H/rr// 1)55 t Vv CORNELL UNIVERSITY UBRARIES Msthematica Library White HafI 3 1924 060 288 960 DATE DUE Sh' 4 139: ^^ mi -^ .yfvu i olIL 1 ry lyyb ^B- -2-^-i§9Q- CAVLORD PRINTEOINU.S.A. The original of tiiis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924060288960 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991. BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF Henrg W, Sage 1891 MATOOMTICS MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth. No. 1. History of Modern Mathematics. By David Eugene Smith. Ji.oo net. No. 2. Synthetic Projective Geometry. By George Bruce Halsted. $i.oo net. No, 3. Determinants. By Laenas Gifford Weld. 1 1.00 net. Hyperbolic Functions. By James Mc- Mauon. Si. 00 net. Harmonic Functions, By William £. Byerly. Ii.oo net. Qrassmann*s Space Analysis. By Edward W. Hyde, ii.oo net. Probability and Theory of Errors. By Robert S. Woodward. Ji.oo net. Vector Analysis and Quaternions. By Alexander Macfarlane. Ji.oo net. Differential Equations. By William Woolsey Johnson. Ji.oo net. The Solution of Equations. By Mansfield Merriman. Ji.oo net. Functions of a Complex Variable. By Thomas S. Fiske. Ji.oo nel. The Theory of Relativity. By Robert D. Carmicuael. Ji.oo net. No. 13, The Theory of Numbers, By Robert D. Carmichael. Ji.oo net. No, 14. Algebraic Invariants, By Leonaso E. Dickson. Ji.2S net. No. 15. Mortality Laws and Statistics. By Robert Henderson. *i.2S nel. No. 16. Diophant'ne Analysis. By Robert D. Carmichael. Si. 25 nel. PUBLISHED BY JOHN WILEY &SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON, No. 4. No. 5. No, 6. No, 7, No, 8. No. 9, No. 10, No. 11. No, 12, MATHEMATICAL MONOGRAPHS EDITED BV MANSFIELD MERRIMAN and ROBERT S. WOODWARD No. 14 ALGEBRAIC INVARIANTS BY LEONARD EUGENE DICKSON Professor of Mathematics in the University of Chicago FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Lhhted 1914 ; 1 1 Copyright, 1914, BY LEONARD EUGENE DICKSON THE SCIENTIFIC PRESS ROBERT DKl"MM(JND AND COMPANY HK(tOKI.^■^■, N > . EDITORS' PREFACE. The volume called Higher Mathematics, the third edition of which was published in 1900, contained eleven chapters by- eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The pubUcation of that volume was discontinued in 1906, and the chapters have since been issued in separate Monographs, they being generally enlarged by additional articles or appendices which either amplify the former pres- entation or record recent advances. This plan of publication was arranged in order to meet the demand of teachers and the convenience of classes, and it was also thought that it would prove advantageous to readers in special lines of mathe- matical Uterature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the demand seems to warrant it. Among the topics which are under con- sideration are those of elliptic functions, the theory of quantics, the group theory, the calculus of variations, and non-Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this Series of Monographs may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. PREFACE This introduction to the classical theory of invariants of algebraic forms is divided into three parts of approximately equal length. Part I treats of linear transformations both from the stand- point of a change of the two points of reference or the triangle of reference used in the definition of the homogeneous coor- dinates of points in a line or plane, and also from the stand- point of projective geometry. Examples are given of invariants of forms / of low degrees in two or three variables, and the vanishing of an invariant of / is shown to give a geometrical property of the locus /=0, which, on the one hand, is inde- pendent of the points of reference or triangle of reference, and, on the other hand, is unchanged by projection. Certain covariants such as Jacobians and Hessians are discussed and their algebraic and geometrical interpretations given; in particular, the use of the Hessian in the solution of a cubic equation and in the discussion of the points of inflexion of a plane cubic curve. In brief, beginning with ample illustra- tions from plane analytics, the reader is led by easy stages to the standpoint of linear transformations, their invariants and interpretations, employed in analytic projective geometry and modern algebra. Part II treats of the algebraic properties of invariants and covariants, chiefly of binary forms: homogeneity, weight, annihilators, seminvariant leaders of covariants, law of reciproc- ity, fundamental systems, properties as functions of the roots, and production by means of differential operators. Any quartic equation is solved by reducing it to a canonical form by means of the Hessian (§33). Irrational invariants are illustrated by a carefully selected set of exercises (§ 35). vi PREFACE Part III gives an introduction to the symbolic notation of Aronhold and Clebsch. The notation is first explained at length for a simple case; likewise the fundamental theorem on the types of symbolic factors of a term of a covariant of binary forms is first proved for a simple example by the method later used for the general theorem. In view of these and similar attentions to the needs of those making their first acquaintance with the s^TnboHc notation, the difficulties usually encountered will, it is beheved, be largely avoided. This notation must be mastered by those who would go deeply into the theory of invariants and its applications. Hilbert's theorem on the expression of the forms of a set linearly in terms of a finite number of forms of the set is proved and appHed to estabHsh the finiteness of a fundamental set of covariants of a system of binary forms. The theory of transvectants is developed as far as needed in the discussion of apolarity of binary forms and its application to rational curves (§§ 53-57), and in the determination by induction of a fundamental system of covariants of a binary form without the aid of the more technical supplementary concepts employed by Gordan. Finally, there is a discussion of the types of S3an- bolic factors in any term of a concomitant of a system of forms in three or four variables, with remarks on line and plane coordinates. For further developments reference is made at appropriate places to the texts in English by Salmon, Elliott, and Grace and Young, as well as to Gordan's Invariantentheorie. The standard work on the geometrical side of invariants is Clebsch- Lindemann, Vorlesungen iiber Geomelrie. Reference may be made to books by W. F. Meyer, Apolaritdl und Rationale Curve, Bericht iiber den gegenwarligen Stand der Invariantentheorie, and Formentheorie. Concerning invariant-factors, elementary divi- sors, and pairs of quadratic or bilinear forms, not treated here, see Muth, Elementartheiler, Bromwich, Quadratic Forms and their Classification by Means of Invariant Factors, and Bocher's Introduction to Higher Algebra. Lack of space prevents also the discussion of the invariants and covariants arising in the PREFACE vii theory of numbers; but an elementary exposition is available in the author's recent book, On Invariants and the Theory of Numbers, published, together with Osgood's lectures on func- tions of several complex variables, by the American Mathematical Society, as The Madison Colloquium. In addition to numerous illustrative examples, there are four- teen sets of exercises which were carefully selected on the basis of experience with classes in this subject. The author is indebted to Professor H. S. White for suggest- ing certain additions to the initial list of topics and for reading the proofs of Part I. Chicago, May, 1914. TABLE OF CONTENTS PART I Illustrations, Geometrical Interpretations and Applications OF Invariants and Covariants PAGE § 1. Illustrations from Plane Analytics 1 § 2. Projective Transformations 4 § 3. Homogeneous Coordinates of a Point in a Line 8 § 4. Examples of Invariants 9 § 5. Example? of Covariants 11 § 6 Forms and Their Classification 14 § 7. Definition of Invariants and Covariants 14 Exercises 15 § 8. Invariants of Covariants 16 § 9. Canonical Form of a Binary Cubic. Solution of Cubic Equations 17 § 10. Covariants of Covariants 18 § 11. Intermediate Invariants and Covariants 19 Exercises 20 § 12. Homogeneous Coordinates of Points in a Plane 20 § 13. Properties of the Hessian 23 § 14. Inflexion Points and Invariants of a Cubic Curve 26 Exercises 28 PART II Theory of Ixv.^riants in Non-symbolic Notation § 15. Homogeneity of Invariants 30 § 16. Weight of an Invariant of a Binary Form 31 § 17. Weight of an Invariant of any System of Forms 32 Exercises 33 § 18. Products of Linear Transformations 33 § 19. Generators of all Binary Linear Transformations 34 § 20. Annihilator of an Invariant of a Binary Form 34 Example and Exercises 36 § 21. Homogeneity of Covariants 37 § 22. Weight of a Covariant of a Binary Form 38 § 23. Annihilators of Covariants 39 Exercises 40 ix X TABLE OF CONTENTS PAGE § 24. Alternants 41 § 25. Seminvariants as Leaders of Binary Covariants 42 § 26. Number of Linearly Independent Seminvariants 43 § 27. Hermite's Law of Reciprocity 45 Exercises 46 §§ 28-31. Fundamental System of Covariants 47 §§32.33. Canonical Form of Binary Quartic; SolutionofQuartic Equations. . 50 § 34. Seminvariants in Terms of tlie Roots 53 § 35. Invariants in Terms of the Roots 54 Exercises 55 § 36. Covariants in Terms of the Roots 56 Exercises 58 § 37. Covariant with a Given Leader 58 § 38. Differential Operators Producing Covariants 5!) Exercises 61 PART III Symbolic Notation §§ 39-41. The Notation and its Immediate Consequences 63 Exercises 65, 66 §§ 42-45. Covariants as Functions of Two Symbolic Types 67 § 46. Problem of Finiteness of Covariants 70 § 47. Reduction to Problem on Invariants 71 § 48. Hubert's Theorem on a Set of Forms 72 §§ 49, 50. Finiteness of a Fundamental System of Invariants 73 § 5i . Finiteness of Syzygies 76 § 52. Transvectants 77 §§ 53. 54. Binary Forms Apolar to Given Forms 78 §§ 55, 56. Rational Plane Cubic Curves 81 § 57. Rational Space Quartic Curves 83 §§ 58, 59. Fundamental System of Covariants of Linear Forms; of a Quadratic Form; Exercises 84 § 60. Theorems on Transvectants; Convolution 85 § 61. Irreducible Covariants Found by Induction 87 § 62. Fundamental System for a Binary Cubic 89 § 63. Results and References on Higher Binary Forms 91 § 64. Hermite's Law of Reciprocity Symbolically 91 § 65. Ternary Form in Symbolic Notation 92 Exercises 93 §§ 66, 67, Concomitants of Ternary Forms 93 § 68. Quaternary Forms 97 Index 99 ALGEBRAIC INVARIANTS PART I ILLUSTRATIONS, GEOMETRICAL INTERPRETATIONS AND APPLICATIONS OF INVARIANTS AND COVARIANTS. 1. Illustrations from Plane Anal3rtics. If x and y are the coordinates of a point in a plane referred to rectangular axes, while x' and y' are the coordinates of the same point referred to axes obtained by rotating the former axes counter-clock- wise through an angle 6, then T: x=x' cos 6—y' sin d, y=x' sin d+y' cos d. Substituting these values into the linear function l = ax+by+c, we get a'x'+b'y'+c, where a' = a cos 6+b sin 6, b' = —a sin d+b cos 6. It follows that a'^+b'^ = a^+b^. Accordingly, a^+b^ is called an invariant of / under every transformation of the type T. Similarly, under the transformation T let L=Ax -i-By+C=A'x'+B'y'+C, so that A' = A cose+Bsine, B' = -A sin d+B cos 0. ALGEBRAIC INVARIANTS By the multiplication * of determinants, we get a' h' ' a b A' B' A B a' -b' a-b B' A' B A cos e — sin e sin d cos 8 = aB-bA, = aA+bB. cos e sin -sin 6 cos 6 The expressions at the right are therefore invariants of the pair of linear functions I and L under every transformation of type T. The straight Hnes represented by / = and L = are parallel if and only if aB — bA=0; they are perpendicular if and only if aA+bB = 0. Moreover, the quotient of aB — bA by aA+bB is an invariant having an interpretation; it is the tangent of one of the angles between the two lines. As in the first example, A^+B^ is an invariant of L. Between our four invariants of the pair /and L the following identity holds: {aA+bBy + {aB-bA)^ = {a2+b^)(A2+B^). The equation of any conic is of the form 5 = 0, where S = ax^+ 2bxy +cy^ + 2kx +2ly+m. Under the transformation T, S becomes a function of x' and y', in which the part of the second degree F = a'x'^+2b'x'y'+c'y'^ is derived solely from the part of 5 of the second degree: f=ax'^+ 2bxy + cy- The coefficient a' of x"^ is evidently obtained by replacing X by cos e and y by sin d in /, while c' is obtained by replacing x by — sin 6 and y by cos 6 in /. It follows at once that a +c =a+c. Using also the value of b', we can show that '>]a'c'-b"^^c-b^, * We shall always employ the rule which holds also for the multiplication of matrices: the element in the rth row and sth column of the product is found by multiplying the elements of the rth row of the first determinant by the cor- responding elements of the sth column of the second determinant, and adding the products. §1] ILLUSTRATIONS FROM ANALYTICS 3 but a more general fact will be obtained in § 4 without tedious multiplications. Thus a+c and d = ac — b^ are invariants of /, and also of 5, under every transformation of type T. When S = represents a real conic, not a pair of straight Unes, the conic is an ellipse if d>0, an hyperbola if d<0, and a parabola if d = 0. When homogeneous coordinates are used, the classi- fications of conies is wholly different (§ 13). If X and y are the coordinates of a point referred to rectan- gular axes and if x' and y' are the coordinates of the same point referred to new axes through the new origin (r, s) and parallel to the former axes, respectively, then t: x = x'+r, y = y'+s. All of our former expressions which were invariant under the transformations T are also invariant under the new trans- formations /, since each letter a, b, . . . involved is invariant under /. But not all of our expressions are invariant under a larger set of transformations to be defined later. We shall now give an entirely difTerent interpretation to the transformations T and /. Instead of considering {x, y) and {x, y') to be the same point referred to difTerent pairs of coordinate axes, we now regard them as difTerent points referred to the same axes. In the case of t, this is accomplished by translating the new axes, and each point referred to them, in the direction from (r, s) to (0, 0) until those axes coincide with the initial axes. Thus any point {x, y) is translated to a new point {x', y'), where x' = x—r, y = y—s, both points being now referred to the same axes. Thus each point is translated through a distance Vr^+s^ and in a direction parallel to the directed line from (0, 0) to (-r, -s). In the case of T, we rotate the new axes about the origin clockwise through angle d so that they now coincide with the initial axes. Then any point (x, y) is moved to a new point {x', y') by a clockwise rotation about the origin through angle e. By solving the equations of T, we get x'=x cos e+y sin 6, y' = —x sin d+y cos 9. 4 ALGEBRAIC INVARIANTS These rigid motions (translations, rotations, and combinations of them) preserve angles and distances. But the transformation x' = 2x, y' = 2y is a stretching in all directions from the origin in the ratio 2:1; while x' = 2x, y'=y is a stretching perpen- dicular to the j»-axis in each direction in the ratio 2:1. From the multiplicity of possible types of transformations, we shall select as the basis of our theory of invariants the very restricted set of transformations which have an interpretation in projective geometry and which suffice for the ordinary needs of algebra. 