Crystalline Reflexion and Refraction. 49 



these tangent planes will be the waves derived from the incident 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 of xz, one side of this rectangle being the distance, on the axis of x, 



between the origin and the common intersection of the tangent planes ; and the 



triangular face of each prism having the same distance for one side, and a point 



of contact for the opposite angle. These prisms, as they have a common 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 

 surface of the wave sphere (described with the radius OS) in the ^, A 



points T,, P, Y, ; and let the right line, in which the plane of 

 the transversals intersects the plane of incidence, meet the sphere ^ 

 in L,. Then the points Y,, P,, L,, being all in the plane of incidence, will be 

 on the same great circle "Y,P,L,; and drawing the great circles T P YT 

 we shall have Y, P,=i,, Y, T.^t^^)' T, P,=e, Y, L^=i;=,^4-k, by (23); whence 

 P,L=.. 



As the transversal t^ is perpendicular to the plane OTP, or to the plane of 

 the great circle T,P,, the cosine of the spherical angle T P Y is the sine 

 of ^2 ; and therefore, from the triangle T,P,Y,, we have 



costj2j= cos<2COS£4-sint2sin£sin6'2, (25) 



which being substituted in ('24), gives 



»?2 sin2(2+2sin\sin0jtan£ 



w, ~ sin2t, 



and comparing this result with (10), we find 



sin^ijtan £ 



(26) 



(27) 



sin ^2 

 whence, and from (20), it follows that 



~ -. sin^tgtane 



VOL. XVIII. ^ /) /14J J ^'? / 



