130 Prof. E. C. Pickering on the Polarization of 



These results are perhaps better shown in figs. 1 and 2, in 

 which abscissae represent values of i, and ordinates percentages 

 of polarization. In fig. 1 the four highest curves represent the 

 polarization of the beams reflected by 1, 2, 8, and 20 surfaces. 

 The other four curves give the corresponding refracted beams. 

 Fig. 2 gives all the curves of Table II. relating to 20 surfaces — 

 the five curves corresponding to A, B, the intensity of the refracted 

 beam, and the polarization of both the reflected and refracted 

 beams. When i=z0, both the reflected and refracted beams are 

 unpolarized. With 10 plates of glass about half the light is re- 

 flected, the transmitted ray being but little brighter than that 

 reflected. With 1 or 2 surfaces the reflected beam increases as 

 i increases ; with 8 surfaces it remains nearly constant up to 50°; 

 while with 20 surfaces a marked diminution is perceived. This 

 very remarkable result may be expressed by saying that 10 plates 

 of glass transmit more light obliquely than normally. The ap- 

 pearance to the eye confirms this result, but it deserves a careful 

 photometric proof. At 57° the reflected ray is, of course, in all 

 cases totally polarized ; but at other angles the amount of polar- 

 ization is greater the less the number of surfaces, instead of the 

 contrary, as might have been anticipated. 



With the refracted ray quite a different law holds. For 1 sur- 

 face the polarization increases from 0° to 90° ; with 2 surfaces 

 it becomes sensibly constant near 90°; while with a larger num- 

 ber a distinct maximum is obtained. It is commonly supposed 

 that the greatest effect is obtained at the angle of total polariza- 

 tion. But the maximum is sensibly beyond this, unless a very 

 large number of plates are employed ; and hence it seems probable 

 that a bundle of plates, polarizing by refraction, would give the 

 best results if set at a greater angle than 57°, as 65° or 70°. 

 The transmitted ray, however, diminishes rapidly for large angles 

 of incidence. A very large number of plates are required to render 

 the polarization nearly complete, which accounts for the light 

 always remaining when even the best polariscopes by refraction 

 are crossed. At 90° all the refracted beams are polarized by the 

 same amount of 41*2 per cent. Or, at grazing incidence, the 

 amount of polarization is independent of the number of plates, 

 one polarizing as completely as a hundred. This number, 41*2, 

 may be obtained as follows. Differentiate the value of A in 

 terms of i and r, and make 2 = 90°, when the refracted beam 

 will equal 1— A = 4tan?-$ = 3*376 di, since when i == 90° 



r = 40° 10 ; *7. In the same way 1— B= -r- — di = S-llodi: and 



J sin 2r 



applying to them the formulae for the polarization of the refracted 

 beam, we find it equal to 41*2. 



To show how far these effects are due to internal reflection, 



