Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornells replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990.Tuis Book Belongs TbA TREATISE ON THE THEORY OF INVARIANTS BY OLIVER E. GLENN, Ph.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA GINN AND COMPANY BOSTON · NEW YORK · CHICAGO · LONDON ATLANTA · DALLAS · COLUMBUS · SAN FRANCISCOCOPYRIGHT, 1915, BY OLIVER E. GLENN ALL RIGHTS RESERVED 215.9 iKfee Stfjenatum GINN AND COMPANY · PRO- PRIETORS · BOSTON · U.S.A.PREFACE The object of this book is, first, to present in a volume of medium size the fundamental principles and processes and a few of the multitudinous applications of invariant theory, with emphasis upon both the nonsymbolical and the symbol- ical method. Secondly, opportunity has been taken to empha- size a logical development of this theory as a whole, and to amalgamate methods of English mathematicians of the latter part of the nineteenth century — Boole, Cayley, Sylvester, and their contemporaries — and methods of the continental school, associated with the names of Aronhold, Clebsch, Gordan, and Hermite. The original memoirs on the subject, comprising an ex- ceedingly large and classical division of pure mathematics, have been consulted extensively. I have deemed it expe- dient, however, to give only a few references in the text. The student in the subject is fortunate in having at his command two large and meritorious bibliographical reports which give historical references with much greater completeness than would be possible in footnotes in a book. These are the article " In varianten theorie ” in the " Enzyklopadie der mathe- matischen Wissenschaften” (I B 2), and W. Fr. Meyer’s " Bericht fiber den gegenwartigen Stand der In varianten- theorie ” in the " Jahresbericht der deutschen Mathematiker- Vereinigung” for 1890-1891. The first draft of the manuscript of the book was in the form of notes for a course of lectures on the theory of inva- riants, which I have given for several years in the Graduate School of the University of Pennsylvania. The book contains several constructive simplifications of standard proofs and, in connection with invariants of finite iiiIV THE THEORY OF INVARIANTS groups of transformations and the algebraical theory of ter- nariants, formulations of fundamental algorithms which may, it is hoped, be of aid to investigators. While writing I have had at hand and have frequently consulted the following texts: Clebscii, Théorie der binàren Formen (1872). Clebscii, Lindemann, Vorlesungen liber Geometrie (1875). Dickson, Algebraic Invariants (1914). Dickson, Madison Colloquium Lectures on Mathematics (1913). I. In- variants and the Theory of Numbers. Elliott, Algebra of Quantics (1895). Faa di Bruno, Théorie des formes binaires (1876). Gordan, Vorlesungen liber Invariantentheorie (1887). Grace and Young, Algebra of Invariants (1903). W. Fr. Meyer, Allgemeine Formen und Invariantentheorie (1909). W. Fr. Meyer, Apolaritàt und rationale Curven (1883). Salmon, Lessons Introductory to Modern Higher Algebra (1859 ; 4th ed., 1885). Study, Methoden zur Théorie der ternâren Formen (1889). O. E. GLENN Philadelphia, Pa.CONTENTS CHAPTER I. THE PRINCIPLES OF INVARIANT THEORY Section 1. The Nature of an Invariant. Illustrations PAGE I. An invariant area..........................................1 II. An invariant ratio.........................................2 III. An invariant discriminant . 4 IV. An invariant geometric relation............................5 Y. An invariant polynomial ....... 6 VI. An invariant of three lines ....... 8 VII. A differential invariant ........ 9 VIII. An arithmetical invariant..................................12 Section 2. Terminology and Definitions. Transforma- tions I. An invariant..............................................14 II. Qualities or forms.........................................14 III. Linear transformations....................................15 IV. A theorem on the transformed polynomial .... 16 V. A group of transformations................................18 VI. The induced group.........................................19 VII. Cogrediency................................................20 VIII. Theorem on the roots of a polynomial.......................21 IX. Fundamental postulate.....................................22 X. Empirical definition .....................................22 XI. Analytical definition.....................................23 XII. Annihilators...............................................25 Section 3. Special Invariant Formations I. Jacobians ......... 27 II. Hessians..................., c f 0 . . 28 III. Binary resultants .... .... 29 vvi THE THEORY OF INVARIANTS PAGE IV. Discriminant of a binary form......................31 V. Universal covariants...............................32 CHAPTER II. PROPERTIES OF INVARIANTS Section 1. Homogeneity of a Binary Concomitant I. Theorem on homogeneity...............................33 Section 2. Index, Order, Degree, Weight I. Definitions..........................., . . .35 II. Theorem on the index.................................35 III. Theorem on weight....................................36 Section 3. Simultaneous Concomitants I. Theorem on index and weight..........................38 Section 4. Symmetry. Fundamental Existence Theorem 39 CHAPTER III. THE PROCESSES OF INVARIANT THEORY Section 1. Invariant Operators I. Polars...............................................42 II. Polar of a product...................................45 III. Aronhold’s polars....................................46 IV. Modular polars.......................................48 V. Operators derived from the fundamental postulate . .49 VI. Transvection.........................................51 Section 2. The Aronhold Symbolism. Symbolical Processes I. Symbolical representation ...........................53 II. Symbolical polars....................................55 III. Symbolical transvectants.............................56 IV. Standard method of transvection......................57 V. Formula for the rth transvectant....................59 VI. Special cases of operation by 12.....................61 VII. Fundamental theorem of symbolical theory . . . .62CONTENTS vii Section 3. Keducibility. Elementary Irreducible Systems - PAGE I. Illustrations...................................,61 IL Reduction by identities.................................66 III. Concomitants of binary cubic. Table I...............68 Section 4. Concomitants in Terms of the Roots I. Theorem on linear factors...............................69 II. Conversion operators.......................................70 III. Principal theorem.............................. . .72 IY. Hermite’s reciprocity theorem..............................76 Section 5. Geometrical Interpretations. Involution I. Involution . . . . . . . . . .78 II. Projective properties represented by vanishing covariants . 80 CHAPTER IY. REDUCTION Section 1. Gordan’s Series. The Quartic I. Gordan’s series............................................83 II. The quartic. Table II.....................................89 Section 2. Theorems on Transvectants I. Monomial concomitant a term of a transvectant ... 92 II. Theorem on the difference between two terms of a trans- vectant . 94 III. Difference between a transvectant and one of its terms . . 96 Section 3. Reduction of Transvectant Systems I. Reducible transvectants of a special type .... 97 II. Fundamental systems of cubic and quartic forms . . .100 III. Reducible transvectants in general......................102 Section 4. Syzygies I. Reducibility of ((ƒ, g), h).............................105 II. Product of two Jacobian s...............................106 III. The square of a Jacobian................................107 IY. Syzygies for the cubic and quartic forms................107 Y. Syzygies derived from canonical forms...................108THE THEORY OE INVARIANTS viii Section 5. Hilbert’s Theorem PAGE I. Theorem........................................... 112 II. Linear cliophantine equations.......................116 III. Finiteness of system of syzygies....................119 Section 6. Jordan’s Lemma I. Representation of a form............................120 II. Jordan’s lemma......................................122 Section 7. Grade 1. Definition..........................................124 II. Grade of a covariant................................124 III. Covariant congruent to one of its terms.............125 IV. Representation of a covariant of a covaviant .... 126 CHAPTER V. GORDAN’S THEOREM Section 1. Proof of the Theorem I. Lemma 1.............................................128 II·. Lemma 2.............................................131 III. Lemma 3.............................................133 IV. Theorem.............................................138 Section 2. Fundamental Systems of Cubic and Quartic I. System of cubic.....................................141 II. System of quartic...................................142 CHAPTER VI. FUNDAMENTAL SYSTEMS Section 1. Simultaneous Systems I. Linear form and quadratic...........................144 II. Linear form and cubic...............................145 III. Two quadratics......................................145 IV. Quadratic and cubic. Table III......................146 Section 2. System of the Quintic I. The quintic. Table IV . . 150CONTENTS IX Section 3. Resultants in Aronhold’s Symbols PAGE I. Linear form and w-ic.................................151 II. Quadratic form and n-ic..............................151 Section 4. Fundamental Systems for Special Groups I. Boolean system of a linear form . . . . . .156 II. Boolean system of a quadratic.......................156 III. Formal modular system of a linear form . . . .157 Section 5. Associated Forms................................158 CHAPTER VII. COMBINANTS AND RATIONAL CURVES Section 1. Combinants I. Definition . ...............................162 II. Theorem on Aronhold operators . . . . . .163 III. Partial degrees . . . . . . . . .165 IV. Resultants are combinants.............................166 V. Bezout’s form of the resultant...................... 168 Section 2. Rational Curves I. Meyer’s translation principle.........................169 II. Covariant curves......................................171 CHAPTER VIII. SEMINVARIANTS. MODULAR INVARIANTS Section 1. Binary Seminvariants I. Generators of binary group............................175 II. Definition...........................................176 III. Theorem on annihilator il.............................176 IV. Formation of seminvariants............................177 V. Roberts’ theorem......................................179 VI. Symbolical representation of seminvariants . . . .180 VII. Finite systems of seminvariants.......................184 Section 2. Ternary Seminvariants I. Annihilators ..... .... . 189 II. Symmetric functions of groups of letters .... 191x THE THEC&Y OF INVARIANTS PAGE III. Semi-discriminants. Table V............................193 IV. Invariants of m-lines. 202 Section 3. Modular Invariants and Covariants I. Fundamental system of modular quadratic form, modulo 3. Table VI...................·.....................203 CHAPTER IX. INVARIANTS ÔF TERNARY FORMS Section 1. Symbolical Theory I. Polars and transvectants .............................209 II. Contragrediency........................................212 III. Fundamental theorem of symbolical theory . . . .213 IV. Reduction identities..............··. . . . . 218 Section 2. Transvectant Systems I. Theorem on monomial concomitants . . , . . . 219 II. The difference between two terms of a transvectknt . . 220 III. Fundamental systems of invariant formations of ternary quadratic and cubic forms. Table VII .... 223 IV. Fundamental system of two ternary quadrics . . . . 225 Section 3. Clebsch’s Translation Principle . . . 228 APPENDIX Exercises and theorems............................. 231-241 INDEX.....................................................243THE THEORY OF INVARIANTS CHAPTER I THE PRINCIPLES OF INVARIANT THEORY SECTION 1. THE NATURE OF AN INVARIANT. ILLUSTRATIONS We consider a definite entity or system of elements, as the totality of points in a plane, and suppose that the system is subjected to a definite kind of a transformation, like the transformation of the points in a plane by a linear trans- formation of their coordinates. Invariant theory treats of the properties of the system which persist, or its elements which remain unaltered, during the changes which are im- posed upon the system by the transformation. By means of particular illustrations we can bring into clear relief several defining properties of an invariant. I. An invariant area. Given a triangle ABC drawn in the Cartesian plane with a vertex at the origin. Suppose that the coordinates of A are (xv y^; those of B (xv y2). Then the area A is A = — #2^1)’ or, in a convenient notation, A = \(xy). Let us transform the system, consisting of all points in the plane, by the substitutions x = \xx' + y = \2x’ + ix2y'. 12 THE THEORY OF INVARIANTS The area of the triangle into which A is then carried will be A'=1(^2 - xWi)=K*Y), and by applying the transformations directly to A, A = (1) If we assume that the determinant of the transformation is unity, then Z) = (V)= 1, A' = A. Thus the area A of the triangle ABC remains unchanged under a transformation of determinant unity and is an in- variant of the transforma- tion. The triangle itself is not an invariant, but is car- ried into abC. The area A is called an absolute in- variant if D = 1. If D =£ 1, all triangles having a vertex at the origin will have their areas multiplied by the same number P-1 under the trans- formation. In such a case A is said to be a relative invariant. The adjoining figure illustrates the transformation of J.(5, 6), J5(4, 6), (7(0, 0) by means of x = xf + yf, y = xr + 2 yf. II. An invariant ratio. In I the points (elements) of the transformed system are located by means of two lines of reference, and consist of the totality of points in a plane. For a second illustration we consider the system of all points on a line EF. We locate a point C on this line by referring it to two fixed points of reference P, Q. Thus C will divide the segment PQ in a definite ratio. This ratio, PC/CQ,THE PRINCIPLES OF INVARIANT THEORY 3 is unique, being positive for points 0 of internal division and negative for points of external division. The point C is ---------£----------2______________2_________£________*f said to have for coordinates any pair of numbers (xv x2) such that „ nn ^=7% (2) x2 CQ where A is a multiplier which is constant for a given pair of reference points P, Q. Let the segment PQ be positive and equal to p. Suppose that the point 0 is represented by the particular pair (jPj, jp2), and let D(jqv q2) be any other point. Then we can find a formula for the length of CD. For, CQ_PC__ PQ = fx p2 \Pl \Pl+Pi XPi+Pa’ and Consequently CD — CQ— DQ = PQ = p Ï2 Mi + ?2 M(qp) (3) Theorem. The anharmonie ratio { CDEF\ of four points C(pv P2fi Ï2>> E(rv ri)' Eisv *2)’ defined by ? CDEF\ = CD · EF OF. ED' is an invariant under the general linear transformation T: x1= XjZj + gtx^ x2 = X2x' + g2x^ (\g) =£ 0. In proof we have from (3) | CDEF\ = ilP)(sr^. ispKqr) But under the transformation (cf. (1)), (?^)=(vx?y)> (3i) (4)4 THE THEORY OF INVARIANTS and so on. Also, O, D, E, F are transformed into the points ?2>> E'(.rV F'(,»'v *2>> respectively. Hence {GI)EE\ = = iy'p'X8'/) = \C'D'E'F'\, and therefore the anharmonic ratio is an absolute invariant. III. An invariant discriminant. A homogeneous quadratic polynomial, ƒ=a0x\ + 2 + a2^, when equated to zero, is an equation having two roots which are values of the ratio xx/x2. According to II we may repre- sent these two ratios by two points C(pv jt>2), 2>( = 4(>0a2 - a?)