On the Intensity of the Light reflected from a Pile of Plates, 489 



mine the angle of maximum polarization, the very simple equation 

 cos0 sin 2 v+ {2m— 1) cos a sin 2 0=0. . . . (24) 



7T . 



For any assumed value of i from vs to pr, 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 a sin a 1 

 2m— 1 = = 



tan d ' sin vp l p 2 ' 

 Now we have seen that both j o 1 and p 2 continually increase, and 



therefore m continually decreases, from z=ar to i=n. At the first 



of these limits /o 2 =0, and therefore m= oo. At the second p l =p 2 =z 1> 

 and therefore m=\. 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 0COSC, (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 

 which the polarization is a maximum. We have, from (25), (22), 

 and (24), 



c?^ i = (sin 2 6 cos o--j-sin 2 a cos 6) dio= —2(m— 1) cos a sin 2 6 dio, 

 and cos a is negative. Hence ^ decreases as u> (and therefore i) de- 



creases, or as m increases. For m=l, 2 = 9 anc ^ X^/^ -2 * ^ or 



m= ao, cos ff=0, and therefore Xi = 0> or the maximum polarization 

 tends indefinitely to become perfect as the number of plates is indefi- 

 nitely increased. 



For a given number of plates the angle of maximum polarization 

 may be readily found from (24) by the method of trial and error. 

 But for merely examining the progress of the functions, instead of 

 tabulating i for assumed values of m, it will serve equally well to 

 tabulate m for assumed values of i. The following Table gives for 

 assumed angles of incidence; decreasing by 5° from 90°, the number 

 of plates required to make these angles the angles of maximum pola- 

 rization of the transmitted light, and the value of ^^ which determines 

 the defect of polarization. 

 i= 90° 85° 80° 7o° 70° 65° 60° 56°40'(=or) 

 m= 1 1'330 1-944 2-913 4-921 9*775 30-372 oo 



Xl =-433 -422 -390 '337 '265 -177 *075 



Jan. 30. — " On the Calculus of Symbols."— Second Memoir. By 

 W. H. L. Russell, Esq., A.B. 



"On Internal and External Division in the Calculus of Svmbols." 

 Bv William Spottiswoode, Esq., M.A., F.R.S. 



Phil. Mag. S. 4. Vol. 24. No. 163. Dec. 1862, 2 K 



