38 



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



These are usually obtained by formulating some theory respecting 

 the character of the ether in the media and the nature of the vibrations 

 in a train of waves, but there is an obvious advantage in directly 

 basing our investigations, if possible, on the known experimental laws 

 of the propagation of a luminous disturbance. 



This has been done by Voigt* in the case of an ordinary isotropic 

 medium by using the principle of interference that lies at the very 

 basis of the science of physical optics, combined with the fact that the 

 propagational speed of light is independent of the direction of the 

 waves ; from the equations thus obtained he then forms an expression 

 that may be regarded as representing the energy of the luminous 

 disturbance, and generalising this he deduces by the principle of least 

 action the equations that refer to other classes of homogeneous media. 



There is, perhaps, something artifical in this extension of the 

 expression for the energy, and it is therefore better, when this can 

 be done, to apply to each separate case the method employed by 

 Yoigt for isotropic media. 



2. This plan of procedure presents no difficulty in the case of 

 ordinary cr3 T stalline media. According to Fresnel's laws of double 

 refraction, the polarisation-vectors of the waves that can be propa- 

 gated in any given direction, are parallel to the axes of the central 

 section of a certain ellipsoid — the ellipsoid of polarisation — parallel to 

 the plane of the waves, and the propagational speeds of the corre- 

 sponding waves are given by the inverse of the lengths of these axes. 



If then the equation of the ellipsoid be 



a n x 2 + a 2 oy' 2 + a 33 z 2 + la^yz + 2a 13 ?x + 2a 12 xi/ =1 (1 ), 



we obtain by the ordinary methods of determining maxima and 

 minima 



(a n -a) 2 )cc + ai 2 f3 + a ls y = F/, "| 



«i 2 a + (« 22 - w 2 )/3 + «23y = Fm, > (2), 



a 1B a + a 23 fi + (^33 - w" 2 ) y = Fn, J 



where 



F = (a n oL + a 12 ft + a l3 y) I + (a 12 a, + a 22 /3 + a 2B y) m + (ci lz a + a 2 $P + a B $y) n 



(3), 



(/, m, n) being the direction-cosines of the normal, co the propagational 

 speed of the wave, and (a, (3, y) the direction-cosines of its polarisation- 

 vector. 



Now if (u, v, w) be the components of the polarisation-vector, the 

 principle of interference is expressed by 



u = 2aD, v = 2/5D, w = 2yD, D = A . Exp.{iK (Jx + my + nz- wt), 



* £ Kompenditmi der Theoretisclien Physik,' vol. 2, part Y, §§6, 7. 



