276 PROFESSOR POWELL ON THE ELLIPTIC POLARIZATION OF LIGHT 



intermediate between the dark and bright systems. In the lower degrees of ellipti- 

 city this is modified by, and passes into, that just described. ' 



(14.) We may illustrate the application of the formula by one or two particular 

 cases : — 



1st. If we suppose at some incidence a=/3 while e<2, then )^=45° will give branches 



with dislocation at •v^=0°, but none at -4^=45°; that is the nearest approach to the 

 dark system. This agrees with formula (15.), where in this case I is a minimum 

 when y^=i4b°, also with (18.). 



2ndly. On the same supposition %=0 will give complementary changes both in 

 the ground and in the rings at%^ = 0, and similar changes, though less conspicuous, at 

 '4/ = 45 ; that is the intermediate system. 



Srdly. For the general values of a and /3 at the maximum cos ^=0 ; and %=0 gives 

 branches with dislocation at -^=0, but no change at 'vf/=45°; or the darkest system. 

 But %=45° will give (since a>j3) a complementary change in the ground at -4/ =45°, 

 and branches with dislocation at 1^=0; or the distorted system. 



But if in this case a=/3, or the polarization be circular, the term (4.) disappears, 

 and there is no distorted system in any position of p(^. 



Observation shows this to be the case in perfectly circular light, and very nearly 

 so in the higher degrees of ellipticity. 



(15.) For the branches, when "4^=0, for x in general we have 



I=a2 gin2 ^-j-j32 cos^ )^ — a/3 sin 2>^ cos ^. 



Hence, on making successively cos f=l, cos ^=0, cos ^= — 1, &c., it is obvious 

 that the intensity of the branches for the maximum ellipticity would be a mean be- 

 tween that in the dark and bright systems of plane polarized light if a and /3 were the 

 same in the respective cases, which we shall see is the case; at all events, this relation 

 of the intensities agrees with observation as far as the eye can judge. 



(16.) Again, for the maximum ellipticity, 



I=/32+(a2-/32)sin2%, 



which can never be =0 ; or the branches are never absolutely dark ; but it is evidently 

 a minimum when %=0, and a maximum when p(^=90°, in which cases respectively 

 I=/32, or I=a2. If the polarization were circular these values would be equal, or 

 the brightness the same in all positions of the analyzer. 



(17.) For incidence 0°, the expression (15.) being made =0, or, 

 I = a2 sin2 %+/32 cos^ p^^ — 2aj3 sin % cos %=0, 

 we have for the position of the analyzer for absolutely dark branches, as in (3.), 



asin;i(i~j3cosx=0, 



or tan %= - tan t 