= 0. If ƒ be transformed by i7, the result is a quadratic poly- nomial in xfv 2?2, or ƒ' = af0xf* + 2 a'ja44 + 442· Now if the points <7, 2) coincide, then the two transformed points <7', Df also coincide. For if (72)== 0, (3) gives (g'jt?) = 0. Then (4) gives (j'jt/) = 0, since by hypothesis (X/*) =£(). Hence, as stated, OfDf = 0. It follows that the discriminant Dr of f must vanish as a consequence of the vanishing of D. Hence Dr = KD. The constant K may be determined by selecting in place of ƒ the particular quadratic fx = 2 xxx2 for which 2) = — 4. Transforming fx by T we have f i== 2 XjX2^i + 2(X1/x2 + X2/Xj)ir^2 if- 2 $THE PRINCIPLES OF INVARIANT THEORY 5 and the discriminant of f[ is Df = — 4(\/i)2. Tlien the sub- stitution of these particular discriminants gives -4 (V)2-----4 AT, K= (Xrf. We may also determine K by applying the transformation T to ƒ and computing the explicit form of f. We obtain < = + 2 4" <*2^2’ al = + ^2^l) 4- <4 = a. Therefore the discriminant of ƒ is a relative invariant of T (Lagrange 1773) ; and, in fact, the discriminant of f is always equal to the discriminant of ƒ multiplied by the square of the determinant of the transformation. . Preliminary Geometrical Definition. If there is associated with a geometric figure a quantity which is left unchanged by a set of transformations of the figure, then this quantity is called an absolute invariant of the set (Halphen). In I the set of transformations consists of all linear trans- formations for which (X/i) = 1. In II and III the set consists of all for which (\/i) =£ 0. IV. An invariant geometrical relation. Let the roots of the quadratic polynomial ƒ be represented by the points 0*1’ **2)’ an(l ^ be a second polynomial, <£ = l0x\ + 2 bxxxx2 + \x% whose roots are represented by (^, q2), (*2, *2), or, in a briefer notation, by (g), (s). Assume that the anharmonic ratio of the four points (p), (g), (r), (a), equals minus one,6 THE THEORY OF INVARIANTS (Sf)(jr ) = _ L (6) (spXqr) The point pairs ƒ =0, = 0 are then said to be harmonic conjugates. We have from (6) 2 h = 2 + 2Pirih = **Vh)· Hence «0 = iW 2 al = - GVl + iV2>» ^2 = ^1’ ¿0 = ?2S2’ 2 = — (?251 + ?ls2)’ ^2 == ?1®1’ and by substitution in (2 A) we obtain A = a0&2 — 2 a1b1 + a2b0 = 0. (7) That A is a relative invariant under T is evident from (6): for under the transformation/, become, respectively, f = OiK - '4p[ ) (x>A - x'A )i 4>' = Oi?2 - 4?l)Oi*2 - X2Sl)’ Pi = P2P1 - PiPv p’i = - \Pi + XPv »1 = Wl - PlrV r2 = - Vl + \r2- Hence (^yXsV) + (s’p'Xq'r’y = OX)2[(?.P)0) + («pX»]· That is’ A' = (X/*)aA. where Therefore the bilinear function h of the coefficients of two quadratic polynomials, representing the condition that their root pairs be harmonic conjugates, ¿s a relative invariant of the transformation T It is sometimes called a joint invariant, or simultaneous invariant of the two polynomials under the transformation. V. An invariant polynomial. To the pair of polynomials ƒ, , let a third quadratic polynomial be adjoined, yfr = «0*1 + 2 Cxxxx^ + c2a| = (2^ - *2«1)(*1^2 - *2Wl)·THE PRINCIPLES OF INVARIANT THEORY 7 Let the points (uv w2) (tq, v2), be harmonic conjugate to the pair (jp), (V); and also to the pair (j), ($). Then Cq. We can prove readily that O is an invariant of the trans- formation T. For we have in addition to the equations (5), Vo = W + 2 ¿1X1X2 4“ ^2^2’ *i = + ^2^1) + ^^2^2’ (9) ¿2 = V*l + 2 Vl^2 + Vi· Also if we solve the transformation equations T for xrv x'2 in terms of xv xT we obtain x\ = (V) -1(/*aa?i - ^2), (10) »2 = (V)_1(— X2*l + ^2)· Hence when ƒ, ' = ^ll^l “b ^12^2 “b ^13^3 ^21^1 "b ^22^2 “b a23^3 a31^1 “b ^32^2 "b ^33^3 all^l "b a\2^2 "b #13^3 a21^1 “b a22^2 + ^23^3 ^31^1 “b a32^2 “b ^33/^3 «11^1 + a12^2 + «13^3 a21pl + «22*2 + «28*8 a31Pl + a32P2 + «33*3 = (\ijlv)D. (14) The latter equality holds by virtue of the ordinary law of the product of two determinants of the third order. Hence D is an invariant. VII. A differential invariant. In previous illustrations the transformations introduced have been of the linear homogeneous type. Let us next consider a type of trans- formation which is not linear, and an invariant which repre- sents the differential of the arc of a plane curve or simply the distance between two consecutive points (x, y) and (x + dx, y 4- dy) in the (x, y) plane. We assume the transformation to be given by = X(x, y, «)»y = y·,«), where the functions X, Y are two independent continuous functions of #, y and the parameter a. We assume (a) that the partial derivatives of these functions exist, and (S) that10 THE THEORY OF INVARIANTS these are continuous. Also (c?) we define AT, Y to be such that when a = a0 -STO* y, a0) = a?, F(», y, a0) = y. Then let an increment 8a be added to aQ and expand each function as a power series in 8a by Taylor’s theorem. This gives x' = X(x, y, a0) + dX(x' ÿ" ao) -|--, oa0 ÿ = Y(x, y, a0) + ^ &a H----- (15) Since it may happen that some of the partial derivatives of AT, Y may vanish for a = a0, assume that the lowest power of 8a in (15) which has a non-vanishing coefficient is (Sa)*, and write (Sa)* = St. Then the transformation, which is in- finitesimal, becomes J x] = X + y' = y + yZt. where f, tj are continuous functions of x, y. The effect of operating I upon the coordinates of a point P is to add infin- itesimal increments to those coordinates, viz. 8x = £St, 8y = 7]8t. (16) Repeated operations with I produce a continuous motion of the point P along a definite path in the plane. Such a motion may be called a stationary streaming in the plane (Lie). Let us now determine the functions £, 77, so that such that when the corresponding quantity ' is constructed for the transformed function/' the equality <£'= M(j> holds. Suppose that M depends only upon the transforma- tions, that is, is free from any relationship with/. Then is called an invariant of ƒ under the transformations of the set. The most extensive subdivision of the theory of invariants in its present state of development is the theory of invari- ants of algebraical polynomials under linear transformations. Other important fields are differential invariants and num- ber-theoretic invariant theories. In this book we treat, for the most part, the algebraical invariants. II. Quantics or forms. A homogeneous polynomial in n variables xv xv ···, xn, of order m in those variables is called a quantie, or form, of order m. Illustrations are f(xv x2) = Vi + 3 axx\x2 + 3 a^c^c\ + azx\, f(Xy> Xy Xq) = #200**! “b ^110^1*^2 ~b ^020*^2 “h ^ + 2 #011^2^3 ~b aQQ2X3· With reference to the number of variables in a quantie itTHE PRINCIPLES OF INVARIANT THEORY 15 is called binary, ternary ; and if there are n variables, w-ary. Thus /(#x, x2) is a binary cubic form; f(xv x2, x%) a ternary quadratic form. In algebraic invariant theories of binary forms it is usually most convenient to introduce with each coefficient a{ the binomial multiplier as in f(xv x2)· When these multipliers are present, a common notation for a binary form of order m is (Cayley) f(xv s2)= (a0, av ···, amJxv z2)m = + ma^-'x^ H-----. If the coefficients are written without the binomial numbers, we abbreviate f(xv ^2) = (a0, av ···, am\xv x2)m = + a^-% H--------. The most common notation for a ternary form of order m is the generalized form of f(xv x2, x3) above. This is )U I f(xv xv x3) = X -T-~T~u p,q,r=o [p \ q where jt), q, r take all positive integral values for which p + q + r = m. It will be observed that the multipliers associated with the coefficients are in this case multinomial numbers. Unless the contrary is stated, we shall in all cases consider the coefficients a of a form to be arbitrary variables. As to coordinate representations we may assume (xv xv x3), in a ternary form for instance, to be homogenous coordi- nates of a point in a plane, and its coefficients apqr to be homogenous coordinates of planes in Af-space, where M + 1 is the number of the as. Thus the ternary form is repre- sented by a point in M dimensional space and by a curve in a plane. III. Linear transformations. The transformations to which the variables in an n-ary form will ordinarily be sub- jected are the following linear transformations called colline- ations:16 THE THEOEY OF INVAEIANTS xi = + ^1*2 + ·· · + x% = \x[ + /j.2x'2 -\---h ct-24, (23) + /*n4 H------H 0\X· In algebraical theories the only restriction to which these transformations will be subjected is that the inverse trans- formation shall exist. That is, that it be possible to solve for the primed variables in terms of the un-primed variables (cf. (10)). We have seen in Section 1, V (11), and VIII (22) that the verification of a covariant and indeed the very exist- ence of a covariant depends upon the existence of this inverse transformation. Theorem. A necessary and sufficient condition in order that the inverse of (23) may exist is that the determinant or modulus of the transformation, Pv vv ■ "’«I /V vr · •,<^2 ·· ···. °-» M = (X/jLV ··· · (!!3■, In proof of this theorem we note that one form of ƒ' is f(fixi + Pixv ^4 + P2X2^' But since ƒ is homogeneous this may be written f = xinf(f 1 + f*lx2/xv \ + ^4/4)· We now expand the right-hand member of this equality by Taylor’s theorem, regarding as a parameter, ff — 4m[/ C^i’ ^2) + ^2) +^@(^)/(Xi,X2j+ + \^[) ('‘flx) ^+ ·*·]· /' =/(\, \y?+ - +p(^0/(xx’ **)*S"-r*$r+ + /(Xi’ where18 THE THEORY OF INVARIANTS Comparison of this result with the above form of f1 involving the coefficients a[ gives (23x). An illustration of this result may be obtained from (5). Here m = 2, and #c == 4- 2 #x\xX,2 4“ ^2^2 = f C^l’ ^2) = ftp ai = «0Xl/il + «l(XlAt2 + X2^l)+a2X2^2=5(M^/(Xl’ X2)’ (24) 1 / d \ 2 a2 = “b ^ a\ix\ll2 “b ^2^2 = 2\ 5\y ^ V. A group of transformations. If we combine two trans- formations, as yand jrr . xi = %ix\ “b Vix2' X2 = %2X1 ~b V2X2'> there results rprpi . X1 = “b fJ'l^2^Xl “b Q^iVl “b ’ x2 = (X2£x + /¿2f2)^1 + (X2^1+ ^2)^2· This is again a linear transformation and is called the prod- uct of T and T. If now we consider \v X2, /ax, /¿2 in y to be independent continuous variables assuming, say, all real values, then the number of linear transformations is infinite, i.e. they form an infinite set, but such that the product of any two transformations of the set is a third transformation of the set. Such a set of transformations is said to form a group. The complete abstract definition of a group is the following : Given any set of distinct operations y, T\ Tn7 ··., finite or infinite in number and such that: («) The result of performing successively any two opera- tions of the set is another definite operation of the set which depends only upon the component operations and the sequence in which they are carried out: (/8) The inverse of every operation T exists in the set;THE PRINCIPLES OF INVARIANT THEORY 19 that is, another operation T~x such that TT~l is the identity or an operation which produces no effect. This set of operations then forms a group. The set described above therefore forms an infinite group. If the transformations of this set have only integral coeffi- cients consisting of the positive residues of a prime number jp, it will consist of only a finite number of operations and so will form a finite group. VI. The induced group. The equalities (24) constitute a set of linear transformations on the variables a0, av a2. Like- wise in the case of formulas (23j). These transformations are said to be induced by the transformations T. If T carries ƒ into f and Tf carries ƒ' into ƒ", then (r = 0, 1, m). This is a set of linear transformations connecting the anr directly with a0, ···, am. The transformations are induced by applying i7, T! in succession to ƒ. Now the induced trans- formations (23x) form a group; for the transformations in- duced by applying T and Tf in succession is identical with the transformation induced by the product TTThis is capable of formal proof. For by (23^ the result of trans- forming/ by TT is l»» v a" = A'/OA + hU + Mrf,), where A.=(\ir)1 + ti1r}2) d d 20 THE THEORY OF INVARIANTS But d (M1 + W2 )g(x^+A^) Vl 3(Aifi + /*1?2) #1 ¿OilS +/*i&) Hence and by the method of (IV) combined with this value of A But this is identical with (24x). Hence the induced trans- formations form a group, as stated. This group will be called the induced group. Definition. A quantic or form, as for instance a binary cubic ƒ, is a function of two distinct sets of variables, e.g. the variables xv xv and the coefficients a0, ···, av It is thus quaternary in the coefficients and binary in the variables xv xr We call it a quaternary-binary function. In gen- eral, if a function F is homogeneous and of degree i in one set of variables and of order © in a second set, and if the first set contains m variables and the second set w, then F is said to be an w-ary-w-ary function of degree-order (¿, ©). If the first set of variables is a0, ···, am, and the second xv ···, xn, we frequently employ the notation VII. Cogrediency. In many invariant theory problems two sets of variables are brought under consideration simul- F=(a0, amy(xv xny.THE PRINCIPLES OF INVARIANT THEORY 21 taneously. If these sets (xv ···, #n), (yv y2, ···, ?/n) are subject to the same scheme of transformations, as (23), they are said to be cogredient sets of variables. As an illustration of cogredient sets we first take the modular binary transformations, Tp:x 1 = X2 = \X'l + ^2*2’ where the coefficients A, /jl are integers reduced modulo p as in Section 1, VIII. We can prove that with reference to Tv the quantities #f, #§, are cogredient to xv xv For all binomial numbers ], where p is a prime, are divisible by p except Hence, raising the equations of Tp to the pth power, we have x\ = + fx\x[P, xl = Wpc'f + (mod jt?). But by Fermat’s theorem, A? = At·, (mod p) (i = 1, 2). Therefore x\ = \xa4P + = X24p + M24P' and the cogrediency of x\, a?§ with oq, #2 under Tp is proved. VIII. Theorem. The roots (r^\ r^), (7*j2), r^2)), ···, (r(f\ y·^1)) of a binary form f=aQxT + ma1x'£-1x2+ ··· + amx™, are cogredient to the variables. To prove this we write ƒ = (ri^ - r^x^Ofxi ~ »f%2) ··· (r!il>x1 - r['n)x2), and transform ƒ by T. There results ƒ' = n [(4% - »·<%>' + OfVj - rfV2)4]· ¿=1 Therefore r!ii) _ ; r'(i) = - - r''V2).22 THE THEORY OF INVARIANTS Solving these we have = xy^ + (V)rf = Xtf-w + Thus the r’s undergo the same transformation as the #’s (save for a common multiplier (\/x)), and hence are cogredi- ent to xv as stated. IX. Fundamental postulate. We may state as a funda- mental postulate of the invariant theory of quantics subject to linear transformations the following: Any covariant of a quantic or system of quantics, i.e. any invariant formation containing the variables xv xv ··· will keep its invariant property unaffected when the set of elements xv xv ··· is replaced by any cogredient set. This postulate asserts, in effect, that the notation for the variables may be changed in an invariant formation pro- vided the elements introduced in place of the old variables are subject to the same transformation as the old variables. Since invariants may often be regarded as special cases of covariants, it is desirable to have a term which includes both types of invariant formations. We shall employ the word concomitant in this connection. Binary Concomitants Since many chapters of this book treat mainly the con- comitants of binary forms, we now introduce several defini- tions which appertain in the first instance to the binary case. X. Empirical definition. Let ƒ = ^ - l)a2^'24 + ··· + amx%, be a binary form of order m. Suppose ƒ is transformed by I7 into ƒ' = af0xf™ + 4- ··· +THE PRINCIPLES OF INVARIANT THEORY 23 We construct a polynomial in the variables and coeffi- cients of ƒ. If this function is such that it needs at most to be multiplied by a power of the determinant or modulus of the transformation (X/x), to be made equal to the same function of the variables and coefficients of ƒ, then (f> is a concomitant of ƒ under T, If the order of in the vari- ables xv x2 is zero, is an invariant. Otherwise it is a co- variant. An example is the discriminant of the binary quadratic, in Paragraph III of Section 1. If (f> is a similar invariant formation of the coefficients of two or more binary forms and of the variables xv xv it is called a simultaneous concomitant. Illustrations are h in Paragraph IV of Section 1, and the simultaneous co variant C in Paragraph V of Section 1. We may express the fact of the invariancy of <£ in all these cases by an equation <*>'=(X/x)*4>, in which f is understood to mean the same function of the coefficients aj, afv ···, and of oJv that is of a0, av ···, and xv xT Or we may write more explicitly = ^*00*; xv xv ·*·’ xn)* (26) An equation such as (25) will be called the invariant rela- tion corresponding to the invariant . XI. Analytical definition.* We shall give a proof in Chapter II that no essential particularization of the above * The idea of an analytical definition of invariants is due to Cayley. Intro- ductory Memoir upon Quantics. Works, Vol. II.24 THE THEORY OF INVARIANTS definition of an invariant 0 of a binary form ƒ is imposed by assuming that 0 is homogeneous both in the a’s and in the x's. Assuming this, we define a concomitant 0 of ƒ as follows : (1) Let 0 be a function of the coefficients and variables of/, and 0' the same function of the coefficients and varia- bles of/'. Assume that it is a function such that 30' a\, dp\ $P 2 (27) (2) Assume that 0' is homogeneous in the sets \v \ ; fiv fi2, and of order k in each. Then 0 is called a concomitant of ƒ. We proceed to prove that this definition is equivalent to the empirical definition above. Since 0' is homogeneous in the way stated, we have by Euler’s theorem and (1) above where k is the order of 0' in \v X2. Solving these, It- = It- = - h OAj C/A^ Hence #' = It- + It" ^ OA»j oA2 Separating the variables and integrating we have 0' (\fl) where O is the constant of integration. To determine (7, let T be particularized to Xx =:: X\i X% “ *^2*THE PRINCIPLES OF INVARIANT THEORY 25 Then a[ = a^i = 0, 1, 2, ···, w), and f= . Also (\fi) = 1. Hence by substitution we arrive at the same result. Hence the two definitions are equivalent. XII. Annihilators. We shall now need to refer back to Paragraph IV (23x) and Section 1 (10) and observe that function of \v \2, fiv has precisely the same effect as some other linear differential operator involving only a[ (i = 0, ···, m) and xrv x2, which would have the effect (29) when applied to ft regarded as a function of a[, xfv xf2 alone. Such an operator exists. In fact we can see by empirical considerations that is such an operator. We can also derive this operator by an easy analytical procedure. For, <£'=(V)*r = 0, \ d/i (< +2 ¿i+-+if, - - °·<2%) The operators (29x), (292) are called annihilators (Sylvester). Since <£ is the same function of a*, o?x, #2, that ' is of a{, aj, #2, we have, by dropping primes, the result: Theorem. A set of necessary and sufficient conditions that a homogeneous function, <£, of the coefficients and variables of a binary form f should be a concomitant is In the case of invariants these conditions reduce to 0 = 0, ilcf) = 0. These operators are here written again, for refer- ence, and in the un-primed variables: n 3 . , i-v 3 3 (J = ma1-----h (m — 1 )a2----h ··· + am----- 3a0 da1 dam_ m—1 dal v/1^2 A simple illustration is obtainable in connection with the invariant 2>1 = «oa2 - a\ (§ Iir)· Here m = 2: n 3 0 3 ^ 0 3 , 3 û> = %-z-----—, 0 = 2a1-------------h a2-— 0^2 3#q da^ Xî-Dj = — 2 -f- 2 = 0, = 2 — 2 axa2 = 0. It will be noted that this method furnishes a convenient means of checking the work of computing any invariant.THE PRINCIPLES OF INVARIANT THEORY 27 SECTION 3. SPECIAL INVARIANT FORMATIONS We now prove the invariancy of certain types of functions of frequent occurrence in the algebraic theory of quantics. I. Jacobians. Let fvf2, ··.,ƒ* be n homogeneous forms in n variables xv x2, ···, xn. The determinant, Jacobian of the n forms. We prove that J is invariant when the forms fi are transformed by (23), i.e. by x{ = \x[ + ^¿4 H-----h . II. Hessians. If ƒ is an w-ary form, the determinant fxyX-yl fEjXj’ f xixn H= X2X11 ·^C2X2, 'if X2Xn (34) f tnxxi f ZnXf "'fxnxn is called the Hessian of ƒ. That H possesses the invariant property we may prove as follows: Multiply H by M= (\\iV ··· O’), and make use of ■ (33). This gives _LJ£ d Jf MH= \ Pi " ^2 P2 *' " ai " *2 H= dx\ dx1 ’ A 3*2 dx2 ’ 3^2 ^*11 ' JLM. dx2 dx2 ’ ’ dx’n d ten dxx Jf dx2 Pn " ■· O'n d_df_ J_Jf ... d JL dx[ dxn' 34 dxn Replacing ƒ by ƒ' as in (82) and writing iii.Af,, tc, dx' dzx dxx dx[ we have, after multiplying again by M, f xxx{ f x2x^ ... ƒ. , ’ ^ xnxl 3PH= f XlX2* f X2Xi> "'ifTnXt f XxXn' f ^fX’n '"if XnxnTHE PRINCIPLES OF INVARIANT THEORY 29 that is to say, and JVisa concomitant of/. It is customary, and avoids extraneous numerical factors, to define the Hessian as the above determinant divided by \nn X (n — 1)". Thus the Hessian covariant of the binary cubic form ƒ_ a^xz + 3 aix*x2 + 3 a2x xx£ + a3x^, is * A = 2 «1*1 + a2X2 axxx + a2xv a2*i + «3*2 ' (35) = 2(>0a2 - af)zf + 2(a0a8 - a1a3)x1x% + 2(a1as - a\)x\. III. Binary resultants. Let/, ’be two binary forms of respective orders, m, n; m ƒ = aQxf + matxf-lx2 ------1- amx% = n(rfxx — rfx2), i=I 4> = 1>qX\ + nbxx\~xx3 H---h bnx2n = Tl(8i,1>x1 — 8(i%2). 