78 JAMIN ON METALLIC REFLEXION. 



these values vary with the number of reflexions ; and experiment 

 shows, that of them there is one for two reflexions, two for three 

 reflexions, and in general a number equal to the number of re- 

 flexions minus one. Sir David Brewster does not seem to have 

 remarked this relation between the number of reflexions and 

 that of the angles of renewed polarization. It is a veiy simple 

 consequence of the manner in which the difference of phase 

 varies, and the reader will shortly be able to recognise it ; for 

 the moment we content ourselves with pointing out the use to 

 be made of this fact. 



In order that two rays polarized at right angles, whose phases 

 ditfer, may, on uniting, constitute a polarized beam, it is neces- 

 sary that the differences between their phases be equal to 



-, or 2 -, or 3 -, 



Therefore, if the polarization has again become rectilinear after 

 a certain number of reflexions efl^ected at the same incidence by 

 the same metal, it is because the difference of phase of the two 

 rectangular rays has become equal to a multiple of a semi-undu- 

 lation, and the whole question is reduced to finding this mul- 

 tiple. Now this is very easy. We know, in fact, that after a 

 single reflexion, the difference of phase goes on increasing from 

 the incidence of 0° when it is nothing, up to that of 90° ; there- 

 fore, for the angle nearest to 0°, which will restore the [)olariza- 

 tion after (m) reflexions, the difference of phase will be the 



smallest multiple ( ^ ) ; for that which comes after, 2 - ; and so 



on to that nearest to 90°, where it will be (m — 1) — . Therefore 



we shall have for a single reflexion at the same angles, the fol- 

 lowing values of the difference of phase : 



L ^ 2^ X 3 X m—1 X 



m ' 2' m ' 2' m ' 2 m '2' 



The differences of phases will be expressed as a function of 



\ n 



— by a fraction — , {n) taking all integer values from (1) up to 



(m — 1), m representing the number of reflexions. 



It follows from this, that (m) and (w) varying, the same value 

 of the fraction will be reproduced frequently for different num- 

 bers of reflexions : thus, after 2, 4, 6, 8 reflexions, we shall have 



