Doubly Refracting Plates and Elliptic Analysers 



47 



an equation which is independent of c. Substituting values, sim- 

 plifying and collecting terms : 



[ i — ( cos 2 r] sin 2 ir A\ cos n — sin 2 rj cos 2irN l ) 2 ']t' 1 

 — ( cos 2 r] sin 2 7r A^ cos w — sin 2 rj cos 2 7r A 7- , ) 2 =o 



which orives 



t=- 



a — dcos?i 



1 i — (a — 6cos)i)' 2 



where «=sin2^cos2 7rA / " 1 , and ^=cos2>;sin2 7rA 7 1 . 

 a- — £cos«=sinx, then: ' 



t=tg x and e=*g}4x 



7T 



These equations are still usable when 2ttN x = — 



(85) 



Letting 



(86) 



This method evidently has all the experimental advantages of 

 Stokes's analyzer, with the added sensibility of a halfshade sys- 

 tem. It has the disadvantage of slightly more complicated for- 

 mulae for unbalanced halfshades, but for balanced halfshades, 

 as has been shown, the formulae are the same. 



10. Sensibility of Halfshade Systems 



Let the intensity of the light emerging from one side of the 

 halfshade system be I, and from the other /'. Let one-half the 

 difference in intensity between the two halves, y 2 (I — I J )=Al, and 

 the average intensity of the two halves, j4(I-\-I')=I m . Then the 

 condition for an observed match is: 



A/ 



T > 1 AfiL^P^ ■ ■ ■ ■ ) 



(87) 



where f(/ m ,a,(3,y, . . . ) is the photometric function. a,/?,y, . . . 

 are constants depending on the sharpness of the dividing line be- 

 tween the two halves of the field, the parallelism and purity of the 



203 



