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 printed in the Philosophical Transactions for 1861 and 1867. They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He established the principles on which the extension to the general case of $$\scriptstyle{n}$$ indeterminates of quadratic forms depends. He contributed also two memoirs to the Proceedings of the Royal Society of 1864 and 1868, in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liouville, relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are deducible by a uniform method from the principles indicated in his paper. Theorems relating to the case of 5 squares were given by Eisenstein, but Smith completed the enunciation of them, and added the corresponding theorems for 7 squares. The solution of the cases of 2, 4, 6 squares may be obtained by elliptic functions, but when the number of squares is odd, it involves processes peculiar to the theory of numbers. This class of theorems is limited to 8 squares, and Smith completed the group. In ignorance of Smith's investigations, the French Academy offered a prize for the demonstration and completion of Eisenstein's theorems for 5 squares. This Smith had accomplished fifteen years earlier. He sent in a dissertation in 1882, and next year, a month after his death, the prize was awarded to him, another prize being also awarded to H. Minkowsky of Bonn. The theory of numbers led Smith to the study of elliptic functions. He wrote also on modern geometry. His successor at Oxford was J. J. Sylvester.

Ernst Eduard Kummer (1810–1893), professor in the University of Berlin, is closely identified with the theory of numbers. Dirichlet's work on complex numbers of the form $\scriptstyle{a+ib}$, introduced by Gauss, was extended by him, by Eisenstein, and Dedekind. Instead of the equation $\scriptstyle{x^4-1=0}$, the roots of which yield Gauss' units, Eisenstein used the equation