On Nov. 7, 1795, Miss Caroline Herschel discovered a small comet, about 5' in diameter, without a nucleus, but showing a slight central condensation of light. Olbers observed it on Nov. 21, but it was too faint to allow of the field being illuminated, and he was obliged to compare it with stars in the same parallel by noting the times of transit across the field of view. It was round, badly defined, and about 3' in diameter. The orbit greatly perplexed the calculators, and Prosperin declared that no parabola would satisfy the observations.
On Oct. 19, 1805, Thulis at Marseilles discovered a small comet faintly visible to the naked eye. Huth stated that on the 20th it was very bright in the centre, though without a nucleus, and was 4' or 5' in diameter. On Nov. 1 a tail 3° long was visible. Several parabolic orbits were calculated, and an elliptic one by Encke to which a period of 12.12 yrs. was assigned.
On Nov. 26, 1818, Pons at Marseilles discovered a small and ill-defined telescopic comet. As it remained visible for nearly 7 weeks a fairly complete series of observations was obtained. Encke, finding that under no circumstances whatever would a parabolic orbit represent them, determined to investigate the elements rigorously according to the method of Gauss then but little practised. So doing, he found that the orbit was certainly an ellipse, with a period of no more than about 312 years. On looking over a catalogue of the comets whose orbits had been calculated up to that time he was struck by the similarity which the elements obtained by him bore to those of the Comets of 1786 (i.), 1795 and 1805, and he was strongly impressed with the idea that all these 4 comets were really one and the same comet, especially as reckoning backwards from 1818 intervals of 312 years, or multiples of that period, would nearly or quite coincide with the perihelion passages of the comets of 1805, 1795 and 1786 (i.). The question could only be settled positively and conclusively by calculating backwards the effects of planetary perturbation and the necessary calculations Encke was able to accomplish by an extraordinary effort in 6 weeks, The result