2. Projective Transformations. All of the points on a straight line are said to form a range of points. Project the Fig. 1. points A, B, C, . . . oi a. range from a point V, not on their line, by means of a pencil of straight lines. This pencil is cut by a new transversal in a range ^i, Bi, Ci, . . . , said to be perspective with the range A, B, C, . . . . Project the points Ai, Bi, Ci, . . . from a new vertex ?; by a new pencil and cut it by a new transversal. The resulting range of points A', B' C, . . ■ is said to be projective with the range A, B, C, . Likewise, the range obtained by any number of projections and sections is called projective with the given range, and §2] PROJECTIVE TRANSFORMATIONS 6 the one-to-one correspondence thus estabhshed between cor- responding points of the two ranges is called a prqjec- tivity. To obtain an analytic property of a projectivity, we apply the sine proportion to two triangles in Fig. 1 and get ^C_ sin^FC ^C_ sin^FC AV sin^CF' BV~sinACV' From these and the formulas with D in place of C, we get AC^AT sin A VC AD^AT sin AVD BC BV' sin BVC BD BV' sin BVD' Hence, by division AC . AD sin AVC . sin AVD BC ' BD "sin BVC ' sin BVD' The left member is denoted by (ABCD) and is called the cross-ratio of the four points taken in this order. Since the right member depends only on the angles at V, it follows that {ABCD) = {AiBiCxDi), if ^1, . . . , I>i are the intersections of the four rays by a second transversal. Hence if two ranges are projective, the cross-ratio of any four points of one range equals the cross- ratio of the corresponding points of the other range. Let each point of the line AB be determined by its dis- tance and direction from a fixed initial point of the line; let a be the resulting coordinate of A, and b, c, x those of B, C, D, respectively. Similarly, let A', B', C, D' have the coordinates a' , b', c', x', referred to a fixed initial point on their line. Then (^5CZ>) = — -f ^^=^-J^-=-^^, = {A'B'C'D'). c — b x — b c —b X —b Hence x' — b' ,x — b , c — a c' — a' -i = k^.' k = -a c-b ' c'-b" 6 ALGEBRAIC INVARIANTS SO that ^ is a finite constant 5^ 0, if C is distinct from A and B, and hence C distinct from A' and B'. Solving for x', we obtain a relation yx+8 In fact. a /3 7 S 9^0. a = h'-ka', P = ka'b — ab', 7 = l-j^, S = bk-a. If we multiply the elements of the first column of A by 6 and add the oroducts to the elements of the second column, we get b'-ka' b'ib-a) 1—k b—a = (6-a) -ka' b' -k 1 = k(b-a){b'-a')9^0, if B and A are distinct, so that B' and A ' are distinct. Hence a projectivity between two ranges defines a linear fractional transformation L between the coordinate a; of a general point of one range and the coordinate x' of the corre- sponding point of the other range. The transformation is uniquely determined by the coordinates of three distinct points of one range and those of the corresponding points of the other range. If the ranges are on the same line and if A'=A, B'=-B, C' = C, then k = l, a = 5, i3 = 7 = 0, and x'=x. Thus (ABCD) = {ABCD') impUes D'=D. Conversely, if L is any given linear fractional transfor- mation (of determinant 5>^ 0) and if each value of x is inter- preted as the coordinate of a point on any given straight line I and the value of x' determined by L as the coordinate of a corresponding point on any second given straight line I', the correspondence between the resulting two ranges is a pro- jectivity. This is proved as follows : Let A, B, C, D he the four' points of I whose respective coordinates are four distinct values xi, xz, xz, Xi of x such that yxi+Sy^O. The corresponding values xi', x% , xz , Xi of § PROJECTIVE TRANSFORMATIONS ic' determine four distinct points A', B' , C , D' of /'. For, if iT^j, Xi -Xj = aX4+/3 aXj-\-p _ A(xi — X}) yXi+8 yXj+S {yXi+5){yXj+8) ^0, {A'B'C'D')=^ X3 -xi . X4 -X] ^ ^^lZ3^^^^Il^ = (ABCn^ xs —x-i Xi —X2 0:3 — :«;2 0:4 — ai;2 since, if k denotes 7X1+5, A /A\./A /-._. hh' I3I2/ \lih/ lih If A'r^A, project the points A', B' , C , D' from any con- venient vertex V on to any line ABi through A and distinct Fig. 2. from /, obtaining the points Ai=A, Bi, Ci, Di of Fig. 2. Let V be the intersection of BBi with CCi and let VDi meet / at P. Then UBCP) = (AiBiCiDi) = {A'B'C'D') = (ABCD). From the first and last we have P=D, as proved above. Holding xi, xz, xa fixed, but allowing Xi to vary, we obtain two projective ranges on / and I'. If ^'=^, we use I' itself as ^5i and see that the ranges are perspective. 8 ALGEBRAIC INVARIANTS If / and /' are identical, we first project the range on /' on to a new line {A 'B' in Fig. 2) and proceed as before. Any linear fractional transformation L is therefore a pro- jective transformation of the points of a line or of the points of one line into those of another line. The cross-ratio of any four points is invariant. 3. Homogeneous Coordinates of a Point in a Line. They are introduced partly for the sake of avoiding infinite coor- dinates. In fact, if 7?^0, the value —S/y of x makes x' infinite. We set ar = a:i/a:2, thereby defining only the ratio of the homogeneous coordinates xi, xi of a point. 'LeX.x' =x\ Ix-i . Then, if p is a factor of proportionality, L may be given the homogeneous form pX\ =aX\-\-^X2^ PX2 =~iX\-\-hX2., a5 — /37?^0. The nature of homogeneous coordinates of points in a line is brought out more clearly by a more general definition. We employ two fixed points A and B of the line as points of reference. We define the homogeneous coordinates of a point P of the line to be any two numbers x, y such that x_ AP y'^PB' where c is a constant 5^0, the same for all points P, while .i4P is a directed segment, so that AP= —PA. We agree to take y = if P = B. Given P, we have the ratio of x to y. Conversely, given the latter ratio, we have the ratio of AP to PB, as well as their sum AP+PB = AB, and hence can find AP and therefore locate the point P. Just as we obtained in plane analytics (c/. § 1) the relations between the coordinates of the same point referred to two pairs of axes, so here we desire the values of x and y expressed in terms of the coordinates i, and -q of the same point P referred to new fixed points of reference A', B'. By definition, there is a certain new constant k^^O such that 1 = ^^ V PB'- §3] HOMOGENEOUS COORDINATES 9 Since A'P+PB'=A'B', we may replace A'P by A'B'-PB' and get p^,_ k^-A'B' Let A have the coordinates ?', ?;', referred to A', B' . Then PA=PB'-AB' = PB'-^^ J;'^'-J:;'^f-'^'^' Similarly, if B has the coordinates Ji, th, referred to A', B', p^_ {r)^i-im)k-A'B' Hence, by division, Since we are concerned only with the ratio of x to y, we may set Since the location of A and B with reference to A' and B' is at our choice, as also the constant c (and hence r and s), the values of tt]' and — r^' are at our choice, likewise sr)i and —s^i. There is,however,the restriction ^5^5, whence v'h^Vi^' Thus a change of reference points and constant multiplier c gives rise to a linear transformation a /3 1 7 5 of coordinates, and conversely every such transformation can be interpreted as the formulas for a change of reference points and constant multiplier. 4. Examples of Invariants. The hnear functions l = ax+by, L = Ax+By become, under the preceding linear transformation T, a{a^+Pv)+b(y^+Sr,)=a'^+b'v, A'^+B'r,, where a' = (ux+by, b'=a0+b8, A' = Aa+By, B'=Ap+BS. x=a^+Pv, y=y^+Sv, A = F^O, 10 ALGEBRAIC INVARIANTS Hence the resultant of the new linear functions is a' h' a h a /3 = A a b A' B' A B 7 5 A B and equals the product of the resultant r=aB—bA of the given functions by A. Since this is true for every linear homogeneous transformation of determinant A, we call r an invariant of I and L of index unity, the factor which multiphes r being here the first power of A. Employing homogeneous coordinates for points on a line, we see that / vanishes at the single point {b, — a) and that L = only at {B,—A). These two points are identical if and only if b : a = B : A, i.e., if r — 0. The vanishing of the invariant r thus indicates a geometrical property which is independent of the choice of the points of reference used in defining coordinates on the line; moreover, the property is not changed by a projection of this line from an outside point and a section by a new line. Thus r = gives a projective property. Among the present transformations T are the very special transformations given at the beginning of § 1. Of the four functions there called invariants of / and L under those special transformations, r alone is invariant under all of the present transformations. Henceforth the term invariant will be used only when the property of invariance holds for aU linear homo- geneous transformations of the variables considered. Our next example deals with the function f=ax^+2bxy+cy^. The transformation T (end of § 3) replaces / by F=A^^+2B^v+Cv^ in which A=ac^+2bay+cy'^, B = aa/3+6(a5+^7) +cy5, C = aP^+2bfid+c8^. li the discriminant d = ac-lP' of / is zero, / is the square of a linear function of x and y, so that the transformed function §4] EXAMPLES OF INVARIANTS 11 F is the square of a linear function of | and »;, whence the discriminant D=AC—B'' of F is zero. In other words, d = implies D = 0. By inspection, the coefl&cient of —IP, the highest power of 6, in the expansion of D is Thus D—A^d is a linear function bq+r of b, where q and r are functions of a, c, a, fi, 7, 5. Let a and_c remain arbitrary, but give to b the values Vac and —Vac in turn. Since d = and Z) = 0, we have 0=Vacq+r, 0= —Vacq-\-r, whence r = q = 0, D = A^d. Thus d is an invariant of / of index 2. Another proof is as follows : A^d-- =D. We just noted that d = Q expresses an algebraic property of /, that of being a perfect square. To give the related geo- metrical property, employ homogeneous coordinates for the points in a line. Then /=0 represents two points which coin- cide if and only if J = 0. Thus the vanishing of the invariant d oi f expresses a projective property of the points represented by/=0. a 7 a b a /3 /3 8 b c 7 s a 7 aa+by a^+b8 A B 6 ba+c 7 bfi+c 8 B C 5. Examples of Covariants. The Hessian (named after Otto Hesse) of a function j{x, y) of two variables is defined to be d^f d'f h = dx^ dxdy dj ay dydx a/ Let/ become F(|, 7;) under the transformation T: x=a^+fiv, y = y^+8r,, L \ = a /3 7 8 9^0. 12 ALGEBRAIC INVARIANTS Multiplying determinants according to the rule in § 1, we have hA = +7 9y «9y dxdy' ,9'/ dxdy a/ /3 ay + 6- aa;93' ay dxdy a/ az* aw aa; dx dv aw ay a)- where, by T, (1) .=.9A+^9f^9/a^+a^a2^aF ^^^a^+,a(^a^. aa; aj- aa; a^ dy a? a^ a^ ay dv By the same rule of multiplication of determinants, •/!A = A\aF 9 ■ a^ ay/a^ ^-9_+aA\9^, dx dy) di 9^9F 9 , a h7- , dx dyj dn dx dy) dv \ Applying (1) with/ replaced by a-f/af for the first column and by dP/dv for the second column, we get A^h = d^F a ^2 d^F a^jaf d^F a^a^? d^F dv" Hence the Hessian of the transformed function F equals the product of the Hessian h of the given function / by the square of the determinant of the linear transformation. Conse- quently, h is called a covariant of index 2 of /. For an interpretation of k=0, see Exs. 4, 5, § 7. In case / is the quadratic function / of § 4, A reduces to 4J, where d is the invariant ac—b^. The functional determinant or Jacobian (named after C. G. J. Jacobi) of two functions f{x, y) and g{x, y) is defined to be d{f,g). d{x, y) a/ a/ dx dy dg dg dx dy EXAMPLES OF COVARIANTS 13 Let the above transformation T replace / by F{i, rj), and g by G(J, 7j). By means of (l), we get 9(^,G). 3(^ n) 4+4 dx dy dx dy dx dy dx dy di di dx dy a fi _^diLg) dg dg ■ It 5 d{x, y) dx dy Hence the Jacobian of / and g is a covariant of index unity of f and g. For example, the Jacobian of the linear functions I and L in § 4 is their resultant r; they are proportional if and only if the invariant r is zero. The last fact is an illus- tration of the Theorem. Two functions f and g of x and y are dependent if and only if their Jacobian is identically zero. First, li g = {f), the Jacobian of/ and g is 9/ dx 9A dy .9f dx 9/ dy = 0. Next, to prove the second or converse part of the theorem, let the Jacobian of / and g be identically zero. If g is a constant, it is a (constant) function of /. In the contrary case, the partial derivatives of g are not both identically zero. Let, for example, dg/dx be not zero identically. Consider g and y as new variables in place of x and y. Thus f=F(g,y) and the Jacobian is dFdg dF dg_^dF dF dg dx dg dy dy dy dg dg dg dg dx ay dx dy 14 ALGEBRAIC INVARIANTS Hence dF/dy is identically zero, so that F does not involve y explicitly and is a function of g only. 6. Forms and their Classification. A function like aa:^+6a;2y, every term of which is of the same total degree in x and y, is called homogeneous in x and y. A homogeneous rational integral function oi x, y, . . . is called a form (or quantic) in x, y, . . . . According as the number of variables is 1, 2, 3, . . . , or q, the form is called unary, binary, ternary, . . . , or q-ary, respectively. Accord- ing as the form is of the first, second, third, fourth, . . . , or ^th order in the variables, it is called linear, quadratic, cubic, quartic, . . . , or p-ic, respectively. For the present we shall deal with binary forms. It is found to be advantageous to prefix binomial coefficients to the literal coefficients of the form, as in the binary quadratic and quartic forms ax^-\-2bxy-\-cy'^, aoO(^+Aaio^y+&a2x'^y^-\-^azxy^-{-aiy^. 7. Definition of Invariants and Covariants of Binary Forms. Let the general binary form / of order p, be replaced by by the transformation T (§5) of determinant Ar^O. If, for every such transformation, a polynomial /(ao, . . . , a^) has the property that I{Aq, . . . ,A;,)=AHiao, . . . , ap), identically in ao, . . . , a„, after the ^'s have been replaced by their values in terms of the a's, then /(ao, . . . , a^) is called an invariant of index X of the form /. §7] DEFINITION OF INVARIANTS 15 If, for every linear transformation T of determin^ant At^O, a polynomial K in the coefficients and variables in / is such that* K{Aq, . . . ,Aj,; k, v)=^^K{ao, . . . ,aj,; x, y), identically in ao, . . . , flp, |, v, after the ^'s have been replaced by their values in terms of the a's, and after x and y have been replaced by their values in terms of | and rj from T, then K is called a covariant of index X of /. The definitions of invariants and covariants of several binary forms are similar. These definitions are illustrated by the examples in §§4, 5. Note that / itself is a covariant of index zero of /; also that invariants are covariants of order zero. EXERCISES 1. The Jacobian ol f=ax''+2bxy+cy' and L=rx-\-sy is J = 2{as-br)x+2ibs—cr)y. If 7 is identically zero, f=lL^, where / is a constant. How does this illustrate the last result in § 5? Next, let / be not identically zero. Let k and / be the values of x/y for which /=0; m that for which L=0 and » that for which J = 0. Prove that the cross-ratio {k, m, I, n)= —1. Thus the points represented by /= are separated harmonically by those repre- sented by Z,= 0,7 = 0. 2. If / is the Jacobian of two binary quadratic forms/ and g, the points represented by 7=0 separate harmonically those represented by /=0 and also those represented by g=0. Thus 7 = represents the pair of double points of the involution defined by the pairs of points represented by/=Oandg=0. 3. If /(x, y) is a binary form of order n, then (Euler) dx '^y Hint: Prove this for /=aA:*y"~* and for /=/i +/2. 4. The Hessian of {ax-\-hyY is identically zero. Hint: It is sufficient to prove this for a;". Why? * The factor can be shown to be a power of A if it is merely assumed to be a function only of the coefficients of the transformation. 16 ALGEBRAIC INVARIANTS 5. Conversely, if the Hessian of a binary form/(j:, y) of order n is iden- tically zero, / is the «th power of a linear function. Hints: The Hessian of / is the Jacobian of df/dx, df/dy- By the last result in § 5, these derivatives are dependent: cx dy where a and b are constants. Solving this with Euler's relation in Ex. 3, we get or p/ iax+by) — =naf, {ax+bv) — — nbf, dx ' dy 3 log/ no 9 log / nb dx ax+by' Qy ax+by' Integrating, log;-» log {ax+by)^(i>(y) = is an imaginary cube root of unity. But x/y+2=i/)). Hence a:/y=— 3, — u— 2, — 0)2— 2. 10. Covaiiants of Covariants. Any covariant of a system of covariants of a system of forms is a covariant of the forms. The proof of this theorem is similar to that used in the following illustrations. We first show that the Jacobian of a binary cubic form / and its Hessian A is a covariant of index 3 of/. We have d(F, H) ^ ^ d(f,A'h) ^ ^,9(/, h) d{i,v) ^{x,y) d{x,y)' As the second illustration we consider the forms /, L in Ex. 1, § 7. Their Jacobian is the double of the covariant K = vx-\-wy of index unity, where v=as — br, w = bs — cr. Thus K and L are covariants of the system of forms/, L. These two linear covariants have as an invariant their resultant / = V w r s = as^ — 2brs+cr^. Under a linear transformation of determinant A, let / become j4|^+- . ., and Z become i?^ +5?). By the covariance of iT, V^+Wv = A(vx+wy), V=AS-BR, W=BS-CR. Thus our transformation replaces the linear form having the coefficients Av and Aw by one having the coefficients V and W. Th- resultant E = Av Aw r s §11] INTERMEDIATE INVARIANTS 19 of this linear form and L is an invariant of index unity. Hence V W = A£, V W = A2 V w R s J R s r s so that 7=115— 'Z£'r is an invariant of index 2 of / and L. From the earlier expression for I, we see that it is the resultant of / and L. We have therefore illustrated also the theorem that the resultant of any two binary forms is an invariant of those forms. 