Polarization of the Liglit of the Sky. 517 



45° + R ; with this it is only necessary to change the sign of 

 cos 2(f>.) 



Then, admitting the formula sin «=tan <£, we have 



B 



sin a= — • 

 A 



We may put this in the form A 2 = A 2 cos 2 a-j-B 2 . Now, if we 

 look at a point on the beam in a direction enclosing an angle 6 

 with the beam in a horizontal plane, we have 



Vertical component ........ =A, 



Component of the horizontal A perpendicular 



to the line of vision =A cos 6, 



Component of B = Bsin#. 



But we cannot assume that the A and B seen from any angle 

 6 are always the same. In the first stage we may assume this 

 with great probability ; in the second it is easy to see that there 

 could be no neutral angle if A and B were always the same ; for 

 then we should have as the condition for the neutral angle 



A 2 = A 2 cos 2 0-fB 2 sin 2 0, 



which can only be satisfied by = 0. Let, therefore, the changed 

 values of A and B in the horizontal plane for angle 6 be repre- 

 sented by A', B'j then the condition for the neutral angle 

 becomes 



A 2 =A' 2 cos 2 + B' 2 sin 2 6>. 



Comparing this with the empirical formula 



A 2 = A 2 cos 2 +B 2 , if 6=u, 

 we get 



B'sin0=B, A=A'. 



Now, if we took B to be a measure of the stage and assumed 

 (what we know is not true) that the stage is so advanced by 

 the increasing thickness of the beam that B' is proportional 

 to the thickness observed, then (since the thickness observed 



at inclination is inversely as sin 0) we might put B'=— : — -^ 



which would reduce all the experimental phenomena to one 

 simple law. Regarding the experiments alone, there is nothing 

 to show that this explanation is not correct, except the pro- 

 bability, derived from the very perfect theory of the first stage, 

 that the size of the particles, and not the thickness of the 

 observed stratum, is connected with the advancement of the 

 stage. But the sky-observations are conclusive, and show at 

 once that this explanation will not hold ; for there, as I have 

 already said, we observe directly the change of stage due to the 



