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ellipse. As these two curves have a common centre, and 

 as the radius of the circle, 5, is of intermediate magnitude 

 to the semiaxes of the ellipse, it follows that they must 

 intersect in four points, as is represented in the annexed 

 diagram. 



Now, when two rays pass within the 

 crystal in any common direction, as 

 OAB, their velocities are represented 

 by the radii vectores of the two parts of 

 the wave, OA and OB ; and their direc- 

 tions, at emergence, are determined by 

 the positions of the tangent planes at the 

 points A andB. But in the case of the 

 ray OP, whose direction is that of the 

 line joining the centre with one of the 

 four cusps, or intersections just mentioned, the two radii 

 vectores unite, and the two rays have the same velocity. 

 There are still, however, two tangents to the plane section 

 at the point P; so that it might be supposed that the 

 rays proceeding with this common velocity within the crystal 

 would still be divided at emergence into two, and two only, 

 whose directions are determined by the tangent planes. This 

 seems to have been Fresnel's view of the case. Sir William 

 Hamilton has shown, however, that there is a cusp at each of 

 the four points just mentioned, not only in this particular 

 section, but in every section of the wave-surface passing 

 through the line OP ; or, more properly, that there is a con- 

 oidal cusp on that surface at the four points of intersection of 

 the circle and ellipse, and consequently an infinite number of 

 tangent planes, which form a tangent cone of the second 

 degree. Hence, a single ray, such as OP, proceeding 

 within the crystal in one of these directions, should be di- 

 vided into an infinite number of rays at emergence, whose 



