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 Charles Eugène Delaunay (born 1816, and drowned off Cherbourg in 1872), professor of mechanics at the Sorbonne in Paris, explained most of the remaining acceleration of the moon, unaccounted for by Laplace's theory as corrected by Adams, by tracing the effect of tidal friction, a theory previously suggested independently by Kant, Robert Mayer, and William Ferrel of Kentucky. George Howard Darwin of Cambridge (born 1845) made some very remarkable investigations in 1879 on tidal friction, which trace with great certainty the history of the moon from its origin. He has since studied also the effects of tidal friction upon other bodies in the solar system. Criticisms on some parts of his researches have been made by James Nolan of Victoria. Simon Newcomb (born 1835), superintendent of the Nautical Almanac at Washington, and professor of mathematics at the Johns Hopkins University, investigated the errors in Hansen's tables of the moon. For the last twelve years the main work of the U.S. Nautical Almanac office has been to collect and discuss data for new tables of the planets which will supplant the tables of Le Verrier. G. W. Hill of that office has contributed an elegant paper on certain possible abbreviations in the computation of the long-period of the moon's motion due to the direct action of the planets, and has made the most elaborate determination yet undertaken of the inequalities of the moon's motion due to the figure of the earth. He has also computed certain lunar inequalities due to the action of Jupiter.

The mathematical discussion of Saturn's rings was taken up first by Laplace, who demonstrated that a homogeneous solid ring could not be in equilibrium, and in 1851 by B. Peirce, who proved their non-solidity by showing that even an irregular solid ring could not be in equilibrium about Saturn. The mechanism of these rings was investigated by James Clerk Maxwell in an essay to which the Adams prize was awarded.