11. Intermediate Invariants and Covariants. From the invariant ac — b^ of the binary quadratic form / = ax^ + 2bxy + cy'^ we may derive an invariant of the system of forms / and /', where f'=^a'x^ + 2b'xy+c'y2. Let any Hnear transformation replace / and /' by If / is any constant, the form/+//' is transformed into F+tF'. By the invariance of the discriminant oif+tf, {A+tA')iC+tC')-{B+tBy^Ama-\-ta')ic+tc')-(b+tby\, identically in /. The equality of the terms free of t states only the known fact that ac — 6^ is an invariant of /. Similarly the equahty of the terms involving fi states merely that a'c' — b'^ is an invariant of/'. But from the terms multiplied by t, we see that (1) ac'+a'c-2bb' is an invariant of index 2 of the system of forms /, /"'. It is said to be the invariant intermediate between their dis- criminants. It was discovered by Boole in 1841. The method is a general one. Let K be any covariant of a form f(x, y, . . .). Let c, 6, ... be the coefficients of /. Let f(x, y, . . .) be a form of the same order with the coeffi- cients a', h', . . . . If in i? we replace a by a+ta', b by b+tb', . . . , and expand in powers of t, we obtain as the 20 ALGEBRAIC INVARIANTS coefficient of any power r of ^ a covariant of the system /, /'. By Taylor's theorem, this covariant is (2) J./a'A+j'|^ + . . V^, r!V da db / ' in which the symbolic rth power of 9/3a is to be replaced by aV3a% etc. EXERCISES 1. For r=l, K=ac-b^, (2) becomes (1). 2. Taking as K the Hessian (2) of cubic (1) in § 8, obtain the covariant (ac' +a'c-2bb')x' + {ad' +a'd-bc' -b'c)xy + {bd' +b'd-2cc')y' of index 2 of a pair of binary cubic forms. 3. If (1) is zero, the pair of points given by /=0 is harmonic with the pair given by/'=0. 12. Homogeneous Coordinates of Points in a Plane. Let Li-. a,x+bty+c, = (f = l, 2, 3) be any three linear equations in x, y, such that fli bi Ci A= 32 ^2 C2 5^0. as ^3 cs Interpret x and y as the Cartesian coordinates of a point referred to rectangular axes. Then the equations represent three straight lines Lt forming a triangle. Choose the sign before the radical in ^'~±Va^+b^ so that pi is positive for a point {x, y) inside the triangle and hence is the length of the perpendicular from that point to Lt- The homogeneous (or trilinear) coordinates of a point {x, y) are three numbers xi, X2, xa such that pXl^klpl, pX2=k2p2, pX3=kzpz, where ki, k2, ks are constants, the same for all points. In view of the undetermined common factor p, only the ratios of xi, X2, X3 are defined. 1 12] HOMOGENEOUS COORDINATES 21 For example, let the triangle be an equilateral one with sides of length 2, base on the .r-axis and vertex on the y-axis. The equations of the sides Li, Li, Lt are, respectively, v3+*=^' vr*=^' ^=«- Take each /t«=l. Then y+V3(«-l) y-Vz{x + \) pXi= , pac2= ;; , pXi=y. -2 -2 The curve ii;iX2=3t3' is evidently tangent to I,i(i.e., »;i=0) at 2= (010), and tangent to Li at ^=(100). Substituting for the Xt their values, we see that the Cartesian equation of the curve is Fig. 3. \\(y-V3)'-3x^} =y' or *' + ( J'+:;^)'=|■ Hence it is a circle with radius CP and center at the intersection C of the normal to Li at P with the normal to ii at Q. Changing the notation for the coefficients of ktpi, call them at, bt, d. Then we have (H) pXt = a{X+biy+Ct, Ap^O (i = l,2,3). 22 ALGEBRAIC INVARIANTS Multiply the ith equation by the cof actor At of at in the determinant A and sum for i = l, 2, 3. Next use as multiplier the cofactor Bi of bt; finally, the cofactor d of C|. We get Ax = pZAiXt, Ay = pXBiXt, A = p^CtXi. Hence x and y are rational functions of xi, X2, xs: /ps _ ^1X1+^2^:2+^3 3:3 ^ ^ix-i +^2X2+^3^:3 Ci3:i+C2a:2+C3a:3' C1X1+C2X2+C3XS Any equation f{x, y)=0 in Cartesian coordinates becomes, by use of (C), a homogeneous equation (i){xi, X2, X3)=0 in homogeneous coordinates. The reverse process is effected by use of (H) . In particular, since any straight line is represented by an equation of the first degree in x and y, it is also rep- resented by a homogeneous equation of the first degree in Xi, X2, X3. For example, the sides of the triangle of reference are xi = 0, 0:2 = 0, xa = 0. Conversely, any homogeneous equation of the first degree in a;i, 0:2, X3 represents a straight line. The degree of 4> is always that of /. Take the y-axis as Li, the «-axis as L2, and let L, recede to infinity by making as and 63 approach zero. Then (H) and (C) become px, = x, pXi=y, pX3=l; x=-, y=-. X3 Xi We are thus led to a very special, but much used, method of passing from homogeneous to Cartesian coordinates and conversely. For a new triangle of reference, let the homogeneous coor- dinates of (x, y) be yi, y2, yz- Then, as in {H), pyi = a\x+h\y+c'i (i = l, 2, 3). Inserting the values of x and y from (C), we get relations like /: Tyt = e,xi+JiX2+gtX3 (i=l,2, 3). Hence a change of triangle of reference and constants ^1, k2, ks gives rise to a linear homogeneous transformation / of coordinates. The determinant of the coefficients in t is not § 131 PROPERTIES OF THE HESSIAN 23 zero, since yi = 0, y2 = 0, >'3 = represent the sides of the new triangle. Conversely, any such transformation t may be interpreted as a change of triangle of reference and con- stants kt. Instead of regarding t as a set of relations between the coordinates of the same point referred to two triangles of reference, we may regard it as defining a correspondence between the points {xi, X2, X3) and (yi, y2, yz) of two different planes, each referred to any chosen triangle of reference in its plane. This correspondence is projective; for, it can be effected by a series of projections and sections, each projection being that of the points of a plane from a point outside of the plane and each section being the cutting of such a bundle of pro- jecting lines by a new plane. Proof will not be given here, nor is the theorem assumed in what follows. It is stated here to show that if / is any invariant of a ternary form / under all linear transformations t, then 7 = gives a projective property of the curve /=C. It is true conversely that any projective transformation between two planes can be effected by a linear homogeneous transformation on the homogeneous coordinates. Thus for three variables, just as for two (§§ 2, 3), the investigation of the invariants of a form under all linear homogeneous transformations is of especial importance. 13. Properties of the Hessian. Let f{xi, . . . , x„) be a form in the independent variables xi, . . . , Xn. The Hessian A of / is a determinant of order n in which the elements of the zth row are dxidxi dx,dX2 ' ' ' ' dXtdXn Let /become (yi, . . . , y„) under the transformation T: Xi = Ciiyi+Ci2y2+. ■ ■+Ci„y„ {i = l, . ■ . , «), of determinant A = |c^|. The product hA is a determinant of order n in which the element in the ith row and ^th column is the sum of the products of the above elements of the ith. &?h = = Hessian of . T,i =1 n 24 ALGEBRAIC INVARIANTS row of h by the corresponding elements of the jth column of A, and hence is dxtdxi 'doct'dX2 dXtdXn _ 9/ 9/ 9^1 I 9/ dX2 . I 9/ dx„ \ 9 9<» Let a' be the determinant obtained from A by interchanging its rows and columns. In the product A'-hA, the element in the rth row and yth coliunn is therefore ^ 9 9<^ , , , 9 9<^ 9 9<^ 92:1 dyj dxn dyi dyr dyj since dr is the partial derivative of xt with respect to y,. Hence dyrdyj Thus A is a covariant of index 2 of /. To make an application to conies, let / be a ternary quad- ratic form. Then h is an invariant called the discriminant of /. Let (ai, 02, as) be a point on /=0 (for example, one with a;3 = 0). For Cii=a« and Ct2, c^ chosen so that Af^O, transformation T makes (x) = (a) correspond to (y) = (100). Hence we may assume that (100) is a point on/=0, so that the term in xi^ is lacking. Consider the terms xil with the factor xi. li 1=0, f involves only X2 and xz and hence is a product of two linear functions, while h=0. In the contrary case, we may introduce Z as a new variable in place of X2- This amounts to setting l = X2, f = XiX2 + aX2^ + bX2X3+CX3^- Replacing xi by Xi — ax2~bx3, we get xiX2 — kxs^, whose Hessian is 2k. Hence /=0 represents two (distinct or coincident) straight lines if and only if the Hessian (discriminant) of / is zero. Moreover, if the discriminant is not zero, then k^^Q and we may replace Vkxs by xz and get a;ia;2-a;3^. Hence all conies, which do not degenerate into straight lines, are equivalent § 13] PROPERTIES OF THE HESSIAN 25 under projective transformation. If the triangle of reference is equilateral and the coordinates are proportional to the per- pendiculars upon its sides, a;ia;2— :C32 = is a circle (§ 12). On the contrary, if we employ only translations and rota- tions, as in plane analytics, there are infinitely many non- equivalent conies ; we saw in § 1 that there are then two invariants besides the discriminant. Next, to make an apphcation to plane cubic curves, let f{xi, X2, X3) be a ternary cubic form. A triangle of reference can be chosen so that P = (001) is a point of the curve /=0. Then the term in xz^ is lacking, so that f = X3jl+X3f2+f3, where /« is a homogeneous function of xi and X2 of degree i. We assume that P is not a singular point, so that the partial derivatives of / with respect to xi, X2, and X3 are not all zero at P. Hence /i is not identically zero and can be introduced as a new variable in place of xi. Thus, after a preliminary linear transformation, we have X3^xi +X3(axi^+bxiX2+cx2'^) -I-/3. Replace 3:3 by a:3—|(a:»;i-|-6a;2). We get F = X3^Xi+eX3X2- +C, where C is a cubic function of xi, X2, whose second partial derivative with respect to Xt and xj will be denoted by C^. The Hessian of F is Cii C12 20:3 H= C12 C22 + 2CT3 2e:r2 2a;3 2e:t;2 2a;i If the transformation which replaced / by F is of deter- minant A, it replaces the Hessian h oi f hy n = A^h. Thus H = represents the sime curve as k = 0, but referred to the same new triangle of reference as F = Q. We may therefore speak of a definite Hessian curve of the given curve /=0. In investigating the properties of these curves we may therefore 26 ALGEBRAIC INVARIANTS refer them to the triangle of reference for which their equations aTeH = 0,F = Q. The coefficient of xs^ in H is evidently — 8e. Thus P is on the Hessian curve if and only if e = 0. If , 4>=\^'' ^'"i, 1 C12 C22 I so that is the Hessian of C. By § 8, = 36{-b'^xi-+axiX2+ bx2^) . Eliminating Xz^ between F = 0, Zr = 0, we get xi^+2C22C = \2{x2^+&bx2^xi^+'^ax2Xi^-Zl^xi*)=0. If a;i=0, then 2:2 = and we obtain the known intersection § 14) INFLEXION POINTS OF CUBIC 27 (001). For the remaining intersections, we may set a;i = l and obtain from each root r of (1) r4+6&r2+4ar- 362 = two intersections (1, r, dox's). For, if x'3 = 0, then C = 0, so that (1) would have a multiple root, whence a^+4ib^ = 0. But the three partial derivatives of F would then all vanish at (2b,— a, 0) or (1,0,0), according as by-^0 or 6 = 0. Hence there are exactly nine distinct points of inflexion. For each of the four roots of (1), the three points of inflexion P and (1, r, dzx'a) are collinear, being on X2 = rxi. Since we may proceed with any point of inflexion as we did with P, we see that there are 9-4/3 or 12 lines each joining three points of inflexion and such that four of the lines pass through any one of the nine points. The six points of inflexion not on a fixed one of these lines therefore he by threes on two new lines; three such lines form an inflexion triangle. Thus there are §12 = 4 inflexion triangles. The fact that there are four inflexion triangles, one for each root ;■ of (1), can also be seen as follows: iTH+rF = {rxi-X2)\x3^-rx2^-{r'+3b)xiX2-{r^+Qbr+3a)xi^]. The last factor equals X3^-^\rx2+Kr^+db)xi\\ r and hence is the product of two linear functions. Corresponding results hold for any cubic curve /=0 without singular points. We have shown that / can be reduced to the special form F by a linear transformation of a certain determinant A. Follow this by the transformation which multiplies xs by A and Xi by A"^^ and hence has the determin- ant A~^ Thus there is a transformation of determinant unity which replaces / by a form of type F, and hence replaces the Hessian h oi f hy the Hessian H of F. Hence there are exactly four values of r for which f = h-\-2ArJ has a linear factor and therefore three hnear factors. These r's are the roots of a quartic (1) in which a and b are functions of the coeflScients 28 ALGEBRAIC INVARIANTS of/. To see the nature of these functions, let Xi — 'KX2—ij^3 be a factor of ^. After replacing xi by 1^2+ nxs in f, we obtain a cubic function of X2 and 0:3 whose four coefl&cients must be zero. Eliminating X and n, we obtain two conditions involving r and the coefficients of / rationally and integrally. The greatest common divisor of their left members is the required quartic function of r. Unless the coefficient of r* is constant, a root would be infinite for certain /'s. The inflexion triangles of a general cubic curve /=0 are given by h+24rf=0, where h is the Hessian of f and r is a root of the quartic (1) in which a and b are rational integral invariants off. The expHcit expressions for these invariants are very long; they are given in Salmon's Higher Plane Curves, §§ 221-2, and were first computed by Aronhold. For their short sj-m- bolic expressions, see § 65, Ex. 4. EXERCISES 1. Using the above inflexion triangle yij'23'3=0, where rXi — X2 = yi, Vrx3±{rx2+kxi) = 2y2, 2yi, k=(r^+3b)/2, r^+k = Ur'+b)^0, as shown by use of (1), we have the transformation Vrx3 = y2+yz, (r'+k)x, = ry,+D, (r'+k)x2= -ky,+rD, where D = yi—y:,. Using (1) to eliminate a, show that o r o Adding the product of the latter by 54 to its Hessian, we get the product of yiy2y3 by 3^(r''+b)/r^. Hence the nine points of inflexion are found by setting yi, y^, y^ equal to zero in turn. 2. By multiplying the y's in Ex. 1 by constants, derive called the canonical form. Its Hessian is 6'A, where h= -a^2(2i»+Z.i»+23') +(a'+2/3»)ziZ2Z3. Thus find the nine inflexion points and show that the four inflexion triangles are ZiZj23=0, 2zi'— 3/ziZi23=0 (/=!, w, u'), §14] INFLEXION POINTS OF CUBIC 29 where u is an imaginary cube root of unity. Their left members are constant multiples of Zh+rf, where r=30', — {la—0)^ are the four roots of (1), with 6 = /3(a'-/3'), 4a = a«-20a'|3'-8^«. 3. The Jacobian of /i(xi, . . . , x„), . . . , f„{x,, 9*1 9.t^ ' " " " dXn Xn) is dJndU 3xi 9x2 ?)Xu Show that it is a covariant of index unity of /i, . . . ,/n. 4. Hence the resultant of three ternary linear forms is an invariant of index imity. 5. If /i /» are dependent functions, the Jacobian is zero. PART II THEORY OF INVARIANTS IN NON-SYMBOLIC NOTATION 15. Homogeneity of Invariants. We saw in § II that two binary quadratic forms / and /' have the invariants d = ac-b'^, s = ac'+a'c-2bb' of index 2. Note that s is of the first degree in the coefficients a, b, c oi f and also of the first degree in the coefficients of/', and hence is homogeneous in the coefficients of each form separately. The latter is also true of d, but not of the invariant s+2d. When an invariant of two or more forms is not homogeneous in the coefficients of each form separately, it is a sum of invariants each homogeneous in the coefficients of each form separately. A proof may be made similar to that used in the following case. Grant merely that s+2d is an invariant of index 2 of the binary quadratic forms / and /'. In the transformed forms (§ 11), the coefficients A, B, C oi F are linear in a, b, c; the coefficients ^ ', B' , C of F' are linear in a' , b', c' . By hypothesis AC'+A'C-2BB'+2{AC~B^) =^^{s^-2d). The terms 2dli? of degree 2 m. a, b, c on the right arise only from the part 2{AC-B'^) on the left. Hence d is itself an invariant of index 2 ; likewise 5 itself is an invariant. However, an invariant of a single form is always homo- geneous. For example, this is the case with the above dis- criminant d of /. We shall deduce this theorem from a more general one. 30 515] HOMOGENEITY OF INVARIANTS 31 Let I be an invariant of r forms /i, . . . ,fr of orders p\, . . . , pT in the same q variables xi, . . . , Xq. Let a particular term / of / be of degree di in the coefficients of /i, of degree ^2 in the coefficients of /2, etc. Apply the special transformation of determinant A =a9. Then/( is transformed into a form whose coefficients are the products of those of /( by a"*. Hence in the function / of the transformed coefficients, the term cor- responding to t equals the product of t by This factor therefore equals A'', if X is the index of the invariant. Thus r •^dipi^\q. i = l Hence "Zdipt is constant for all the terms of the invariant. For the above two quadratic forms, r=pi = p> = 2. For invariant d, we have rfi = 2, d:=0, Zdipi=4 = 2\. For 5, we have d, = d-2=l, 2(fi/ii=4. Again, the discriminant (§ 8) of the binary cubic form is of constant degree 4 and index X = 6 ; we have ^dtpt = 4 • 3 = 2X. If, as in the last example, we take r = l, we see that an invariant of index X of a single q-ary form of order p is of constant degree d, where dp = X^, and hence is homogeneous. 