3 =1 It will be well known to students of the higher algebra that the following symmetric function of the roots (rj'\ rj0), ($^\ <£) caHe(l the resultant of ƒ and <£. Its vanishing is a necessary and sufficient condition in order that ƒ and $ should have a common root. n m R(J, it will be sufficient to recall that the roots (rv r2), (sr $2) are cogredient to xv x2. Hence when ƒ, are each transformed by T\ R undergoes the transformation = X/f + f^r^, (X/i)rf =X2/(i) + + rf'N (MHi} = Vi(i’ + rf*’ * Throughout this book the notation for particular algebraical concomitants is that of Clebsch.30 THE THEORY OF INVARIANTS in which, owing to homogeneity the factors (\/a) on the left may be disregarded. But under these substitutions, — r^s[j) = (X/i)~1(rj(i)S2(y) “ r20MO))· Hence #(f, fj=(vr«(/, 4>), which proves the invariancy of the resultant. The most familiar and elegant method of expressing the resultant of two forms/, in terms of the coefficients of the forms is by Sylvester’s dialytic method of elimination. We multiply ƒ by the n quantities xJ-1, a^~2a?2, ···, a^_1 in succes- sion, and obtain a0Xjt+n~1 4- ma1x^l+n~2x2 + ··· + amx\~lx a^x^n~2x2 + ··· 4- + amx\~2x2Yl, (37) a0x^xg 1 4* ··· + mam_1xyx^+n 2 + amx%+n h Likewise if we multiply by the succession a^'1, x™~2xv ···, a^"1, we have the array b0x™+n-1 + nb1xf+n~, viz. «0 ma1 · · ■■■ am 0 0 .. . 0 CO * O · 0 ao ··· ··· mam-x am o·. .· 0 Sh £ 0 0 0 . a, A % 2 ■ 0 nbl 0 0 0 b. A particular case of a resultant is shown in the next para- graph. The degree of i2( ƒ, <£) in the coefficients of the two forms is evidently m 4- n.THE PRINCIPLES OF INVARIANT THEORY 31 IV. Discriminant of a binary form. The discriminant D of a binary form ƒ is that function of its coefficients which when equated to zero furnishes a necessary and sufficient condition in order that ƒ = 0 may have a double root. Let f=f(xv = a0x™ 4- maxx™ % H-+ amx™, and let fx/xv z2) = fjxv «2) = J7· Then, as is well dx„ known, a common root of ƒ = 0, / = 0 is a double root of ƒ = 0 and conversely. Also dxx hence a double root of ƒ = 0 is a common root of ƒ = 0, = 0, — = 0, and conversely; or D is equal either to the dxt dx2 eliminant of ƒ and or to that of ƒ and Let the roots oxx dx2 of #2) = 0 be 8(^)(i = 1, ···, m— 1), those of fh (xv x2) = 0, (if\ f)(i = 1, ·.·, m— 1), and those of ƒ = 0 be (Vf\ r^(J = 1, 2, ···, m). Then a0d=/(41\ 41})/(42)’ 4W_1)> Now Of(xv x^) = X\ , Qf(xv x2)=x2'^''> where 0 and il are the annihilators of Section 2, XII. Hence 0D = 4a)) = «¿1))/Oi2\ 42)) «T“1))=0· Thus the discriminant satisfies the two differential equations OD = 0, ilD = 0 and is an invariant. Its degree is 2(m — 1). An example of a discriminant is the following for the binary cubic/, taken as the resultant of oxl ox232 THE THEORY OF INVARIANTS ~\R = 2 a, 2 a, 0 2 a, 2 a, 0 (39) d = = (X/A)(a/ÿ'). (40) — (ctçfl3 — «i^)2 “ ^Ca0a2 al)(a1^3 aD* V. Universal covariants. Corresponding to a given group of linear transformations there is a class of invariant forma- tions which involve the variables only. These are called universal covariants of the group. If the group is the infinite group generated by the transformations T in the binary case, a universal covariant is d = = - x&v where (y) is cogredient to (x). This follows from xi^i + \xi + \y\ + ny'v +Nf% If the group is the finite group modulo p, given by the trans- formations Tp, then since x\, x% are cogredient to xv xv we have immediately, from the above result for c?, the fact that L = x\x2 - xxxl (41) is a universal covariant of this modular group.* Another group of linear transformations, which is of con- sequence in geometry, is given by the well-known trans- formations of coordinate axes from a pair inclined at an angle © to a pair inclined at an angle ©' = /3 — a, viz. x __ sin Q - a) r, | sinp-ff) 1 “ 1 sin© sin© v2 ’ Xo =- sin a sin /3, %i H : ^2* sin © sin © Under this group the quadratic, x\ + 2 xtx2 cos © 4- x\ is a universal covariant, f * Dickson, Transactions Amer. Math. Society, vol. 12 (1911). t Study, Leipz. Ber. vol. 40 (1897). (42) (43)CHAPTER II PROPERTIES OF INVARIANTS SECTION 1. HOMOGENEITY OF A BINARY CONCOMITANT I. Homogeneity. A binary form of order m ƒ = a0z™ + ma1xT?~1x2 H--------------l· amx™, is an (m +1)-ary-binary function of degree-order (1, m). A concomitant of ƒ is an (m + 1)-ary-binary function of de- gree-order (z, G>). Thus the Hessian of the binary cubic (Chap. I, § 3, II), A = 2(a0a2 — af)xf + 2(a0a3 - axa^)xxx3 + 2(axa% - af)x% (44) is a quaternary-binary function of degree-order (2, 2). Likewise ƒ + A is quaternary-binary of degree-order (2, 2), but non-homogeneous. An invariant function of degree-order (i, 0) is an invariant of ƒ. If the degree-order is (0, g>), the function is a universal covariant (Chap. I, § 3, V). Thus a^a2 — a\ of degree-order (2, 0) is an invariant of the binary quadratic under T\ whereas x\x2 —- xxz% of degree-order (0, p + 1) is a universal modular covariant of Tr Theorem. If (7=(a0, av ···, amy(xv x2)ta is a concomitant of f = (a0, .··, am')(xv z2)mi its theory as an invariant function loses no geyierality if we assume that it is homogeneous both as regards the variables xv x2 and the variables «0, ···, am. Assume for instance that it is non-homogeneous as to xv xv Then it must equal a sum of functions which are separately homogeneous in xv z2. Suppose (7= Cj -j- C2 A- —V Q$, 3334 THE THEORY OF INVARIANTS where O} =(a0, av ···, am)i'(xv = 1, 2, ·.·, Suppose now that we wish to verify the covariancy of (7, directly. We will have Cf = (a', ···, x^** = (45) in which relation we have an identity if a[ is expressed as the appropriate linear expression in a0, ·.., am and the x[ as the linear expression in xv x2, of Chapter I, Section 1 (10). But we can have y=i j=1 identically in xv x2, only provided , A, w. Consider the form ƒ and a corresponding concomitant relation Of =). (48) And this relation is true of invariants, i.e. (48) holds true when (D = 0. To prove this we transform ƒ = + maxx^~lx2 ------h a,mx™, by the following special case of T: xx = \xfv x2 = \x2.36 THE THEORY OF INVARIANTS The modulus is now X2, and aj= Xma?· (j = 0, ·.·, m). Hence from (47), (X«a0, \»av = X2*(a0, av .·, x2y. (49) But the concomitant C is homogeneous. Hence, since the degree-order is (i, g>), X*m “(^0’ ***’ = ^*(^0’ ***’ ®*»)*(^ 1’ *^2)*°* Hence 2 A = tW— &). III. Theorem. Every concomitant C of f is isobaric and the weight is given by w = \(im + ©), (50) where (i, ft)) is the degree-order of C, and m the order of f The relation is true for invariants, i.e. if <0= 0. In proof we transform ƒ by the special transformation Xx = aj, = Xz'. (51) Then the modulus is X, and aj = X>a?· (ƒ = (), 2, ..., m). Let r = agafa2 · · · be any term of O and rf the corresponding term of C\ the transformed of O by (51). Then by (47), Thus or Tf =X3+2r+‘"+^-“ag0fa£ ... = X*t. w — ft) = h = |(im — ft)), w = l(im + a)). Corollary 1. index, The weight of an invariant equals its w=k = im. Corollary 2. The degree-order (i, cd) of a concomitant (7 cannot consist of an even number and an odd numberPROPERTIES OF INVARIANTS 37 except when m is even. Then i may be odd and oo even. But if m is even oo cannot be odd. These corollaries follow directly from (48), (50). As an illustration, if C is the Hessian of a cubic, (44), we have i = 2, oo = 2, m = 3, w = \(2 · 3 + 2) = 4, ¿ = 1(2.3 -2) = 2. These facts are otherwise evident (cf. (44), and Chap. I, § 3, II). Corollary 3. The index k of any concomitant of ƒ is a positive integer. For we have w — oo — Jc, and evidently the integer w is positive and oo __ w. SECTION 3. SIMULTANEOUS CONCOMITANTS We have verified the invariancy of two simultaneous concomitants. These are the bilinear invariants of two quadratics (Chap. I, § 1, IV), yjr = a0xf + 2 atxtx2 + a2x\, = b0xf + 2 bxxxx2 + b2x% viz. h = a0b2 — 2 axbx 4- a2b0, and the Jacobian 0 of yfr and (cf. (8)). For another illustration we may introduce the Jacobian of <£ and the Hessian, A, of a binary cubic/. This is (cf. (44)) A = |/o(^o^3 ^1^2) 2 ^10*0^2 *l)>l + 2 lh(alaZ - aD- K{a0a2 - al)]^2 H" [2 ^1) ^(^0^3 «1^2)]% and it may be verified as a concomitant of <£> and f=a0x ?+ ....38 THE THEORY OE INVARIANTS The degree-order of J is (3, 2). This might be written (1 + 2, 2), where by the sum 1 + 2 we indicate that J is of partial degree 1 in the coefficients of the first form <£ and of partial degree 2 in the coefficients of the second form ƒ. I. Theorem. Let f , ··· be a set of binary forms of respective orders mv m2, m2, ···. Let C be a simultaneous concomitant of these forms of degree-order (h + h 4- i8+ ···, ®). Then the index and the weight of 0 are connected with the numbers m, i, co by the relations k = l(^ixm j — û>), (52) w = ^Çlilm1 4- «), and these relations hold true for invariants (i.e. when co = 0). The method of proof is similar to that employed in the proofs of the theorems in Section 2. We shall prove in detail the second formula only. Let ƒ = a0zT' + = b0xf* -|-, \/r = c ^)t(^n4)e = (V)lA?+'£(^0’ ·*·’ 0^*1^)* (55) = (X/a)*+‘jBr. Now cfj is the same function of the ar0, ···, arm that Cj is of a0, ···* #w. When the ens and c’s in (55) are replaced by their values in terms of the a’s, we have Kf = [aj, ···, 4)e= (VO^K’ —* am]il(vv x%>* (66) = (\fjiyk+«K, where, of course, [a0, .·., am]h(xv x2y considered as a func- tion, is different from (a0, ..., am)u(xv x2y. But (56) is a covariant relation for a covariant of ƒ. This proves the theorem. The proof holds true mutatis mutandis for concomitants of an ii-ary form and for simultaneous concomitants. The index of K is p = i · \ (ini — ©) + l(i(o — e) = \(iim — e), and its weight, w = \(iim + e). Illustration. If ƒ is a binary cubic, ƒ = a0^f + 3 axx\x2 + 3 a^xx\ + a2a|, then its Hessian, A = 2 [(a0a2 — a\)x[ + (a0as — axa^xxx^ + (axaz — a|)a|], is a covariant of ƒ. The Hessian 2 R of A is the discrimi- nant of A, and it is also twice the discriminant of/, 2 R = 4[- O0«3 - «i«2)2 + 4Oofl2 - «‘0(«i«8 - «!)]·CHAPTER III THE PROCESSES OF INVARIANT THEORY SECTION 1. INVARIANT OPERATORS We have proved in Chapter II that the system of invari- ants and covariants of a form or set of forms is infinite. But up to the present we have demonstrated no methods whereby the members of such a system may be found. The only methods of this nature which we have established are those given in Section 3 of Chapter I on special invariant formations, and these are of very limited application. We shall treat in this chapter the standard known processes for finding the most important concomitants of a system of quantics. I. Polars. In Section 2 of Chapter I some use was made of the operations + X2 Pi ~~ + /¿2 —“ * Such opera- d/i^ d\2 tors may be extensively employed in the construction of in- variant formations. They are called polar operators. Theorem. Let f = a0x™ + ··· be an n-ary quantic in the variables xv ···, xn, and a concomitant of f the corresponding invariant relation being Then if yv yv ···, yn are cogredient to xv xv ···, a?n, the function <£'=(«;, :.y(x' = (\fi ··· 0-)*(«. (57) is a concomitant off. 42THE PROCESSES OF INVARIANT THEORY 48 It will be sufficient to prove that (58) the theorem will then follow directly by the definition of a covariant. On account of cogrediency we have Xi = \iXr1 + fiiXf2+ ··· +(TiXrn, (59) yi = ^iy[ +m2+ ··· +<^nO' = i, ···> n). Hence _d_ dx[ dx[ d dxx ^ _d_ dx^ _j_ dxx dx\ dx2 dx\ dxn dx\ = Xi ldx, ■ + \o dXo - + + Xn— » dxn d d <9 dx' dx. + Mn dXn d d d = . It is convenient, however, and avoids adventitious numerical factors, to define as the polar of <£ the expression times a numerical factor.44 THE THEORY OF INVARIANTS We give this more explicit definition in connection with polars of ƒ itself without loss of generality. Let ƒ be of order m. Then \m rf d ==( ' m \ oxi yndx)f f*' (61) the right-hand side being merely an abbreviation of the left- hand side, is called the rth ÿ-polar of ƒ. It is an absolute covariant of ƒ by (60). For illustration, the first polars of ƒ = a0xZ -f- 3 axxfx2 4- 3 a2xxx\ -f a3z|, 9 = ^200*^1 4" 2 <^110^1^2 4" a020^i 4" 2 a\o\x\x% 4- 2 #011^2^3 4" ^002^3’ are, respectively, fv = (v? + 2 «^2 + a2xl)yx 4- (atxf 4- 2 a2xxx2 4- azx\)yv 9v= (^200^14- ^110^2 4" **101^3)^14" C^io^i 4“ ^020^2 4" ^011^3)^2 4“ (^101^14" ^011^2 4“ ai 4- («i^f 4- 2 a2yxy2 4- a$%)x2 If g = 0 is the conic O of the adjoining figure, and (y) = (yr y2, y3) is the point P, then gy = 0 is the chord of contact AB, and is called the polar line of P and the conic. If P is within the conic, ?p gy= 0 joins the imaginary points of contact of the tangents to O from P. We now restrict the discussion in the remainder of this chapter to binary forms. We note that if the variables (y) be replaced by the variables (x) in any polar of a form f the result is f itself i.e. the original polarized form. This follows by Euler’s theorem on homogeneous functions, sinceTHE PROCESSES OF INVARIANT THEORY 45 In connection with the theorem on the transformed form of Chapter I, Section 2, we may observe that the coefficients of the transformed form are given by the polar formulas a0 ^2) f0' =«/ojo ^2-A* **> a = fofj.ni· (63) The rth y-polar of ƒ is a doubly binary form in the sets (yv y2), x%) degree-order (r, m~r). We may how- ever polarize f a number of times as to (if) and then a number of times as to another cogredient set (z) ; This result is a function of three cogredient sets (x), (j/), (2). Since the polar operator is a linear differential operator, it is evident that the polar of a sum of a number of forms equals the sum of the polars of these forms, (ƒ+ <£+ ··· V = ƒ/+ <£y + ···· II. The polar of a product. We now develop a very im- portant formula giving the polar of the product of two binary forms in terms of polars of the two forms. If F(xv x2) is any binary form in xv x2 of order M and (y) is cogredient to (x), we have by Taylor’s theorem, k being any parameter, F(xx + kyv x2 + kyf) -jw *>+*(,£)*+ = F+(^jFJi + (^jF^ + - +{f^jF^+ .... (65) Let F=f(xv x2)(l)(xv #2), the product of two binary forms of respective orders m, n. Then the rth polar of this prod- uct will be the coefficient of kr in the expansion of f(x 1 + kyv x2 + ky2) x(xt + kyv x2 + ky2f46 THE THEORY OF INVARIANTS ( 771 1 71/\ divided by ( ^ 1, by (65). But this expansion equals f + (^)fyk + (^)fvM+ - +(^)fAr+ - j4>+(^)4>yk +(2)^+ - +(;)*/*+ ···} Hence by direct multiplication, This is the required formula. The sum of the coefficients in the polar of a product is unity. This follows from the fact (cf. (62)) that if (y) goes into (#) in the polar of a product it becomes the origi- nal polarized form. An illustration of formula (66) is the following : Letf=a0x\ + ·.·, = b^\ + .... Then — + | fy^y + ify&j2' III. Aronhold’s polars. The coefficients of the transformed binary form are given by a'r x2)(r = 0, ···, m). These are the linear transformations of the induced group (Chap. I, § 2). Let be a second binary form of the same order as ƒ, = btfc™ 4- mb1xf~'[x2 + ....THE PROCESSES OF INVARIANT THEORY 47 Let be transformed by 7 into m T-y +i-£-Ail (67) That is, [b —) is an invariant operation. It is called the daJ Aronhold operator but was first discovered by Boole in 1841. Operated upon any concomitant of ƒ it gives a simultaneous concomitant of ƒ and . If m = 2, let Then 1= a0a2 - a\. + bli1+^ ~2 aA+ This is h (Chap. I, § 1). Also 2(jAj/=4(J062-if), the discriminant of . In general, if ^ is any concomitant of/, ^ = (i*Q, ···, a!n^)i(xyt ***’ then {¥ ¿)V= 0* O = 0, x' ■’ 0 <68> are concomitants of ƒ and <£. When r — i, the concomitant is %=(A)’ ***’ dm)l{Xyi X%)U>· The other concomitants of the series, which we call a series of Aronhold’s polars of i/r, are said to be intermediate to yfr48 THE THEORY OF INVARIANTS and x, and of the same type as yjr. The theory of types will be referred to in the sequel. All concomitants of a series of AronholcTs polars have the same index k. Thus the following series has the index k = 2, as may be verified by applying (52) of Section 3, Chapter II to each form (f=a(p% + ···; = 4* •••): (aQa2 - a\)x\ + (aQas - axa^)x^ + (axas - a\)x% VJ^r)H=(aA ~ 2 «1*1 + «2^)4 +(«0*3+ «3*0 ~aA~ a1^3 — 2 «2*2 + «8*l)4’ = (a0, a^^x^ x2)0> is any invariant function, i,e. a concomitant of a binary form/, then d=(«0» —> - ¿¡¡r)w (7°) is an invariant operator (Boole). If this operator is operated upon any covariant of ƒ, it gives a concomitant of ƒ, and50 THE THEOEY OF INVAEIANTS if operated upon a covariant of any set of forms g, A, ···, it gives a simultaneous concomitant of ƒ and the set. This process is a remarkably prolific one and enables us to construct a great variety of invariants and co variants of a form or a set of forms. We shall illustrate it by means of several examples. Let ƒ be the binary quartic and let be the form ƒ itself. Then dd> = df= aa— and — 4 a* + 6a2 d4 dx^dxf -4 do d* dx2dx% a**’ 2\ ¥'f= 2Ooa4 - 4 ala3 + 3 «i)= *· This second degree invariant i represents the condition that the four roots of the quartic form a self-apolar range. If this process is applied in the case of a form of odd order, the result vanishes identically. If H is the Hessian of the quartic, then ÔH= Oo«2 2(«o«3 - + («o«4 + 2 axa% - 3 a|) - 2(alai - a2as) d* And + («2a4-«|)^ bx\bx\ dx2dxf j12 dH · ƒ = 6(a0«2rt4 + 2 — afa4 — a^a\ — = «7. (70i) This third-degree invariant equated to zero gives the con- dition that the roots of the quartic form a harmonic range. If TTis the Hessian of the binary cubic ƒ and 9 = bm - +"·+(- + - +(-l)maJ0. (71) This, the bilinear invariant of ƒ and g, is the simplest joint invariant of the two forms. If it is equated to zero, it gives the condition that the two forms be apolar. If m = 2, the apolarity condition is the same as the condition that the two quadratics be harmonic conjugates (Chap. I, § 1, IV). VI. The fundamental operation called transvection. The most fundamental process of binary invariant theory is a differential operation called transvection. In fact it will subsequently appear that all invariants and covariants of a form or a set of forms can be derived by this process. We proceed to explain the nature of the process. We first prove that the following operator i2 is an invariant: 12 = A A dxt dx2 d d ’ where (y) is cogredient to (x). In fact by (70), (72) 12' = . d d d . d Xl^r + X2T7’ Ldx, dx0 dxA dXc ^ d ^ d d d + ^2 ^ ’ MlT---b — dVi dV\ dVi which proves the statement. Evidently, to produce any result, X2 must be applied to a doubly binary function. One