1902.] 



of Fresnel's Polarisation-vector. 



39 



where k = 2tt/A, A. being the wave-length, and the differential 

 equations of the vector are obtained by eliminating the exponentials 

 and the direction-cosines from these expressions by the aid of 

 equations (2). 



This gives at once 



9 8 8\/3 8o 8 8o 8 30 



o o ) 



v dx dy dz/ \dx du^ dy dv ' 82 3 

 where 



20 = a n w 2 + a-22V 2 + a 33 w 2 + 2a 2 svw + 2ai 3 wu + 2fti 2 wv (5). 



If we introduce a new vector sr, the time-gradient of which is denned 



by 



^^3_^^3o_33o_380_3 3o N 

 K dy dw dz dv dz du dx dw dx dv dy du; 



equation (4) may be written 



(6), 



. . x / dw$ 3-ST2 dw\ 8^3 3^ dw\\ 



. -\%--&> (7) - 



Thus Fresnel's theory of double refraction leads to the consideration 

 of three vectors — 



(1.) The polarisation-vector D with components u, v, iv. 



(2.) A vector E with components d&/du, 80/3#, 30/8w. 



(3.) A vector -w, such that & = curl E, 

 and D and w are connected by the relation, £) = - curl w. 



Also the vectors D and w are perpendicular to one another and in 

 the plane of the wave, and the vector E is perpendicular to the vector 

 vr and in the direction of the normal to the ellipsoid of polarisation at 

 the point in which the polarisation-vector meets it — that is, it is per- 

 pendicular to the ray. 



We thus see brought out quite clearly the connection between 

 Fresnel's theory and the electromagnetic theory for crystalline media. 



The boundary conditions that must be satisfied at the passage 

 between two crystalline media follow at once, if we assume that the 

 transition takes place by a rapid but continuous change of the pro- 

 perties of the one medium into those of the other, and that the above 

 equations hold within the region where this variation occurs. Taking 

 the interface as the plane x ■— 0, we see that these conditions are the 

 continuity of ^r 2j ^3, 30/3v, 30/3w ; and since the curl of a vector has 



