Adams started with the assumptions, not improbable, that the orbit of the unknown planet was a circle, and that its distance from the sun was twice that of Uranus. This latter assumption was in accord with the so-called "Bode's Law," which taught that a simple numerical relationship exists between the planetary distances (4, 7, 10, 16, 28, 52, 100, 196), and that the planets as they lie more remote from the sun tend to be more nearly double the distance of the next preceding. Adams was encouraged, by his first attempt, to undertake a more precise determination.
On his behalf Professor Challis of Cambridge applied to Astronomer Royal Airy, who furnished the _Reductions of the Planetary Observations_ made at Greenwich from 1750 till 1830. In his second endeavor Adams assumed that the unknown planet had an elliptical orbit. He approached the solution gradually, ever taking into account more terms of the perturbations. In September, 1845, he gave the results to Challis, who wrote to Airy on the 22d of that month that Adams sought an opportunity to submit the solution personally to the Astronomer Royal. On the 21st of October, 1845, the young mathematician, twice disappointed in his attempt to meet Airy, left at the Royal Observatory a paper containing the elements of the new planet. The position assigned to it was within about one degree of its actual place.
On November 5 Airy wrote to Adams and, among other things, inquired whether the solution obtained would account for the errors of the radius vector as well as for those of heliocentric longitude. For Airy this was a crucial question; but to Adams it seemed unessential, and he failed to reply.
By this time a formidable rival had entered the field. Leverrier at the request of Arago had undertaken to investigate the irregularities in the tables of Uranus. In September of the same year Eugène Bouvard had presented new tables of that planet. Leverrier acted very promptly and systematically. His first paper on the problem undertaken appeared in the _Comptes Rendus_ of the Académie des Sciences November 10, 1845. He had submitted to rigorous examination the data in reference to the disturbing influence of Jupiter and of Saturn on the orbit of Uranus. In his second paper, June 1, 1846, Leverrier reviewed the records of the ancient and modern observations of Uranus (279 in all), subjected Bouvard's tables to severe criticism, and decided that there existed in the orbit of Uranus anomalies that could not be accounted due to errors of observation. There must exist some extraneous influence, hitherto unknown to astronomers. Some scientists had thought that the law of gravitation did not hold at the confines of the solar system (others that the attractive force of other systems might prove a factor), but Leverrier rejected this conception. Other theories being likewise discarded he asked: "Is it possible that the irregularities of Uranus are due to the action of a disturbing planet, situated in the ecliptic at a mean distance double that of Uranus? And if so, at what point is this planet situated? What is its mass? What are the elements of the orbit which it describes?" The conclusion reached by the calculations recorded in this second paper was that all the so-called anomalies in the observations of Uranus could be explained as the perturbation caused by a planet with a heliocentric longitude of 252° on January 1, 1800. This would correspond to 325° January 1, 1847.