555 



zation improves. When the plates are very numerous, i/> 2 = 1 at the 

 polarizing angle, and on both sides of it decreases rapidly, whereas ^ lf 

 which is always small, suffers no particular change about the polar- 

 izing angle. Hence in this case x must be a minimum a little beyond 

 the polarizing angle. Let us then seek the angle of incidence which 

 makes x a minimum in the case of an arbitrary number of plates. 

 We have from (20) and (2), 



sin 2 <7 sin 2 sin 2 a cos 2 0+ (2m 1) sin 2 cos 2 o 

 sin 2 o-+(2z I)sin 2 0' sin 2 a cos 2 sin 2 cos 2 a 



^_ sin 2 a cos 2 + (2m I ) sin 2 cos 2 <r__ l _ _ 2m _ 



I)sin 2 cosec 2 + (2m- 1) cosec 2 a* * ' 



Hence x is a minimum along with cosec 2 + (2m 1) cosec 2 a. Differ- 

 entiating, and taking account of the formulae (22), we find, to deter- 

 mine the angle of maximum polarization, the very simple equation 



I)cos<7sin 2 0=0. . . . (24) 



For any assumed value of t from ta to > this equation gives at 



once the value of m, that is, the number of plates of which a pile must 

 be composed in order that the assumed incidence may be that of 

 maximum polarization of the transmitted light. The equation may 

 be put under the form 



~ _ ,_ tan or sin a _ 



tan0'sin0~~ V/pTpa* 



Now we have seen that both p l and p 2 continually increase, and 



7T 



therefore m continually decreases, from i='& to z=o' ^ t ^ ie ^ rs * 

 of these limits p 2 = 0, and therefore m= GO. At the second p 1 =p 2 = 1, 

 and therefore m=l. Hence with a single plate the polarization of 

 the transmitted light continually improves up to a grazing incidence, 

 but with a pile of plates the polarization attains a maximum at an 

 angle of incidence which approaches indefinitely to the polarizing 

 angle as the number of plates is indefinitely increased. 

 Eliminating m from (23) and (24), we find 



X = cos cos or, ...... (25) 



which determines for any pile Xi> the defect of maximum polariza- 

 tion of the transmitted light, in terms of the angle of incidence for 



