Reflexion and Refraction. 103 



the preservation of vis viva ; the other three are obtained from 

 the principle of equivalent vibrations, by resolving the vibra- 

 tions, 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 sur- 

 face of the crystal ; and in the third equation they are resolved 

 parallel to the intersection of these two planes. When the angles 

 0i, 02, 03 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 ii, 2 , 3 are those 

 of incidence, refraction, and reflexion ; but, for the sake of sym- 

 metry, 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 t 3 is the supplement of *i. Attending to this 

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



T! COS 0i - r 3 COS 3 = r 2 COS 2 



n a 



TI COS 0i + T 3 COS (7 3 = T 2 COS 



COS ti 



sin e 2 



sin <! 

 and, by adding and subtracting these, we find 



cos 2 sin (/i + * 2 ) ->l 



V (6) 



mAoa-fti-n) I 



which values if we substitute in equations (1) and (2), observing 

 that m 3 = m h as is evident, we shall get 



sin 2 (<i + 2 ) sin 2 (i t - t a ) m z sin 2 2^ 



cos 2 0! cos 2 3 mi cos 2 



(7) 