16. Weight of an Invariant / of a Binary Form f . Give to / and/ the notations in § 7. Let / = cao<'»ai<^' . . . a/f be any term of /, and call w = ei+2e2+^e3 + . . .+pei, the weight of t. Thus w is the sum of the subscripts of the factors c« each repeated as often as its exponent indicates. We shall prove that the various terms of an invariant of a binary form are of constant weight, and hence call the invariant isobar ic. For example, aox^+2aixy+a2y^ has the invariant aoa2— ai^, each of whose terms is of weight 2. 32 ALGEBRAIC INVARIANTS To prove the theorem, apply to / the transformation X=J, y=(xrt. We obtain a form with the literal coefficients Ao = ao, A\=a\a, A2 = a2ot^, ■ ■ ■, Ap = a^'''. Hence if I is of index X, 7(ao, axa, . . . , apa;P)=a''/(ao, ai, . . . , Cp), identically in a and the a' a. The term of the left member which corresponds to the above term / of 7 is evidently coao^" . . . aj,^vdy. Hence w = \. The weight of an invariant of degree J of a binary ^-ic is thus its index and hence (§ 15) equals \dp. 17. Weight of an Invariant of any System of Forms. Let /i, . . . ,/„ be forms in the same variables X\, . . . , Xq. We define the weight of the coefficient of any term of ft to be the exponent of x^ in that term, and the weight of a product of coefficients to be the sum of the weights of the factors. For q = 2, this definition is in accord with that in § IG, where the coefficient at of xi''"*X2* was taken to be of weight k. Again, in a ternary quadratic form, the coefficients of Xi^, x\X2, and x^^ are of weight zero, those of x\Xz and x^xz of weight unity, and that of x-^ of weight 2. Under the transformation of determinant a, U becomes a form in which the coefficient c' corresponding to a coefficient c of weight k in j\ is ca^. If / is an invariant, I(c')=c^I{c), identically in a. Hence every term of / is of weight X. Thus any invariant of a single form is isoharic; any invariant of a system of two or more forms is isobaric on the whole, but not necessarily isobaric in the coefficients of each form separately. The index equals the weight and is therefore an integer ^ 0. 518] PBODUCTS OF LINEAR TRANSFORMATIONS 33 EXERCISES 1. The invariant aofl'z+ajo'o— 2aia'i of aox'+2aixy+aiy^, a'ox^ +2a'ixy +a'iy' is of total weight 2, but is not of constant weight in Oo, fli, 02 alone. 2. Verify the theorem for the Jacobian of two binary linear forms. 3. Verify the theorem for the Hessian of a ternary quadratic form. 4. No binary form of odd order p has an invariant of odd degree d. ^0, ^0, 18. Products of Linear Transformations. The product TT' of a /3 7 5 ' a' a p y 5 is defined to be the transformation whose equations are obtained by eliminating | and rj between the equations of the given transformations. Hence ,. I x=a"X+p"Y, y = y"X+5"Y, ■ \a"=aa'+py',p"=aP'+^S',y" = ya'+Sy\S" = y0' + Sd'. Its determinant is seen to equal AA' and hence is not zero. By solving the equations which define T, we get V 5 |3 —y , a i = -rx--y, v=—-x+-y. A A A A These equations define the transformation T~^ inverse to T; each of the products TT~^ and T~^T is the identity trans- formation x = X, y = Y. The product of transformation Tg, defined in § 1, by Tg' is seen to equal Te+e', in accord with the interpretation given there. The inverse of Te is T -e'. i=xcos e+ysin 8, ?;= — x sin 9+y cos 9. Consider also any third linear transformation Ti: X=aiU+PiV, Y = yiU+5iV. To prove that the associative law (TT')Ti = TirTi) 34 ALGEBRAIC INVARIANTS holds, note that the first product is found by eliminating first $, Ti and then X, Y between the equations for T, T' , T\, while the second product is obtained by eliminating first X, Y and then ^, 77 between the same equations. Thus the final eliminants must be the same in the two cases. Hence we may write TT'Ti for either product. 19. Generators of All Binary Linear Transformations. Every binary linear homogeneous transformation is a product of the transformations T„: x=^+nr), y = v; St: x=^, y = kri (^5^0); V: x=-v, y=i. From these we obtain * F-' = F^: x-=r,, y=-i,\ V-'T-nV = T'„: x = x', y = y'+nx'; V-'StV =S\: x = kx', y=y' (k^O). For d9^0, the transformation T in § 18 equals the product For 5 = 0, so that 187 ?^0, T equals 20. Annihilator of an Invariant of a Binary Form. The binary form in § 7 may be written as either of the sums Transformation V, of determinant unity, replaces the second sum by i^(^^a,-t(-iy^r>-ir,i. Comparing this with the first sum we see that an invariant of /must be unaltered when (1) a, is replaced by (-l)'a,-< (i = 0,l, . . . ,p), • The 7"'s are of the nature of translations, and the 5's stretchings. §20] AXNIHILATOR OF INVARIANTS 35 By § 16, a function I{ao, . . . , a^) is invariant with respect to every transformation St if and only if it is isobaric. Finally, the function must be invariant with respect to every T„; under this transformation let Differentiating partially with respect to n, we get Tji+l since ri=y is free of n, while ^ = x — nn. The total coefficient of P-'tj^ is the second term being absent ify = 0. But Hence] dn 9m , , d KAo,. .,Aj,) 9/ . 9/ ,oA dl , ,.j 9/ (2) =Ao — -+2Ai—-+6A2—:r +. .+pAp-i — — . ^ ^ 9m dAi 9^2 9^3 dA^ Now I{ao, . . . , Gp) is invariant with respect to every transformation T„, of determinant unity, if and only if 7(^0, . . . , Ap)=I{ao, . . . , ap), identically in n and the a's. This relation evidently implies dljAo, . . ■,Ap) _^ 9m Conversely, the latter impHes that I{Ao, . . . , Ap) has the same value for all values of n and hence its value is that given by n = 0, viz., I(ao, . ■ • , cip). Hence / has the desired property if and only if the right member of (2) is zero identically in n and the a's. But this is the case if and only if ni{ao, . . . , ap) = 0, 36 ALGEBRAIC INVARIANTS identically in the a's, where li is the differential operator i2 = ao--+2ai-^+3a2:^+. . .+paj,.i^. oai 9^2 9^3 oflp In other words, / must satisfy the partial differential equation 127 = 0. In Sylvester's phraseology, / must be anni- hilated by the operator Q. From this section and the preceding we have the important Theorem. A rational integral function I of the coefficients of the binary form f is an invariant of f if and only if I is iso- baric, is unaltered by the replacement (1), and is annihilated by Q. EXAMPLE An invariant of degree d of the binary quartic (§6) is of weight 2d (end of § 16). For d=l, the only possible term is kch] since 0=a{ka2) = 2/feai, we have /fe=0. For d=2, we have I=raoa< +saiai +102*, Ql=(s+4r)aoa,+i^t+3s)aiat = 0, i=— 4r, /=3r, /=r(aoa«-4aiOi+3 of K, KM, B,...;^,v,-- .)=A^Ki{a, b, . . . ; x, y, . . .), and A'l is a covariant. In this way, K = Ki-\-K2 + . . . . Henceforth, we shall restrict attention to covariants which are homogeneous in the variables, and hence of constant order. A covariant K of constant order o> of a single form f is homo- geneous in the coefficients, and hence of constant degree d. For, let / have the coefficients a, b, . . . and order p, and apply the transformation x=a^, y=ar], .... The coefficients of the resulting form are A=aPa, B=aPb Thus K(a^a,a^b,. . . ; a-^x,a-^y,. . .) = {a'^YK{a,b, . . .■,x,y, . . .), identically in a, a, b, . . . , x, y, . . . , since the left member 38 ALGEBRAIC INVARIANTS equals K{A, £,...; |, 77, .. .)• Now K is homogeneous in X, y, . . . , of order co; thus a-"KiaPa, aPb, . . .; X, y, . . .)=a9^K{a, b, . . . ; x, y, . . .). Thus if K has a term of degree din a, b, . . . , then tt-".Q:Pd=a9>', pd-w = q\, so that d is the same for all terms of K. If f is a form of order p in q variables and if K is a covariant of degree d, order w and index X, tken pd — w=q\. 22. Weight of a Covariant of a Binary Form. In f = aox''+paixP-'^y+. . . + (\]atxP-iyi+. . .+0^7'' the weight of at is k. We now attribute the weight 1 to x and the weight to y, so that every term of / is of total weight p. Apply to/ the transformation x=^, y^at). The Hteral coefficients of the resulting form are Ao = aQ, Ai=aai, . . ., Ap=aPap. If ^ is a covariant of degree d, order oi, and index X, then K{Ao, . . . ,Ap] ^, ri)=a^Kiao, . . . , a,,; x, y). Any term on the left is of the form cAo'oAi'i . . . A/p^r,"-' (eo+ei+ . . . +e„=d). This equals cao'oai'i . . . a/px^-y^-'a"^-" {W = r +ei+2e2+ . . . +pe„). This must equal a term of the right member, so that JF — a) = X. But W is the total weight of that term. Hence every term of K is of the same total weight. A covariant of index X and order u of a binary form is isobaric and its weight is co+X. For a form /of order pinq variables, we attribute the weight 1 to Xi, Xi, . . ■ ,x,_i and the weight to Xq-, then (§17) every term of / is of total weight p. By a proof similar to the above, a covariant of index X and order o) of / is isobaric and its weight is u+X. §23] ANNIHILATORS OF COVARIANTS 39 Consider a covariant K homogeneous and of total order u in the variables Xi, . . . , Xq of two or more forms /,. As in § 15, K need not be homo- geneous in the coefficients of each form separately, but is a sum of covariants homogeneous in the coefficients of each. Let such a isT be of degree dt in the coefficients of /(, of order ^j. As in ^21, Epidi—o> = q\. The total weight oi K is ai+\. For example, ii pi = p2 = q=2, fi = aox'+2aixy+a2y\ fi = box'+2biXy+b2y\ The Jacobian of /i and fi is 4X, where K = {aobi—aibo)x' +{aob2— aibojxy +{aibi— a,bi)y^. Here di = di=l, 01=2, X= 1, and K is of weight 3. 23. Annihilators of Covariants K of a Binary Form. Pro- ceeding as in § 20, we have instead of (2) 9« ^' «> // j=odA, dn d^dn dndn and obtain the following result: K is covariant with respect to every transformation x = ^+nri, y = v, if and only if it is annihilated by * (1) n-y^ /n = ao-^ + . . .+/'ap-.^ dx \ dai dap The binary form is unaltered if we interchange x and y, di and Op-, for i = 0, 1, . . . , p. Hence K is covariant with respect to every transformation x = ^, y = ri+n^, if and only if it is annihilated by (2) O-x^- (o^pa^^+(p-l)a2^--+. . .+a,-^). dy \ dao dai dap-i/ Denote a covariant of order u of the binary p-ic by K=Sx"+Sixf-^y+. . .+S^r- * For another derivation, see the corollary in § 47. 40 ALGEBRAIC INVARIANTS By operating on K by (2), we must have {OS-Si)o(f+{OSi-2S2)x-'-'y+. . .+(05,_i-a,5j:cr-' identically in x, y. Hence K becomes (3) K=SxfJrOSx<'-'^y+\0''Sx"-^y^^. . .+ i-O-Jy, whilt.by O5„ = 0, (4) 0"+'5=0. Hence a covariant is uniquely determined by its leader S. (Cf. § 25). Similarly, K is annihilated by (1) if and only if (5) fi5 = 0, fi5i = w5, i2S2=(a)-l)5i, ..., n5„ = 5„-i. The function 5 of oo, . . . , a^ must be homogeneous and isobaric (§§21, 22). If such a function 5 is annihilated by fi, it is called a seminvariant. If we have Su, we may find •51,-1 by (5), then S^-2, ■ ■ ■ , and finally Si. But if K is a covariant, we can derive S^ from 5. For, by § 20, the transformation x=—r], y=^ replaces / by a form in which Ai = ( — iyap-,; by the covariance of K, S{A)^+. . =S{A)r+- ■ .^S(a)xf+. . .+Sja)r, so that S^{a)=S(A). Hence S^ is derived from 5 by the replacement (1) in § 20. When the seminvariant leader S is given, and hence also a> (see Ex. 1), the function (3) is actually a covariant of /; likewise the function whose coefficients are given by (5). Proof will be made in § 25. In the following exercises, indirect verification of the covariance is indicated. EXERCISES 1. The weight of the leader 5 of a covariant of order u of a binary form / is W— ;-i^=2 (k + l)a,-^, = 2 {p-j+l)aj-^ = ^\p-k)at+i^ of invariants of a binary form. We have k=o 1 9at+i y=i 9at9ojJ The terms involving second derivatives are identical. Hence n0-0fi = 2 {i+l){p-i)ai— -Xi(p-i + l)ai-^ i=Q 9fli •=! 3ot = 2(/>-2f)af-^- i =0 Odt since the first sum is the first sum in flO with j replaced by i+1, and the second is the first sum in OQ with k replaced by X — 1 . If 5 is a homogeneous function of oo, • • . , flp of total degree d and hence a sum of terms cao'oai'^ . . . ap'v (eo+ei+- • •+ep=d), we readily verify Euler's theorem: I ^ = dS. i=o 9ai ^ ALGEBRAIC INVAfilANTS If 5 is isobaric, it is a sum of terms t=cao'oai'i . . . ap'v (ei+2e2+. . .■\-pep='w) where w is constant; then Xia,— =X ie,t = wt, 2 la.— = wS. «=o 9ai t=o •=o 9fl« Hence i/" S is both homogeneous (of degree d) and isobaric {of weight w) inao, . . . , a^,, then (1) (nO-On)S = wS, co = pd-2w. A covariant with the leader S has the order w. (Ex. 1, § 23.) Since OS is of degree d and weight w+1, we have iQCP-0'^Q)S=inO-OQ:)OS+OiQO-02)S = {w-2)OS+wOS = 2{w-l)OS. Hence for r = 1 and r = 2, we have (2) (00-0'0)5 = r(co-r+l)0'-i5. To proceed by induction, note that (2) impHes iW+'-0'+'Q)S=iW-0'Q)OS+0'-{QO-OQ)S =r(a)-2-r+l)0'-5+ajO5 = (r+l)(w-r)0'-5, so that (2) holds also when r is replaced by r+1- 25. Seminvariants as Leaders of Binary Covariants. Lemma. // 5 is a seminvariant, not identically zero, of degree d and weight w, of a binary p-ic, then dp — 2wl,0. Suppose on the contrary that 5 is a seminvariant for which w<0, where oi = dp — 2w. By the definition of a seminvariant, QS = 0. Hence, by (2), § 24, (1) n0'-5=r(co-r+l)0^-i5 (r = l, 2, 3, . . .) and no one of the coefficients on the right is zero. But Qdp-«. +15=0, being of degree d and weight dp+1; in fact, the largest weight of a function of ao, • • • , dp of degree d is dp, the weight of a/. Then (l) for r=dp-w+l gives O''''-"'5 = 0. Then (1) §25) SEMIN VARIANT LEADERS OF CO VARIANTS 43 for r=dp-w gives O'*"-"'-* 5 = 0, etc. Finally, we get 5 = 0, contrary to hypothesis. Theorem. There exists a covariant K of a binary p-ic whose leader is any given seminvariant S of the p-ic. The covariant iiT is in fact given by (3), § 23. By (l), for r = a)+l, no- +'5=0. Hence O'^^S is a seminvariant of degree d and weight w'=w+u+\=pd—w-\-\. Then dp-2w' = -{pd-2w)-2 is negative. Hence (4), §23, follows from the Lemma. Thus K is annihilated by the operator (2), § 23. Next, in («-^s)^- the coefficient of xf y is in0^5-^^(a,-r+l)0'-'5=-,ln0^5-r(a,-r+l)0:-'51, r! {r — \)\ r\ which is zero by (1). Hence K is covariant with respect to all of the transformations Tn and T'n of § 19. Now T-iT\T.x = V: x=-Y, y = X, as shown by eliminating ^, t], Ji, tji between y= V, \ v = vi + ^i, ^i = X-Y, m= Y. Since K is of constant weight, it is covariant with respect to every St (§ 16). Hence, by § 19, K is covariant with respect to all binary linear transformations. 26. Niunber of Linearly Independent Seminvariants. Lemma. Given any homogeneous isobaric function 5 of oo, . . . , ap of degree d and weight w, where O3=dp — 2w>0, we can find a homogeneous isobaric function 5i of degree d and weight w+1 such that n5i=5. •14 ALGEBRAIC INVARIANTS In (2), § 24, replace S by 2''~'^S, whose degree is d and weight is w — r + l, so that its w is u+2r — 2. We get QQ-Q'-^S-0'-Q'S = rii^+r-l)0'-^n''-^S. Multiply this by r!co(o) + l) . . . (co+r-1)' The new right member cancels the second term of the new left member after r is replaced by r — 1 in the latter. Hence if we sum from r = l to r=w+l, the terms not cancelling are those from the first terms of the left members, that from the right member for r = l, and that from the second term on the left for r=w+l. But the last is zero, since fi"'+'5=0, O^S being of weight zero and hence a power of oo. Hence we get QSi=S, where r=ir\u{w + l) . . . {co+r-1) Theorem.