When we watch a great comet approaching the sun the nucleus is first seen to become brighter and more clearly defined; at a later stage luminous matter appears to be projected from it towards the sun, often in the shape of a fan or a jet, which sometimes oscillates to and fro like a pendulum. In the head of Halley's comet, for instance, Bessel observed in October, 1835, that the jet in the course of eight hours swung through an angle of 36 deg.. On other occasions concentric arcs of light are formed round the nucleus, one after another, getting fainter as they travel further from the nucleus. Evidently the material of the fan or the arcs is repelled by the nucleus of the comet; but it is also repelled by the sun, and this latter repulsive force compels the luminous matter to overcome the attraction of gravitation, and to turn back all round the nucleus in the direction away from the sun. In this manner the tail is formed. (_See_ Plate XII.) The mathematical theory of the formation of comets' tails has been developed on the assumption that the matter which forms the tail is repelled both by the nucleus and by the sun. This investigation was first undertaken by the great astronomer Bessel, in his memoir on the appearance of Halley's comet in 1835, and it has since been considerably developed by Roche and the Russian astronomer Bredichin. Though we are, perhaps, hardly in a position to accept this theory as absolutely true, we can assert that it accounts well for the principal phenomena observed in the formation of comets' tails. Professor Bredichin has conducted his labours in the philosophical manner which has led to many other great discoveries in science. He has carefully collated the measurements and drawings of the tails of various comets. One result has been obtained from this preliminary part of his enquiry, which possesses a value that cannot be affected even if the ulterior portion of his labours should be found to require qualification. In the examination of the various tails, he observed that the curvilinear shapes of the outlines fall into one or other of three special types. In the first we have the straightest tails, which point almost directly away from the sun. In the second are classed tails which, after starting away from the sun, are curved backwards from the direction towards which the comet is moving. In the third we find the appendage still more curved in towards the comet's path. It can be shown that the tails of comets can almost invariably be identified with one or other of these three types; and in cases where the comet exhibits two tails, as has sometimes happened, then they will be found to belong to two of the types.