206 DOUBLE REFRACTION AND CIRCULAR POLARIZATION 



In equations (29), we are to read 4- or in the second members, 

 according as the ray is right-handed or left-handed. (See 16.) It 

 follows that if the value of is positive, the greater velocity will 

 belong to a right-handed ray, and the smaller to a left-handed, but if 

 the value of <f> is negative, the opposite is the case. Except when 

 = 0, and the polarization is linear, there will be one right-handed 

 and one left-handed ray for any given wave-normal and period. 



18. When U 1 = U 2 , equations (29) give 



V2 rT 2 -f-0. 

 Pi Pz> - u y 



where U represents the common value of 'U l and U 2 . The polariza- 

 tion is therefore circular. The converse is also evident from equations 

 (29), viz., that a ray can be circularly polarized only when the 

 direction of its wave-normal is such that U 1 = \J Z . Such a direction, 

 which is determined by a circular section of the ellipsoid (24) precisely 

 as an optic axis of a crystal which conforms to Fresnel's law of double 

 refraction, may be called an optic axis, although its physical pro- 

 perties are not the same as in the more ordinary case.* If we write 

 V B and V L , respectively, for the wave- velocities of the right-handed 

 and left-handed rays, we have 



-, V L 2 =IP--; (33) 



R V L 



whence 



R v L/ R v L 



and V R V L = . (34) 



R L 



The phenomenon best observed with respect to an optic axis is the 

 rotation of the plane of linearly polarized light. If we denote by 

 the amount of this rotation per unit of the distance traversed by the 

 wave-plane, regarding it as positive when it appears clockwise to the 



* Our experimental knowledge of circularly or elliptically polarizing media is confined 

 to such as are optically either isotropic or uniaxial. The general theory of such media, 

 embracing the case of two optic axes, has however been discussed by Professor von Lang 

 ("Theorie der Circularpolarization," Sitz.-J3er. Wiener Akad., vol. Ixxv, p. 719). The 

 general results of the present paper, although derived from physical hypotheses of an 

 entirely different nature, are quite similar to those of the memoir cited. They would 



become identical, the writer believes, by the substitution of a constant for r- or ^ 



in the equations of this paper. (See especially equations (18), (20), (28).) 



That a complete discussion of the subject on any theory must include the case of 

 biaxial media having the property of circular or elliptical polarization, is evident from 

 the consideration that it must at least be possible to produce examples of such media 

 artificially. An isotropic or uniaxial crystal may be made biaxial by pressure. If it 

 has the property of circular and elliptic polarization, that property cannot be wholly 

 destroyed by the application of small pressures. 