* The number of linearly independent seminvariants oj degree d and weight w of the binary p-ic is zero if pd—2w<0, but is (w; d, p)-{w-l; d, p), if pd—2w^0, where (w; d, p) denotes the number of partitions of w into d integers chosen from 0, 1, . . . , p, with repetitions allowed. If p^4, (4; 2, p) = 3, since 4+0, 3+1, 2+2 are the partitions of 4 into 2 integers. Also, (3; 2, p) = 2, corresponding to 3+0, 2+1. Hence the theorem states that every seminvariant of degree 2 and weight 4 of the binary p-ic, p^i, is a numerical multiple of one such (see the Example in § 20). The literal part of any term of a seminvariant 5 specified in the theorem is a product of d factors chosen from oo, ai, . . . , Op, with repetitions allowed, such that the sum of the subscripts of the d factors is w. Hence there are {w; d, p) possible terms. Giving them arbitrary coefficients and oper- ating on the sum of the resulting terms with £2, we obtain a linear combination S' of the {w—\; d, p) possible products * Stated by Cayley; proved much later by Sylvester. §27] LAW OF RECIPROCITY 45 of degree d and weight w — \. By the Lemma there exists* an S for which 05 is any assigned S' . Thus the coefficients of our 6" = 05 are arbitrary and hence are hnearly independent functions of the {w; d, p) coefficients of 5. Hence the con- dition 05 = imposes {w—\; d, p) hnearly independent linear relations between the coefficients of 5 and hence determines {w — \; d, p) of the coefficients of 5 in terms of the remaining coefficients. Thus the difference gives the number of arbitrary constants in the general seminvariant 5, and hence the number of linearly independent seminvariants 5. 27. Hermite's Law of Reciprocity. Consider any partition m; = wi+W2 + - • ■+«« of w into 8^d positive integers such that p^ni^n2 ... ^ Wj. Write Ml dots in a row; then in a second row write W2 dots under the first «2 dots of the first row; then in a third row write W3 dots under the first «3 dots of the second row, etc., until IV dots have been written in 5 rows. Now count the dots by columns instead of by rows. The number wi of dots in the first (left-hand) column is 5; the number W2 in the second column is ^ mi ; etc. The number of columns is ni £ p. Hence we have a partition w = mi+m2+. . .+m, of w into TT ^ p positive integers not exceeding d. Hence to every one of the {w; d, p) partitions of the first kind corresponds a unique one of the {w; p, d) partitions of the second kind. The converse is true, since we may begin with an arrangement in columns and read off an arrangement by rows. The correspondence is thus one-to-one. Hence {w; d,p) = {w; p, d). By two applications of this result, we get (w; d, p)-{w-l; d, p) = (w; p,d)-{w-l; p, d). Hence, by the theorem of § 26, the number of linearly independent ♦Provided />(i-2(i£'-l)>0, which holds if /'(i-2a'^0. But if pd-2w where \ / \"/ a 2=0.2 — ao' a'a = as - _3aia2 • > and the roots off = Oare, ai+ai/ao (^ = 1, ...,p) Since ao at — 2a, P (at— a] .) + . . P . + (ai-ap) » each root of /' = is a linear function of the differences of the roots of /=0 and hence is unaltered by every transformation r„. The same is true of a'2/ao, a'z/ao, . . . , which equal numerical multiples of the elementary symmetric functions of the roots of/' = 0. Hence the polynomials A2= aoa'2 = 30^2 — di^, A3 = flora's = floras — 3aoaia2+2ai^, Ai=ao^a'i = ao^ai — 4ao^aia3+6aoai^a2 — 3ai* are homogeneous and isobaric,* and are invariants of / with respect to all transformations T„. By definition they are, therefore, seminvariants of / provided the subscript of each A in question does not exceed p. * This is evident for A^, A3, Ai. Further A's will not be employed here. A general proof follows from § 34. 48 ALGEBRAIC INVARIANTS Since /' was derived from / by a linear transformation of determinant unity, any semin variant S of / has the property Siao, . . . ,a„)=5(ao,0,a'2, . • . ,a'j)=5(flo, 0, —,..., —^ ). Hence any rational integral seminvariant is the quotient of a polynomial in ao, ^2, • • . , ^p by a power of oo- For /> ^ 4, we shall find which of these quotients equal rational integral functions of ao, • • • , dv and hence give rational integral seminvariants. The method is f'ue to Cayley. For /> = 1 , 5 is evidently a numerical multiple of a power of ao. Since ao is the leader of the covariant /=ooa;+ai)' of /, we conclude that every covariant of a binary linear form / is a product of a power of / by a constant; in particular, there is no invariant. 29. Binary Quadratic Form. Since A2 does not have the factor ao, we conclude that every rational integral seminvariant is a polynomial in ao and A2- Now A2 is an invariant of / (§4), andao is the leader of the covariant/ of/. Hence a fundamental system of rational integral covariants of the binary quadratic form f is given by f and its discriminant A 2 ■ We express in these words our result that any such covariant is a rational integral function of/ and Ao. 30. Binary Cubic Form. We seek a polynomial P(ao, ^2,^3) with the implicit, but not explicit, factor Oq. Write A\ for the terms of At free of ao: (1) A'2 = -ai^, A'3 = 2aiK We desire that P(0, A'2, ^'3) = 0, identically in ai. Now 4^V+^V = 0, (2) iA2^+A3^ = aoW, where D is the discriminant of the cubic form, D = ao'^a-.r - Gaoaia2a3 +4aoa2''' +4:ai^a3 - 3ai^a2^. §311 FUNDAMENTAL SYSTEM OF COVARIANTS 49 By means of (2) we eliminate A-^ and higher powers of Az from F(ao, A2, Az) and conclude that any seminvariant is of the form x/ao*, where tt is a polynomial in ao, ^2, Az, D, of degree 1 or in ^3- If ^>0, we may assume that not every term of x has the expHcit factor oq. In the latter case, tt does not have the implicit factor ao- For, if it did, 7r'=Tr(0, A'2, A'z, Z)')=0, D' = Aai^az-2>ai^a2^. Since az occurs in D' , but not in A'2 or A'z, ir' is free of D' . By (1), the first power of A'z is not cancelled by a power of A'2- Hence tt' is free of A'z and hence of A'2- A Jundamental system of rational integral seminvariants oj the binary cubic is given by ao, A2, Az, D. They are connected by the syzygy (2). A fundamental system of rational integral covariants of the binary cubic f is given by f, its discriminant D, its Hessian H, and the Jacobian J of f and H. They are connected by the syzygy (3) 'im+J^=fW. The last theorem follows from the first one and (2), since oq, A2, Az are the leaders of the covariants/, H, J- 31. Binary Quartic Form. We first seek polynomials PiflQ, A 2, Az, Ai) with the impHcit, but not explicit, factor 00. Thus i" = P(0,^'2, ^'3, ^'4)^0, A'2=-ai^, A'z = 2ai^ A'i=-3ai\ The simplest P' is evidently 3A'2^+A'i. We get ^14+3^2^=00^/, / = 0004 — 4aia3 +3^2^. We drop A^ and consider polynomials ir{ao, A2, Az, I) with the implicit, but not explicit, factor, ao- Such a polynomial is given by (2), § 30. For ao = 0,D= -ai^I = A'2l. We have A2l—D=aoJ, J = (20O2O4 — ooos^ +2aia203 — 01^04 — 02^. Eliminating D between this relation and (2), § 30, we get (1) aoV-ao2^ 2/ + 4^ 2^+^32 = 0. 50 ALGEBRAIC INVARIANTS In view of their origin, / and / are seminvariants of the quartic /. Since they are unaltered by the replacement (1), §20, they are invariants of/ (c/. §20, Example and Ex. 1). In view of (l), tt equals a polynomial in ao, A2, A3, I, J, of degree or 1 in Az- Suppose that does not have the explicit factor Oq- Then the equal function of oq, . . . , ^4 is not divisible by Aq- For, if it were, 0(0, — ai^, 2ai3, 3a2^— 4aia3, —ar^ai+. . .)=0. In view of the term a^, cannot involve J, and hence not /. Nor can be hnear in .4 3 in view of the odd power ai^. Hence is free of ^3 and hence of ^42- A fundamental system of rational integral seminvariants of the binary quartic is given by ao, A2, A3, I, J. They are con- nected by the syzygy (l). A fundamental system of rational integral covariants of the binary quartic f is given by f, its invariants I and J, its Hessian H and the Jacobian G of f and H. They are connected by the syzygy (2) fU-fmi+iH^+G^ = 0. The second theorem follows from the first one, since ao, A 2, A3 are the leaders of the covariants/, H, G. It would be excessively laborious, if not futile, to apply the same method to the binary quintic, whose fundamental system is composed of 23 covariants,* most of which are very complex. The symbolic method is here superior both as to theory and as to compact notation (see Part III.). Canonical Form of Binary Quartic. Solution of Quartic Equations 32. Theorem. A binary quartic form f, whose discrim- inant is not zero, can be transformed linearly into the canonical form (1) X* + F4+6wZ2F2. *Faa di Bruno, Thcorie der Bindren Formen, German tr. by Walter, 1881, pp. 199, 316-355. Salmon, Modern higher Algebra, Fourth Edition, 1885, p. 227, p. 347. §32) CANONICAL FORM OF QUARTIC 51 The reason there is here a parameter m lies in the existence of two invariants / and J of weights (and hence indices) 4 and 6, and hence a rational absolute invariant P/J-, i.e., one of index zero, and consequently having the same value for/ and any form derived from / by linear transformation. Since / vanishes for four values of x/y and hence is the product of four linear functions, it can be expressed (in three ways) as a product of two quadratic forms, say those in the right members of the next equations. To prove our theorem it suflSces to show that there exist constant p, q, r, s (each 5^0) and a, /3 (a?^/3) such that p{x+ayy+q(x+0y)^ = ax'^ + 2bxy+cy', r(x+ayy+s{x+^yy = gx'^ + 2hxy+ky^. For, the product/ of these becomes (1) by the transformation X=<^{x+ay), Y = <^{x+py), of determinant p^O. The conditions for the two identities are p+q = a, pa+qfi = b, pa^+q0^ = c, r+s = g, ra+sP = h, ra^+s^^ = k. The first three equations are consistent if 1 1 a a P b ^(/3-a)=c-6(a+/3)+aa/3 = 0. li p = 0, or if q=0, the same equations give b^ = ac, so that the first quadratic factor of / and hence / would have a double root. Similarly, the last three equations have solutions r?^0, 59^0, if k-h{a+P)+ga^ = 0. If the determinant ah — bg is not zero, the last two relations determine a+/3 and a^, and hence give a and /3 as the roots of * {ah — bg)z-—{ak—cg)z+bk — ch = 0. * Its left member is obtained by setting x/y=—z in the Jacobian of the two quadratic factors of/. 52 ALGEBRAIC INVARIANTS If its roots were equal, the two relations would give c-2ba+ac^ = 0, k-2ha+ga^ = 0, and the two quadratic factors of / would vanish for x/y = -a. If ak — bg = 0, but ch — bk9^0, we interchange x with y and proceed as before. If both determinants vanish, either Jr^O and the second quadratic factor is the product of the first by h/b, or else & = and hence h = and no transfor- mation of/ is needed. 33. Actual Determinatioii of the Canonical Quartic. Let A denote the determinant of the coefficients of x, y in A', Y. Then /, its invariants I and / and Hessian H are related to the canonical form, its invariants and Hessian, as follows: f=X* + Y*+6mX'^Y^, Z = A«(1+3ot2), / = a6(w-ot3), Thus A^TW may be found from the resolvent cubic equation 4(A%)3-7(A2w)+/ = 0. Then A* may be found from I. We may select either square root as A^ and hence find m. In fact, by replacing .Y by XV — 1 in /, the signs of A^ and m are changed. By elim- inating X'^+Y'^, we get A2w/-F=A2(9w2- 1)X2F2. If 9m^=l, f is the square of X^dzY'^ and the discriminant of / would vanish. Hence we obtain XY by a root extraction. Thus X and Y are determined up to constant factors t and i"'. We may find / by comparing the coefficients of x* and x^y in / and the expansion of its canonical form, or by use of the Jacobian G of / and H : G = A^il-9m^-)XYiX*-Y*), and combining the resulting X^ — Y"^ with the earlier A'^ + F^. Or from J and XY we can find X'^ + Y^ and then Xd=Y. To solve /=0, we have only to find the canonical form §34] SEMINVARIANTS IN TERMS OF THE ROOTS 53 Seminvariants, Invariants, and Covariants of a Binary Form/ AS Functions of the Roots of/=0, §§ 34-37. 34. Seminvariants in Terms of the Roots. Give / the nota- tion used in § 28, so tliat ai, . . . , ap are the roots of /=0. After removing possible factors oq from a given seminvariant of /, we obtain a seminvariant S not divisible by oq. Let 5 be the degree of the homogeneous function S of the a's. Thus 5 is the product of ao* by a polynomial in ai/ao, . . . , aj/ao of degree 8. The latter equal numerical multiples of the ele- mentary s}-mmetric functions of ai, . . . , a,,, each of which is linear in every root. Hence our polynomial equals a sym- metric polynomial c in ai, . . . , ap of degree 5 in every root. Since 5 is of constant weight w and since at/ao equals a function of total degree i in the roots, a is homogeneous in the roots and of total degree w in them. Besides being homogeneous and isobaric in the a's, a sem- invariant must be unaltered by every transformation T„ of § 28. Under that transformation, each root is diminished, byw(§28). Since ai=ai + {ai — ai) (^" = 2,. • • , p) we can express a- as a polynomial P(ai) whose coefficients are rational integral functions of the differences of the roots. If P(ai) is of degree ^ 1 in ai, we have P{ai) = P{a\—n) , for all values of n. But an equation in n cannot have an infinitude of roots. Hence P(ai) does not involve ai, so that a equals a polynomial in the differences of the roots. Multiplying by the factors oq removed, we obtain the theorem : Any seminvariant of degree d and weight w of the binary form aox'+. . . equals the product of ao" by a rational integral symmetric function a of the roots, homogeneous {of total degree w) in the roots, of degree ^d in any one root, and expressible as a polynomial in the differences of the roots. Conversely, any such product can be expressed as a poly- nomial in the a's and this polynomial is a seminvariant. ^4 ALGEBRAIC INVARIANTS Since the factor c is symmetric in the roots, and is of degree = J in any one root, its product by ao'^ equals a homogeneous polynomial in the a's whose degree is d. This polynomial is isobaric since c is homogeneous, and is unaltered by every transformation r„, since o- is expressible as a function of the differences of the roots. The importance of these theorems is due mainly to the fact that they enable us to tell by inspection (without com- putation by annihilators) whether or not a given function of the roots and ao is a seminvariant. A like remark applies to the theorem in § 35 on invariants and that in § 36 on covariants. EXAMPLE The binary cubic has the seminvariant 3 = aoM(Sai)='-32aia2}=aoM (— " ) ~H~] i = -9(ao(72-ai'). 35. Invariants in Terms of the Roots. A seminvariant of / is an invariant of / if and only if it is unaltered by the trans- formation x=—r),y = i{%20). For the latter, so that ttr is replaced by —I/oit, and hence a,— a, by ar—a. The coefl&cient of ^ in the transformed binary form is Ao = { — '0''aioi2 • ■ • apUo. By § 34, any seminvariant of / is of the type Co'' Sc< (product of w factors like ar — as). Hence this is an invariant if and only if it equals (_l)p'*(aj . . . ap)''ao''2ci(product of the te; corresponding ^^^ — ^1, §35] INVARIANTS IN TERMS OF THE ROOTS 55 and hence if iai"^ ... a/ equals the product of the factors aru, in the denominators. This is the case if and only if each root occurs exactly d times in every term of the sum and if pd is even. By the total number of as, pd = 2-w. Any invariant of degree d and iveight 10 of the binary form aox^+ ■ ■ ■ equals the product of a,,'^ by a sum of products of constants and certain dijferences of the roots, such that each root occurs exactly d times in every product; moreover, the sum equals a homogeneous symmetric function of the roots of total degree w. Conversely, the product of any such sum by ao'' equals a rational integral invariant. EXERCISES 1. coH«i— "2)' is an invariant of the binary quadratic form. Any- invariant is a numerical multiple of a power of this one. 2. ao'2;(ai— a2)Ha3— a«)' is an invariant of the binary quartic. .3 3. ao'2(oi— a2)(ai— as) is not an invariant of the binary cubic. s 4. If we multiply«(i^'''~" by the product of the squares of the differences of the roots of the binary p-ic f, we obtain an invariant (discriminant of /). Also verify that /»rf=2w. 5. The sum of the coefficients of any seminvariant is zero. Hint: Use/= (x+y)'', whose roots are all equal. 6. Every invariant of the binary cubic is a power of its discriminant. 7. A function which satisfies the conditions in the theorem of § 35 except that of symmetry in the roots is called an irrational invariant. If ai, . . . , 04 are the roots of a binary quartic /, and «= («i — c»4)(a2— aa), V= iai — at)(a3 — ai), K)= (ai— a2)(a3— a*), why are aou, a^v, aow irrational invariants of /? They are the roots of z'— 12/z— 5 = 0, where 5' is the product of Oo* by the product of the squares of the difTerences of the roots and hence is the discriminant of /. Hints: tt+D+w=0, and s=uv-\-uw-\-vw is a symmetric function of au ■ ■ , «< in which each a^ occurs twice in every product of differences, so that a^'^s is an invariant of degree 2. By the Example in §20, ao's=cI, where c is a constant. To determine c, take ai= l,a2= — 1, 03 = 2, «<= — 2, so that /=(a;J-y2)(x2-4y''), 7=73/12, m=-9, d=1, w = 8, i=-73. Hence c= — 12. As here, so always an irrational algebraic invariant is a root of an equation whose coefficients are rational invariants. 