Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical.
In addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.
Such quantics have been termed by Cayley multipartite.
+a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch.