42 



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



Hence, from the principle of interference expressed by 

 u = 2aD, v = 2/iD, w = 2yD, D = A exj).{a< (Ix + my + nz - o)t)} r 

 we obtain, as in §2, 

 (ii, v, w) 



= ^ 2 /d_ 3_ _8\ fi _/_8 _a_ _8\ aa + _s_ ^ +J 3 a? 



3v ' (W \3z' 3#' \3x ' 3^ 8y " 3^ 3? ' 3w, 

 /3w 8^ 3w 3w dv_ 9tt\ , 

 \3y 3^ ' 3^ 3z ' 3a? 3y/ 



which may be written in the form 



D = - curl w 9 "or = curl E + />D, 



D, E, and ft having the same significance as in § 2. 



The boundary conditions, obtained as in the former case, are the 

 continuity of *r 2 , -s^, d&/dv, d&/dw, the interface being a; = 0, together 

 with the continuity of u and vr\ - pti, since within the transition-layer 

 div. D = 0, div. (yr - pD) = ; the two latter conditions are not inde- 

 pendent of the previous four, as 



dy dz 

 *r 1 -pu=d_ 312 3 30 

 dy dw dz dv 



4. When we come to the consideration of magnetically active 

 media, our position is still more uncertain, but the following is 

 suggested as an extension of Fresnel's theorem, being a generalisa- 

 tion of results that appear to be established for isotropic media. 



In any direction within a magnetically active crystal two oppositely 

 polarised streams can be propagated that have their planes of maxi- 

 mum 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 speed represented by the reciprocal of the 

 length of either axis of the section by an amount that is inversely 

 proportional to the period of the vibrations and directly propor- 

 tional to the length of the axis, to the ratio of the axes of the elliptic 

 vibration perpendicular and parallel to the axis, and to the compo- 

 nent perpendicular to the section of a vector dependent upon the 

 intensity of the magnetic field. 



Thus if a-, o-' be the axes of the section, l } m, n the direction- 

 cosines of its normal, b v 6 2 » h the components of the vector B deter- 