56 ALGEBRAIC INVARIANTS 8. If ai, 012 are the roots of the binary quadratic form /, and 03, en the roots of /' in § 11, the simultaneous invariant ac' +a'c — 2bb' = aa'laiat+aiai— i{ai+a2){a3+ai)\ = lao(u — v), if the product //' is identified with the quartic in Ex. 7. Hence a simul- taneous invariant of the quadratic factors of a quartic is an irrational invar- iant of the quartic. Why a priori is the invariant three-valued? 9. The cross-ratios of the four roots of the quartic are —v/u, etc. These six are equal in sets of three if /= 0. For, if 5= 0, , , . , —V —u —w vw=u{—v—w) = u', uiv=v{—u—w) = v', — = — = . U W V The remaining three are the reciprocals of these and are equal. 10. By Ex. 3, § 11, one of the cross-ratios is — 1 if ac' -|- . . . =0. Why does this agree with Ex. 8? 11. The product of the squares of the differences of the roots of the cubic equation in Ex. 7 is known " to be -i{-12l)^-27S^ = ao^iu-vy{u-wy{v-w)K Also,* «2= 256(7' -27/2). Hence the left member becomes 3«-4«72. Thus 33.42/= ^ao^{u — v){u—w)(v—w). Using J from § 31, and the special values in Ex. 7, show that the sign is plus. Verify that the cross-ratios equal —1, —1, 2, 2, 5, 5, if 7 = 0. 36. Covariants in Terms of the Roots. Let K {oo, . . . ,ap;x,y) be a covariant of constant degree d (in the coefficients) and constant order 01 (in the variables) of the binary f orm/ = ocx" +. . . Then where k is a polynomial in x/y and the roots ai, . . . , a^ of /=0. Under the transformation T„ in §28, let / become ^0^"+ • ■ • , with the roots a'l, . . . , a p. Then / ai= a (, ar—a,=aT—ai y V Making use of the identities -=( aij+ai, at={ai — ai)+ai, X y ' Cf. Dickson, Elementary Theory of Equations, p. 33, p. 42, Ex. 7. §36] COVARIANTS IN TERMS OF THE ROOTS 57 we see that k equals a polynomial P{ai) whose coefficients are rational integral functions of the differences of x/y, ai, . . . , ap in pairs. Since K{Ao, . . . , Ap-, ^, ri)=K{ao, . . . ,aj,; x,y), Ao = ao, n = y. we have K(a'i, . . . , a'p, -I =K\a\, . . . , up, -j. The left member equals P(a'i) since a.\={ai—ai)-\-a'\, -=( ai)+a'i. ■n \y I Hence for every n. Hence ai does not occur in P(ai), and k is a polynomial in the differences of x/y, ai, . . . , ap. 'Let W be the weight of K and hence of the coefficient of y'^. Then k is of total degree W in the a's and of degree to in x/y. Thus (c = Semiproduct of u differences like — ar I y •{product of W—u differences like a,—as\. Hence iir = ao'^2Cjlproduct of co differences like x—a,y\ ■ (product of PF — CO differences like ar—a,]. Next, for x=-v, y = ^, f becomes F = Aoi''+ . . . with a root — l/ttr corresponding to each root ar of /. The function K for F is i4o''SCi I product of o> differences like IH — v = ^^ [ ar —ar ■ \ product olW—u differences like ar—as [ arOs Using the value of i4o in § 35, we see that the factor {-l)^ai'' . . .ap-i must be cancelled by the —a, and the ara, in the denominators. 58 ALGEBRAIC INVARIANTS Thus each term of the sum involves every root exactly d times. The signs agree since dp = oi-\-2{W-iS), as follows by counting the total number of a's. Any covariani of degree d, order u and weight W of ooix-aiy) . . . (x—apy) equals the product of ao'' by a sum of products of constants and 03 differences like x — ary and W — w differences like ur — as, such that every root occurs in exactly d factors of each product; more- over, the sum equals a symmetric function of the roots. Conversely, the product of ao'^ by any such sum equals a rational integral covariant. EXERCISES 1. /=OoJ:'+3ai«'>i+3a2xy2+a3}'' has the covariant K = ao'^'Lix-aiyYiai-azY. 3 Show that the coe£Bcient of x'' in K equals — 18(ao02— fli'). Why may we conclude that K= — 18H, where H is the Hessian of/? 2. The same binary cubic has the covariant ao'2(x—aiy)(x — Cfiy){ai—a3){a3—a,) = 9H. 3 3. Every rational integral covariant of the binary quadratic / is a prod- uct of powers of / and its discriminant by a constant. 37. Covariant with a Given Leader S. If the seminvariant S has the factor ao, and S = aoQ, and if Q is the leader of a covariant K of /, then, since ao is the leader of /, 5 is the leader of the covariant fK. Hence it remains to consider only a seminvariant S not divisible by ao. If 5 is of degree d and weight w, 5' = ao'^2Ci(product of w factors like a^— as), where each product is of degree at most d in each root, and of degree exactly d in at least one root (§34). If each product is of degree d in every root, 5 is an invariant (§ 35) and hence is the required covariant. In the contrary case, let a2, for example, enter to a degree less than d; we supply enough factors x~a2y to bring the degree in a2 up to d. Then ao'^ §38] DIFFERENTIAL OPERATORS 59 multiplied by the sum of the total products is a covariant with the leader S. For example, 00^2(02—03)^, ao^X(a2 — a3){a3—ai) 3 3 are the leaders of the covariants in Exs. 1, 2, § 36, of the binary cubic. The present result should be compared with the theorem in § 25. We may now give a new proof of the lemma in § 25 that dp — 2w^0 for any seminvariant 5 of degree d and weight w of the binary ^-ic. Whether 5 has the factor qq or not, the first term of the resulting covariant K is Sx", where u = dp — 2w. For, in each product in the above S, the roots ai, . . . , ap occur 2w times in all. In K each root occurs d times. Hence we inserted dp — 2w factors x—aym. deriving K from 5. 38. Differential Operators Producing Covariants. Let the transformation replace /(a;, y) by 4>{^, v)- Then 9^_9/; 9x 5/; 93)^^0/; a/; a? dx 9? dy 8? "dx dy' 9* = M 9^ -(- 9[ sy = g 0/ + 5 a/ dv dx dv dy dv dx dy Solving, we get dy dv d^ dx dv 9? or df=D4>, dif=Di(t>, if we introduce the differential operators d^A rf:=-Af, D = al-p}. D,=y^~5^ dy dx dv di dv 9C As usual, write d^dif for d{d(dif)\. Since the result of operating with d on df is ths same as operating with D on the equal function D (r+5 = co). 60 ALGEBRAIC INVARIANTS The right member is the result of operating on (p with the operator obtained by substituting D for d/dn and Z>i for — 3/9^ in -■iiJH. (r+s = u), whose terms are partial derivatives of order u. Hence, if the form l{x,y) = ^Crsx'y' {r+s = u) becomes X(^, rj) under the transformation T, our right mem- ber is the result of operating on (p with \id/dv, —d/dO- The left member is the result of operating on / with V dy dx/ \dy dx, Hence if T replaces the forms f{x, y), l{x, y) by (p{^, rj), X({, ?/), then /(^,-9 fix, y) L \dy dx/ is a consequetice of the equations for T, if w is the order of l{x, y). Let / and / be covariants of indices m and n of one or more binary forms ft with the coefficients ci, co, . . . . Under T let the transformed forms have the coefficients Ci, C2, .... Then /(C; ?, 7,)=A'"/(c; x, y), 1{C; ^ „)=A"/(c; x, y). But {i, 7,)=A-'"/(C; i, v), X(?, v)=A-%C; ?, ,). Inserting these into the formula of the theorem, and mul- tiplying by A"*"*"", we get The function in the right member is therefore a covariant of index oi+m+n of the ft. We therefore have the theorem of Boole, one of the first known general theorems on covariants: §38] DIFFERENTIAL OPERATORS 61 Theorem. // / and f are any covariants of a system of binary forms, we obtain a covariant {or invariant) of the system of forms by operating on f with the operator obtained from I by replacingxby d/dy and y by-^/dx, i-e., x'y' byX-lYd^'^'/dy'^dx'. EXERCISES 1. Taking l=f=ax^+2bxy+cy', obtain the invariant i{ac—b^) of /. 2. If /=/ is the binary quartic, the invariant is 2 -4! / of § 31. 3. Using the binary quartic and its Hessian, obtain the invariant /. 4. Taking l=aox''+. . . ,f=box''+. . ., obtain their simultaneous invariant 2(-l)'r.)u,ip_<. i=o \-i If also /=/, we have an invariant of/, which vanishes if p is odd. For p = 2 and/>=4, deduce the results in Exs. 1, 2. 5. A fundamental system of covariants of a quadratic and cubic Q==Ax^+2Bxy+Cy\ f=ax^+3bx'y+5cxy'+dy' is composed of 15 forms. We may take Q and its discriminant AC—B'; f, its discriminant and Hessian h, given by (5) and (2) of § 8, the Jacobian yof/andff: J = {a'd-3abc+2b'')x' +3(abd+b'c-2ac')x^y +Z{2b'd-acd-bc')xy^ + (3bcd-ad'~2c^)y'; the Jacobian of / and Q: (Ab-Ba)x^+{2Ac-Bb-Ca)x'y+{Ad+Bc-2Cb)xy^+{Bd-Cc)y'; the Jacobian of Q and h : {As-Br)x^ + {At-Cr)xy + iBl-Cs)y'; the result of operating on / with the operator obtained as in the theorem from l=Q: Li = iaC+cA-2bB)x + {bC+dA-2cB)y; the result of operating on Q with the operator obtained from ii: Lt= \aBC-b{2B'+AC) +3cAB-dA'}x + {aO-SbBC+c(AC+2B^)-dAB]y; 62 ALGEBRAIC INVARIANTS the result Li, of operating on J with Q and the result Li of operating on Q with Zs (so that Ls and L, may be derived from ii and Li by replacing a, . . . , d by the corresponding coefficients of J) ; the intermediate invariant At+Cr—2Bs of Q and It (§ 11) ; the resultant of Q and /: a2C3-6aJ£0+CacC(2B2_^C) +ad{6ABC-SB') +9b'-AC' -18bcABC+6bdA(2B^-AC)+Qc\4^C-6cdBA'+d''A'; the resultant of Li and Z4 (= resultant of Li and £3), obtained at once as a determinant of order 2. SaXxaon, Modern Higher Algebra, § 198, gives geo- metrical interpretations. Hammond, Amer. Jour. Math., vol. 8, obtains the syzygies between the 15 covariants. PART III SYMBOLIC NOTATION The Notation and its Immediate Consequences, §§ 39-41 39. Introduction. The conditions that the binary cubic (1) f = aoXi^ + 3aiXi^X2+3a2XiX2^ +a3X2^ shall be a perfect cube (2) (aiXi +0:2X2)^ are found by eliminating ai and 02 between (3) ai^ = ao, ai^a2 = ai, aia.2^ = a2, a^ = a3, and hence the conditions are (4) a^)a2 = al\ aiaz=a2^. Thus only a very special form (l) is a perfect cube. However, in a symbolic sense * any form (1) can be rep- resented as a cube (2), in which ai and aa are now mere symbols such that (3') ai^, ai^aa, aia2^, 02^ are given the interpretations (3), while any linear combination of these products, as 2ai^ — 7a2^, is interpreted to be the cor- responding combination of the a's, as 2ao — Tas. But no inter- pretation is given to a polynomial in ai, a2, any one of whose terms is a product of more than three factors a, or fewer than three factors a. Thus the first relation (4) does not now follow from (3), since the expression a-^a-^ (formerly equal to both * Due to Aronhold and Clebsch, but equivalent to the more complicated hyperdeterminants of Cayley. 63 6-i ALGEBRAIC INVARIANTS 00^2 and ai^) is now excluded from consideration; likewise for a{^a2* and the second relation (4). In brief, the general binary cubic (l) may be represented in the symbolic form (2) since the products (3') of the symbols ai, a2 are in effect independent quantities, in so far as we permit the use only of linear combinations of these products. But we shall of course have need of other than linear functions of ao, • • • , as- To be able to express them sym- bolically, we represent / not merely by (2), but also in the symbolic forms (5) {fiiXi+^2X2f, (71^1 +723:2)^, . . . , so that (6) |3i3 = ao, |3i2;32 = ai, fii^2^ = a2, fi2^ = az; 71^ = 00, .... Thus aoa2 is represented by either ai^^i^2^ or /3i^aia2^, while neither of them is identical with the representation ai^a2Pi^02 of ai^. Hence aoa2-ai2 = K«i^/3i/322+^i^ai«22-2ai2a2/3i2/32) = |ai/3i(ai/32-a2)3i)2. We shall verify that this expression is a seminvariant of /. If Xi=Xi+tX2, a-2=Z2, then / becomes F = ^oXi3 + . . ., where ^0 = 00, Ai=ai+tao, A2=a2 + 2tai+Pao, A3 = a3+dta2+3fiai+fiao. Hence, by (3), F = {aiXi-\-a'2X2y, a'2=a2-{-tai. Similarly, the transform of (5i) is (^1X1 +^'2X2)3, P'2=fi2+tPl. Hence we obtain the desired result AoA2-Ar' = ^aMai0'2-a'20iy = oai/3i (q:i/32 — a2/3i)2 = 00^2 — fli^. §40] SYMBOLIC NOTATION 65 40. General Notations. The binary «-ic f=aoXi''+naiXi"-^X2 + . . . + (")atXi"-''x2''+. . .+a„X2" is represented symbolically as ax" = /3j" = . . . , where az = aiXi+a2X2, 0z = 01X1+^2X2, ■ ■ ■ , ai" = ao, ai"~''a2 = ai, • • . , ai"~*a2* = ai, . . ., a2"- = a„; /3i" = ao, .... A product involving fewer than n or more than n factors ai, a2 is not employed except, of course, as a component of a product of n such factors. The general binary linear transformation is denoted by T: a;i = |iXi+7jiZ2, a;2 = ?2^i + ';2X2, {iv)^0 where (^17) = ?i'?2 — ?2';i. It is an important principle of com- putation, verified for a special case at the end of § 39, that T transforms oi" into the »th power of the linear function {oci^i+a2h)Xi + (aiiii+a2V2)X2=a(Xi+a^X2, which is the transform of uj by T. Further, (1) ? ? = ? ? • f ''M = M(?'^), «£ a. ai a2 li ')i /3£ ^ ^1 ^2 «2 172 where {a0) = a 1^2 — a2/3 1 = - (/3a) . Thus (aj^;-a,^{)" = (?7,)"(a;8)", SO that (a/3)" is an invariant of ax" = /3i" of index n. Since (/3a)" represents the same invariant, the invariant is identically zero if n is odd. EXERCISES 1. (a0)' is the invariant 2(ao02— Oi') of az^=0x^. 2. (a0)* is the invariant 2/ of ax*=3i* (§ 31). 3. (apy (0yy (yay is the invariant 6/ of ax*=Px*=yx* (§ 31). 4. The Jacobian of ax"' and /3i" is max'"~^ai max"'~'^a: = mn{ap)ax"'~'-fix m — lo n — 1 66 ALGEBRAIC INVARIANTS 5. The quotient of the Hessian of ax^ = 0i^ by «'(n— 1)= equals n-2 , OCX ar ccj" aia2 _'/3/-2^,2 Pi ^iA ^x"~^/3,ft p/'-W n-2 Olx OliOi c^x^'-a..' ai Pi 71 012 & 72 oczPz 7i one-half of the sum of which equals J oi" ^/3i" ^{ap)'. 6- = M)7i + (07)ai+(7a)/3i=O. 41. Evident Covariants. We obtain a covariant 2ir of by taking a product of w factors of type ax and X factors of type (a/3), such that a occurs in exactly n factors, /3 in exactly n factors, etc. On the one hand, the product can be inter- preted as a polynomial in ao, . . . , a„, xi, X2- On the other hand, the product is a covariant of index X of /, since, by (1), §40, [ABYiACYiBCy . . . ^J5^a . . . ^ihYiafiyiayYiPyy . . . ax-^xh.' • • • , ifX = r+s+/ + . . .and Ax = AiXi+A2X2, Ai=ai, ^2=a„ iAB)=AiB2-A2Bi, etc. The total degree of the right member in the a's, (3's, . . . is 2\ + u = 7id, if d is the number of distinct pairs of symbols ai, 0:2; j3i, /32; . . .in the product. Evidently d is the degree of K in ao, ai, . . . , and to is its order in xi, X2- Any linear combination of such products with the same CO and X, and hence same d, is a covariant of order co, index X and degree d of /. EXERCISES 1. (a/3)(a7)aj:'/3iVi'and (apy{ay)ax'^fii^y\ arecovariantsofai^=^i5=7j5. 2. (aPYa:^ ""■ ^i" "*■ is a covariant of a^", &"». 3. If m = n, /3i"=oi'' and r is odd, the last covariant is identically zero. 4. 0(^1^+201X1X^+02X2'' and boXi''+2bix,X2+b2X2' have the invariant (ap)' = n„b;-2aib,+a2bo. §4;] SYMBOLIC NOTATION 67 COVARIANTS AS FUNCTIONS OF TwO SYMBOLIC TyPES, §§ 42-45 42. Any Covariant is a Polsmomial in the ai, (aj3). This fundamental theorem, due to Clebsch, justifies the symbolic notation. It shows that any covariant can be expressed in a simple notation which reveals at sight the covariant property. While a similar result was accomplished by expressing covariants in terms of the roots (§36), manipulations with sjinmetric functions of the roots are usually far more complex than those with our symbolic expressions. The nature of the proof will be clearer if first made for a special case. The binary quadratic a? has the invariant of index 2. Under transformation T of § 40, a^ becomes (a|Xi+a,X2)2=^oXi2 + . . ., ^o=a£^ ^i=af«„ ^2=a,2. Hence AqA2—A^ equals a£2/3,2-af^ja,;8, = (^'?)'^. We operate on each member twice with (1) y^^l 9L_, and prove that we get &{a0)^ = 12K, so that K is expressed in the desired symbolic form. We have (?';) = 6'?2— ?2'?i, dv2 aiiov2 dvi 9?23')i Vi^vy = 6ih), F2(^7,)2 = 12, since V{^v) = 2, by inspection. Next (2) Faj/3,= F(aih+«2?2)(|3i7ji+|32'72)=ai<32-a2/3i = (a/3). 68 ALGEBKAIC INVARIANTS Hence = ^ja,(a/3)+a£/3,(^a), F^ajjSfa,^, = (^a) (a^) + (aP) (/3a) = - 2(a^)2. The difference of the expressions involving V^ is 6(a/3)2 Hence if (l) operates twice on the equation preceding it, the result is 6(a/3)2 = 12if, K = i{apy. 43. Lemma. F"(|r;)" = («+l)(w!)2. We have proved this for « = 1 and n = 2. If w ^ 2, 9^ -(^,,)'' = «(f7,)"-l+«(«-l)(^^)''-^2fl. 9?i9';2 Similarly, or by interchanging subscripts 1 and 2, we get Subtracting, we get It follows by induction that, if r is a positive integer, Vr{^r,)" = (n + l)\n{n-l) . . . in-r+2)\^{n-r + l){^r,y-\ The case r = n yields the Lemma. 