Crystalline Reflexion and Refraction. 



45 



tions, by resolving the vibrations, or transversals, in three rectangular directions. 

 In the second equation, the transversals are resolved perpendicular to the plane 

 of incidence ; in the fourth, perpendicular to the surface of the crystal ; and in 

 the third equation they are resolved parallel to the intersection of these two 

 planes. When the angles 6^, 9^, 6^ begin, the transversals are in the plane of 

 incidence in such a relative position, that if they were turned round together in 

 that plane through a right angle, they would point each in the direction of its 

 own wave's progress. , These angles increase on the same side of the plane of 

 incidence, and range through the whole circumference. The angles i^, i^, i^ 

 are those of Incidence, refraction, and reflexion ; but, for the sake of symmetry, ; 

 they are taken to be the angles which the wave normals, drawn from the origin 

 in the direction of each wave's motion, make with the perpendicular to the 

 surface, this perpendicular being directed towards the interior of the crystal. 

 Thus it happens that i^ is the supplement of t^. Attending to this circum- 

 stance, equations (3) and (4) give us 



COS(, 



r cos^j ■ 



-r.cos^.=r cos^. 



cos I, 



T^COsd^-\-TfOsO=:TCOSd, 



and by adding and subtracting these, we find 



» sinti 



(5) 



COS0, sin(«,+(^) 

 '^i~''cos6>, sin2j, ' 



cosO, sin((,-0. _ . '// - / 



T,-=T. 



T.=:T 



(6) 



which values if we substitute in equations (1) and (2), observing that mz=m^, 

 as is evident, we shall get 



(7) 

 (8) 



cos'^0, COS'^03 7n^ COS^0^ ' 



•'' sin(<j+tjtan0j— sin(«j— <Jtan03=sln2<jtan02- 



Subtracting from (7) the Identity 

 / \ '"^J // J \ J- sinS^i+f )— sinYt,— t )=sin2t,sin2t. j 



— ^ Z l/!U 



'/Tu vV c:^ 







