1 n 3 . n 2 = n 1 (n ] n 2 ) . n 2 = a 12 n 1 . n 2 , where 

 a 12 = n 1 n 2 — cos (n b n 2 ). In short, the "product" of any 

 number of dyads is always a dyad, including an ordinary 

 scalar factor. In what follows we shall employ the general 

 notation 



niii ; -= cos (n;, xL 1 ) = a i j = aj i (5) 



The incident ray or operand r need not be written out in 

 each case ; it is enough to deal with the operators fl; and 

 with their resultants, i. e. with the reflectors themselves. 

 All of the operations involved being associative as well as 

 distributive, the multiplication will be done as in ordinary 

 algebra, the only precaution (owing to non-commutativity) 

 being the preservation of order. 



Double mirror. — The unit normals of the component 

 mirrors being n b n 2 , and n 1 n 2 = a 12 = a the cosine of their 



Phil. Mag. S. 6. Vol. 32. No. 191. JSov. 1916. 2 L 



