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 the modulus is a divisor of their resultant. Joseph Liouville (1809–1882), professor at the Collége de France, investigated mainly questions on the theory of quadratic forms of two, and of a greater number of variables. Profound researches were instituted by Ferdinand Gotthold Eisenstein (1823–1852), of Berlin. Ternary quadratic forms had been studied somewhat by Gauss, but the extension from two to three indeterminates was the work of Eisenstein who, in his memoir, Neue Theoreme der höheren Arithmetik, defined the ordinal and generic characters of ternary quadratic forms of uneven determinant; and, in case of definite forms, assigned the weight of any order or genus. But he did not publish demonstrations of his results. In inspecting the theory of binary cubic forms, he was led to the discovery of the first covariant ever considered in analysis. He showed that the series of theorems, relating to the presentation of numbers by sums of squares, ceases when the number of squares surpasses eight. Many of the proofs omitted by Eisenstein were supplied by Henry Smith, who was one of the few Englishmen who devoted themselves to the study of higher arithmetic.

Henry John Stephen Smith[90] (1826–1883) was born in London, and educated at Rugby and at Balliol College, Oxford. Before 1847 he travelled much in Europe for his health, and at one time attended lectures of Arago in Paris, but after that year he was never absent from Oxford for a single term. In 1861 he was elected Savilian professor of geometry. His first paper on the theory of numbers appeared in 1855. The results of ten years' study of everything published on the theory of numbers are contained in his Reports which appeared in the British Association volumes from 1859 to 1865. These reports are a model of clear and precise exposition and perfection of form. They contain much original matter, but the chief results of his own discoveries were