66 Mr. A. W. Ward. On the Magnetic Rotation [May 9, 



X = C COS a. COS (27r/X) (vt — z) 



y = c sin a. cos {2ir\X) {vt—z) 



where c 2 is the intensity of the light, and the other symbols have their 

 nsnal meanings. 



Let ft be the angular retardation in passing through the crystal. 

 Then the equation of the emergent elliptically polarised light is 



x = c cos a. cos (27r/\) (vt—z) 



y = c sin x cos (27r/X-) (vt—z + \ftl27r) 



The inclination w of the axis of this ellipse to the axis of x is given 



by 



tan 2u) = tan 2oc cos J3. 



In this equation ft is a function of X and viz., (2irzj\) O^— /* 2 ) where 

 «i and /i% are the refractive indices along the axes of x and y respec- 

 tively. We may, therefore, put ft equal to Jcz where h is a constant 

 for the same medium and wave-length. Hence w is a function of z t 

 and we can find the increase in to due to an increase dz in z. It is 

 given by 



2dtn = —cos 2 2u) tan 2« sin ft . dft, 



or dto = — \~k sin 4w tan hz . dz. 



This equation gives us the rotation of the plane of polarisation due 

 to the doubly refracting nature of the medium, while the light passes 

 through a thickness dz. Let us suppose the effect of the magnetic 

 rotation on the light traversing the element dz, may be represented by 

 an additional rotation of these axes 



du — m dz, 



where mis a constant depending on the nature of the medium and 

 the strength of the magnetic field. Hence, when both these small 

 effects are superposed we get 



du) = m dz — \h sin 4to tan hz dz. 



Let us denote by w 1 the value of u> when m is positive, and w 3 its 

 value when m is negative. Then the apparent magnetic rotation is 

 Wj — w 2 , Q say. 



We have 



dQ, = 2m dz—\h. sin 20 cos 2 + w 2 ) tan hz dz. 

 This equation is easily integrated when Q is small. In that case 



