r 487 ] 





LI. On Multiple Reflexion. By L. Silberstein, Ph.D., 

 Scientfiic Adviser to Adam LLilger, Ltd.* 



THE purpose of the present paper is to give a very simple 

 method of dealing with reflexions from any number of 

 plane mirrors. The subject has been taken up in connexion 

 with some technical problems concerning the construction of 

 the kind of triple mirrors known as central mirrors. 



Consider first a single plane mirror. Let the unit vector 

 iii represent its normal, drawn away from the reflecting side. 

 Let the direction of the incident ray be given by the vector r, 

 and that of the reflected ray by r lm The tensors r, r± of 

 these vectors are irrelevant. It will be convenient, however, 

 to make them equal. If both are taken as unit vectors, then 

 their scalar product r^ will give at once the cosine of the 

 angle included between the incident and the reflected rays. 



Now, by the fundamental law of reflexion, r x - r = AB has 



(A) 



the direction x^ and the size — 2to.iY, that is, 



r 1 = r-2n 1 (n i r), .... 

 or, using the dot as separator, 



r 1= = [1-211!. n 1 ]r=H 1 r. 



Thus, the linear vector operator or the dyadic, 



H 2 = 1 — 2x1! . i^, 



when applied f to the incident ray r, gives the reflected 

 ray r x . In view of this property the operator XI x can be 

 called the reflector belonging to the mirror in question. It 

 is a pure versor, i. e. it leaves intact the tensor of the operand. 



* Communicated by the Author. 



t It may be applied to the operand either as prefactor, r 1 = i2 1 r, or as 

 a postfactor, t x = tQ. x , the operator for a simple mirror being self-conjugate 

 or symmetrical. The operator for a multiple mirror, however, is not 

 symmetrical, and to avoid confusion we shall use it always as a prefactor. 



