in the Neighbourhood of the Polarizing Angle. 11 

 In order that the light may vanish we must have both 



T+ S tan ol tan cos 7 = 0, .... (11) 



M + Stanatan/3sin 7 = (12) 



In the neighbourhood of the polarizing angle, M, S vary 

 slowly, but T varies rapidly. Hence, if 7 be given, we may 

 regard (12) as determining tan a tan /3, while (11) gives T, 

 and thence the precise angle of reflexion for the dark spot. 

 If there be no ellipticity, M = ; whence tanatan/3 = 0, 

 T=0, indicating, as was to be expected, that the dark spot 

 occurs at the Brewsterian angle. 



But this law is not universal. For if there be no compen- 

 sator, 7=0, and we have as the expression for the intensity, 



(T cos a cos + S sin a sin /3) 2 + M 2 cos 2 a cos 2 j3. 



Hence, if a and j3 are small, the second term cannot be made 

 to vanish, and the brightness is a minimum when 



T=— Stan i tan 0. 



The position of the nearly dark spot is thus dependent upon 

 a, /3, and assumes the Brewsterian position only when either 

 « or /3 vanishes. 



In the case of a quarter-wave-plate, 7= + 2 7r ? an d the 

 equations become 



T = 0, &=tanatan/3=+M/S. . . . (13) 



The dark spot thus occurs at the Brewsterian angle, while 

 tan a tan j3 gives the value of M/S, viz. the k of Jamin. 

 Accordingly if yS be set to any convenient angle*, and F be 

 then adjusted so as to bring the dark spot to the central 

 position, the product of the tangents of a and /3, each mea- 

 sured from the zeros obtained in the preliminary adjustments, 

 gives k. 



But the following procedure not only affords greater 

 delicacy, but makes us comparatively independent of the 

 positions of the zeros. Set /3, e. g.,to + 30°, and find a ; then 

 reset (3 to —30°. The new value of a would coincide with 

 the old one if there were no ellipticity ; and the difference of 

 values measures a upon a doubled scale. If a! be the second 

 value, so that the difference is a / — «, then 



k = tan 30° tan i (a! -a), 



* In my apparatus it was convenient to throw the fine adjustment 

 upon 06. 