such type of function is a y-polar of a binary form. But52 THE THEOEY OF INVAKIANTS Theorem. The remit of operating il upon any y-polar of a binary form f is zero. For, if ƒ = a0x™ 4- ···, and this vanishes by cancellation. If il is operated upon another type of doubly binary form, not a polar, as for instance upon^, where ƒ is a binary form in xv x2 and g a binary form in yv yv the result will generally be a doubly binary invariant formation, not zero. Definition. If f(x) — a0xf+ ··· is a binary form in (V) of order m, and g(jj^) = b0y\ + ··· a binary form in (y) of order w, then if yv y2 be changed to xv x2 respectively in \m — r \n — r \m\n ilrKx)9(.y)^ (73) after the differentiations have been performed, the result is called the rth transvectant (Cayley, 1846) of f(x) and g(x)· This will be abbreviated (ƒ, gf, following a well-established notation. We evidently have for a general forjnula ( ƒ. ay = | m 9f(*) dx^-sdx^ &g(x) dx\dxr2s (74) We give at present only a few illustrations. We note that the Jacobian of two binary forms is their first transvectant. Also the Hessian of a form ƒ is its second transvectant. ForTHE PROCESSES OF INVARIANT THEORY 53 11= m\m — l)2 (I m — 2)2 (|m)2 (ƒX\X\ft2 X2 ^ S\\X<1 “H ƒtiX2%fXiX±) •As an example of multiple transvection we may write the following covariant of the cubic/: Q = (/(ƒ>/)2)1= («0*8 - 3 a0«l«2 + 2 *1>1 + 3(*0*1*3 - 2 *0*2 + *1*2)^2 (74i) — 3(a0a2a3 — 2 + 1 - («0«| - 3 axa2as + 2 a§)a|. If ƒ and g are two forms of the same order m, then (ƒ, g)m is their bilinear invariant. By forming multiple transvections, as was done to obtain Q, we can evidently obtain an un- limited number of concomitants of a single form or of a set. SECTION 2. THE ARONHOLD SYMBOLISM. SYMBOLICAL INVARIANT PROCESSES I. Symbolical representation. A binary form ƒ, written in the notation of which ƒ = a+ 3 + 3 a2xxx^ + azx\ is a particular case, bears a close formal resemblance to a power of linear form, here the third power. This resem- blance becomes the more noteworthy when we observe that the derivative bears the same formal resemblance to the bxx derivative of the third power of a linear form : = 3O0£f + 2 axxxx2 + a2x$). That is, it resembles three times the square of the linear form. When we study the question of how far this formal resemblance may be extended we are led to a completely54 THE THEORY OF INVARIANTS new and strikingly concise formulation of the fundamental processes of binary invariant theory. Although/ = a0x™ -f- ··· is not an exact power, we assume the privilege of placing it equal to the rath power of a purely symbolical linear form axxx 4- a2x2^ which we abbreviate ax. ƒ = (oqsq + a2x2yn= a™ = aQxf + ···. This may be done provided we assume that the only defined combinations of the symbols av a,2, that is, the only combina- tions which have any definite meaning, are the monomials of degree ra in av a2 ; oc™ = (Iq-, cc™ lct2 = flq, ···, cc^ = dm·, and linear combinations of these. Thus a™ + 2 ccf~2a$ means a0 + 2 a2. But of~2cc2 is meaningless ; an umbral expression (Sylvester). An expression of the second degree like a0as cannot then be represented in terms of as alone, since am . am-zas — a2m-3a3 js undefined. To avoid this difficulty we give ƒ a series of symbolical representations, f=< = /3? = Yil= -, wherein the symbols (av o^), (/3V y@2), (7^ 72), ··· are said to be equivalent symbols as appertaining to the same form/. Then «r=/3r=Yii= ··· =«0* «r1«2=^r1^2=7r1Y2= ··· — Now a0a3 becomes (_3/S|) and this is a defined combina- tion of symbols. In general an expression of degree i in the a’s will be repre- sented by means of i equivalent symbol sets, the symbols of each set entering the symbolical expressions only to the rath degree ; moreover there will be a series of (equivalent) symbolical representations of the same expression, as a0a3 = af/3™-3/3! = = ····THE PROCESSES OF INVARIANT THEORY 55 Thus the discriminant of ƒ = «I = /3f = ··· = a0x\ + 2 axxxx2 + a2z\ 18 I) = 4(a0a2 - a?) = 4(af/3| - = 2(af/8| — 2 + a|/3jf), or 2) = 2(«/S)2, a very concise representation of this invariant. Conversely, if we wish to know what invariant a given symbolical expression represents, we proceed thus. Let ƒ be the quadratic above, and 9 — Px = ax— · ·· = V? + 2 hxix2 + hxv where p is not equivalent to a. Then to find what J = which evidently contains the symbols in defined combinations only, represents in terms of the actual coeffi- cients of the forms, we multiply out and find J= («1 Pi- “iPl)(alxl + “2*2)01*1 + PiX2> = (tfPlPi - “l«2/>l)*f + (“fpi - “l/>f)*i*2 + («1 «2P2 - aiPlP2)Xl· = («0*1 - aA)Xl + (aA - aih)XlX2 + («1*2 - «2*1)*!· This is the Jacobian of/and g. Note the simple symbolical form J = (ap)a*/v II. Symbolical polars. We shall now investigate the forms which the standard invariant processes take when expressed in terms of the above symbolism (Aronhold, 1858). For polars we have, when ƒ = a™ = fi™ = ··, &-=“î',(î'i)(vi+■ <"'v Hence Z,r=<-r<· (75) The transformed form of ƒ under ^will be f = [“i(Xi*! + Mi*0 + «2(^*1 + ^2*2)]“ = [(«^ + a2X2)*i + («i/ij + «2M2)4]“56 THE THEORY OF INVARIANTS or ƒ = («X + IV. Standard method of transvection. We may derive transvectants from polars by a simple application of the fundamental postulate. For, as shown in section 1, if/ = a0x\ H------- axi fyr = \m - \m \_dx\yi + (0 drf day~1dx2 yTxy* + (82) d d Now (y) is cogredient to (a;). Hence------, ---are cogredient dx2 dxt to yv yv If we replace the y's by these derivative symbols and operate the result, which we abbreviate as dfyr, upon a second form g = b% we obtain = (a6)^rr*rr=(/^)r· (83) When we compare the square bracket in (82) with a™~r times the square bracket in (83), we see that they differ precisely in that yv y2 has been replaced by 62, — br Hence we enunciate the following standard method of transvection. Let ƒ be any symbolical form. It may be simple like ƒ in this paragraph, or more complicated like (78), or howsoever complicated. To obtain the rth transvectant of ƒ and = b% we polarize fr times, change yv y2 into &2, — bt respectively in the result and multiply by bnx~r. In view of the formula (77)58 THE THEORY OF INVARIANTS for the polar of a product this is the most desirable method of finding transvectants. For illustration, let F be a quartic, F = a% = b*, and ƒ its Hessian, ƒ = (aJ)2a?6|. Let Then 9 = «!· (ƒ, ^)2 = (a5)2<#lj x «* y2; y=a (abf |a25f+ axaybxby + ( )a2i| x a, Jy=a (84) = l(a6)2(6a)2a|«iC4· f (ab)\aa)(ba)axbxax + ±(ab)2(aa)2b%ax. Since the symbols a, 6 are equivalent, this may be simplified by interchanging a, b in the last term, which is then identical with the first, (ƒ, 9f= \(abf(bafalax + l(ab)\aa)(ba)axbxax. By the fundamental existence theorem this is a joint co- variant of F and g. Let ƒ be as above and g = (ay£)af/3x, where a and ¡3 are not equivalent. To find say, in a case of this kind we first let g = ( a/3)a%0x = 4a) - 4m\ <7 = ... /9<*\ in which af = afxx + a^x2, = $fxx 4- We can obtain a formula of complete generality for the transvectant (ƒ, ;/)r by applying the operator SI directly to the product^. We have a2 dx-fiy; dx2dy{ -/F=2“W) Í9 fff «W60 THE THEORY OF INVARIANTS Subtracting these we obtain I m \n Repetitions of this process, made as follows: (/^>2= \m — 2 \n — 2 \m\n ^ («<«>£ ^—1 , (88) lead to the conclusion that the rth transvectant of ƒ and g, as well as the mere result of applying the operator il to fg r times, is a sum of terms each one of which contains the product of r determinant factors (a/3), m — r factors a*, and n — r factors ¡3X. We can however write (ƒ, g)r in a very simple explicit form. Consider the special case ƒ ’X 9 <7 = /W>· Here, by the rule of (88), (f,gy= ](a<»l3M)(a(»/3<»)a(f + (cfl>0a)Xa<«>0 (2,)42) + + («/3(2)Xa(s,iS(1))«i:2> + («(2)y3(i))(a«,y3( 2>)<* + (a<2>/3(1,X«(1,/3(2,)43> (89) + (a<2)£<2,)(a(3)/8a))41) + («®/3(2,j(«(I)i9(1))«® + 4- («(S)/3(1)X«(2)/3<2))41) + («'3)iS(2))(«(l)/gU))42) + («<3)/3(2))(a<2^(l>)41* | -r- [2|3, in which occur only six distinct terms, there being a repetition of each term. Now consider the general case, and the rth transvectant. In the first transvectant one term contains i1 = («(i>y3(1))42) ··· <4'n)/3<2) ··· ^/8(1>)(a<2>/3<2))<43) ··· /3y!) ··· arising from ilfx, and another term u1 arising from il<2, where ^Sa(a«)/8(2))a<.1,48) ··· 4m)/8<,1)/9<8)... /3(yn). Thusitx (y=x) and likewise any selected term occurs just twice in (ƒ, X«<2)/3(2>X«(3,^(3))«i4) - ··· will occur in (ƒ, ^)® as many times as there are ways of permuting the three superscripts 1, 2, 3 or |3 times. Finally in (ƒ, ) y/ (Mi) This formula holds true if m = n=j, that is, for il* (xyy. VII. Theorem. Every monomial expression (j> which con- sists entirely of symbolical factors of two types, e.g. determinants of type (a/3) and linear factors of the type ax, and which is a de- fined expression in terms of the coefficients and variables of a set of forms fig, ··· is a concomitant of those forms. Con- versely, every concomitant of the set is a linear combination of such monomials. Examples of this theorem are given in (78), (84), (87). In proof of the first part, let =(aPy(wf)* ··· where ƒ = a™; and /3, 7, .·. may or may not be equivalent to a, depending upon whether or not appertains to a single form ƒ or to a set/, g, ···. Transform the form/, that is, the set, by T. The transformed of/ is (76) f = (aA^ + Hence on account of the equations of transformation, But Hence (j>f = («A — «A)p(«a 7„ — ! = (, (92) which proves the invariancy of . Of course if all factors of the second type, a*, are missing in , the latter is an m- variant. To prove the converse of the theorem let <£ be a concomi-THE PROCESSES OF INVARIANT THEORY 63 tant of the set ƒ, g, ··· and let the corresponding invariant relation be written (a0, — ; xv z2) (®i + "2 = C94) where P is a monomial expression consisting of factors of the type aA only and Q a monomial whose factors are of the one type c^. But the inverse of the transformation T (cf. (10)) can be written L (X/x) x2 — & (V)’ where ^1= — x2, f2 = Then (94) becomes 2(- 1 y*PQ?iï7 = (V)*+". (95) We now operate on both sides of (95) by where _ d2 d2 We apply (90) to the left-hand side of the result and (91) to the right-hand side. The left-hand side accordingly be- comes a sum of terms each term of which involves neces- sarily a) + & determinants (a/3), (a£). ln fact, since the result is evidently still of order a> in xv xv there will be in each term precisely 00 determinant factors of type («£) and k of type (a/3). There will be no factors of type ax or |A re- maining on the left since by (91) the right-hand side becomes a constant times <£>, and does not involve X, /x. We now replace, on the left, (a£) by its equivalent ax, Q3£) by /3X, etc. Then (95) gives, after division by the constant on the right, = 2a(a/3)p(ay)g ··· ap/3* ···, (96) where a is a constant; which was to be proved.64 THE THEORY OF INVARIANTS This theorem is sometimes called the fundamental theorem of the symbolical theory since by it any binary invariant problem may be studied under the Aronhold symbolical representation. SECTION 3. REDUCIBILITY. ELEMENTARY COMPLETE IRREDUCIBLE SYSTEMS Illustrations of the fundamental theorem proved at the end of Section 2 will now be given. I. Illustrations. It will be recalled that in (96) each sym- bolical letter occurs to the precise degree equal to the order of the form to which it appertains. Note also that Jc + a>, the index plus the order of the concomitant, used in the proof of the theorem, equals the weight of the concomitant. This equals the number of symbolical determinant factors of the type (a/3) plus the number of linear factors of the type ax in any term of <£. The order co of the concomitant equals the number of symbolical factors of the type ax in any term of . The degree of the concomitant equals the number of distinct symbols a, /S, ··· occurring in its symbolical representation. Let = («£)p(«Y)4(/3y)r ··· ... be any concomitant formula for a set of forms ƒ = a™, g = .... No generality will be lost in the present dis- cussion by assuming to be monomial, since each separate term of a sum of such monomials is a concomitant. In order to write down all monomial concomitants of the set of a given degree i we have only to construct all symbolical products involving precisely i symbols which fulfill the laws p + q + ··· + p = m, p + r + ··· + = (aft)p(ay)q(/3y)TuP/3°'yrx, P + ? + P = 2, p + r + A) = (ab)2(cb)axc2. ~ (ƒ, Q')2 = $(ab)2(bc) [axc2 + 2 cxcyay]y=d x dx (1020 = \[(ab)2(cd)2(bc)azdx + 2(ab)2(ad)(cd)(bc)cxdx]. But by the interchanges a ~ <#, b^c (ab)2(cd)2(bc)axdx = (dc)2(ba)2(cb)axdx = 0. By the interchange c~d the second term in the square bracket equals (ab)2(cd)cxdx[(ad)(bc) 4- (ca)(bd)~\, or, by (100) this equals (ab)z(ed)2cxdx = 0. Hence (ƒ, Q)2 vanishes. We may note here the result of the transvection (A, A)2; B = (A, A)2 = (ab)2(cd)2(ac)(bd).68 THE THEORY OF INVARIANTS III. Concomitants of binary cubic. We give below a table of transvectants for the binary cubic form. It shows which transvectants are reducible in terms of other concomitants. It will be inferred from the table that the complete irredu- cible system for the binary cubic ƒ consists of ƒ, A, Q, R, one invariant and three covariants, and this is the case as will be proved later. Not all of the reductions indicated in this table can be advantageously made by the methods intro- duced up to the present, but many of them can. All four of the irreducible concomitants have previously been derived in this book, in terms of the actual coefficients, but they are given here for convenient reference: ƒ = -f 3 a^c\x^ + 3 a2xxx\ + azz%, A = 2(a0a2 - aj)xj + 2(a0as - axa^)xxx^ + 2(axa3 - a\)x\ (cf. (85)), Q = (aga3 — 3 -f 2 + 3(a0a1a3 — 2 a0a| -4- a\a^)x\x2 - 3<>0a2a3 — 2 a\az + axa$)xxx\ - (a0a§ — 3 axa2az + %a%)x\ (cf. (39)), R = 8(&q#2 ®l) 0*1^8 <*i) ^(^0^3 (cf. (741))· The symbolical forms are all given in the preceding Paragraph. TABLE I First Transv. Second Transv. Third Transv. (ƒ.ƒ)=<> (/,/)2 = a (/,/)3 = o (ƒ. A)= Q (ƒ, A)* = 0 (A, A) = 0 (A, A)* = R (ƒ, <2)=-iA* (ƒ, «)2 = « if, Q)s = R (A, Q)=\Rf (A, Q)2 = 0 (<2, «) = o , {Q, §)2 = iUA («, Q)3 = oTHE PROCESSES OF INVARIANT THEORY 69 SECTION 4. CONCOMITANTS IN TERMS OF THE ROOTS Every binary form ƒ = a™ = i™'= ··· is linearly factorable in some field of rationality. Suppose ƒ = (r§)Xl - r^x^) (rf)xl - rf xz) · · · {r(2m)x1 - r^x^). Then the coefficients of the form are the elementary sym- metric functions of the m groups of variables (homogeneous) **2y)) (/ = 1, 2, m). These functions are given by ... ru)ru+1) ... rr'o'+i> ··. (103,) Let be a concomitant of ƒ, and let the corresponding in- variant relation be <¿>'=0',..., a'my(x'v z'2y = (\/j.)\a0, ···, amy(xv x2y=(\ix)k4>. When the primed coefficients in f are expressed in terms of the roots from (103!) and the unprimed coefficients in in this invariant relation are expressed in terms of the roots from (103), it is evident that cf>f is the same function of the primed r’s that is of the unprimed r’s. This proves the theorem.70 THE THEORY OF INVARIANTS II. Conversion operators. In this Paragraph much advan- tage results in connection with formal manipulations by in- troducing the following notation for the factored form of/: ƒ=4X)42) ··· 4m)* (105) Here a(xj) = a[j)x1 + a(|}o;2 0*= ···? w). The a’s are related to the roots (r^-), of the previous Paragraph by the equations /yO) — /yO) — i* al — '2 ’ a2 — '1 » that is, the roots are (+4y), — a^) 0 = 1? ···, wi). The umbral expressions av a2 are now cogredient to a[J\ (Chap. I, § 2, VII, and Chap. Ill, (76)). Hence, is an invariantive operator by the fundamental postulate. In the same way and [A6c.] = [i>a][A][i>c]··· are invariantive operators. If we recall that the only degree to which any umbral pair av a2 can occur in a symbolical concomitant, cf> = Sn&(a5)(ac) ··., of ƒ is the precise degree it is evident that [Da6c...] operated upon gives a concomitant which is expressed entirely in terms of the roots (4\ — aj^) /· Illustrations follow. Let 2 ^ be the discriminant of the quadratic /=a2 = J2= ..., ^ = (^)2. Then (a<1 fla)* = 2(“a>5Xai); Hence [2)o6]^=-2(«<1)«(2))2. (106)THE PROCESSES OF INVARIANT THEORY 71 This result is therefore some concomitant of ƒ expressed entirely in terms of the roots of ƒ. It will presently appear that it is, except for a numerical factor, the invariant <\> it- self expressed in terms of the roots. Next let be the co variant Q of the cubic ƒ = ···· Then Q = (aby(ae)bxcl l[i>J(Q = (aa)j)(a(2)5)(m(3))Jx44.(a(i)5)(«(8)J)(^(2))MB + (a(8)i)(«(2)i)(^(1))^ l[Dab] Q = (aa)«i2))(«(2)«a))(^(8))48)^ H-(a(1)a(8))(a(2)«(1))(tf«(8>)42)4-+ (a(1)«(2))(a(2)«(8))()(i«(2))43)c| + (ic(1)a(3))(ot(3)a(1))(i?a(2))42)i?| -h(aa)a(30(«(3)«(2\)(^(2041)4 + («(3)«(10(^(2)ot(3))(^(1))42)^ -h(a(8)«(2))(a(2)a(l))(éîaa))aW)tf2+(a(3)a(2))(a(2)a(3))(ia(l))aa)i?|i [i>a6c] Ç = - 252(«d)«(2))2(aa)cc(3))43)2i,(2)i (107) wherein the summation covers the permutations of the superscripts. This is accordingly a covariant of the cubic expressed in terms of the roots. Now it appears from (104) that each coefficient of ƒ = a™ = ··· is of degree m in the as of the roots (a^\ — ai?))· Hence any concomitant of degree i will be of degree im in these roots. Conversely, any invariant or covariant which is of degree im in the root letters a will, when expressed in terms of the coefficients of the form, be of degree i in these coefficients. This is a property which invariants enjoy in common with all symmetric functions. Thus [Da6]<£ above is an invariant of the quadratic of degree 2 and hence it must be the discriminant itself, since the latter is the only invariant of ƒ of that degree (cf. § 3). Likewise it appears from Table I that Q is the only co variant of the cubic of degree-order (3, 3), and since by the present rule \_Date]Q is of degree-order (3, 3), (107) is, aside from a numerical multiplier, the expression for Q itself in terms of the roots.72 THE THEORY OF INVARIANTS It will be observed generally that [Dab...