rv 



48 Mr. Mac Cullagh on the Laws of 



L normal, between the directions of -^-x and — y, makes with the direction of —y 



B 



an angle w which obviously has - for its tangent; and therefore, by (21), 



v=i,-\-^: (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 /i, which depends on the relative ethereal masses 

 put in motion by the incident and refracted waves. Conceiving the incident and 

 refracted rays to be cylindrical 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, which are the wave lengths, 

 and of the areas of their bases. The altitudes are evidently as sini, to sin(^. 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 t at 

 which the wave normal is inclined to the ray. The perpendicular 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 i^^. ; putting i for the angle which the 

 refracted ray makes with the negative direction of^. The second base is greater 

 than the perpendicular section of the refracted pencil in the proportion of unity 

 to cose. Therefore compounding all these ratios, we find 



(24) 



w, sin£jCOSt,cosE 



The same result may be otherwise obtained by observing that, in a system of 

 waves, the corresponding masses are proportional to the ordinates y of the points 

 where the rays meet their wave surfaces. By a system of waves, I mean an 

 incident wave with all that are derived from it by reflexion or refraction at the 

 ■^ \ \y same surface of the crystal, or at parallel surfaces. If at the point where the 

 incident" ray intersects its spherical wave surface, we apply a tangent plane inter- 

 secting the plane of xz in a right line parallel to z, through which right line 

 r other planes are drawn touching the wave surface of the crystal in four points, 



