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 generally, and who applied it to the theory of orthogonal substitution. Cayley set himself the problem to determine a priori what functions of the coefficients of a given equation possess this property of invariance, and found, to begin with, in 1845; that the so-called "hyper-determinants" possessed it. Boole made a number of additional discoveries. Then Sylvester began his papers in the Cambridge and Dublin Mathematical Journal on the Calculus of Forms. After this, discoveries followed in rapid succession. At that time Cayley and Sylvester were both residents of London, and they stimulated each other by frequent oral communications. It has often been difficult to determine how much really belongs to each.

James Joseph Sylvester was born in London in 1814, and educated at St. Johns College, Cambridge. He came out Second Wrangler in 1837. His Jewish origin incapacitated him from taking a degree. In 1846 he became a student at the Inner Temple, and was called to the bar in 1850. He became professor of natural philosophy at University College, London; then, successively, professor of mathematics at the University of Virginia, at the Royal Military Academy in Woolwich, at the Johns Hopkins University in Baltimore, and is, since 1883, professor of geometry at Oxford. His first printed paper was on Fresnel's optic theory, 1837. Then followed his researches on invariants, the theory of equations, theory of partitions, multiple algebra, the theory of numbers, and other subjects mentioned elsewhere. About 1874 he took part in the development of the geometrical theory of link-work movements, originated by the beautiful discovery of A. Peaucellier, Capitaine du Génie à Nice (published in Nouvelles Annales, 1864 and 1873), and made the subject of close study by A. B. Kempe. To Sylvester is ascribed the general statement of the theory of contravariants, the