~\ preserves not only the degree-order (i, g>) of <£, but also the weight since w — \ (im 4- ©). If then in any case happens to be the only concomitant of/of that given degree-order (i, g>), the expression is precisely the concomitant (j> expressed in terms of the roots. This rule enables us to derive easily by the method above the expressions for the irreducible system of the cubic ƒ in terms of the roots. These are ƒ = 4i)a(2)a(3) ; a%. A = 2(a(1)a'2))2<43)2; (ab)2axbx. Q = ; (aby(ac)bxcl R = (a(1)a(2))2(a(2)a(3) )2(a(3)a(1))2; (ab')\cd)\ac)(bd'). Consider now the explicit form of Q: Q = (a(1)a(2))2(a(1)a(3))af)2a^2) + (a(2W3))2(a(2)a(1))41)243> + (a(3)a(1))2(a(3)a(2))42)M1) + (a(3)a(2))2(a(3)«(1))41)242) + (a(2)a(1))2(a(2)a(3))43)241) + (a(1)43))2(a(1V2)}42)243)· It is to be noted that this is symmetric in the two groups of letters (aj·^, a^)· Also each root (value of ƒ) occurs in the same number of factors as any other root in a term of Q. Thus in the first term the superscript (1) occurs in three factors. So also does (2). III. Principal theorem. We now proceed to prove the principal theorem of this subject (Cayley). Definition. In Chapter I, Section 1, II, the length of the segment joining C(xv #2), and y2) ; real points, was shown to be qj) _ Wyv) where X is the multiplier appertaining to the points of reference P, (J, and fi is the length of the segment PQ. If the ratios xxixv y^y^ are not real, this formula will not represent a real segment CD, But in any case if (r^, r^), (4*°, are any two roots of a binary form ƒ = a™, real orTHE PROCESSES OF INVARIANT THEORY 73 imaginary, we define the difference of these two roots to be the number |>ov*n = \fj.(rU)rlk)) . J (Xrjy) + r^jy)(\r{f + r(2k)) We note for immediate use that the expression n)(Xff > + rf) ... (Xrf > + rj^) is symmetric in the roots. That is, it is a symmetric func- tion of the two groups of variables (Vjy\ (ƒ = 1, ···, m). In fact it is the result of substituting (1, —X) for (xv #2) in ƒ = (- iy(r^xYr^x) ... r), and equals n(r) = (a0 — maiX+ ··· 4-( — l)maTOXm). Obviously the reference points P, Q can be selected * so that (1, — X) is not a root, i. e. so that II (r) 0. Theorem. Let f be any binary form, then any function of the two types of differences [>], [r(%] = \/jL(rij)x)/(Xr^ + r(2jy)(Xx1 + x2), which is homogenous in both types of differences and symmetric in the roots (r^\ r^'}) (j = 1, ···, m) will, when expressed in terms of xv x2 and the coefficients of f and made integral by multiplying by a power of II (V) times a power of (\xx -i-#2), be a concomitant if and only if every one of the products of differences of which it consists involves all roots (r[J), r£;)) ([values of f) in equal numbers of its factors. Moreover all concomitants of f are functions cf> defined in this way. If only the one type of difference [r(^} r(A)] occurs in (f), it is an invari- ant, if only the type [r0):r], it is an identical covariant,— a power of f itself and if both types occur, is a covariant. [Cf. theorem in Chap. Ill, § 2, VII.] * If the transformation T is looked upon as a change of reference points, the multiplier X undergoes a homographic transformation under T.74 THE THEORY OF INVARIANTS In proof of this let the explicit form of the function de- scribed in the theorem be == ^ [V(1) y(2) Ja*|V(l) J Pk ··· k where al + ^1 + = a2 + @2 + *·· = ***’ ft ··· =/32+i72 + ··’ = **·’ and <£ is symmetric in the roots. We are to prove that is invariantive when and only when each superscript occurs in the same number of factors as every other superscript in a term of . We note first that if this property holds and we express the differences in in explicit form as defined above, the terms of 2 will, without further algebraical manip- ulation, have a common denominator, and this will be of the form n(r)u(Xa;1+ %)». Hence H(r)“(Xa-1 + x2Y4> is a sum of monomials each one of which is a product of determinants of the two types (rU) (rulx). But owing to the cogrediency of the roots and variables these determinants are separately invariant under T7, hence Il(r')u(\xl + x^)v is a concomitant. Next assume that in $ it is not true that each superscript occurs the same number of times in a term as every other superscript. Then although when the above explicit formulas for differences are introduced (\xx+x^) occurs to the same power v in every de- nominator in 2, this is not true of a factor of the type (Xr^-f-r^). Hence the terms of 2 must be reduced to a common denominator. Let this common denominator be H(r')u(\x1 +x%)v. Then n(r)M(Xa;1 4- #2)v is of the form <£1=SlI(My> + r2i>),,j,k(ril)r<2))a*(ril)r(8))ft *’* x (ra) #)p*(r(%)°·* where not all of the positive integers ujk are zero.THE PROCESSES OF INVARIANT THEORY 75 Now^ is invariantive under T. Hence it must be unaltered under the special case of T\ x1 = — xf2^x2 = xfr From this rti) = _ rKj)^ rU) = rKj)t Hence a-sn (Ar^} — r^))w^(/,(1V(2))ayi:(raV(3))^ ··· (ra)x)pk ♦ ··, k j and this is obviously not identical with on account of the presence of the factor n. Hence x is not a concomitant. All parts of the theorem have now been proved or are self- evident except that all concomitants of a form are expres- sible in the manner stated in the theorem. To prove this, note that any concomitant of/, being rational in the coeffi- cients of ƒ, is symmetric in the roots. To prove that need involve the roots in the form of differences only, before it is made integral by multiplication by Yi(r^u(\xl -f- #2)v, it is only necessary to observe that it must remain unaltered when ƒ is transformed by the following transformation of determinant unity : — Xl ~f" CX2, X2 — X'^’t and functions of determinants (r(i)r(W), (rU)x) are the only symmetric functions which have this property. As an illustration of the theorem consider concomitants of the quadratic ƒ =(rn)^)(r(2)^). These are of the form = ^ [r(1)ri2)]aA[ra)^]pyfc[r(2)^])x) (r{2)x). Here and That is, Hence = [r(i>r(2)]«* [r<2>r(8)]0*[r<8)r]7* k ak + Ik = ak + fik = ftk + 7 Jfc· ak — fik — Ik ~ ^ a. n(V)4a =(?J(r(3)r(2))2(r(2)r(3))2(r(3)ra))2Ja= QRa Thus the discriminant B and its powers are the only invariants. IV. Hermite’s reciprocity theorem. If a form f = a™ = bmx = · · · of order m has a concomitant of degree n and order o>, then a form g = anx = ♦·· of order n has a concomitant of degree m and order a. To prove this theorem let the concomitant of ƒ be I^^k(ah')p(ac)q ··· arxb*x ··· (r + s+ ··· = to), where the summation extends over all terms of I and k is numerical. In this the number of distinct symbols a, J, ··· is n. This expression I if not symmetric in the n letters a, 5, c?, ··· can be changed into an equivalent expression in theTHE PEOCESSES OF INVAEIANT THEOEY 77 sense that it represents the same concomitant as I, and which is symmetric. To do this, take a term of J, as k(ab)p(ac)q ··· arxhsx ···, and in it permute the equivalent symbols a, J, ··· in all |n possible ways, add the \n resulting monomial expressions and divide the sum by \n. Do this for all terms of I and add the results for all terms. This latter sum is an expression J equivalent to I and symmetric in the n symbols. Moreover each symbol occurs to the same degree in every term of J as does every other symbol, and this degree is precisely m. Now let g = aPa<*> and in a perfectly arbitrary manner make the following re- placements in J: id ^ ^ 1 £ 9 I \ \a(1), ««>, «(»), ..., «<»>ƒ • The result is an expression in the roots (<4i}, — a^) of <7, ... a£)rag)kI((io? ·*·)· Any covariant of ƒ equated to zero gives rise to a “ derived ” point range connected in a definite manner with the range ƒ = 0, and this connecting relation is projective. The identical evanescence of any covariant implies projec- tive relations between the points of the original range ƒ = 0 such that the derived point range obtained by equating the covariant to zero is absolutely indeterminate. The like remarks apply to covariants or invariants of two or more forms, and the point systems represented thereby. I. Involution. If ƒ — (ao’ av *‘*$*^1’ 9==(J)o‘ K -1*11 are two binary forms of the same order, then f + kg = (a0 + £60, ax + kbv ^2)w, where k is a variable parameter, denotes a system of qualities which are said to form, with ƒ and g, an involution. TheTHE PROCESSES OF INVARIANT THEORY 79 single infinity of point ranges given by k, taken with the ranges ƒ = 0, g = 0 are said to form an involution of point ranges. In Chapter I, Section 1, V, we proved that a point pair ((V), (v)) can be found harmonically related to any two given point pairs ((jt?), (r)), ((/), («))· If the latter two pairs are given by the respective quadratic forms f, g, the pair ((u), (v)) is furnished by the Jacobian O of/, g. If the éliminant of three quadratics ƒ, g, h vanishes identically, then there exists a linear relation ƒ “h kg + lh = 0, and the pair A = 0 belongs to the involution defined by the two given pairs. Theorem. There are, m general, 2(m — 1) quantics be- longing to the involution f -f kg which contain a squared linear factor, anc? ¿Ae set comprising all double roots of these quantics is the set of roots of the Jacobian of f and g. In proof of this, we have shown in Chapter I that the dis- criminant of a form of order m is of degree 2(m —1). Hence the discriminant of ƒ+ kg is a polynomial in k of order 2(m — 1). Equated to zero it determines 2(m — 1) values of k for which ƒ + kg has a double root. We have thus proved that an involution of point ranges contains 2(m — 1) ranges each of which has a double point. We can now show that the 2(m— 1) roots of the Jacobian of ƒ and g are the double points of the involution. For if aqw2 — x^ux is a double factor of f+ kg, it is a simple factor of the two forms ÈÎ-jl.Tc^L ÈL + k*2- dxj dXi dxf dXz and hence is a simple factor of the k éliminant of these forms, which is the Jacobian of f g. By this, for instance, the points of the common harmonic pair of two quadratics80 THE THEORY OF INVARIANTS are the double points of the involution defined by those quadratics. The square of each linear factor of C belongs to the involution ƒ -f- kg. In case the Jacobian vanishes identically the range of double points of the involution becomes indeterminate. This is to be expected since ƒ is then a multiple of g and the two fundamental ranges ƒ = 0, g = 0 coincide. II. Projective properties represented by vanishing covari- ants. The most elementary irreducible covariants of a single binary form ƒ = (a0, av ··· x^)m are the Hessian JY, and the third-degree covariant Î7, viz. We now give a geometrical interpretation of each of these. Theorem. A necessary and sufficient condition in order that the binary form f may be the mth power of a linear form is that its Hessian H should vanish identically. If we construct the Hessian determinant of (r2xx — rxx^)m, it is found to vanish. Conversely, assume that AT= 0. Since H is the Jacobian of the two first partial derivatives the equation H= 0 implies a linear relation dxx dx, 2 dxx 1 dx2 Also by Euler’s theorem and df , df . x'£l+x‘T^=wf' ox1 dx2 Expansion of the éliminant of these three equations gives df-m d(.KjX% + K^Xy) THE PROCESSES OF INVARIANT THEORY 81 and by integration ƒ=(* 1*1 + and this proves the theorem. Theorem. A necessary and sufficient condition in order that a binary quartic form ƒ = a0#f 4- ··· should be the product of two squared linear factors is that its sextic covariant T should vanish identically. To give a proof of this we need a result which can be most easily proved by the methods of the next chapter (cf. Appendix (29)) e.g. if i and J are the respective invariants of ƒ, i = 2 (a0a4 — 4 axaz + 3 a£), then «0 «1 a2 «1 «2 az «2 a3 (5T, Î7)6 = ^j(is — 6 J2). We also observe that the discriminant of ƒ is 6 J"2). Now write al as the square of a linear form, and ƒ=«??! = <4 =&l = ···· Then iT=(a|i2, a|)2 = i[(««)2?l + (?a)2«! + 4 (aa)(0a)a,i Ja2 = H30)2?*2 + 3( minus terms which contain the factor (aby+l(xy). We now use (111) as a recursion formula, taking k = m, m — 1, ···. This gives (ab)mbny-m (aby-'aybj)^-”1 — Cnm hn>\m K^x) uxJyii-mi __(nm hn\m-1 — ^#9 ux)yU-m+l 1 n—m+2 (G&yyjwM ll2> We now proceed to prove by induction that Caby+1a^ib^-l‘-1b”-”‘ = «0( a?, £)*«*_, + b%)'‘yilk_.i(xy)+ + «#(«” blŸy^j_l(xyy+ - (US) + «*_*-!«, Ky^_m(xy)^-\ where the «’s are constants. The first steps of the induction are given by (112). Assuming (113) we prove that the rela- tion is true when k is replaced by k — 1. By Taylor’s theorem £*-i+ {*"*+ - +f+l = th_lQ-l)»-1+iA_2(£-l)»-2+ - +#1(£-l) + i0· Hence (ahJ>y - <&£) = th-iiabyixyY + th^2(ab')h-\xy')h-\bx+ — + th_i(^ab')h~i+\xyy-i+1a^-ibi-1+ ... +t0(ab)(xy)ahy-lbhx-1. (114)REDUCTION 85 Hence (111) may be written («“ b”Yyn_k m—k h h=l i=1 in which the coefficients Ahi are numerical. But the terms Thi=(aby^+k+la^-k-h+i-lbf-k-h+i-lb^-m(m—k'k h > 1, i < K) for all values of A, i are already known by (112), (113) as linear combinations of polars of transvectants; the type of expression whose proof we seek. Hence since (115) is linear in the Tu its terms can immediately be arranged in a form which is precisely (113) with k replaced by k — 1. This proves the statement. We now substitute from (113) in (109) for all values of k. The result can obviously be arranged in the form axhy = ffoOC, Kyyn + 0i«, biyyn-i(xy} + ... (116) + C](ax\ ^yyn-K^y+ ··· + Om(a™, h^n_m(xyy. It remains to determine the coefficients Cy. By (912) of Chapter III we have, after operating upon both sides of (116) by il] and then placing y = æ, I m\n \}\m + n—) + 1 ----=4=----; Qabya^-jbn~j = Cj , : m—j | n—j \m 4- n — 2j + 1 (abya™-ibn-i Solving for (7?·, placing the result in (116) (j = 0, 1, and writing the result as a summation, » (“Y = . iN ¡3, (m + n-j + V\ (xyÿ(a™, bnxyyn-i. , m), (117) This is Gordan’s series. To put this result in a more useful, and at the same time86 THE THEORY OF INVARIANTS a more general form let us multiply (117) by (aby and change m, n into m — r, n — r respectively. Thus (abya%-rb;-r (m7rTlr) -2bS"-- (118) ) f d \k ( d \k If we operate upon this equation by \ z — j , \y —J , we ob- tain the respective formulas (abyarr^~r~k f m — r\f n — r — Jc J A J -S r< (m + n — 2r — /+ v v *’ 3 r (us) -j-r+k (120) V 3 (abya™~r-kakybny-r lm — r — k\in — r\ 3 A 3 + — j 0 It is now desirable to recall the standard method of transvec- tion ; replace yx by c2, y2 by — ex in (119) and multiply by ep-n+r+^ with the result = X /—-—^ -4-—4r OyV(a?, W* y /m-f n-2r-j+ly K x xjyn~< (aby(hc)n-r-ka™- rbkc%~n+r+k im — r\fn —r — Jc = ^ 1)/A._^_ A_________l 2r—y+1^ («, c%y-^-k. (121) Likewise from (120) (aby(bcy-r(ac')ka%-r-ke%-n+r-k fm— r—k\in~r\ = ? (-1)4—J a' ■ ¿(Cag.K)i+r,c*y-i-r+k. ^m + n — 2r —y+1^ (122)REDUCTION 87 The left-hand member of equation (121) is unaltered in value except for the factor (—l)n_* by the replacements a~c, m~p, r~n — r - k; and likewise (122) is unaltered except for the factor (— 1)M+* by the replacements a~c, m~p, r~n — r. The right-hand members are however altered in form by these changes. If the changes are made in (121) and if we write f=b%,g = a™,h = eg, ax = 0, a2 = n — r — k, a3 = r, we obtain m — ot-y — a3\/ X J^T ¿0 ¿7IT C(/» 9Tt+i, hy^-i ^ fm +n ■—Za3—j+ Y\ -(-u'S/, ' ( where we have (p - «1 - «2^ i ln+p-2ai-j +1\^ ((ƒ, hy-'+>\ (i23) «2 4- «3 > «3 + aj >m, «! + «2 (124i) together with ax = 0. If the corresponding changes, given above, are made in (122) and if we write cq = k, o^ = n — r, cq = r, we obtain the equation (123) again, precisely. Also relations (124j) repro- duce, but there is the additional restriction cq + az = n. Thus (123) holds true in two categories of cases, viz. (1) a1==0 with (124^, and (2) a2 + a3 = n with (124j). We write series (123) under the abbreviation ƒ 9 h n m P «1 «2 «î (Î) «! = 0, (ii) «j + a2 = »· «2-t-«3>M, «3+ <*!>»*, a1 + a2>^, It is of very great value as an instrument in performing reductions. We proceed to illustrate this fact by proving certain transvectants to be reducible.88 THE THEORY OF INVARIANTS Consider (A, Q) of Table I. (A, . But (A, A) = 0, (A, ƒ)2= 0, (A, A)2 = E. Hence (A, where f = (ƒ, ƒ )4. Hence when wi > 4 this transvectant is always reducible.REDUCTION 89 II. The quartic. By means of Gordan’s series all of the reductions indicated in Table I and the corresponding ones for the analogous table for the quartic, Table II below, can be very readily made. Many reductions for forms of higher order and indeed for a general order can likewise be made (cf. (124)). It has been shown by Stroh* that certain classes of transvectants cannot be reduced by this series but the simplest members of such a class occur for forms of higher order than the fourth. An example where the series will fail, due to Stroh, is in connection with the decimic/= aJP. The transvectant is not reducible by the series in its original form although it is a reducible covariant. A series discovered by Stroh will, theoretically, make all reductions, but it is rather difficult to aPPty’ and moreover we shall presently develop powerful methods of reduction which largely obviate the necessity of its use. Stroh’s series is derived by operations upon the identity (ah)cx + (bc)ax + (ca)bx = 0. TABLE II r = 1 r = 2 r = 3 r = 4 (ƒ,ƒ)- 0 H 0 i (ƒ, ny T i if 0 J (ƒ, T)' À«fa-6fP») 0 i(Jf-iH) 0 (S, HY 0 i(2 Jf-iH) 0 i*2 (H, TY UJP-ifH) 0 0 (T, Ty 0 — 7W2f2+V iH2-l2JfH) 0 0 We infer from Table II that the complete irreducible sys- tem of the quartic consists of ƒ, H, T, i, J. * Stroh ; Mathematische Annalen, vol. 31.90 THE THEORY OF INVARIANTS This will be proved later in this chapter. Some of this set have already been derived in terms of the actual coefficients (cf. (TOj)). They are given below. These are readily derived by non-symbolical transvection (Chap. Ill) or by the method of expanding their symbolical expressions and then expressing the symbols in terms of the actual coeffi- cients (Chap. Ill, § 2). ƒ = aQx\ + 4 axx\x2 + 6 a2x \x\ + 4 a3xxa% + JY= 2[(a0a2 - a\)x{ -f- 2(a0az - axa^)x\x2 4* (#o#4 4* 2 3 -f-2 (æ4æ4 4” (^2^4 ^3)^2] ’ T= (alaz — 3 a0axa2 4- 2 a^)x\ + (a§a4 4- 2 — 9 a0a\ 4- 6 a\a^)x\x2 4- 5(a0a1a4 — 3 a0a2a3 4- 2 4- 10(afa4 — aa<2))^a)2aw)2 7= 2(a(1)a(2))2(a(1)a(3))a^2)a^.3)2a^4)3, (126) i=2(a-*, X j=o ((#92 THE THEORY OF INVARIANTS Substitution of the values of the transvectants (.ff,/)r, (AT, II)4 gives Off Ty = h(-zJH+iy). The series for (7, 7)2 = ((ƒ, H), 7)2 is f/ H 7) 4 4 6 , .0 2 1, or ((/, #), Tf+((/, id* ?)=((/, 7>2, ¿o+KCO 7)3, -ff)°. But ((ƒ, 5)2, T) = (i if, 7) = l i(f T) = YV OY2 - 6 i52). Hence, making use of the third line in Table II, (I7, T)* = - tVOY2 + 6 i52- 12 JHf ), which we wished to prove. The reader will find it profit- able to perform all of the reductions indicated in Table II by these methods, beginning with the simple cases and pro- ceeding to the more complicated. SECTION 2. THEOREMS ON TRANSVECTANTS We shall now prove a series of very far-reaching theorems on transvectants. I. Theorem. Every monomial expression, , in Aronhold symbolical letters of the type peculiar to the invariant theory, i.e. involving the two types of factors (ab), ax; = n(abytacy — afJbyfx ·.