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 method, and showed that by his process the quintic could be transformed to any one of the four trinomial forms. Hamilton defined the limits of its applicability to higher equations. Sylvester investigated this question, What is the lowest degree an equation can have in order that it may admit of being deprived of $$\scriptstyle{i}$$ consecutive terms by aid of equations not higher than $\scriptstyle{i}$th degree. He carried the investigation as far as $\scriptstyle{i=8}$, and was led to a series of numbers which he named "Hamilton's numbers." A transformation of equal importance to Jerrard's is that of Sylvester, who expressed the quintic as the sum of three fifth-powers. The covariants and invariants of higher equations have been studied much in recent years.

Abel's proof that higher equations cannot always be solved algebraically led to the inquiry as to what equations of a given degree can be solved by radicals. Such equations are the ones discussed by Gauss in considering the division of the circle. Abel advanced one step further by proving that an irreducible equation can always be solved in radicals, if, of two of its roots, the one can be expressed rationally in terms of the other, provided that the degree of the equation is prime; if it is not prime, then the solution depends upon that of equations of lower degree. Through geometrical considerations, Hesse came upon algebraically solvable equations of the ninth degree, not included in the previous groups. The subject was powerfully advanced in Paris by the youthful Evariste Galois (born, 1811; killed in a duel, 1832), who introduced the notion of a group of substitutions. To him are due also some valuable results in relation to another set of equations, presenting themselves in the theory of elliptic functions, viz. the modular equations. Galois's labours gave birth to the important theory of substitutions, which has been greatly advanced by C. Jordan of Paris, J, A, Serret (1819–1885) of the Sorbonne in Paris, L. Kronecker (1823–1891) of