CONICAL AND CYLINDRIC REFRACTION. 229 



tangents to the two nappes intersect each otlier, the direction CN will be the direc- 

 tion of a refracted ray, which will correspond 

 to different incident, rays for the two nappes. 

 The annexed figure illustrates ^these proposi- 

 tions. Let ZZ'XX' be the section of the wave 

 surface through x, z, or a, c. Let APQ be the 

 linear projection of the common tangent plane 

 of both nappes, and N the umbilical point. 

 Draw NA', tangent at N, to the circular sec- 

 tion, and NA", tangent also at N, to the ellip- 

 tical section. The radii of the wave surface 

 being the measures of ray velocity in their 

 several directions, as related to an assumed 

 unit, which is the uniform velocity of light in vacuo, take AD, equal to that 

 assumed unit, and from A, A', and A", where the several tangents cut the 

 axis ZZ' produced, describe the arcs aa, a'a', a" a". From draw tangents to 

 these arcs, CD, CD', CD". And from C again draw the perpendiculars, CR, 

 CR', CR", to these tangents. 



RC is the direction which an incident ray must have upon the surface ZZ' 

 of a crystal cut perpendicularly across the line intermediate between its optic 

 axes, (which is the axis of its greatest elasticity,) in order that it may be re- 

 fracted to P and Q ; — a case in which, as we have seen, it will be refracted 

 within the crystal in a hollow cone. At emergence (if the second surface is 

 parallel to ZZ') the emergent light will resume its original direction; and, as 

 this will happen for every point of the circular base of the cone, the emergent 

 beam will be a hollow cylinder. 



The lines R'C, R"C are the directions of incidence of two rays, of which the 

 first will send a refracted ray to N, belonging to the nappe whose section is 

 circular ; and the second will send another refracted ray to the same point, be- 

 longing to the nappe whose section is elliptical. Each of these will have a com- 

 panion refracted ray which will not go to N. The companion of the first will take 

 the direction Cn, found by drawing the tangent K'n to the ellipse ; and that of the 

 second will take the direction Cn', found by drawing the tangent A"w' to the circle. 

 The rays refracted to N will, on emergence, resume their parallelism to the 

 incident rays R'C, R"C, and will therefore be divergent. Now, if it be con- 

 sidered that the umbilical points, N, are conoidal, it will be perceived that any 

 plane passing through CN, will furnish two tangents like A'N, A"N, and there- 

 fore two incident rays, which will send corresponding refracted rays to N. It 

 will accordingly be understood that a conical pencil of convergent rays, incident 

 at C. will produce a conical pencil of divergent rays at its emergence from the 

 opposite and parallel face of the crystalline plate. Also, though the incident 

 cone of light be a solid cone, the emergent cone will be hollow. For, from the 

 graphic construction by which the direction of refracted rays is determined, it 

 is evident that none of the rays of the solid incident cone are refracted to N, 

 except only those whose incident direction is R'C, R"C, &c., in the several 

 azimuths around CN. 



These propositions were deduced by Sir William Hamilton from the equation 

 of the wave surface, before any phenomena of the kind had been observed or 

 even suspected. At his request Dr. Lloyd made a carei'ul stiidy of a crystal of 

 arragonite cut in the manner just supposed ; and the restilt of his examination 

 confirmed the theory in every particular. The success of the observation requires 

 very delicate adjustments. Mr. Soleil, of Paris, has since constructed a small 

 apparatus to facilitate the observation. 



When the emergent cylinder or cone of rays is observed with an analyzer 

 like Nicol's prism, one radius of the circle disappears. As the analyzer is turned 

 in azimuth, this dark radius changes position, advancing in azimuth twice as 