·, is a term of a determinate transvectant. In proof let us select some definite symbolical letter as a and in all determinant factors of which involve a set a1= — y2, a2 — yv Then <£ may be separated into three factors, i.e. ' = PQa%,REDUCTION 93 where Q is an aggregate of factors of the one type ¿y, Q = bsycl ···, and P is a symbolical expression of the same general type as the original but involving one less sym- bolical letter, P = (bc)u(bd}v ··· b%crx ···. Now does not involve a. It is, moreover, a term of some polar whose index r is equal to the order of Q in y. To obtain the form whose rth polar contains the term PQ it is only necessary to let y = x in PQ since the latter will then go back into the original polarized form (Chap. Ill, § 1, I). Hence is a term of the result of polarizing (PQ}y=x r times, changing y into a and multiplying this result by ag. Hence by the standard method of transvec- tion is a term of the transvectant ((P£)y=x, a^y (r+p=m). (127) For illustration consider cf) = (ab)%ac)(bc)azbxe%. Placing a ~ y in (aò)2(a' = — · «X· Placing y inf we obtain Thus is a term of A = (-(be)b*c%, aff. In fact the complete transvectant A is + A = - fa(be)(ca)*aj>% - ^(bc)(caf(ba)axblcx - i^{bc)ica)(bayaxbxcl - ^(be)(ba)zax4 and $ is its third term. Definition. The mechanical rule by which one obtains the transvectant from the product ajij, consist- ing of folding one letter from each symbolical form af, b™94 THE THEORY OF INVARIANTS into a determinant (&5) and diminishing exponents by unity, is called convolution. Thus one may obtain (aby^ac^af^cl from by convolution. II· Theorem. (1) The difference between any two terms of a transvectant is equal to a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the forms in the original transvectant by convolution. (2) The difference between the whole transvectant and one of its terms is equal to a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the original forms by convolution (Gordan). In proof of this theorem we consider the process of con- structing the formula for the general rth transvectant in Chapter III, Section 5. In particular we examine the structure of a transvectant-like formula (89). Two terms of this or of any transvectant are said to be adjacent when they differ only in the arrangement of the letters in a pair of symbolical factors. An examination of a formula such as (89) shows that two terms can be adjacent in any one of three ways, viz.: (1) P(a<«^)(a<«/8(*>) and (2) ptaWfi^aW and P(^h)^)a(j\ (3) P(«(0^)/8i« and P(a^/3^)^(J\ where P involves symbols from both forms ƒ, g as a rule, and both types of symbolical factors. The differences between the adjacent terms are in these cases respectively (1) P(a (2) P(a^a^)/3^\ (3) POS^/S^X0· These follow directly from the reduction identities, i>c. from formulas (99), (100).REDUCTION 95 Now, taking/, where A and B involve only factors of the first type (78). Then formula (90) holds true ; ( ƒ, gY = n r (a«)/8)(a«))8(2)) ... «¿1)42) ··· 4r)£m2) - ££■> and the difference between any two adjacent terms of (ƒ, gy is a term in which at least one factor of type (<*£) is re- placed by one of type'(«a') or of type (/3/3'). There then remain in the term only r — 1 factors of type (a/3). Hence this difference is a term of a transvectant of lower index of forms obtained from the original forms/, g by convolution. For illustration, two adjacent terms of ((a6)2a|8|, c%)2 are (ai)2(ac)26J<;2, (aby(ac)(bc)axbx4. The difference between these terms, viz. (ab)\ac)bxc% is a term of ((abfaj)^ be any monomial expression in the symbols of a single form ƒ, and let some symbol a occur in precisely r determinant factors. Then equals a linear combination of transvectants of index < r of a™ and forms obtained from (P#)y=sa. (cf. (127)) by convolution. For illustration 4>=(«&)20)2«f*i= ((aiyalll - ((abfaxbz, 4) +*((«*/> 4)°· It may also be noted that (PQ)y=x and all forms obtained from it by convolution are of degree one less than the degree of in the coefficients of ƒ. Hence by reasoning induc- tively from the degrees 1, 2 to the degree i we have the result : Theorem. Every concomitant of degree i of a form f is given by transvectants of the type (^i-H ƒ )( where the forms are all concomitants of f of degree i — 1. (See Chap. Ill, § 2, VII.) SECTION 3. REDUCTION OF TRANSVECTANT SYSTEMS We proceed to apply some of these theorems. I. Reducible transvectants (0*_i,/y. The theorem given in the last paragraph of Section 2 will now be amplified by another proof. Suppose that the complete set of irreducible concomitants of degrees < i of a single form is known. Let these be f 72’ ***’ 7*i98 THE THEORY OF INVARIANTS and let it be required to find all irreducible concomitants of degree i. The only concomitant of degree unity is f=a%. All of degree 2 are given by (//)r=(«Or«rrJr% where, of course, r is even. A covariant of degree i is an aggregate of symbolical products each containing i symbols. Let Ct be one of these products, and a one of the symbols. Then by Section 2 0* is a term of a transvectant (C'i-i, O', where O^i is a symbolical monomial containing i—T sym- bols, i.e. of degree i — 1. Hence by Theorem II of Section 2, Ci=(cufy+2(t-i =/a7i1- 7?· Hence (7,· is a sum of transvectants of the type and since any covariant of/, of degree i is a linear combina- tion of terms of the type of (7* all concomitants of degree i are expressible in terms of transvectants of the type (^ = po-, where p, i-»fy=<&*?, <%y- Then in the ordinary way by the standard method of trans- yection we have the following: = Kp(*,fy+ ···. (131) Hence if p2 now represents ( ƒ)> so that (^ ƒ y = PP2 + ƒ y* C ƒ < ƒ)· (132) Evidently p, p2 are both covariants of degree < i and hence are reducible in terms of ƒ, yv ···, yk. Now we have the right to assume that we are constructing the irreducible con- comitants of degree % by proceeding from transvectants of a stated index to those of the next higher index, i.e. we assume these transvectants to be ordered according to in- creasing indices. This being true, all of the transvectants (<^._i, ƒ y at the stage of the investigation indicated by (132) will be known in terms of ƒ, yv ···, yk or known to be irreducible, those that are so, since ƒ *_!, /y for which no factor of order ^ j occurs in Thus for instance if m = 4, (/2,/)? is re- ducible for all values of j since ƒ2 contains the factor ƒ of order 4 and j cannot exceed 4. We note that if a form y is an invariant it may be omitted when we form <£>t·^, for if it is present (^f_w/y will be re- ducible by (80).100 THE THEORY OF INVARIANTS II. Fundamental systems of cubic and quartic. Let w = 3 (cf. Table I). Then ƒ = a3 is the only concomitant of degree 1. There is one of degree 2, the Hessian (ƒ, ƒ )2 = A. Now all forms 2 of (02’/ V are included in <¿2 and either a = 2, ¡3 = 0, or a = 0, /3 = 1. But ( ƒ 2, ƒ y is re- ducible for all values of j since ƒ2 contains the factor ƒ of order 3 and j > 3. Hence the only transvectants which could give irreducible concomitants of degree 3 are (A ,/y 0 = 1,2). But (A,/)2 = 0 (cf. Table I). In fact the series [ƒ ƒ ƒ} 3 3 3 1 2 1 gives *((ƒ. ƒ)»,ƒ)» = - ((ƒ, ƒ )2, ƒ )2 = - (A,/)2 = 0. Hence there is one irreducible covariant of degree 3, e.g. (A,/) = - Q. Proceeding to the degree 4, there are three possibilities for 03 in (<£3, fy. These are 03 =/3, /A, Q. Since /> 3 (/3, ƒ y, (/A, ƒ y 0 = 1, 2, 3) are all reducible by Section 3,1. Of (#, ƒ y 0 = 1, 2, 3), (<>, ƒ )2 = 0, as has been proved before (cf. (102)), and (#,/) = ^A2 by the Gordan series (cf. Table I) If A ƒ] 3 2 3. Oil· Hence (#, ƒ )3 = — B is the only irreducible case. Next the degree 5 must be treated. We may have 04=/*,/2A,/ 3 in (04, ƒ y the only possibility for anREDUCTION 101 irreducible form is (A2, ƒ /, and this is reducible by the prin- ciple of I if j < 3. But (A*/)* = (S§a* af)3=(Sa)2(S'«)S; = (S£ (&»)*«,)-0. For (8a)2ax = (A,/)2 = 0, as shown above. Hence there are no irreducible concomitants of degree 5. It immediately follows that there are none of degree > 5, either, since 05 in (<£>6,/y is a more complicated monomial than <£4 in the same forms/, A, Q and all the resulting concomitants have been proved reducible. Consequently the complete irreducible system of concom- itants of ƒ, which may be called the fundamental system (Salmon) of ƒ is ƒ, A, Ç, R. Next let us derive the system for the quartic ƒ; m = 4. The concomitants of degree 2 are (ƒ, ƒ)2 = H, ( ƒ, ƒ)4 = i· Those of degree 3 are to be found from («ƒ)' 0'=1,2,3,4). Of these (ƒ, üT) = ^ and is irreducible ; (ƒ, By = Jis irre- ducible, and, as has been proved, (iT, /)2 = ^i/ (cf. (124)). Also from the series ƒ / ƒ 4 4 4, 1 3 i (i?, /)3 = 0. For the degree four we have in (03, ƒ)* 03 =/3,/^ T, all of which contain factors of order ^ƒ >1 except Î7. From Table II all of the transvectants (7,/)? O = 1» 2, 3, 4) are reducible or vanish, as has been, or may be proved by Gordan’s series. Consider one case ; (î*,/)4· Applying the series ƒ B ƒ 4 4 4, 1 3 1102 THE THEORY OF INVARIANTS we obtain ((ƒ, H). ƒ)* = -((ƒ, H)\ff - &((ƒ, But ((ƒ, H)\ ff= \ *(ƒ, ff = 0 ; and (ƒ, #ƒ = 0 from the proof above. Hence ((f,H),fy=(T,fy = o. There are no other irreducible forms since <£4 in (<£4, ƒ/ will be a monomial in/, 5", I7more complicated than 3. Hence the fundamental system of/consists of ƒ, H, % i, J. It is worthy of note that this has been completely derived by the principles of this section together with Gordan’s series. III. Reducible transvectants in general. In the trans- vectants studied in (I) of this section, e.g. ƒ/, the second form is simple,/ = a™, of the first degree. It is pos- sible and now desirable to extend those methods of proving certain transvectants to be reducible to the more general case where both forms in the transvectants are monomials in other concomitants of lesser degree. Consider two systems of binary forms, an (A) system and a (i?) system. Let the forms of these systems be (A) : Av A2, ···, Ak, of orders av a2, ···, ak respectively; and (1?) : Bv B2, ···, Bb of orders bv J2, ···, bt respectively. Suppose these forms expressed in the Aronhold symbolism and let = A*'A** ··· Ayk, = B&BP* ··· 2?f*. Then a system ( 0) is said to be the system derived by trans- vection from (A) and (i?) when it includes all terms in all transvectants of the type (<*>, (133)REDUCTION 103 Evidently the problem of reducibility presents itself for analysis immediately. For let (f> = pcr, yjr = fiv, and suppose that j can be separated into two integers, j=j\+Jr such that the transvectants (p, t*y\ ( (ƒ> (f> 02> (ƒ. *2)3» CA *3)6· The reader will find it very instructive to find for other and more complicated (J.) and (i?) systems the transvectants of (<7) which do not contain reducible terms. It will be found that the irreducible transvectants are in all cases finite in number. This will be proved as a theorem in the next chapter. SECTION 4. SYZYGIES We can prove that m is a superior limit to the number of functionally independent invariants and covariants of a single binary form ƒ = a™ of order m. The totality of in- dependent relations which can and do subsist among the quantities Xp X2'> XP *^2’ ^t 0 = ***» ^0» ^2’ f^P P2’ O'/*) are m + 4 in number. These are di = 0* = 0, ···, m); aq = x2 = X2xi + ^2 5 M=\1fx2 — \fAv When one eliminates from these relations the four variables Xr Xg, fiv /i2 there result at most m relations. This is the maximum number of equations which can exist between a{, (i = 0, ···, ra), aq, xv xfv x2, and M. That is, if a greater number of relations between the latter quantities are as- sumed, extraneous conditions, not implied in the invariant problem, are imposed upon the coefficients and variables. But a concomitant relation ***’ dm£)»ƒ)---o (*1 9?f + —r 2 W+jP — Z n+P’—A ((^ ;0> ƒ) -((y,/),»+„*·; ¿)2/· m + — 2 We add these equations and divide through by 2, noting that (ƒ, (/)= ~(g,t"), and obtain ((¿¿0. A)» n — m 2(wi + w — 2) + h'fg—\ (g, hff. (134) This formula constitutes the proof of the theorem. It may also be proved readily by transvection and the use of reduction identity (101). II. Theorem. If e = al\ ƒ =6* g = <%, h = d% are four Unary forms of orders greater than unity, then (e,f )(g, h) = -%(e,gffh + \(e, h)2fg+ l(ƒ, g)2eh-£(ƒ, hfeg. (135) We first prove two new symbolical identities. By an elementary rule for expanding determinants Hence = 2 (ab)(be) (ca) (de) (ef) (fd) (ad)2 (ae)2 (of)2 = (bd)2 (be) 2 (bf) 2. (136) («O2 (ee)2 (C/)2 In this identity set et = — xv c2=xv f\= —xv /2 = xv a\ axa2 a\ b\ bf2 b2 u2 = — (ab)(bc)(ca) c\ C1C2 c2 C2 «1 a^a2 «1 d\ —2d2dx <*? b\ bf>2 ft2 e\ 2 e2ej ef c\ ctc2 s& ('2 fl -2 /2/x /lREDUCTION 107 Then (136) gives the identity. (137) 0,/)(4S A) = (ai)(cd)a“-1i5-1c?-1dr1 (ae)2 (acT)2 a2 = i<-2&r2 ƒ = & 0 = ƒ h = g. 8=2Q* + AS + Rf*=0. (139)108 THE THEORY OF INVARIANTS Hence we obtain S=2 T* + + $Jf*=0. (i40) This is the syzygy connecting the members of the funda- mental system of the quartic. Of the twenty-three members of a system of the quintic nine are expressible as Jacobians (cf. Table IV, Chap. VI). If these are combined in pairs and substituted in (135), and substituted singly in (138), there result 45 syzygies of the type just derived. For references on this subject the reader may consult Meyer’s “ Bericht ueber den gegenwartigen Stand der Invariantentheorie ” in the Jahresbericht der Deutschen Mathematiker-Vereinigung for 1890-91. V. Syzygies derived from canonical forms. We shall prove that the binary cubic form, ƒ = a^x\ + 3 axx\x2 + 3 a^xxx\ + azx^ may be reduced to the form, f=X*+ F3, by a linear transformation with non-vanishing modulus. In general a binary quantic ƒ of order m has m + 1 coefficients. If it is transformed by T l Xj -f- *^2 = ^2*^1 4” /^2*^2’ four new quantities \v /jlv \2, are involved in the coeffi- cients of ƒ'. Hence no binary form of order m with less than m — 3 arbitrary coefficients can be the transformed of a general quantic of order m by a linear transformation. Any quantic of order m having just m— 3 arbitrary quanti- ties involved in its coefficients and which can be proved to be the transformed of the general form ƒ by a linear trans- formation of non-vanishing modulus is called a canonical form of/. We proceed to reduce the cubic form ƒ to the canonical form Xs + F3. Assume ƒ= aüxf + · · · = p1(xl + a^o)3+p2(xi + «2^2)3 = -X”3 + Yz- (140i)REDUCTION 109 This requires that/be transformable into its canonical form by the inverse of the transformations S:X= p\xx + p\a^x2, Y = p\xx 4- p\^xv We must now show that pv pv av a2 may actually be de- termined, and that the determination is unique. Equating coefficients in (HOj) we have Pi+P* = av «1^1 + a2p2 = av (140,) ^lPl = ^2’ ^lPl ”b ^1^2 = ^3* Hence the following matrix, M, must be of rank 2 : 1 ai «? «f M= 1 a2 «1 a2 «o ai «2 as From M= 0 result 1 «1 «1 1 «X «1 1 «2 «1 = 0, 1 «2 «1 «0 «1 «2 «1 «2 a% Expanding the determinants we have JP(Iq 4· -f- Ra2 = 0, * Pax 4- Qa2 4- Ras = 0. Also, evidently JP + Qct-i 4· Rwi =0 (i = 1, 2). Therefore our conditions will all be consistent if av are determined as the roots, ^ s- f2, of * 0 a a2 l az % -tt* % = o. This latter determinant is evidently the Hessian of/, divided by 2. Thus the complete reduction of ƒ to its canonical formno THE THEORY OF INVARIANTS is accomplished by solving its Hessian covariant for the roots av av and then solving the first two equations of (1402) for pv pv The inverse of S will then transform ƒ into X3 + Ys. The determinant of S is and D =jfc 0 unless the Hessian has equal roots. Thus the necessary and sufficient condition in order that the canonical reduction be possible is that the discriminant of the Hessian (which is also the discriminant, R, of the cubic/) should not vanish. If R = 0, a canonical form of ƒ is evidently X2Y. Among the problems that can be solved by means of the canonical form are, (a) the determination of the roots of the cubic ƒ = 0 from X3+F3 = (X + F)(Af+a>F)(Ar+a>2F), (o being an imaginary cube root of unity, and (J) the deter- mination of the syzygy among the concomitants of ƒ. We now solve problem (ò). From Table I, by substituting ao = as = l,-«i = a2 = 0, we have the fundamental system of the canonical form : X3+F3, 2X7, X3-F3, -2. Now we may regard the original form ƒ to be the transformed form of X8 + Y3 under S. Hence, since the modulus of S is we have the four invariant relations / = X3+F3, A = 2 D2X7, ^ = i>3(X3- 73), R = - D6.2. It is an easy process to eliminate i), AT, Y from these four equations. The result is the required syzygy : f2R+ 2Q2 + A3 = 0.KEDUCTION 111 A general binary quartic can be reduced to the canonical form (Cayley) X4 + Y±+ 6mX*Y*; a ternary cubic to the form (Hesse) X3 + Ys + ^3 + 6 mXYZ. An elegant reduction of the binary quartic to its canonical form may be obtained by means of the provectant operators of Chapter III, § 1, V. We observe that we are to have identically /=(a0, av · ··, a^xv x2Y = X\ + X\ + 6mX\X\, where Xv X2 are linear in xv x2 ; Xl = alXl + = Plxl + £**2- Let the quadratic X^2 be q = (A0, Av A2\xv x2)2. Then dq ■ Xj = (A„ Av A,- ±yxf = 0 a = 1, 2). dx2 dxx 6 mdq . X\X\ = 12.2(4 A0A2 - A\)ml1l2 = 12 X XxX2. Equating the coefficients of x\, xxxv x\ in the first equation above, after operating on both sides by dq, we now have AqCl2 — Axax + A2a0 = XA0, A0a3 — Axa2 + A2ax = | XAj, A0a4 - Axas 4- A2a2 = \A2. Forming the éliminant of these we have an equation which determines X, and therefore m, in terms of the coefficients of the original quartic/. This éliminant is a a2-\ &2 ~~“ X a<2 + 2 ^ a% = 0, or, after expanding it, X3 — ^X — «/" = 0,112 THE THEORY OF INVARIANTS where i, J are the invariants of the quartic ƒ determined in Chapter III, § 1, V. It follows that the proposed reduction of ƒ to its canonical form can be made in three ways. A problem which was studied by Sylvester,* the reduction of the binary sextic to the form X\ + X* + X% + 30 mX\X\X% has been completely solved very recently by E. K. Wakeford.f SECTION 5. HILBERT’S THEOREM We shall now prove a very extraordinary theorem due to Hilbert on the reduction *of systems of quantics, which is in many ways closely connected with the theory of syzygies. The proof here given is by Gordan. The original proof of Hilbert may be consulted in his memoir in the Mathematische Annalen, volume 36. I. Theorem. If a homogeneous algebraical function of any number of variables be formed according to any definite laws, then, although there may be an infinite number of functions F satisfying the