44. Lemma. If the operator V is applied r times to a product of k factors of the type aj and I factors of the type j3„ there results a sum of terms each containing k—r factors aj, I— r factors /?„ and r factors (a/3) . The Lemma is a generalization of (2), § 42. To prove it, set .!=«{( W . . . «{(*', 5=^,"^;3,(2) . . . ^,«. §44] SYMBOLIC NOTATION 69 Then 9^297)1 ,=1 t=i ' "' «{''' ^,''^' Subtracting, we get Hence the lemma is true when r = l. It now follows at once by induction that (1) V'AB = 22(a(si)^('i)) . . . (aMpitr)) ^ ^ ^ ^ ^ ^ a{(«i) . . . a{(sO |3,('i) . . . /3,(«')' where the first summation extends over all of the ^(^ — 1) . . . {k — r-\-\) permutations Si, . . . , St oi 1, . . . , k taken r at a time, and the second summation extends over all of the /(/ — I) . . . (/— r+1) permutations /i, . . . , 4 of 1, . . . , / taken r at a time. Corollary. The terms of (l) coincide in sets of r\ and the number of formally distinct terms is k\ II 1 /A/zy^j {k-r)l il-r)l r\ \r/\r For, we obtain the same product of determinantal factors if we rearrange Si, . . . , Sr and make the same rearrangement of /i, . . . , tr. 45. Proof of the Fundamental Theorem in § 42. Let K be a homogeneous covariant of order co and index X of the binary form /in § 40. By § 40, the general linear transformation replaces /=o!i" by 2 f")^tXi"-*X2* = (ajZi+a,X2)». Hence (1) ^t=aj'-V (^ = 0,1,. . .,«). 70 ALGEBRAIC INVARIANTS By the covariance of K, (2) K{Ao, . . .,A„; Xi, X2) = {hyK(ao, . . . , a„; xi, X2). By (1) the left member equals S -ZABXi^-'Xi', • =o in which the inner summation extends over various products AB, where ^ is a product of a constant and factors of type aj, and 5 is a product of a constant and factors of type a,. Let xi=y2, and X2 = —yi. Then, by solving the equations of T, § 40, X, = yJ{^n), X2=-yJ{h)- Hence the equation (2) becomes 2 i:{-\yABy,''-'yt={kvY+''K. Since the right member is of degree X+co in ^1, I2, and of degree X+'>' in rn, 7)2, we infer that each term of the left mem- ber involves exactly X+oj factors with subscript ^ and X+w factors with subscript ?;. Operate with V'^'^" on each member. By § 43, the right member becomes cK, where c is a numerical constant ?^0. By § 44, the left member becomes a sum of products each of X+co determinantal factors of which w are of type {ay)^az, and hence X of type (a/3) . The last is true also by the definiton of the index X of K. Hence K equals a polynomial in the symbols of the types a^, (a/3). To extend the proof to covariants of several binary forms ai", Ti", . . . , we employ, in addition to (l),Ct = 7{'"~*7,* and read aj, 7^, . . . for aj in the above proof. fi J FiNITENESS OF A FUNDAMENTAL SYSTEM OF COVAEIANTS, §§ 46-51 46. Remarks on the Problem. It was shown in §§28-31 that a binary form / of order <5 has a finite fundamental system of rational integral covariants K\, ■ ■ ■ , Ks, such therefore that any rational integral covariant of / is a poly- § 47] FINITENESS OF COVARIANTS " 71 nomial in Ki, . . . , Ks mth. numerical coefficients. We shall now prove a like theorem for the covariants of any system of binary forms of any orders. The first proof was that by Gordan; it was based upon the symbolic notation and gave the means of actually constructing a fundamental system. Cayley had earlier come to the conclusion that the fundamental system for a binary quintic is infinite, after making a false assumption on the independence of the syzygies between the covariants. The proof reproduced here is one of those by Hilbert; it is merely an existence proof, giving no clue as to the actual covariants in a fundamental system. 47. Reduction of the Problem on Covariants to one on In- variants. We shall prove that the set of all covariants of the binary forms fi, . . . , fn is identical with the set of forms derived from the invariants / of /i, . . . , ft and l=xy' — x'y by replacing x' by x and y' by y in each /. It is here assumed (§ 15) that T is homogeneous in the coefficients of I and that the covariants are homogeneous in the variables. Let the coefficients of the /'s he a, b, ... , arranged in any sequence. Let A, B, . . .be the corresponding coefficients of the forms obtained by applying the transformation in § 5. The latter replaces I by ^n'-^'v, where T,'=ay'-yx', i' = &x'-§y'. Solving these, we get i^x' =ai' +§yi' , ^y'=yi' + br,'. Let I{a, b, . . . ; x'-, y') be an invariant of / and the /'s. Then I{A,B,. . .; i',v')=^H{a,h,. . .■,x',y'). Since / is homogeneous, of order u, in x\ y', the right member equals A'' -"/(a, b, . . . ; Ax', Ay'). Hence we have the identity in ^', t\ : I{A, B,...;^', V)=A^-"/(a, b, . . . ; a^'+Pv', y^'+Sv'). 72 ALGEBRAIC INVARIANTS Thus we may remove the accents on ^ , ^ . Then, by our transformation, I{A,B,. . .; ^„)=A^-"/(a, i, . . .; x,y). Hence I(a, 6, . . . ; at, y) is a covariant of /i, . . . , /t of order ij> and index X — «. The argument can be reversed. Note that the sum of the order and the index of a covariant is its weight (§ 22) and hence is not negative. Corollary. A covariant of the binary form / has the annhilators in § 23. For, an invariant of/ and xy' —x'y has the annihilators Q-y'\ 0-x'^. dx" dy' 48. Hubert's Theorem. Any set S of forms in xi, . . . , x„ contains a finite number of forms Fi, . . . , Ft suck that any form F of the set can be expressed as F=fiFi + . . .+/tFt, where /i, . . . , fk are forms in xi, . . . , x„, but not necessarily in the set S. For w = l, 5 is composed of certain forms cix^', C2x'^, . Let e, be the least of the e's, and set Fi=Csx"^. Then each form in 5 is the product of Fi by a factor of the form cx^, e ^ 0. Thus the theorem holds when w = l. To proceed by induction, let the theorem hold for every set of forms in w — 1 variables. To prove it for the system 5, we may assume, without real loss of generality,* that 5 contains a form Fq of total order r in which the coefficient of Xn"" is not zero. Let F be any form of the set S. By division we have F = FoP+R, where i? is a form whose order in x„ * Let 7^ be a form in 5 not identically zero and let the linear transformation ^i = cuy,+ctiy2+. . .+Cinyn (; = 1,- . ■ , n) replace f(a:i, . . , Xn) by /C(yi, . . . , yn). In the latter the coefficient of the term involving only yn is obtained from F by setting Xf = ctn and hence is F(cin. tin, ■ ■ ■ , Cnn), which is not zero for suitably chosen c's (Weber's Algebra, vol. 1, p. 457; second edition, p. 1-47). But our theorem will be true for 5 if proved true for the set of forms K. §^8] FINITENESS OF COVARIANTS 73 is "P= 2CrZ)"-"+T7', r=0. where Co, . . ■ , Cm are constants. First, we have VDP = P+ 7,2— + a^ +D- ^'^ dv2 9^1 9fi9'72 -(-/>-a||-,. I^+T)-!^) ='2+X+m)P+Z?FP, 911 0K20T)l/ by Euler's theorem for homogeneous functions (§24). If P is replaced by D"~^P, so that X and n are increased by « — 1, we get F7)"P = (X+M+2«)7)" -^P+DVD^ "'P- 76 ALGEBRAIC INVARIANTS Using this as a recursion formula, we get FZ)"P={»(X+M)+M(« + l))Zy"-'P+Z)''FP, which reduces to the result in § 43 if P=l, whence X = m = 0. Hence (l) holds when m = l. To proceed by induction from m to m + 1, apply V to (l). Thus m r=0 In the result for VD^P, replace n by n — m+r and P by V^P, and therefore diminish X and m by r. We get where tr={n—m+r){\+fjL—r+n~m + l). Hence, changing r+1 to r in the second summand, we get m+l ym+l^np^ V (Crtr+Cr-l)D''-'"+'-^VP, r=0 with Cm+i = 0, C_i = 0. Thus (1) is true for every m. 51. Finiteness of Syzygies. Let 7i, . . . , 7„ be a funda- mental system of invariants of the binary forms /i, . . . , fa. Let S(zi, . . . , Zm) be a polynomial with numerical coefficients such that S{Ii, . ■ . , /m),^when expressed as a function of the coefficients c of the /'s, is identically zero in the c's. Then 5(7) = is a syzygy between the invariants. By means of a new variable z^+i, construct the homogeneous form 5'(zi, . . . , Zm+i) corresponding to 5. By § 48, the forms S' are expressible linearly in terms of a finite number S'l, . . . , 5't of them. TakeZm+i = l. Thus (1) S = CiSi+. . .+CtSt, where Ci, . . . , Ct are polynomials in Zi, . . . , z^. Take zi =7i, . . . , Zm=Im- Hence there is a finite number of syzygies 5i = 0, . . . , St = 0, such that any syzygy 5 = imphes a relation (l) in which Ci, . . . , G are invariants. In particular, every syzygy is a consequence of 5i = 0, . . . , St = 0. § 52] TRANSVECTAXTS 77 52. Transvectants. Any two binary forms have the covariant (1) {f,y=(afiya.''-%-% called the rth transvectant (Ueberschiebung) of / and , and due to Cayley. It is their product if r = 0, their Jacobian if r = l, and their Hessian ii j= and r = 2, provided numerical factors are ignored (Exs. 4, 5, § 40). It may be obtained by differentiation and without the use of the symbolic notation. In fact, a special case of (1), § 44, is so that if / is of order k and <> of order /, (2) m), u)y=^-^^ ^[F'/(«)0(,)i,.f. After /(^i, |2)-<^(i7i, 172) is operated on by V'', we set 171 = ^1, For example, Iet/(j)=ajflj, <^(f) = Tj', -P=«j'3j7,'. Then a'P d'^P The difference is VP. Taking m= iu 12= £2, we get 3i-.j(ft7-i-/32Ti)+/3j(o:i72-«27i)}7j^ The numerical factor in (2) is here 1/6. Hence (3) (ajSj, 7j')'=5(/37)a£7£^+Ka7)/3f7j'. In general, consider the two forms }=a^^^W^^ . . . a^"', = /3£('>/3£'2) . . . ^^W. Then by (1), § 44, and the Corollary, and by (2), \r I \r where the summation extends over all the combinations of the '8 ALGEBRAIC INVARIANTS a's r at a time, and over all the permutations of the jS's r at a time. Thus the number of terms in the sum is the reciprocal of the factor preceding 2. If the a's are identified and also the ^'s, (4) becomes (1). If * = 2, /=3, r=l, we have one-sixth of a sum of six terms; then if the 0's are identified we have two sets of three equal terms and obtain (3). Since F is a differential operator, (2) gives (5) ( 2c/,, 2*;<^;) - = 2 2c,^,(/,, <^,)^ Apolarity; Rational Curves, §§ 53-57 53. Binary Forms Apolar to a Given Form. Two binary quadratic forms are called apolar if their lineo-linear invariant is zero; then they are harmonic (Ex. 3, § 11). In general, the binary forms /=a."= 2 (^\aiXx^-^xoi, <^=;8."= 2 \V\hxx'' -'X2\ i=0\t/ i=0\t/ of the same order, are called apolar if (1) (a0)" = ^2^(-l)'(")a,6„_, = O. In particular, / is apolar to itself if n is odd (Ex. 4, § 38). Let the actual Hnear factors of be /3x'^', . . . , /Sj^"'. By (1), (4), § 52, (a/3)'' = (aA ^."> . . . ^,(''')" = («^'") • • ■ (a^""). But ^i''^ vanishes if xi and X2 equal respectively Thus (a/3'''') =aiy/'"'+a2y2'''' =%(>•). Hence if <^ vanishes for xi=yi^''\ X2=y2^'^ ('■ = li • • • > w), a is apolar to f if and only if uyd) a^(2) . . , ay(n) = 0. Thus / is apolar to an actual nth power (y2Xi—yiX2)'' if and only if ay" = 0, i.e., if yi, y2 is a pair of values for which 1=0. § 53] APOLARITY 79 If no two of the actual linear factors I, of / are propor- tional, / is apolar to n actual wth powers k" and these are readily seen to be linearly independent. Then their linear combinations give all the forms apolar to/. For, if /is apolar to <^i, . . . , 0„, it is apolar to ^i<^i-|-- • ■+knn, where ki, . . . , k„ are con- stants, since, by (5), § 52, {f,kin)'' = ki(J,n)'' = 0. Moreover,/ is not apolar to n+l linearly independent forms For, if so, we have n+l equations like (1), in which the deter- minant of the coefficients of ao, • • ■ , dn is therefore zero. But this implies a linear relation between the <^'s. /// is the product of n distinct linear factors It, a form

is apolar to f. In particular, if r and s are the distinct roots of f=ax^-\-2bx+c=Q, the only quadratics harmonic to / are g{x—rY-\-h{x—sY. In case h, . . . , h are identical, while li9^li{i>r), we may replace /i", . . . , l^in the above discussion by h", li"~\ ■ • ■ , l^n-r+iy-i^ where X is any linear function of xi and xz which is linearly independent of h. In fact, after a linear trans- formation of variables, we may set li=X2, \ = xi. Then the above r forms have the factor X2"~'"^^ and hence are of type (j} with bi = 0{i ^n—r). Also, / now has the factor X2^, so that ai = 0(ia2XiX2^ +azX2^ , h = &3 = hQX^^^h^x^^X2^'ih2X^X2^+hzX2^, jz=-i£'=CisXt'^-ZCiXiH2-\-'iC2XiX'^-^CzX'^. Each is apolar to the cubic form <^ = (a)3)(a7)((3T)ax&Ti- For, by (4), § 52, and the removal of a constant factor by (5), (0, 6.3)3 = (a/3)(aT)(^7)(a5)(/35)(7«), which is changed in sign if 5 is interchanged with a, /3, or 7, §54] APOLARITY 81 a\^ aia2 a2^ /3i2 W2 ^2^ 71^ 7172 72^ and hence is zero if 6/ is one of the /(. Hence each /« is apolar to 0. Now (a^)(a7)(i37) = In fact, the determinant vanishes if {a0) = as may be seen by setting j8i = cai, /32 = ca2. Moreover, the two members are of total degree six and the diagonal term of the determinant equals the product of the first terms q:i/32, etc., on the left. Since a],^ax=ai^xi+ai^a2X2 = aoXi+aiX2, etc., we find, by multiplying the members of the last equation by ux^xix, = aoXi+aiX2 boXi+biX2 C0X1+C1X2 aiXi+a2X2 biXl+b2X2 C1X1+C2X2 020:1+03^2 b2Xl + b3X2 C2X1 +C3X2 where = [012]a:i3 + [013]a:i2x2 + [023]xiX22 +[l23]a;23, m= bi Ct bj Ci at bt Ct If

is the unique form apolar to three linearly independent cubic forms fi. f 2, fz- The extension to n binary w-ics is readily made. 55. Rational Plane Cubic Curves. The homogeneous coor- dinates ?, rj, f of a point on such a curve are cubic functions of a parameter /. We may take t=xi/x2 and write P$=/l, P'7=/2, P?=/3, where p is a factor of proportionality and the /'s are the cubic forms in § 54. We may assume that the/'s are linearly independent, since otherwise all of the points (|, tj, f) would lie on a straight line. 82 ALGEBRAIC INVARIAMTS There is a unique cubic form apolar to fi, J2, fs (§ 54). This cubic form, denoted by (j> = (j>/, is fundamental in the theory of the cubic curve. TJiree points determined by the pairs of parameters xi, X2; A'l, yo', and zi, 22, are collinear if and only if (1) 4>x^,z = Q. For, if the three points lie on the straight line (2) /£+wr, + nf = 0, the three pairs of parameters are pairs of values for which (3) C(X1, X2)^//l+w/2+w/3=0. Since C is apolar to 0, (l) follows from the first italicized theorem in § 53. Conversely, (1) implies that the cubic C which van- ishes for the three pairs of parameters is apolar to and hence (§ 53) is a linear combination of /i, f-i, fz, say (3); the corre- sponding three points lie on the straight line (2) . Since (2) meets the curve in three points the ratios Xi/x2 of whose parameters are the roots of (3), the curve is of the third order. We restrict attention to the case in which the actual linear factors ax, fix, yz of are distinct. Since any cubic apolar to Tnbolic, expression of/i, etc. §56] RATIONAL PLANE CUBIC CURVE 83 case there are therefore exactly three inflexion points and they are collinear. 56. Any Rational Plane Cubic Curve has a Double Point. Let Pi denote the point (^, r;, s") determined by the pair of parameters Xi, X2. If the ratios xi/xo and yi/y2 are distinct and yet Pi coincides with P^, then Pi is a double point. For, any straight line (2), § 55, through Pi meets the curve in only the three points whose pairs of parameters satisfy the cubic equation (3), and since two of these pairs give the same point Pi, the line meets the curve in a single further point. Hence there is a double point Pi — Py if and only if there are two distinct ratios x\/x2 and y\/y2 such that (1) holds identically in z\, Z2. Let Q be the quadratic form which vanishes for the pairs of parameters xi, X2 and y\, y2 giving a double point. By (1), and the first theorem in § 53, Q is apolar to 4>x^4>t for zi, ^2 arbitrary. Write V as a symbolic notation for 4>, alter- native to 4>x^ Applying the argument made in § 54 for three cubics to two quadratics, we see that the unique quadratic (apart from a constant factor) which is apolar to both i^4>, and 'x^^'u> is their Jacobian J = {(j>(t> )i(t> w- Since and ' are equivalent symbols, their interchange must leave J unaltered. Hence The quantity in brackets equals {'){zw) by (1), § 40. Dis- carding the constant factor ^{zw), we may take Q = (x'i as the desired quadratic form. This is the Hessian of - Conversely, the pairs of values for which Q vanishes are the pairs of parameters of the unique double point of the curve. 