Photographic Impressions of Polarized Rings. Ill 



On working out this expression, and writing x for 2r 2 , we find 



COS ( -xTTX \ COS ( -7TX ) 



1 = 1 + 



tf 2 -l 



(3) 



As the full discussion of this formula presents no difficulty it 

 may be left to the reader. The last factor in the numerator of 

 the fraction is that where fluctuations correspond to the rings. 

 Whenever x passes through an odd integer greater than 1 the 

 first factor changes sign, and there is a dislocation or displace- 

 ment of half an order, but when x passes through the value 1 

 the denominator changes sign along with both factors of the 

 numerator, and there is no dislocation. When x becomes con- 

 siderable the denominator x 2 — l becomes very large, and the 

 fluctuations of intensity become insensible. 



The following table contains the values of I calculated from 

 the formula (3) for 16 values of x in each of the first 7 orders 



of rings. In passing from one ring to its consecutive the angle 



5 



pirx increases by 2tt, and therefore x by 0*8. The sixteenth 



part of this, or 0*05, is the increment of x in the table. Each 

 vertical column corresponds to one order. The value of x cor- 

 responding to any number in the table will be found by adding 

 together the numbers in the top and left-hand columns. 



A curve of intensity might easily be constructed from this 

 table by taking ordinates proportional to the numbers in the 

 table, and abscissse proportional to the values of r, and therefore 

 to the square roots of the numbers 0, 1, 2, 3, 4, &c. But the 

 form of the curve will be understood well enough either from 



