Reflexion and Refraction. 107 



portion of unity to cos e. Therefore, compounding all these 

 ratios, we find 



m 2 sin 2 cos t( 2 ) 



sm d costi COSE 



(24) 



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 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 intersecting the plane of xz in a right 

 line parallel to z, through which right line other planes are 

 drawn touching the wave surface of the crystal in four points, 

 these tangent planes will be the waves derived from the inci- 

 dent wave which touches the sphere ; and the points of contact, 

 including that on the sphere, will be the points where the rays 

 meet the wave surfaces. Then the corresponding masses will 

 be represented by prisms having a common rectangular base in 

 the plane xz, one side of this rectangle being the distance, on 

 the axis of z, between the origin and the common intersec- 

 tion of the tangent planes ; and the triangular face of each 

 prism having the same distance for one side, and a point of con- 

 tact for the opposite angle. These prisms, as they have a com- 

 mon base, will be proportional to their altitudes, which are the 

 ordinates y of the points of contact, The expression (24) may 

 be easily deduced from this relation. 



Let OT, OP, and the negative direction of y meet the sur- 

 face of the wave sphere (described with the L, 

 radius OS] in the points T^ P /} T / ; and let 

 the right line, in which the plane of the trans- 

 versals intersects the plane of incidence, meet 

 the sphere in L,. Then the points F,, P,, L^ Fig. is. 

 being all in the plane of incidence, will be on the same great 

 circle I r / P / Z / ; and drawing the great circles T 4 P^ Y t T t , we 



