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Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00Also such ideas as those of invariants, groups and of form, have modified the entire science.

00The 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.

00The 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.

00The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.

00An important notion in the theory of linear operators in general is that of MacMahon's multilinear operator (" Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond.

00x i, x 2) is said to be a covariant of the quantic. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with.

00Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text.

00This 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.

00For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.

00It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products.

00All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

00If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4).

00An instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the coefficients.

00- An important class of invariants, of several binary forms of the same order, was discovered by Sylvester.

00The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

00and Gordan, in fact, takes the satisfaction of these conditions as defining those invariants which Sylvester termed " combinants."

00A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

00two invariants, two quartics and a sextic. They are connected by the relation 212 = 2 i f?0 - D3 -3 jf 3.

00The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.

00There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3.

00Further, in the case of invariants, we write A= (1, i') 2 and take three new forms B = (i, T) 2; C = (r, r`) 2; R = (/y).

00Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

00- (aa) and then expresses the coefficients, on the right, in terms of the fundamental invariants.

00Hence, solving the cubic, R 2 j = (S -m i a) (S - m 2 a) (S - m3a) wherein m 1 m2, m 3 are invariants.

00The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

00f= ai; the Hessian H = (ab) 2 azbx; the quartic i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 .

00Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.

00For an algebraic solution the invariants must fulfil certain conditions.

00When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.

00The whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o.

00As regards invariants.

00Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.

00Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

00Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

002 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.

00Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.

00Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

00Young, The Algebra of Invariants (Cambridge, 1903).

00For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

00In the course of the ensuing ten years he published a large amount of original work, much of it dealing with the theory of invariants, which marked him as one of the foremost mathematicians of the time.

00At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for algebraical equations up to and including the fifth degree.

00The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.

00If, however, the geometrical property requires two or more relations between the coefficients, say A = o, B = o,&c., then we must have between the new coefficients the like relations, A' = o, B' = o, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u.

00The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.

00algebraic invariants was successful far beyond the hopes of its originators.

00formless invariants in the array from a picture could not be put into words anyway.

00To distinguish groups we compute invariants of the given groups.

00N has many abelian invariants 3. N is a direct product 3a.

00The wonder of it all is the theory of algebraic invariants was successful far beyond the hopes of its originators.

00Invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action.

00To be familiar with some geometric invariants of groups of transformations of the plane.

00The set of atomic invariants contains the instances of static predicates which are always true.

00invariants of a knot.

00Defines knots invariants: group of a knots invariants: group of a knot, Alexander polynomial, Jones polynomial.

00polynomial invariants of finite groups which come from a linear group action.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00Also such ideas as those of invariants, groups and of form, have modified the entire science.

00The 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.

00The 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.

00For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ...

00The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.

00An important notion in the theory of linear operators in general is that of MacMahon's multilinear operator (" Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond.

00x i, x 2) is said to be a covariant of the quantic. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with.

00Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text.

00This 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.

00and we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since AE, B, Ce,...

00For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.

00It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products.

00All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

00If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4).

00An instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the coefficients.

00- An important class of invariants, of several binary forms of the same order, was discovered by Sylvester.

00The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

00and Gordan, in fact, takes the satisfaction of these conditions as defining those invariants which Sylvester termed " combinants."

00A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

00Of any form az there exists a finite number of invariants and covariants, in terms of which all other covariants are rational and integral functions (cf.

00two invariants, two quartics and a sextic. They are connected by the relation 212 = 2 i f?0 - D3 -3 jf 3.

00The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.

00There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3.

00Further, in the case of invariants, we write A= (1, i') 2 and take three new forms B = (i, T) 2; C = (r, r`) 2; R = (/y).

00Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

00- (aa) and then expresses the coefficients, on the right, in terms of the fundamental invariants.

00Hence, solving the cubic, R 2 j = (S -m i a) (S - m 2 a) (S - m3a) wherein m 1 m2, m 3 are invariants.

00The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

00f= ai; the Hessian H = (ab) 2 azbx; the quartic i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 .

00There are 5 invariants: (a, b) 6, (i, i) 4', (1, l'), (f, l 3) 6, ((f, i),14)8; 6 of order 2.: 1, (i,1) 2, (f, 12) 4, (1,1 2) 3, (.f,1 3) 5, ((f, i), 13)6; 5 of order 4: i, (f,1) 2, (1, 1), (f,12)3, ((f, i), 12)4; 5 of order 6 :f, p = (ai) 2 axi 2 x, (f, 1), ((f, i), 1) 2, (p, l); 3 of order 8: H, (f, i), (H, 1); 1 of order 10: (H, i); 1 of order 12: T.

00Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.

00For an algebraic solution the invariants must fulfil certain conditions.

00When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.

00The whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o.

00As regards invariants.

00Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.

00Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

00Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

002 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.

00Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.

00Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

00Young, The Algebra of Invariants (Cambridge, 1903).

00For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

00In the course of the ensuing ten years he published a large amount of original work, much of it dealing with the theory of invariants, which marked him as one of the foremost mathematicians of the time.

00At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for algebraical equations up to and including the fifth degree.

00It appears that X2+Y2+Z2 and LX+MY+NZ are absolute invariants (cf.

00The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.

00If, however, the geometrical property requires two or more relations between the coefficients, say A = o, B = o,&c., then we must have between the new coefficients the like relations, A' = o, B' = o, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u.

00The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.

00

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