determining the position of the Plane of Polarization. 135 



light are given by the displacements along the axes of the 

 ellipse, and by the inclination of an axis of the ellipse to some 

 direction fixed in space. Let the displacements £ and rj be 

 parallel to the axes of the ellipse, and let the axes of x and y 

 be fixed in space, z being the axis along which the light 

 travels ; and let co be the angle between the axes of x and f . 

 If then c 2 be the intensity of the light, tan y the ratio of the 

 axes of the ellipse, the vibrations of the light are given by the 

 equations 



t — c cos y cos — (vt—; 



(i) 



7] = c sin 7 sin — (yt—z). 



-»).T 



The angles <y and co are known whenever we know the 

 history of the light ; how it became converted from plane- 

 polarized into elliptically-polarized. If, for instance, the 

 change took place in passing through a doubly refractive 

 medium whose axes are those of x and y, then 



tan 2co = tan 2a cos ft, *) 

 sin 2y = sin 2a sin jS, J 



(2) 



where /? is the total angular retardation, and a the inclination 

 of the initial plane of vibrations to that of xz. In these equa- 



tions /3 is a function of X, viz. — (/a —/j, 2 )z, where /x x and /jl 3 



are the indices of refraction along the axes of x and y respect- 

 ively. If a is small, variations in co due to X are not important; 

 but if a is large this is no longer the case, as we shall even- 

 tually see. 



Let us now pass the light (1) through a biquartz which is 

 such that the plane of polarization of light, of wave-length X, 

 is turned through an angle cf>. This rotation will simply turn 

 the ellipse as a whole, and not affect the ratio of the axes. 

 Hence for upper half of the biquartz co becomes co + cf> } and for 

 lower half co — cp>. 



Let the light be now ana- 

 lysed by a Nicol whose plane 

 of vibrations makes an angle 6 

 with the plane of xz. If then 

 h 2 be intensity of light passing 

 through the upper half of bi- 

 quartz, and k 2 that of light 

 passing through the lower half, 

 we have, as usual. 



