covery of the partial differential equations satisfied by the invariants and covariants of binary quantics, and the subject of mixed concomitants. In the American Journal of Mathematics are memoirs on binary and ternary quantics, elaborated partly with aid of F. Franklin, now professor at the Johns Hopkins University. At Oxford, Sylvester has opened up a new subject, the theory of reciprocants, treating of the functions of a dependent variable and the functions of its differential coefficients in regard to , which remain unaltered by the interchange of and . This theory is more general than one on differential invariants by Halphen (1878), and has been developed further by J. Hammond of Oxford, McMahon of Woolwich, A. R. Forsyth of Cambridge, and others. Sylvester playfully lays claim to the appellation of the Mathematical Adam, for the many names he has introduced into mathematics. Thus the terms invariant, discriminant, Hessian, Jacobian, are his.
The great theory of invariants, developed in England mainly by Cayley and Sylvester, came to be studied earnestly in Germany, France, and Italy. One of the earliest in the field was Siegfried Heinrich Aronhold (1819–1884), who demonstrated the existence of invariants, and , of the ternary cubic. Hermite discovered evectants and the theorem of reciprocity named after him. Paul Gordan showed, with the aid of symbolic methods, that the number of distinct forms for a binary quantic is finite. Clebsch proved this to be true for quantics with any number of variables. A very much simpler proof of this was given in 1891, by David Hilbert of Königsberg. In Italy, F. Brioschi of Milan and Faà de Bruno (1825–1888) contributed to the theory of invariants, the latter writing a text-book on binary forms, which ranks by the side of Salmon's treatise and those of Clebsch and Gordan. Among other writers on invariants are E. B. Chris-
