of Polarization Plane in an Aeolotropic Medium. 229 



the axis is possible, and will suffer rotation in the direction in 

 tvhich the first spiral goes round the axis. The rotation is one 

 radian for every distance h travelled by the wave. 



It will thus be seen that for a point infinitely near the axis 

 the first and second spirals (and therefore the corresponding 

 axes) are infinitely nearly inclined at an angle of 45° to the 

 horizon. As we recede to an infinite distance from the axis, 

 the inclination of the first axis continuously diminishes to 

 zero, the second axis being, of course, always at right angles 

 to the first. Also, infinitely near the axis the permittivity and 

 inductivity are infinitely nearly isotropic, and as we recede to 

 infinity the first permittivity and inductivity continuously 

 increase to infinity, while the second continuously decrease to 

 zero. The geometrical mean of the first and second permit- 

 tivities is always the constant third, and similarly for the 

 inductivities. 



We thus see that strictly within the four corners of Max- 

 well's theory we find room for the explanation of the rotation 

 of the plane of polarized light in crystals. We may instruc- 

 tively picture (however far from the real truth the picture 

 may be) such a substance as quartz to be made up of a bundle 

 of parallel ropes (as they may be called) , each rope being such 

 a medium as just described. To make the theory strictly 

 applicable the average diameter of a rope should be large 

 compared with the wave-length of light. There seems little 

 doubt, however, that even if the average diameter were com- 

 parable or even small compared with the wave-length there 

 would be a rotation of the plane of polarization. 



To construct the above medium, first note that for an 

 immovable medium equations (3) and (4) are precisely equi- 

 valent to equations (5) and (6) . Suppose the permittivity 

 and inductivity referred to the standard position (c and /n) 

 are constant scalars c and fi . Then many solutions of 

 equations (3) and (4) are known. But p f may be taken as 

 any given function of p. Hence we have corresponding- 

 solutions — which are fully known — for the actual position. 



Remembering the connexions of intensities and fluxes with 

 the position of matter, we see, among other things, that the 

 line-integrals of W and H x referred to the actual position are 

 the line-integrals of E and H referred to the standard position. 

 In any particular case this fact enables us to see at once how 

 W and H / are distributed in the actual space when the solution 

 for E and H is known. For instance, the above statements 

 about the rotation of the plane of polarization in the medium 

 described are seen at once to follow from the following con- 

 struction of that medium : — 



