The best instances of singular exceptions are furnished by the science of optics. It is a general law that in passing through transparent media the plane of vibration of polarised light remains unchanged. But in certain liquids, some peculiar crystals of quartz, and transparent solid media subjected to a magnetic strain, as in Faraday’s experiment (pp. 588, 630), the plane of polarisation is rotated in a screw-like manner. This effect is so entirely sui generis, so unlike any other phenomena in nature, as to appear truly exceptional; yet mathematical analysis shows it to be only a single case of much more general laws. As stated by Thomson and Tait,5 it arises from the composition of two uniform circular motions. If while a point is moving round a circle, the centre of that circle move upon another circle, a great variety of curious curves will be produced according as we vary the dimensions of the circles, the rapidity or the direction of the motions. When the two circles are exactly equal, the rapidities nearly so, and the directions opposite, the point will be found to move gradually round the centre of the stationary circle, and describe a curious star-like figure connected with the molecular motions out of which the rotational power of the media rises. Among other singular exceptions in optics may be placed the conical refraction of light, already noticed (p. 540), arising from the peculiar form assumed by a wave of light when passing through certain double-refracting crystals. The laws obeyed by the wave are exactly the same as in other cases, yet the results are entirely sui generis. So far are such cases from contradicting the law of ordinary cases, that they afford the best opportunities for verification.
In astronomy singular exceptions might occur, and in an approximate manner they do occur. We may point to the rings of Saturn as objects which, though undoubtedly obeying the law of gravity, are yet unique, as far as our observation of the universe has gone. They agree, indeed, with the other bodies of the planetary system in the stability of their movements, which never diverge far from the mean position. There seems to be little doubt that these rings are composed of swarms of small meteoric stones; formerly they were thought to be solid continuous rings, and mathematicians proved that if so constituted an entirely exceptional event might have happened under certain circumstances. Had the rings been exactly uniform all round, and with a centre of gravity coinciding for a moment with that of Saturn, a singular case of unstable equilibrium would have arisen, necessarily resulting in the sudden collapse of the rings, and the fall of their debris upon the surface of the planet. Thus in one single case the theory of gravity would give a result wholly unlike anything else known in the mechanism of the heavens.
It is possible that we might meet with singular exceptions in crystallography. If a crystal of the second or dimetric system, in which the third axis is usually unequal to either of the other two, happened to have the three axes equal, it might be mistaken for a crystal of the cubic system, but would exhibit different faces and dissimilar properties. There is, again, a possible class of diclinic crystals in which two axes are at right angles and the third axis inclined to the other two. This class is chiefly remarkable for its non-existence, since no crystals have yet been proved to have such axes. It seems likely that the class would constitute only a singular case of the more general triclinic system, in which all three axes are inclined to each other at various angles. Now if the diclinic form were merely accidental, and not produced by any general law of molecular constitution, its actual occurrence would be infinitely improbable, just as it is infinitely improbable that any star should indicate the North Pole with perfect exactness.
In the curves denoting the relation between the temperature and pressure of water there is, as shown by Professor J. Thomson, one very remarkable point entirely unique, at which alone water can remain in the