40 



Mr. J. Walker. The Differential Equations [Feb. 8, 



no divergence anywhere, we may add to these the continuity of u and 

 -Sri ; but these conditions are clearly not independent of the former. 



3. In the case of active crystals we are on less sure ground : we are 

 without the guidance of Fresne'l, who only considered the passage of 

 light along the axis of an active uniaxal crystal, and our knowledge 

 of the laws of propagation of light in such media is less definite. 



It is, however, established that in any direction within an active 

 uniaxal crystal there are two streams of permanent type that are 

 oppositely polarised with their planes of maximum polarisation parallel 

 and perpendicular respectively to the principal section, and Gouy* has 

 shown that, neglecting small terms of the second order, the existence 

 of these "privileged" streams may be accounted for by a super- 

 position of the effects of ordinary double refraction and of an inde- 

 pendent rotary power possessed by the medium. 



If this be so, it is easily shown that to the same degree of approxi- 

 mation we have the following extension of Fresnel's theorem respecting 

 the ellipsoid of polarisation : — 



In any direction within an active crystalline medium two oppositely 

 polarised streams can be propagated with their planes of maximum 

 polarisation parallel respectively to the axes of the central section of 

 the ellipsoid of polarisation parallel to the plane of the waves ; and 

 the propagational speeds of these waves are respectively in excess or 

 defect of the speeds represented by the reciprocal of the length of 

 either of these axes by an amount directly! proportional to the period 

 of the vibrations of the polarisation-vector and to the ratio of the axes 

 of the elliptic vibrations perpendicular and parallel to that axis of 

 the section. 



Thus if cTj <r' be the axes of the section, L, L' the axes of the elliptic 

 vibration parallel respectively to these axes, and (o h w 2 the propagational 

 speeds of the waves, 



1 TO L' , TO L 



4?r L 47r L 



rp L ,_ 1 rp L' 

 4^17 = °~ ~ IttTT ' 



whence, approximately, 



J± = Wo 2 + Z.ii 

 ki L - k 2 L' 



where k = 27r/\. 



* 1 Journ. de Phys.,' (2), vol. 4, p. 142 (1885). 



f By taking the excess or defect inversely, instead of directly, proportional to 

 the period, a (8/8^) 2 is introduced in front of the rotary terms in (12). 