conditions laid down, nevertheless a finite number Fv Fv Fr can always be found so that any other F can be written in the form F=AXFX + A2F2 + ... + ArFr, where the Ays are homogeneous integral functions of the variables but do not necessarily satisfy the conditions for the F's. An illustration of the theorem is the particular theorem that the equation of any curve which passes through the in- tersections of two curves Fx = 0, F2 = 0 is of the form F = AXFX + A2F2 = 0. Here the law according to which the ^’s are constructed is that the corresponding curve shall pass through the stated * Cambridge and Dublin Mathematical Journal, vol. 6 (1851), p. 293. t Messenger of Mathematics, vol. 43 (1913-14), p. 25.REDUCTION 113 intersections. There are an infinite number of functions sat- isfying this law, all expressible as above, where Av A2 are homogeneous in xv x2, x3 but do not, as a rule, represent curves passing through the intersections. We first prove a lemma on monomials in n variables. Lemma. If a monomial xfyxfy ··· xkf, where the k's are positive integers, he formed so that the exponents kv ···,/"„ satisfy prescribed conditions, then, although the number of products satisfying the given conditions may be infinite, never- theless a finite number of them can be chosen so that every other is divisible by one at least of this finite number. To first illustrate this lemma suppose that the prescribed conditions are 2 k, + 3 — ko — k, — 0, 78 7 77 (141) kj 4~ k^ = k2 k3· Then monomials satisfying these conditions are /y«2/yi2/y‘5 /y»2/y*3^ ry* sy* /y»2 sy*2^y* /y4 ^3 __ and all are divisible by at least one of the set x\r\xA, x2x3x\. Now if w = l, the truth of the lemma is self-evident. For all of any set of positive powers of one variable are divisible by that power which has the least exponent. Proving by induction, assume that the lemma is true for monomials of n — 1 letters and prove it true for n letters. Let K = x^xfy ··· xfr be a representative monomial of the set given by the prescribed conditions and let P = x^x(^ ··· xann be a specific product of the set. If iTis not divisible by P, one of the numbers k must be less than the corresponding num- ber a. Let kr < ar. Then kr has one of the series of values 0, 1, 2, ar- 1, that is, the number of ways that this can occur for a single exponent is finite and equal to N = ax 4- a2 4- ··· 4- an.114 THE THEOEY OF INVAEIANTS The cases are Aj equals one of the series 0,1, ···, a1 — 1; (dq cases), k2 equals one of the series 0, 1, ···, a2 — 1 ; (a2 cases), (142) etc. Now let Jcr=m and suppose this to be case number p of (142). Then the n — 1 remaining exponents kv Tc2, ···, kr_v Jcr+V ···, kn satisfy definite conditions which could be obtained by making kr = m in the original conditions. Let Kp = ··· Xf ··· X*n = X™Kp be a monomial of the system for which kr = m. Then Kp contains only n — 1 letters and its exponents satisfy definite conditions which are such that x™K'p satisfies the original conditions. Hence by hypothesis a finite number of mono- mials of the type Ksay, Lv L2, ···, Xap, exist such that all monomials Kp are divisible by at least one L. Hence Kp = x™Kfp is divisible by at least one P, and so by at least one of the monomials = x?Lv =* x™Lv ··, M(pap} = x™Lap. Also all of the latter set of monomials belong to the orig- inal system. Thus in the case number p in (142) K is divisible by one of the monomials Now suppose that AT is not divisible by P. Then one of the cases (142) certainly arises and so K is always divisible by one of the products M?\ M are ordered so that if the exponents a, l of S and T are read simultaneously from left to right the term first to show an exponent less than the exponent in the corresponding posi- tion in the other term occurs farthest to the right. If the normal order of $, 7 is ($, 7), then T is said to be of lower rank than S. That is, the terms of tj are assumed to be arranged according to descending rank and there is a term of highest and one of lowest rank. By hypothesis the r¡ functions are formed according to definite laws, and hence their first terms satisfy definite laws relating to their expo- nents. By the lemma just proved we can choose a finite number of tj functions, tjv rj2, ···, r)p such that the first term of any other 7) is divisible by the first term of at least one of this number. Let the first term of a definite tj be divisible by the first term of 7jmx and let the quotient be Pv116 THE THEORY OF INVARIANTS Then r] — i\Vrrh is an ?/ function, and its first term is of lower rank than the first term of >/. Let this be denoted by V = Pi Vm, + Va)■ Suppose next that the first term of r)a) is divisible by ; thus, V(V = Ptf m*+ V(2\ and the first term of T?i2) is of lower rank than that of r)a). Continuing, we obtain = PrVmr + · Then the first terms of the ordered set 71 71O) -r)(2) (?) ... 7h 7 i V i ·> '1 i are in normal order, and since there is a term of lowest rank in rj we must have for some value of r Vlr> = Pr+\Vmr+l· That is, we must eventually reach a point where there is no t) function rjir+1) of the same order as 77 and whose first term is of lower rank than the first term of r}(r). Hence V = P\Vmx + P2Vn,2 + ·■· + Pr+lVmr+1 (144) and all 77’s on the right-hand side are members of a definite finite set VV ^2’ ’ * Vp· But by the original theorem and (143), every F is itself an rj function. Hence by (144) F = AXFX + A2F2 + ·. · + ArFr, (145) where Fi(i = 1, ···,/·) are the F functions involved linearly in rjv 7}v ···, 7]p. This proves the theorem. II. Linear Diophantine equations. If the conditions im- posed upon the exponents k consist of a set of linear Dio- phantine equations like (141), the lemma proved above shows that there exists a set of solutions finite in number by meansREDUCTION 117 of which any other solution can be reduced. That is, this fact follows as an evident corollary. Let us treat this question in somewhat fuller detail by a direct analysis of the solutions of equations (141). The second member of this pair has the solutions kv ¿2' kp k (1) 0 0 1 1 (2) 0 1 0 1 /^N CO 1 0 1 0 O) 1 1 0 0 (5) 1 1 1 1 (6) 2 1 1 0 Of these the fifth is obtained by adding the first and the fourth ; the sixth is reducible as the sum of the third and the fourth, and so on. The sum or difference of any two solutions of any such linear Diophantine equation is evi- dently again a solution. Thus solutions (1), (2), (3), (4) of k1 + ki = k2 + kz form the complete set of irreducible solutions. Moreover, combining these, we see at once that the general solution is (I) kx = x + y, k2 = x + s, kz = y 4- w, k± = z + w. Now substitute these values in the first equation of (141) 2 kt 4- 3 &2 — kz — k± = 0. There results bx + y + 2z=2w. By the trial method illustrated above we find that the irre- ducible solutions of the latter are a;=2, w = 5; y = 2,w=l; z = 1, w = 1; x = 1, y = 1, w=3, where the letters not occurring are understood to be zero. The general solution is here (II) # = 2 a 4- <2, y = 2 b + d, z = c, w = 5a + b + c+3d,118 THE THEORY OF INVARIANTS and if these be substituted in (I) we have kx = 2 a + 2b + 2d —2 a + c + d fag = 5 a -j- 3 b -f- c -j- 4 d k^ = 0 a -f- b 2 c Q d Therefore the only possible irreducible simultaneous solu- tions of (141) are kv ^2’ ^S‘ h (1) 2 2 5 5 (2) 2 0 B 1 (3) 0 1 1 2 (4) 2 1 4 3 But the first is the sum of solutions (3) and (4) ; and (4) is the sum of (2) and (3). Hence (2) and (3) form the com- plete set of irreducible solutions referred to in the corollary. The general solution of the pair is = 2 Of, &2 — = 3 Ct -|- Jc^ = OC -p 2 ¡3. The corollary may now be stated thus: Corollary. Every simultaneous set of linear homogeneous Diophantine equations possesses a set of irreducible solutions, finite in number. A direct proof without reference to the present lemma is not difficult.* Applied to the given illus- tration of the above lemma on monomials the above analysis shows that if the prescribed conditions on the exponents are given by (141) then the complete system of monomials is given by where a and /3 range through all positive integral values independently. Every monomial of the system is divisible by at least one of the set 'ï’2t'3'Î' 'Y> sy ryi * Elliott, Algebra of Quantics, Chapter IX.SEDUCTION 119 which corresponds to the irreducible solutions of the pair (141). III. Finiteness of a system of syzygies. A syzygy S among the members of a fundamental system of concomitants of a form (cf. (140))/, Iv -^2’ "’i Ifi,i Kv ··· is a polynomial in the Ts formed according to the law that it will vanish identically when the J’s are expressed ex- plicity in terms of the coefficients and variables of ƒ. The totality of syzygies, therefore, is a system of polynomials (in the invariants J) to which Hilbert’s theorem applies. It therefore follows at once that there exists a finite number of syzygies, Sv $2’ ***’ such that any other syzygy 8 is expressible in the form /S = 0181 C2S2 + ··· -f- 0V8V. (146) Moreover the (7’s, being also polynomials in the J’s are themselves invariants of/. Hence Theorem. The number of irreducible syzygies among the concomitants of a form f is finite, in the sense indicated by equation (146). SECTION 6. JORDAN’S LEMMA Many reduction problems in the theory of forms depend for their solution upon a lemma due to Jordan which may be stated as follows: Lemma. If u1 + u2 + uz = 0, then any product of powers of uv uv uz of order n can be expressed linearly in terms of such products as contain one exponent equal to or greater than | n. We shall obtain this result as a special case of a consider- ably more general result embodied in a theorem on the representation of a binary form in terms of other binary forms.120 THE THEOEY OF INVAEIANTS I. Theorem. If ax, bx, cx, ··· are r distinct linear forms, and A, B, C, ··· are binary forms of the respective orders a, /3, 7, ·.. where a + /3 + 7 + ··· = n— r + 1, then any binary form f of order n can be expressed in the form f = an~°-A + bx~&B + C£-yC+ and the expression is unique. As an .explicit illustration of this theorem we cite the case n = 3, r = 2. Then a + /3 = 2, a = /3 = 1. ƒ = «I Ooo*i + ^01*2) + Oio^i + Pnxi) · (147) Since ƒ, a binary cubic, contains four coefficients it is evi- dent that this relation (147) gives four linear nonhomo- geneous equations for the determination of the four unknowns jt?00, Pov> Piw Piv Thus the theorem is true for this case pro- vided the determinant representing the consistency of these linear equations does not vanish. Let ax — a^x^ + a2x2, bx — \xx + ¿2^2’ and Z> = «A - aj>v Then the aforesaid determinant is a\ 0 n 0 «2 h\ «1 2 axa2 H 20A 0 «1 0 This equals IP, and D 0 on account of the hypothesis that ax and bx are distinct. Hence the theorem is here true. In addition to this we can solve for the p{j and thus deter- mine A, B explicitly. In the general case the number of unknown coefficients on the right is « + /3 + Y+ ··· + r = n + 1. Hence the theorem itself may be proved by constructing the corresponding consistency determinant in the general case ; * but it is perhaps more instructive to proceed as follows: * Cf. Transactions Amer. Math. Society, Vol. 15 (1914), p. 80.REDUCTION 121 It is impossible to find r binary forms A, B, C, *·· of orders oc, /3, 7, ·.. where a + /3 +7+ ... = n — r + 1, such that, identically, + + — = o. In fact suppose that such an identity exists. Then operate upon both sides of this relation a + 1 times with 3 3 A = «2 ~ «1 -JT Ox = «1*1 + «2*2)· Let gnx be any form of order n and take a2 = 0. Then Aa+vs = ^(«i*^)a+Vra-1 = ^1^+Vra_vr14"a"1 + VrVra”Vr2^ra·2^ + ··· + K-a<*l¥l0%X%~a~\ where the &’s are numerical. Hence Aa+1g% cannot vanish identically in case a2 = 0, and therefore not in the general case a2=£ 0, except when the last n — a coefficients of g% vanish: that is, unless gnx contains anfa as a factor. Hence Aa+lB = + c^-a-y~1Cf + ··., where B!, C are of orders /3, 7, ··· respectively. Now Aa+1B is an expression of the same type as j&, with r changed into r — 1 and n into n — a — 1, as is verified by the equation + 7+ ··· = (n — a—1) — (r — l)+l = ft — r + 1 — a. Thus if there is no such relation as J3=0 for r— 1 linear forms a# 6X, ···, there certainly are none for r linear forms. But there is no relation for one form (r = l) save in the vacuous case (naturally excluded) where A vanishes identi- cally. Hence by induction the theorem is true for all values of r. Now a count of coefficients shows at once that any binary form ƒ of order n can be expressed linearly in terms of w-f 1122 THE THEORY OF INVARIANTS binary forms of the same order. Hence ƒ is expressible in the form f=araA + br?B+cl-yl)+ .... That the expression is unique is evident. For if two such were possible, their difference would be an identically vanish- ing expression of the type E= 0, and, as just proved, none such exist. This proves the theorem. II. Jordan’s lemma. Proceeding to the proof of the lemma, let us = — (ux -f w2)> supposing that uv u2 replace the variables in the Theorem I just proved. Then uz, uv u2 are three linear forms and the Theorem I applies with r = 3, « + # + 7 = n — 2. Hence any homogeneous expression ƒ in uv uv u2 can be expressed in the form u?'aA + u!S-fiB + ul-yC, or, if we make the interchanges in — a n — 0 n — y\ \ X M V } in the form u$A+ u^B + u\C, (148) where ^ + i»=2w-f- 2. (149) Again integers \, /a, v may always be chosen such that (149) is satisfied and x>§W, /*>!n, v>ln. Hence Jordan’s lemma is proved. A case of three linear forms u{ for which ux 4- u2 +us = 0 is furnished by the identity (aV)cx + (bc)ax + (ca)bx = 0. If we express A in (148) in terms of uv u2 by means of ui + ^2 + uz = 0» in terms of u2, u3, and C in terms of w3, uv we have the conclusion that any product of order n of (aV)cx^ (bc)ax, (cd)bx can be expressed linearly in terms ofREDUCTION 123 (ab')nc%, (aby 1(bc)cnx 1aa., (aJ)r* 2(bc)2c% 2af, ···, (al>y(j)cy~*c%a%r\ (bc)na% (bcy~\ca^a,iy‘1bafl (bc^n~\ca)2an~2b% ···, (bey^cay-^b^ (150) (ca/ij (cay-\abybn-2c% ···, (caycaby-^cr% where f w. It should be carefully noted for future reference that this monomial of order n in the three expressions (aJ)^, (bc)ax, (ca)bx is thus expressed linearly in terms of symbolical products in which there is always present a power of a deter- minant of type (a6) equal to or greater than J n. The weight of the coefficient of the leading term of^a covariant is equal to the number of determinant factors of the type («6) in its symbolical expression. Therefore (150) shows that if this weight w of a covariant of ƒ does not exceed the order of the form ƒ all covariants having leading coefficients of weight w and degree 3 can be expressed linearly in terms of those of grade not less than | w. The same conclusion is easily shown to hold for covariants of arbitrary weight. SECTION 7. GRADE The process of finding fundamental systems by passing step by step from those members of one degree to those of the next higher degree, illustrated in Section 3 of this chapter, although capable of being applied successfully to the forms of the first four orders fails for the higher orders on account of its complexity. In fact the fundamental system of the quintic contains an invariant of degree 18 and consequently there would be at least eighteen successive steps in the process. As a proof of the finiteness of the fundamental system of a form of order n the process fails for the same reason. That is,124 THE THEORY OF INVARIANTS it is impossible to tell whether the system will be found after a finite number of steps or not. In the next chapter we shall develop an analogous process in which it is proved that the fundamental system will result after a finite number of steps. This is a process of passing from the members of a given grade to those of the next higher grade. I. Definition. The highest index of any determinant factor of the type («&) in a monomial symbolical concomitant is called the grade of that concomitant. Thus (a6)4(ae)25Je4 is of grade 4. The terms of covariants (84), (87) are each of grade 2. Whereas there is no upper limit to the degree of a con- comitant of a form ƒ of order w, it is evident that the maximum grade is n by the theory of the Aronhold symbolism. Hence if we can find a method of passing from all members of the fundamental system of/of one grade to all those of the next higher grade, this will prove the finiteness of the system, since there would only be a finite number of steps in this process. This is the plan of the proof of Gordan’s theorem in the next chapter. II. Theorem. Every covariant of a single form f of odd grade 2 X — 1 can be transformed into an equivalent covariant of the next higher even grade 2 X. We prove, more explicitly, that if a symbolical product contains a factor (a6)2A_1 it can be transformed so as to be expressed in terms of products each containing the factor (aJ)2\ Let A be the product. Then by the principles of Section 2 A is a term of ((«j)2A- hr1-2^1-2*, 4>)y· Hence by Theorem III of Section 2. A = ((a6)2A_1aJ+1~2A5£+1~2A, y 4-SiT((a6)2A-1<+1-2A^+1"2N (151)EEDUCTION 125 where 7' < 7 and is a concomitant derived from by con- volution, AT being numerical. Now the symbols are equiva- lent. Hence = (a5)2A_1a2+1~2A62+1“2A = — (a6)2A_1^+1“2A5?+1"2A = 0. Hence all transvectants on the right-hand side of (151), in which no convolution in i/r occurs, .vanish. All remaining terms contain the symbolical factor (a6)2A, which was to be proved. Definition. A terminology borrowed from the theory of numbers will now be introduced. A symbolical product, A, which contains the factor (#5)r is said to be congruent to zero modulo (ab)r; • A = 0 (mod (ab)r). Thus the covariant (84) C = J(a5)2(6«)2aX + %(ab)%aa)(ba)axbxax gives C = %(ab)\aa)(fia)axbxax(mod (6a)2). III. Theorem. Every covariant of f = a£=6£ = ... which is obtainable as a covariant of (ƒ, f)2k = 9\x~ik=z (ab)2kal~2kbn~2k (Chap. II, § 4) is congruent to any definite one of its oivn terms modulo (a6)2*+1. The form of such a concomitant monomial in the g sym- bols IS A= (gigj*(gig,)« .. · .... Proceeding by the method of Section 2 of this chapter change gx into y; i.e. gn = yv <712= — Vr Then A becomes a form of order 2 n — 4 k in y, viz. a2”-4* = yS2”-4* = ···. Moreover A = (a2/”4*, 5fg-4*)2n-4* = (a2n-U (aJ)2*a»-2*5^2*y2n-U by the standard method of transvection. Now this transvec- tant A is free from y. Hence there are among its terms ex- pressed in the symbols of/only two types of adjacent terms, viz. (cf. § 2, II) (da) (eh) P, (db) (ea) P.126 THE THEORY OF INVARIANTS The difference between A and one of its terms can therefore be arranged as a succession of differences of adjacent terms of these two types and since P involves (ab)2* any such dif- ference is congruent to zero modulo (&6)2*+1, which proves the theorem. IY. Theorem. If n > 4 Jc, any covariant of the covariant g^n-ik __ (abykal-2kl7f~2k is expressible in the form ZC2M+(ab^(bcf(cayT, (152) where C2M represents a covariant of grade 2 k + 1 at least, the second term being absent (T = 0) if n is odd. Every covariant of g2f~*k of a stated degree is expressible as a linear combination of transvectants of gf^k with covari- ants of the next lower degree (cf. § 2, III). Hence the theorem will be true if proved for T= (g2f~*k, the covariants of second degree of this form. By the fore- going theorem T is congruent to any one of its own terms mod (abf1M. Hence if we prove the present theorem for a stated term of T.