170 F. E. Wright — IVansmission of Light through 



D, cos 8, sin v a + R'„ cos p'„ sin r'„ + R" a COS p„ sin ?'",= (21) 



cos p 5 sin r" s = " D' 2 cos 8' 2 sin i 



D a S1 - 2 *!«/sin 8 S («,, cos r,-a„ sin rj + o ia cos 8 a ) + 



sin r'[~ i , , i . • , n 



_ — « bid P , (a n cos r ,-a,, sin r ,) + *„ cosp' a + 



f 2 L _J 



T^-i |^sin p" 2 (a, n cos r" — a, a sin r" 2 ) + a„ cos pO = 



R„ sin »•'„ 



„ sin 



D' 2 sin o' 2 cos i sin i. 



These equations (19), (20), (21), agree with the fundamental 

 equations of Neuman, MacCullagh and Kirchhoff derived from 

 the mechanical theory of light. They are, however, exceed- 

 ingly complicated, and in their solution certain auxiliary geo- 

 metric and analytic relations are used which simplify and 

 facilitate the practical calculations considerably. The most 

 important of these aids are the index surface introduced by 

 MacCullagh ('), Potier's ( 3 ) generalization of the Neumann- 

 MacCullagh relation, and the conception of the uniradial azi- 

 muth as given by MacCullagh( 3 ) and Neumann. ( 4 ) 



The Newman-Mac Cullagh-Potier relation. 



The index surface, whose radii vectors are proportional to 

 the reciprocal wave normal velocities or directly to the refrac- 

 tive indices for the direction of propagation, is best adapted to 



Fig. 3. 



Q: Q„ 



present graphically the relations between refracted and re- 

 flected waves. It is derived from the index ellipsoid in the same 

 manner that the ray surface is derived from Fresnel's ellip- 

 soid. The index surface (I) of a crystal is the reciprocal of its 

 ray surface (2), just as the index ellipsoid (I) is the reciprocal 



1 J. MacCullagh, Trans. Roy. Irish Acad., xvii, 252, 1833. 

 s Jour. Phys. (2), x, 349, 1891. 



3 Trans. Eoy. Irish Acad., xviii, 31, 1837. 



4 Berliner Akad. Abh., Math. Abt. 144, 1835 ; Pogg. Ann., xlii, 9, 1837. 



