SMITH, HENRY JOHN STEPHEN (1826-1883), mathematician, was born in Dublin on 2d November 1826 and was the fourth child of his parents. When Henry Smith was just two years old his father died, whereupon his mother left Ireland for England. Mrs Smith taught her children herself, and until Henry was over eleven he was under her exclusive care and teaching; after that he was educated by private tutors till he went to Rugby in 1841. Whilst under the first of these tutors, in nine months he read all Thucydides, Sophocles, and Sallust, twelve books of Tacitus, the greater part of Horace, Juvenal, Persius, and .several plays of .iEschylus and Euripides. He also got up six books of Euclid and some algebra, besides reading a considerable quantity of Hebrew and learning the Odes of Horace by heart. On the death of his elder brother in September 1843 Henry Smith left Rugby, and in the end of 1844 gained a scholarship at Balliol College, Oxford. He won the Ireland scholarship in 1848 and obtained a first class in both the classical and the mathematical schools in 1849. He gained the senior mathematical scholarship in 1851. He was elected fellow of Balliol in 1850 and Savilian professor of geometry in 1861, and in 1874 was appointed keeper of the university museum. He was elected F.R.S. in 1861, and was an LL.D. of Cambridge and Dublin. He served on various royal commissions, and from 1877 was the chairman of the managing body of the meteorological office. He died at Oxford on 9th February 1883.

After taking his degree he wavered between classics and mathematics, but finally chose the latter. After publishing a few short papers relating to theory of numbers and to geometry, he devoted himself to a thorough examination of the writings of Gauss, Lejeune-Dirichlet, Kummer, &c., on the theory of numbers. The main results of these researches, which occupied him from 1854 to 1864, are contained in his Report on the Theory of Numbers, which appeared in the British Association volumes from 1859 to 1865, This report contains not only a complete account of all that had been done on this vast and intricate subject but also original contributions of his own. Some of the most important results of his discoveries were communicated to the Royal Society in two memoirs upon Systems of Linear Indeterminate Equations and Congruences and upon the Orders and Genera of Ternary Quadratic Forms (Phil. Trans., 1861 and 1867). He did not, however, confine himself to the consideration of forms involving only three indeterminates, but succeeded in establishing the principles on which the extension to the general case of 72 indeterminates depends, and obtained the general formultr, thus effecting what is probably the greatest advance made in the subject since the publication of Gauss's Disquisitiones ilrithmetiem. A brief abstract of Smith's methods and results appeared in the Proc. Roy. Soc. for 1864 and 1868. In the second of these notices he gives the general formula without demonstrations. As corollaries to the general formuke he adds the forinuhe relating to the representation of a number as a sum of five squares and also of seven squares. This class of representation ceases when the number of squares exceeds eight. The cases of two, four, and six squares had been given by Jacobi and that of three squares by Eisenstein, who had also given without demonstration some of the results for five squares. Fourteen years later the French Academy, in ignorance of Smith's work, set the demonstration and completion of Eisenstein's theorems for five squares as the subject of their "Grand Prix des Sciences Mathematiques." Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general theorems so far as was required to prove the results for the special case of fire squares. A month after his death, in March 1883, the prize of 3000 francs was awarded to him. The fact that a question of which Smith had given the solution in 1867, as a corollary from general formulae governing the whole class of investigations to which it belonged, should have been set by the French Academy as the subject of their great prize shows how far in advance of his contemporaries his early researches had carried him. Many of the propositions contained in his dissertation are general; but the demonstrations are not supplied for the case of seven squares. He was also the author of important papers in which he extended to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the Proc. Land. Math. Soc. and elsewhere. At the time of his death he was engaged upon a memoir on the Theta and Omega Functions, which he left nearly complete. In 1868 he was awarded the Steiner prize of the Berlin Academy for a geometrical memoir, Sur quelques problemcs cubiqucs of biquadratiqucs. He also wrote the introduction to the collected edition of ClifFord's Mathematical Papers (1882). The three subjects to which Smith's writings relate are theory of numbers, elliptic functions, and modern geometry; but in all that he wrote an "arithmetical" mode of thought is apparent, his methods and processes being arith- metical as distinguished from algebraic. He had the most intense admiration of Gauss. He was president of the mathematical and physical section of the British Association at Bradford in 1873 and of the London Mathematical Society in 1874-76. A memorial edition of his collected mathematical works is being (1887) printed by the Oxford university press.

An article in the Spectator of 17th February 1883, written by Lord Justice Bowen, gives perhaps the best idea of Smith's extraordinary personal qualities and influence, his sound judgment, perfect temper, gentle and LeTian wisdom, sweetness of character, delicate gaiety of spirit, and brilliant conversational power, which made him one of the most accomplished and attractive ornaments of any educated company in which he moved.

For further details relating to Henry Smith, reference should be made to the Fortnightly Review for May 1583 and to the "Monthly Notices" of the Roy. Ast. Soc., vol. xliv.