_Differences of Orbit._—When three trustworthy observations of a comet’s place have been made, its orbit may be computed. This may be either an ellipse, a parabola, or hyperbola. If an _ellipse_ the comet is periodical, and the period depends upon the degree of eccentricity. If a _parabola_ the comet will not be seen again, because this form of orbit does not reunite; it consists of branches equally divergent and uniting at perihelion, but extending outwards indefinitely in nearly parallel lines and without convergence. If a _hyperbola_, the comet is also not returnable; the branches of the orbit are widely divergent, and show no tendency to parallelism. These several forms of orbit are somewhat different as applied to various comets, but they are the same in effect. Thus Tempel’s Comet of 1867 revolves in an ellipse having an eccentricity of about 0·4630, while that of Halley’s Comet is 0·9674. No doubt some of the parabolic orbits applied to comets really represent very eccentric ellipses; but the parabola is a convenient form of orbit for computation, and unless ellipticity is very decided it indicates the path with sufficient accuracy. _Discoveries of Comets._—In the latter part of the last century Messier, Mechain, and Miss Herschel shared nearly all the cometary discoveries between them. Then Pons entered the field, and he may be said to have monopolized this branch during the period from 1802 to 1827, for he was the first to announce thirty comets. Pons died in 1831, but the search was actively continued by others. In about 1843 a great rise became apparent in the rate of these discoveries; and we find Di Vico, Mauvais, and Brorsen very successful at this period. Later on, the work was sustained with the same prolific results by Klinkerfues, Bruhns, and Donati, and subsequently by Winnecke, Tempel, and Coggia. Swift and Borrelly also assisted materially to swell our knowledge; while during the last few years Barnard and Brooks have exhibited a surprising amount of zeal in this department. Since 1881 no less than twenty-six comets are to be enumerated as the fruits of their endeavours, and they are still engaged in nightly explorations of the sky with similar ends in view. Their diligent pursuit of these fugitive bodies will doubtless result in many further additions during ensuing years.