379 



tan 20 ~ --;: — -—. rr-, 



2/+(v' + v)cos2a 



sin 2/3 = -, =—777 -r- ; 



in which y and ^- are constant quantities given by the expres- 

 sions „ ( ^\ ( . ^\ ' I \ 

 / = (^M _ - j cos X, ^ = (^M + - j sm X, (b) 



and V, v are quantities depending on the angle of incidence 

 i, in the following way. Let i' be an angle such that 



sin i M , . 



sini' ~ cosx' 



and put 



cos^ 



Mj 



(!>) 



then will 1 > _ f -^g M 



fl V 



The angles d and j3 are given by immediate observation with 

 the instrument ; and from their values at any incidence, and 

 for any azimuth a of the plane of primitive polarization, we 

 can find the constants m and x> which we may afterwards 

 use to calculate the values of and j3 for all other incidences 

 and azimuths, in order to compare them with the observed 

 values. It is indifferent, in the formulae, whether B be re- 

 ferred to the major or the minor axis of the elliptic vibration, 

 as also whether tan /3 be the ratio of the minor to the major 

 axis, or the reciprocal of that ratio; but in what follows 

 we shall suppose Q to be the inclination of the plane of inci- 

 dence to that axis, which, when a is 45° or less, is always 

 the major axis ; and /3 shall be supposed less than 45°, in 

 order that its tangent may represent the ratio of the minor 

 axis to the major. 



When the azimuth a is equal to 45°, the formulae (a) become 



tan 20=^, sin2j3 = -^; (f) 



from which we may deduce the remarkable relation 



cos 20"/' ^^^ 



