io6 On the Laws of Crystalline 



we find 



B tam 2 + tan K 



- = tan-(c 8 + K). (21) 



A 1-tan/otanfc 



But if s = in (16), we have 



for the equation of the right line in which the plane of the trans- 

 versals intersects the plane of incidence. This right line, lying, 

 like the refracted wave normal, between the directions of + x 

 and - ?/, makes with the direction of - y an angle v which ob- 



Tt 



viously has ^ for its tangent; and therefore, by (21), 



v = < 2 + K ; (23) 



which shows that the intersection of the two planes is inclined 

 to the refracted wave normal at an angle equal to K. 



We must now find the value of /?, which depends on the rela- 

 tive ethereal masses put in motion by the incident and refracted 

 waves. Conceiving the incident and refracted rays to be cylin- 

 drical pencils, having of course a common section in the plane 

 of xz, which is the surface of the crystal, let each pencil be cut 

 by a pair of planes parallel to its wave plane, and distant a 

 wave's length from each other ; then the cylindrical volumes so 

 cut out will represent the corresponding masses, since, by our 

 second hypothesis, the densities are equal. These volumes are 

 to each other in the compound ratio of their altitudes, whieh are 

 the wave lengths, and of the areas of their bases. The altitudes 

 are evidently as sin *i to sin t a . The first base is a perpendicular 

 section of the incident pencil ; the second base an oblique section 

 of the refracted one, the obliquity being equal to the angle c at 

 which the wave normal is inclined to the ray. The perpendi- 

 cular sections are to each other as the cosines of the angles which 

 they make with the common section of the cylinders, or as cos i } 

 to cos t( 2 ); putting ( 2 ) for the angle which the refracted ray makes 

 with the negative direction of y. The second base is greater 

 than the perpendicular section of the refracted pencil in the pro- 



