ALGEBRAIC on Linear Indeterminate Equations, Phil. Trans, cli. ; R. S. Ball, Theory of Screws, Dublin, 1876 ; and papers in Phil. Trans. clxiv. and Trans. P. Ir. Ac. xxv. ; W. K. Clifford, on Biquaternions, Proc. L. 31. S. iv. ; A. Buchheim, on Extensive Calculus and its applications, Proc. L. M. S. xv.-xvii. ; H. Taber, on Matrices, Amer. J. M. xii. ; K. Weierstrass, “ Zur Theorie der aus n Haupteinheiten gebildeten complexen Grbssen,” Gbtting. Nachr. 1884 ; G. Frobenius, on Bilinear Forms, Crelle, Ixxxiv. and Perl. Bcr. 1896 ; L. Kronecker, on Complex Numbers and Modular Systems, RerL Ber. 1888; G. Scheffers, “ Complexe Zahlensysteme,” Math. Ann. xxxix. (this contains a bibliography up to 1890) ; S. Lie, Vorlesungen iiber continuirliche Gruppen,
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Leipzig, 1893 (chap. 21). For a more complete account of the literature, and a general view of the subject, the reader may consult H. Hankel, Theorie der complexen Zahlensysteme, Leipzig, 1867 ; 0. Stolz, Vorlesungen iiber allgemeine Arithmetik, ibid. 1883 ; A. N. Whitehead, A Treatise on Universal Algebra, with Applications, vol. i. Cambridge, 1898 (a very comprehensive work, to which the writer of this article is in many ways indebted) ; and the Encyclopadie d. math. Wissenschaften, vol. i. Leipzig, 1898, &c. §§ A 1 (H. Schubert), A 4 (E. Study), and B 1 c (G. Landsberg). For the history of the development of ordinary algebra M. Cantor’s Vorlcsungen uber Geschichte der Mathematik is the standard authority. (g. B. M.)
ALGEBRAIC THE subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables. The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics. They are the chief instruments of research, and have themselves much benefited by being so employed. When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables. The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants. As far back as 1773 Lagrange, and later Gauss, had met with simple cases of such functions; Boole, in 1841 (Camb. Math. Journ. iii. pp. 1-20), made important steps, but it was not till >845 that Cayley (Coll. Math. Papers, i. pp. 80-94, 95-112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically. The subject was carried on over a long series of years by himself, Sylvester, Salmon, Hesse, Aronhold, Hermite, Brioschi, Clebsch, Gordan, &c. The year 1868 saw a considerable enlargement of the field of operations. This arose from the study by Klein and Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions. The invariant theory then existing was classified by them as appertaining to “ finite continuous groups.” Other “ Galois” groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of equations. Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed “ discontinuous,” and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry. The effect of this was to co-ordinate many branches of mathematics and greatly to increase the number of workers. The subject of transformation in general has been treated by Sophus Lie in the classical work Theorie der Transformationsgruppen. The present article is merely concerned with algebraical linear transformation. Two methods of treatment have been carried on in parallel lines, the unsymbolic and the symbolic; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the Continental schools have principally restricted themselves to the latter. The two methods have been conducted so as to be in constant touch, though the nature of the results obtained by the
FORMS
FORMS.
one differs much from those which flow naturally from the other. Each has been singularly successful in discovering new lines of advance and in encouraging the other to renewed efforts. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions. This enabled Hilbert to produce a very simple unsymbolic proof of the same theorem. So the theory of the forms appertaining to a binary form of unrestricted order was first worked out by Cayley and MacMahon by unsymbolic methods, and later Stroh, from a knowledge of the results, was arble to verify and extend the results by the symbolic method. At the moment of writing no English work exists on the symbolic methods, so that it has been judged proper to present to English readers a short resume of those processes, and to refer them for other information to the existing English treatises. The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. The theory brought forward has not yet found a place in any systematic treatise in any language, so that it has been judged proper to give a fairly complete account of it. I. The Theory of Determinants. Let there be given ?i2 quantities ^11 ^12 ^13 ^21 ^22 ^23 • • • ^31 ^32 ^33
- 712 ^w3 • • • #«»
and form from them a product of n quantities “la “2/3 “3y "• anv "Where the first suffixes are the natural numbers 1, 2, 3, ... rt taken in order, and a, f3, y, ... v is some permutation of these n numbers. This permutation by a transposition of two numbers, say a, (3, becomes /3, a, y, ... v, and by successively transposing pairs of letters the permutation can be reduced to the form 1, 2, 3, ... Let k such transpositions be necessary ; then the expression 2(-)*“la“2/3“3y-“Hv> the summation being for all permutations of the n numbers, is. called the determinant of the n2 quantities. The quantities ffia, k®2/3 • • • are called the elements of the determinant; the term ( — ) ayi.aipazy...anv is called a member of the determinant, and there are evidently n! members corresponding to the n! permutations of the n numbers 1, 2, 3, ... n. The determinant is usually written “ll “l2 “l3 • • • “in ^21 ^22 ^23 ••• ^2n A = ^31 ^32 ^33 ••• Clnl &"n2 (X"n3 • • • ®nn the square array being termed the matrix of the determinant. A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant. A theory of matrices has been constructed by Cayley in connexion particularly with the
