8 , PROCEEDINGS OF THE AMERICAN ACADEMY 



able result may be expressed by saying that 10 plates of glass transmit 

 more light obliquely than normally. The appearance to the eye con- 

 firms 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 polarization 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 surface 

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

 sensibly constant near 90° ; while with a larger number a distinct maxi- 

 mum is obtained. It is commonly supposed that the greatest effect is 

 obtained at the angle of total polarization. But the maximum is sen- 

 sibly beyond this, unless a very large number of plates is employed ; 

 and hence it seems probable that a bundle of j^lates, polarizing by re- 

 fraction, 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 is 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 refi'acted 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 jjolarizing as 

 completely as a hundred. This number 41.2 may be obtained as fol- 

 lows. Differentiate the value of A in terms of i and r, and make i = 

 90°, w^hen the refracted beam will equal 1 — A =.4: tang r di^ 3.37 Q 

 di, since when i = 90°, r = 40° 10'.7. In the same way 1 — B = 



8 . 



-: — ~ di = 8.115 di, and applying to them the formulae for the jDolar- 



ization of the refracted beam, we find it equal to 41.2. 



The last portion of Table VI. was determined in part by graphical 

 interpolation, and hence the results are less accurate than those of the 

 other tables. It is believed, however, that the errors are much less 

 than those of any known method of testing them experimentally, and 

 hence give all the accuracy needed for practical purposes. 



To show how far these effects are due to the internal reflection. 

 Table VII. has been computed for the same number of surfaces, sup- 

 posing that no secondary reflection takes place. The first column 

 gives the angle of incidence, the next three the polarization of the 

 reflected, and the last three that of the refracted beam. A comparison 

 with Table VI. shows that while the reflected beam is affected but 

 little, a great change takes place in the transmitted light. A simi)le 

 method of expressing the difference between Tables VI. and VII. is to 