57. Rational Space Quartic Curve. Such a curve is given by 84 ALGEBRAIC INVARIANTS where the four binary quartics are Unearly independent. By § 54, there is a unique quartic = gives the four points at which the osculating plane meets the curve in four consecutive points. It may be shown that the values a:i''\ 3:2''^ for which the Hessian of vanishes give the four points P^t') on the curve the tangents at which meet the curve again. Fundamental Systems of Covariants of Binary Forms §§ 58-63 58. Linear Forms. A linear form ax is its own symbolic representation. If ax — Px, then (a/S) = 0. Hence the only covariants of ax are products of its powers by constants. A fundamental system of covariants of n linear forms is evidently given by the forms and the |w(w— 1) invariants of type (a/S), where ax and 0x are two of the forms. 59. Quadratic Form. A covariant iiT of a single quadratic may have no factor of type (a0) and then it is or may have the factor (aj3) and hence the further factor (a/S), (a7)(/35), (ay) fix, or axfix, including the possibility 5=7. In the first case, K=(afiyKi, where 7vi is a covariant to which the same argument may be applied. Now (ay)=ay if yi=72, -y2=— 71. Hence in the last three cases, K has a factor of the type e=iafi)ay0„ where a„ is either ax or a new mode of writing (07), and similarly /3j is either fix or a new mode of writing (fid). Interchanging the equivalent symbols a and 8, we get e = ifia) fi^, = \ {afi) {ayfi, - fi,a^) = \ iafiYiy^) , §00] FUNDAMENTAL SYSTEM OF COVARIANTS 85 by (1), § 40. We are thus led to the first case. Hence the fundamental system of covariants of / is composed of / and its discriminant. EXERCISES 1. The fundamental system for f=ax^ = bx' and l=ax=Px is /, /, {ah)'', {oaY, {aa)ax. 2. The fundamental system for f~ax''=bz^ and =az'' = ^' is/, , {ab)% (a^)^ {aa)\ {aa)axOLx. Hint: {ool) (a^)a,/3„ = (oa) 2/3„/3j- \ (a/3) ^ayCt, as proved by multiplying together the identities (Ex. 6, § 40) {a0)ay = {a0)ay-{aa)0^, {a0)a,={ap)at-{aa)^,, and noting that a and /3 are equivalent symbols. 60. Theorems on Transvectants. In the expression (4), § 52, for a transvectant, each summand taken without the prefixed nimierical factor is called a term of the transvectant. In the first transvectant (3), § 52, the difference of the two terms is i (/37)a£ - («7)iS£!t{^ = l(/3a)T£l7«^, by Ex. 6, § 40, and is the negative of the 0th transvectant (viz., product) of {a0) and y^^ The act of removing a factor aj and a factor jS^ from a product and multiplying by the factor (a/3) is called a convolution {Faltung). We have therefore an illustration of the following Lemma. The difference between any two terms of a trans- vectant equals a sum of terms each a term of a lower transvectant of forms obtained by convolution* from the two given forms. Consider the rth transvectant of / = Pa£(l' . . . a,«, <^ = (2/3j"> . . . ^i«, where P and Q are products of determinantal factors. Then PQ is a factor of each term of the transvectant. Any two terms T and T' differ only as to the arrangements of the a's and the ^'s. Hence T' can be derived from T by a permuta- * Including the case of no convolution, as Tj' from itself, in the above example. 86 ALGEBRAIC INVARIANTS tion on the as and one on the /3's, and hence by successive interchanges of two as and successive interchanges of two ;3's. Any such interchange is said to replace a term by an adjacent term. For example, the two terms of (3), § 52, are adjacent, each being derived from the other by the inter- change of a with /3. Between T and T' we may therefore insert terms Ti, . . . , r» such that any term of the series T, Ti, T2, . . . , Tn, T' is adjacent to the one on either side of it. Since r-r=(r-ri)+(ri-r2)+. . .+(r„_i-r„)+(r„-r), it sufl&ces to prove the lemma for adjacent terms. The interchange of two a's or two /3's affects just two factors of a term of (4), § 52. The types of adjacent terms are * C{a'^'){a"n, C{a'nW'^'); C{a'&'W„ Cia'n^',; where /3' and /3" were interchanged. The difference of the last two terms is seen to equal C{p"fi')a\ by the usual identity. The latter is evidently a term of the (r — l)th transvectant of / and (/3"/3')V{i8"{/3'{!, which is obtained from 4> by one convolution. The difference of the first two adjacent terms equals C{a'a'Wn, since a la iP 1 p 1 a la 1 p 1 p 1 ' " a' a" aza 2P2P 2 as shown by Laplace's development. The same relation follows also from the identity just used by taking ?i=— a"2, |2=a"i- The resulting difference is a term of the (r — 2)th transvectant of a ja { P (P I which are derived from / and (^ by a convolution. * A pair C(a'P')<*"£i C(d'0)a^, obtained by interchanging a' and a", is essentially of the second type. §60] FUNDAMENTAL SYSTEM OF CO VARIANTS 87 The Lemma leads to a more important result. By the proof leading to (4), § 52, the coefficient of each term of a transvectant is \/N, if N is the number of terms. Just as S = \{Ti-\-T2) implies 5 — ri= 1(72 — Ti), so implies S-Ti=h{T2-T{) + . . MTn-T,)]. Hence the diference between a transvectant and any one of its terms equal a sum of terms each a term of a lower transvectant of forms obtained by convolution from the two given forms. Each term of a lower transvectant may be expressed, by the same theorem, as the sum of that transvectant and terms of still lower transvectants, etc. Finally, when we reach a 0th transvectant, i.e., the product of the two forms, the only term is that product. Hence we have the fundamental Theorem. The di^fference between any transvectant and any one of its terms is a linear function of lower transvectants of forms obtained by convolution from the two given forms. For example, from (3), § 52, and the result preceding the Lemma, we have and (aj3) is derived from ajjSj by one convolution. 61. Irreducible Covariants of Degree m Found by Induction. Let /=«." = /3." = . . .=X," be the binary w-ic whose fundamental system of covariants is desired. Since a term with the factor (a/3) is of degree at least two in the coefficients of /, the only covariants of degree unity are kf, where ^ is a numerical constant. We shall say that / is the only irreducible covariant of degree unity, and that/, ill, . ■ . , K, form a complete set of irreducible covariants of degrees n, in view of the order of /. Moreover, it suffices by (5), § 52, to employ the Cm-\ which are products of powers oif,K\,. . . , Ks- Hence the covariants of degree m are hnear functions of a finite number of transvectants. , In the examination of these transvectants {Cm-i, fY, we first consider those with ^ = 1, then those with k = 2, etc. We may discard any {Cm-\,fY for which Cm-i has a factor. , of order ^k, which is a product of powers of f, Ki, . . . , Ks, and of degree ,fY, and if Cm-i=q4>, then T is obtained by k convolutions of <^/, and qT by the same k convolutions of q4>f, not affecting q. Hence qT is a term of (?> /)*• Hence (C„_i, fY = qT+~^c,{Cm -ijy. J=0 § G2) FUNDAMENTAL SYSTEM OF COVARIANTS 89 But the terms of the last sum have by hypothesis been con- sidered previously, while the covariants q and T are of degree * is of degree , /)* involves fewer than m— 1+1 symbols a, ^, . . . , and hence is of degree < m. 90 ALGEBRAIC INVARIANTS of which the first two may be discarded as before. It remains to consider (/,/)*, for /fe = l, 2, 3. By § 52, (/,/)=(a^)2(a7)(fe7x^^/) Replacing {fiS)yx by (y5)lix + (l3y)dx, and noting that {aP)\ay)ifiy)y.5/ = (HJ)^-f = 0, (/,/) = (a/3)2(aT)(75)A7.5x^. Interchange 7 and S. Hence (7,/)=K«)3)2(76)/3x7.5.{(a7)5. + (3a)7x{. The quantity in brackets equals — (7 5)ax. Hence (JJ)- -|M)2(75)-a.^x7x3.= -1^2. Denoting H by hj^ = h'^z, we have (J, / )2 = (ha) m (a^)a.& +c((fc)2a„ / ) , by the theorem in § 60. Here J = {hcifcix = {H, JY = Q. Since the first term is changed in sign when a and /3 are interchanged, we have (/, fy = 0. For the third case, (/, /)3 = ((ai3)2(a7)/J,7x2, Sx'Y = {amay){p5){y5Y =D, an invariant, evidently equal to {H, HY, the discriminant of H. Thus D is the discriminant of / (§§8, 30) and is not identically zero. Hence D is the only irreducible covariant of degree four. We can now prove by induction that /, E, J and D form a complete set of irreducible covariants of degree %. m ^ 5. Let this be true for covariants Cm-\ of degree = ot — 1. We may discard [Cm-u fY if Cm-i has the factor/ or /, each of which is of order 3 ^ ^ and of degree (1 or 3) less than m — l; and evidently also if it has the factor D. Hence Cm-i = n% e = 2. If ife ^ 2, it has the factor H of order 2'^k and degree 22, E^ has the factor § 64] LAW OF RECIPROCITY 91 H^ of order 4^3 and degree 4=ai^ = fij' = . . .=aQ{xi — piX2){xi — p2X2) . . . {X\ — PpX2) have a covariant of degree d, K=-aQ'^-Z{pi-p2y{pi-pzy{p2-pzY . . . (a;i-piX2)'' . . . {xi-pj,X2y'>, so that each of the roots pi, . . . , pp occurs exactly d times in each product. Consider the binary d-ic J=ax'^ = bt^ = . . . = co{xi—riX2) . . . {xi—raXo). ♦Gordan, Invarianientheorie, vol. 2 (1887), p. 236, p. 275. C/. Grace and Young, Algebra of Invarianls, 1903, p. 122, p. 128, p. 150. t Malhematische Annalen, vol. 17 (1S80), vol. 31 (1888). X For an introduction to it, see Elliott, Algebra of Quantics, 1895, p. 165, p. 247. 92 ALGEBRAIC INVARIANTS To the various powers, whose product is any one term of K, (Pl — P2)', (pi — psV, (P2-P3)*, • • •, (a;i-piX2)'', {xi-p2X2)^, ■ ■ ., we make correspond the symbolic factors {aby, (acY, (be)", . . ., aj\ bj\ . . . of the corresponding covariant of/: C^iabYiacYibcy . . . a.H.hJ' . . ., of degree p (since there are p symbols a, b, c, . . . , cor- responding to Pl, . . ■ , Pp) and having the same order li+h+h + - . . a.s K. Conversely, C determines X'. EXAMPLES Let p = 2. To A' = ao"(pi— Pi)^' corresponds the invariant C= (aft) ^* of degree 2 of f=az^^ = bi^^. Again, to the covariant K(t>' of 4, corresponds the covariant (a6)2« ajbj of the form az^^+' = b:,^^+'. Concomitants of Ternary Forms in Symbolic Notation, §§ 65-67 65. Ternary Form in Symbolic Notation. The general ternary form is ft f='2-rrr, arsiXi''X2'x3', where the summation extends over all sets of integers r, s, t, each = 0,. for which r+s+t = n. We represent / symbolically by / = a;," = j3x" . . . , aj: = aiXi+a2X2+a3X3, .... Only polynomials in ai, 02, as of total degree n have an inter- pretation and ai^a2^az' — drsi- Just as ai)32-a2^i was denoted by (a/3) in §39, we now write ai a2 as /3i /32 /33 71 72 73 §66] CONCOMITANTS OF TERNARY FORMS 93 Under any ternary linear transformation T: x^= ^.Xi + 7,^2 + UXz {i = l, 2, 3) ai becomes aeXi+a^2+afA'3, and/ becomes n -ArsiX^iX'^X's = {a^Xi +a^X2+a[Xz) rlsUl Thus a, behaves like a covariant of index zero of /. Also aj a, ttf fi( /3, /3f ={aPy)UvO, 7{ 7, Tj- so that (ajSi) behaves like an invariant of index unity of/. EXERCISES 1. The discriminant of a ternary quadratic form a^' is I (a^r)'. 2. The Jacobian of a^K /Si"*, 71° is Imn {cc0y)aj~^^x"'~'-yx"~^. 3. The Hessian of ai» is the product of (aP7)'ai"~^^i"'"Si"~^ by a constant. 4. A ternary cubic form az'=/3j'=. . . has the invariants (a0y) (a0 &)(ayd)(0yS), (a0y) (a0 5) (07 «) (^7 «) ( « ««) '• 66. Concomitants of Ternary Forms. If mi, M2, "3 are constants, Mi = wia;i +U2X2 +M3X3 = represents a straight line in the point-coordinates xi, X2, xz- Since ui, U2, M3 determine this line, they are called its line- coordinates. If we give fixed values to xi, X2, X3 and let the line-coordinates mi, mz, M3 take all sets of values for which «i = 0, we obtain an infinite set of straight lines through the point {xi, X2, X3). Thus, for fixed x's, zix = is the equation of the point (xi, X2, X3) in line-coordinates. Under the linear transformation T, of § 65, whose deter- minant (|77f) is not zero, the line Mi = is replaced by Ux = UiXi + U2X2 +U3X3 = 0, in which 333 C/i = S |«M4, C/2 = 2 v'223-y3=2, X2 by yszi-jiss, and X3 by yiZ2—y2^i, we get (?'?f)A'3=y{Z,-y^{. Our relation for a covariant K of order to now becomes 2(product of factors a^, y^, zj, a„, . . . , Zi) = {^r,^Y+^K{a, x), each term on the left having X+co factors with the subscript ?, etc. Apply the operator V to the left member. We obtain a sum of terms with one determinantal factor {afiy), (aPy) or {ayz)=ax, and with X+o)— 1 factors with the subscript f, etc. The result may be modified so that the undesired factor (afiy) shall not occur. For, it must have arisen by applying V to a term with a factor hke a{/3,yf and hence (by the formulas for the Xi) with a further factor z, or zj. Consider therefore the term Ca£/3,yj-2, in the initial result. Then the term — Caf/3,y^f must occur. By operating on these with V, we get C{aPy)z^, —C{aPz)y,„ respectively, whose sum equals C\ (/3yz)a, - (ays)/3,! ^ C(fea, -a.^,) , as shown by expanding, according to the elements of the last row, ai Pi yi Zi Oi2 P2 y2 Z2 0:3 /Ss ys Z3 «i ^1 y>> 2, =0. The modified result is therefore a sum of terms each with one factor of type (afiy) or ax and with X+w— 1 factors with subscript f, etc. Applying V in succession X+co times and modifying the result at each step as before, we obtain as a new left member a sum of terms each with X+u factors of the types (afiy) and ax only. From the right member we obtain nK, where n is a number 5^0. Hence the theorem is proved. §68] QUATERNARY FORMS 97 68. Quaternary forms. For ai=ai:ci+. . .+043:4, „, n Q n _, n c n = 0(1 =Pi = 7i = Oi has the determinant (aPyd) of order 4 as a symbolic invariant of index unity. Any invariant of / can be expressed as a polynomial in such determinantal factors; any covariant as a polynomial in them and factors of type az. In the equation «i = of a plane, ui, . . . , Ui are called plane-coordinates. The mixed concomitants defined as in § 66 are expressible in terms of m, and factors like ax, (a/37 5) , (aPyu) . For geometrical reasons, we extend that definition of mixed concomitants to polynomials P{c, x, u, v), where Di, . . . ,va as weU as mi, . . . , M4 are contragredient to a;i, . . . , 3:4. There may now occur the additional type of factor (a0uv) = (ai/32 -a2/3i) (usVi-UiVa) +. . . + (aafii -atfiz) (uiV2 -U2V1) . These six combinations of the m's and v'a are called the line- coordinates of the intersection of the planes Ux = 0, Vx = 0. For instance, (aPuvY = is the condition that this line of inter- section shall touch the quadric surface ax^=0. We have not considered concomitants involving also a third set of variables wi, . . . , wt, contragredient with the x's. For, in U\Xi + . . .-j-«4^4 = 0, I^lXl-l-. . .+ViXi = 0, W1X1 + . . .+W4Xa = 0, xi, . . . , Xi are proportional to the three-rowed determinants of the matrix of coefficients, so that (auvw) is essentially a,. INDEX (The numbers refer to pages) Absolute invariant, 51 Alternants, 41 Annihilators, 34, 39, 72 Apolarity, 78-84 Binary form, 14, 91 Canonical form of cubic, 17 quartic, 50 ternary cubic, 28 Concomitants, 93, 97 Conic, 2, 21, 24, 94 Contragredient, 94 Contravariant, 94 Convolution, 85 Covariant, 12, 15, 66 — in terms of roots, 56 symbolic factors, 67, 95 — as invariant, 71 Cross-ratio, 5, 15, 56 Cubic curves, 25-29, 81 — form, 14, 16, 48, 80, 89, 93 Degree, 30 Differential operators, 36, 59, 95 Discriminant of binary cubic, 17, 36 quadratic, 10 p-ic, 55 ternary quadratic, 24 Double point, 83 Euler's theorem, 15, 41 Finiteness of covariants, 70-76 syzygies, 76 Forms, 14 Functional determinant, 12 Fundamental system, 48, 61, 84-91 Harmonic, 15, 20, 78 Hermite's law of reciprocity, 45, 91 Hessian, 11, 15-18, 23-28, 58, 66, 84, 93 — curve, 25 Hubert's theorem, 72 Homogeneity, 14, 30, 37 Homogeneous coordinates, 8, 20 Identity transformation, 33 Index, 10, 14, 15, 31, 32 Inflexion point, 26-28, 82 — tangent, 26, 82 — triangle, 27 Intermediate invariant, 19 Interpretation of invariants, 2, 10, 23 Invariant, 1, 10, 14, 28 • — in terms of roots, 54 Inverse transformation, 33 Irrational invariant, 55 Irreducible covariant, 87 Isobaric, 31, 32, 38, 42 Jacobian. 12. 15, 18. 29, 65, 83, 93 Leader of covariant. 40. 43, 58 Line coordinates, 93, 97 ■ — equation of conic, 94 Linear form, 9, 14, 33, 84 — fractional transformation, 6., 22 — transformation, 3, 9, 22. 33. 34, Mixed concomitant, 94, 97 Order, 14 Partitions, 44, 45 Perspective, 4 99 100 INDEX Plane coordinates, 97 Product of transformations, 33 Projective, 4, 23 — property, 10, 11, 23 Projectivity, 5, 6 Quadratic form, 10, 14, 48, 84 Quartic, 14, 36, 49, 83 Quaternary form, 97 Range of points, 4 Rational curves, 81 Reciprocity. See Hermite. Resultant, 10, 18, 19 Seminvariant, 40, 42-50, 64 — in terms of roots, 53 Singular point, 25 Solution of cubic, 17 quartic, 52 Sjonbolic notation, 63 Syzygy, 49, 50, 76 Ternary form, 14, 24, 25, 92 Transformation. See Linear. Transvectants, 77, 85 Unary form, 14 Weight, 31, 32, 38 SEP 19 1991 MATHEMATICS LIBRARY