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 such close relations between the different elements of the phenomenon are conclusive in favour of the hypothesis on which they are based."

The question as to the correctness of Fresnel's construction was discussed for many years afterwards. A striking consequence of it was pointed out in 1832 by William Rowan Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who remarked that the surface defined by Fresnel's equation has four conical points, at each of which there is an infinite number of tangent planes; consequently, a single ray, proceeding from a point within the crystal in the direction of one of these points, must be divided on emergence into an infinite number of rays, constituting a conical surface. Hamilton also showed that there are four planes, each of which touches the wave-surface in an infinite number of points, constituting a circle of contact: so that a corresponding ray incident externally should be divided within the crystal into an infinite number of refracted rays, again constituting a conical surface.

These singular and unexpected consequences of the theory were shortly afterwards verified experimentally by Humphrey Lloyd, and helped greatly to confirm belief in Fresnel's theory. It should, however, be observed that conical refraction only shows his form of the wave-surface to be correct in its general features, and is no test of its accuracy in all details. But it was shown experimentally by Stokes in 1872, Glazebrook in 1879, and Hastings in 1887, that the construction of Huygens and Fresnel is certainly correct to a very high degree of approximation; and Fresnel's final formulae have since been regarded as unassailable. The dynamical substructure on which he based them is, as we have seen, open to objection;