\ the conclusion will follow. In order to select a term from T we first find T by the standard trans- vection process (cf. Chap. Ill, § 2). We have after writing s = n — 2 k for brevity, and asxb% = o?x v f*y « \ T= (aby*(cdykX 7 e r^rg+*· (153) - (V) Now the terms of this expression involving a may be obtained by polarizing a2? t times with respect to y, = (aby2k(bc')°(cd') 2kan~ 2kby2k ~'Jcn~2k ~ adnx ~2k. Assume n ]> 4&. Then by Section 6, i/r = CabyXboyQcay^-4^-2^^-2^ can be expressed in terms of covariants whose grade is greater than 2 k unless k, we have from Section 6 that is of grade n n n > § · 3&, i.e. >2 or else contains (ab')2(be')2(ca)2, i.e. = 202*+! + (a6)2(6i?)2(m)T. (155) (II) Suppose then a^k. Then in since i = 2 k has been treated under yfr above, we have either (a) i ^ k, or (J) 2 k — i > In case (a) (155) follows directly from Section 6. In case (6) the same conclusion follows from the argument in (I). Hence the theorem is proved.CHAPTER V GORDANS THEOREM We are now in position to prove the celebrated theorem that every concomitant of a binary form ƒ is expressible as a rational and integral algebraical function of a definite finite set of the concomitants of ƒ. Gordan was the first to ac- complish the proof of this theorem (1868), and for this rea- son it has been called Gordan’s theorem. Unsuccessful attempts to prove the theorem had been made before Gordan’s proof was announced. The sequence of introductory lemmas, which are proved below, is that which was first given by Gordan in his third proof (cf. Vorlesungen iiber Invariantentheorie, Yol. 2, part 3).* The proof of the theorem itself is somewhat simpler than the original proof. This simplification has been accomplished by the theorems resulting from Jordan’s lemma, given in the preceding chapter. SECTION 1. PROOF OF THE THEOREM We proceed to the proof of a series of introductory lemmas followed by the finiteness proof. I. Lemma 1. If (A): Av A2, ···, Ak is a system of binary forms of respective orders av a2, ···, and (i?): Bv Bv ···, Bx, a system of respective orders bv bv ···, bx, and if = A\A? ... Afc yfr = B?'B$>... Bfi * Cf. Grace and Young; Algebra of Invariants (1903). 128GORDAJSPS THEOREM 129 denote any two products for which the as and the ft's are all positive integers (or zerof then the number of transvectants of the type of T= (0, y which do not contain reducible terms is finite. To prove this, assume that any term of r contains p sym- bols of the forms A not in second order determinant com- binations with a symbol of the B forms, and a symbols of the B's not in combination with a symbol of the A’s. Then evidently we have for the total number of symbols in this term, from (A) and (S') respectively, aiai + a2a2 -f- · · · + = P + J, (irr\ + ^2^2 + “ * + bi@i = cr To each positive integral solution of the equations (156), considered as equations in the quantities a, /3, p, , i/r and a definite index y, and hence a definite transvectant r. But as was proved (Chap. IV, § 3, III), if the solution corresponding to (, yfry is the sum of those corresponding to («¡¡q, and (2, ^2)?3’ Ihen t certainly contains reducible terms. In other words trans- vectants corresponding to reducible solutions contain re- ducible terms. But the number of irreducible solutions of (156) is finite (Chap. IV, § 5, II). Hence the number of transvectants of the type t which do not contain reducible terms is finite. A method of finding the irreducible trans- vectants was given in Section 3, III of the preceding chapter. Definitions. A system of forms (A) is said to be com- plete when any expression derived by convolution from a product of powers of the forms (A) is itself a rational integral function of the forms (A). A system (A) will be called relatively complete for the modulus Gl· consisting of the product of a number of sym- bolical determinants when any expression derived by con-130 THE THEORY OF INVARIANTS volution from a product 0 is a rational integral function of the forms (A) together with terms containing Gc as a factor. As an illustration of these definitions we may observe that /=a|= A = (ah'faj)x, Q = (a6)2(ca)6^?J, B = (aby(cd)\acXbd) is a complete system. For it is the fundamental system of a cubic/, and hence any expression derived by convolution from a product of powers of these four concomitants is a rational integral function of/, A, Q, JR. Again ƒ itself forms a system relatively complete mod- ulo (a6)2. Definition. A system (A) is said to be relatively com- plete for the set of moduli Grv 6r2, ··· when any expression derived from a product of powers of A forms by convolution is a rational integral function of A forms together with terms containing at least one of the moduli Grv Cr2, ··· as a factor. In illustration it can be proved (cf. Chap. IV, § 7, IV) that in the complete system derived for the quartic H = (abfalb% any expression derived by convolution from a power of H is rational and integral in H and G1==(aby, G2 = (bcy(cay(aby. Thus H is a system which is relatively complete with regard to the two moduli G-1 = (aJ)4, 6r2 = (bcy(cay(aby. Evidently a complete system is also relatively complete for any set of moduli. We call such a system absolutely complete. Definitions. The system ((7) derived by transvection from the systems (A), (5) contains an infinite number ofGORDAN’S THEOREM 131 forms. Nevertheless ((7) is called a finite system when all its members are expressible as rational integral algebraic functions of a finite number of them. The system ((7) is called relatively finite with respect to a set of moduli (xrx, 6r2, ··· when every form of ((7) is expressible as a rational integral algebraic function of a finite number of the forms ((7) together with terms containing at least one of the moduli Grv (?2, ··· as a factor. The system of all concomitants of a cubic ƒ is absolutely finite, since every concomitant is expressible rationally and integrally in terms of ƒ, A, Q, R, II. Lemma 2. If the systems (-4), (B) are both finite and complete, then the system ((7) derived from them by transvec- tion is finite and complete. We first prove that the system ((7) is finite. Let us first arrange the transvectants t = (<*>, f y in an ordered array Tl’ ^2’ ***’ Tr? ***’ (157) the process of ordering being defined as follows : (a) Transvectants are arranged in order of ascending total degree of the product (f>yjr in the coefficients of the forms in the two systems (JL), (i?). (5) Transvectants for which the total degree is the same are arranged in order of ascending indices j; and further than this the order is immaterial. Now let tf be any two terms of t. Then (/, yjr are rational and integral in the forms A, B respectively,132 THE THEOEY OF INVAEIANTS Therefore (<£, yfry can be expressed in terms of transvec- tants of the type r (i.e. belonging to ((7)) of index less than j and hence coming before r in the ordered array (157). But if we assume that the forms of (O) derived from all tran svectants before r can be expressed rationally and integrally in terms of a finite number of the forms of ((7), Ci, then all (7’s up to and including those derived from r = o, can be expressed in terms of Ov Ov ·, C„ t. But if r contains a reducible term t = t^tv then since tv t2 must both arise from transvectants before t in the ordered array no term t need be added and all (7’s up to and includ- ing those derived from r are expressible in terms of Cv Ov Or. Thus in building by this procedure a system of (7’s in terms of which all forms of (C) can be expressed we need to add a new member only when we come to a transvectant in (157) which contains no reducible term. But the number of such transvectants in (C) is finite. Hence, a finite num- ber of O's can be chosen such that every other is a rational function of these. The proof that (0) is finite is now finished, but we may note that a set of <7’s in terms of which all others are expres- sible may be chosen in various ways, since t in the above is any term of r. Moreover since the difference between any two terms of t is expressible in terms of transvectants be- fore t in the ordered array we may choose instead of a single term t of an irreducible t = (, i/ry, an aggregate of any number of terms or even the whole transvectant and it will remain true that every form of (O) can be expressed asGORDAN’S THEOREM 133 a rational integral algebraic function of the members of the finite system so chosen. We next prove that the finite system constructed as above is complete. Let Op C2, ···, Or be the finite system. Then we are to prove that any ex- pression X derived by convolution from X= Gfr is a rational integral algebraic function of Ov ···, Or. Assume that X contains p second-order determinant factors in which a symbol from an (A) form is in combination with a symbol belonging to a (jB) form. Then X is a term of a transvectant (, where con- tains symbols from system (A) only, and yfr contains symbols from (JB) only. Then must be derivable by convolution from a product of the As and yfr from a product yfr of B forms. Moreover _ x = (i fy+z(j>, fy o' < />), and 0, ifr having been derived by convolution from <£, fa respectively, are ultimately so derivable from , \fr. But 4> = F(A), and so X is expressed as an aggregate of transvectants of the type of t = (4>, yfry. But it was proved above that every term of r is a rational integral function of Ov Cr. Hence X is such a function; which was to be proved. III. Lemma 3. If a finite system of forms (A), all the members of which are covariants of a binary form f includes f and is relatively complete for the modulus G!; and if in addi- tion, a finite system (i?) is relatively complete for the modulus134 THE THEORY OF INVARIANTS Gr and includes one form Bx whose only determinantal factors are those constituting Gr\ then the system ((7) derived by transvection from (A) and (2?) is relatively finite and complete for the modulus Gr. In order to illustrate this lemma before proving it let (A) consist of one form ƒ = a% = ···, and (2?) of two forms A =(abyajbx, ^ = (a6)2(ac?)(WX^)2. Then (A) is relatively complete for the modulus Gr' = (a&)2. Also B is absolutely complete, for it is the fundamental system of the Hessian of ƒ. Hence the lemma states that ((7) should be absolutely complete. This is obvious. For ((7) consists of the fundamental system of the cubic, / A, is at most equal to the degree of P. Here and ^ are products of powers of forms (A), (5) respectively, and by the statement of the lemma (A) con- tains only covariants of/and includes ƒ itself. This fact is evident when the degree of P is zero. To establish an inductive proof we assume it true when the degree of P is < r and note that *=(P, (*'<·> and, inasmuch as P and P are derived by convolution from a power of/, P = F(A) + afY=F (A) (mod (7'), P = Ff(A) + Cr' Y' = F(A) (mod (P).GORDAN’S THEOREM 135 Also yfr = 3>(jB) + GZ = (i?) (mod 6r). Hence t contains terms of three types (&), (6), (c). (a) Transvectants of the type (P(M), i>(5))( the degree of P(A) being r, the degree of P. (i) Transvectants of type ((¡r'F, ^r)*, Cr'JT being of the same degree as P. (^r, assuming the transvectants to be of the type r = (<£, (b) Those for which the degree of yjr is the same shall be arranged in order of ascending degree of (f>. (tf) Transvectants for which both degrees are the same shall be arranged in order of ascending index j; and further than this the ordering is immaterial. Let tr be any two terms of t. Then t'-t = 2(0, 00»' (ƒ < /). Also by the hypotheses of the lemma $ = F(A')+GfY, ^ = <*>(5)+ QZ. Hence 4>0B)y (mod tf). Now transvectants of the type (^(A), <£>(.#))>' belong before t in the ordered array since f < j and the degree of is the same as that of . Again (6r'T, (J9))?' can by the above corollary (158) be expressed in the form 2(0', iry, where the degree of r is less than that of GrfY and hence less than that of . Consequently t* — t can be written a -1 = £(", yy + 2(f, y/r'y (mod where the degree of n is the same as that of and where ƒ < j, and where the degree of cf>f is less than that of . Therefore if all terms of transvectants coming before t = (, ^yGORDAN’S THEOREM 137 in the ordered array are expressible rationally and integrally in terms of Ov Ov C„ except for terms congruent to zero modulo Cr, then all terms of transvectants up to and including r can be so expressed in terms of 0V 0^ '••i Cg, t, where t is any term of t. As in the proof of lemma 2, if t contains a reducible term t does not need to be added to Ov Ov c„ since then tv t2 are terms of transvectants coming before t in the ordered array. Hence, in building up the system of (7’s in terms of which all forms of (<7) are rationally ex- pressible modulo Cr, by proceeding from one transvectant r to the next in the array, we add a new member to the sys- tem only when we come to a transvectant containing no reducible term. But the number of such irreducible trans- vectants in (V) is finite. Hence ((7) is relatively finite modulo Cr. Note that Cv ···, Oq may be chosen by select- ing one term from each irreducible transvectant in ((7). Finally we prove that ((7) is relatively complete modulo Cr. Any term X derived by convolution from x= c>, is a term of a transvectant (, ^)p, where, as previously, is derived by convolution from a product of A forms and yfr from a product of B forms. Then X = (, ^)e + 2(£, ^)p' pf < p. That is, X is an aggregate of transvectants (<£, yjry, <£ = P can be derived by convolution from a power of/, and yjr = <&(B) (mod Cr).138 THE THEORY OF INVARIANTS Thus, X= 2(P, <&(P))" (mod (?) ee2(P, yfry (mod (?) = £(, y/rfi (mod (?) where is of degree not greater than the degree of P, by the first part of the proof. But all transvectants of the last type are expressible as rational integral functions of a finite number of (7’s modulo (?. Hence the system ((7) is relatively complete, as well as finite, modulo (?. Corollary 1. If the system (P) is absolutely complete then ((7) is absolutely complete. Corollary 2. If (P) is relatively complete for two moduli Grv (?2 and contains a form whose only determi- nantal factors are those constituting (?', then the system ((7) is relatively complete for the two moduli (?r (?2. IV. Theorem. The Bystem of all concomitants of a Unary form ƒ = anx = ··· of order n is finite. ·* The proof of this theorem can now be readily accomplished in view of the theorems in Paragraphs III, IV of Chapter IV, Section 7, and lemma 3 just proved. The system consisting of ƒ itself is relatively complete modulo (a£>)2· It is a finite system also, and hence it satis- fies the hypotheses regarding (J.) in lemma 3. This system (A) = ƒ may then be used to start an inductive proof con- cerning systems satisfying lemma 3. That is we assume that we know a finite system Ak_x which consists entirely of covariants of ƒ, which includes ƒ, and which is relatively complete modulo (a£>)2fc. Since every covariant of ƒ can be derived from ƒ by convolution it is a rational integral func- tion of the forms in Ak_x except for terms involving the factor (a6)2*. We then seek to construct a subsidiary finite system Bk_x which includes one form P2 whose only deter- minant factors are (&6)2*= (?', and which is relatively com- plete modulo (ab)2k+2 = (?. Then the system derived byGORDAN’S THEOREM 139 transvection from Ak_x and Bk_x will be relatively finite and complete modulo (ab)2k+2. That is, it will be the system Ak. This procedure, then, will establish completely an inductive process by which we construct the system concomitants of ƒ relatively finite and complete modulo (ab)2k+2 from the set finite and complete modulo (ab)2k, and since the maximum grade is n we obtain by a finite number of steps an abso- lutely finite and complete system of concomitants of/. Thus the finiteness of the system of all concomitants of ƒ will be proved. Now in view of the theorems quoted above the subsidiary system Bk_x is easily constructed, and is comparatively simple. We select for the form Bx of the lemma Bx = (ab)2kar2kbl-2k=hk. Next we set apart for separate consideration the case (c) n = 4 Jc. The remaining cases are (a) n>4 &, and (b) w<4 Jc. (a) By Theorem IV of Section 7 in the preceding chapter if n>4:Jc any form derived by convolution from a power of hk is of grade 2 k +1 at least and hence can be transformed so as to be of grade 2& + 2 (Chap. IV, § 7, II). Hence hk itself forms a system which is relatively finite and complete modulo (ab)2k+2 and is the system Bk_x required. (b) If n 0 then r is reducible unless 7 = 0, since B is an invariant. Hence, we have to consider which trans- vectants t = (ƒ% My are irreducible. But in Chapter IV, Section 3 II, we have proved that the only one of these transvectants which is ir- reducible is Q = (ƒ, Aj). Hence, the irreducible members of Ax consist of -^1= 1/ M' Qi or in the notation previously introduced, A^\f9 A, Q,R\. But B0 is absolutely complete and finite. Hence these irreducible forms of At constitute the fundamental system of/. II. System of the quartic. Let ƒ = a|= b% = ···. Then A0 = \f\. Here B0 is the single form hx = and B0 is relatively finite and complete (modd (a6)4, (ab)2(bc)\ca)2y The system Ct of transvectants t=(/% h\y is relatively finite and complete (modd (a6)4, Qaby2(be')\ea')2'). In r if j > 1, r contains a term with the factor Qab^Qac)2 which is congruent to zero with respect to the two moduli.GORDAN’S THEOREM 143 Hence j = 1, and by the theory of reducible transvectants (Chap. IV, § 3, III) 4a—4<)<4«, or a = 1, 0 = 1. The members of 01 which are irreducible with respect to the two moduli are therefore hv (ƒ, Ax). Then A.1 = j ƒ, hv (ƒ, ^)? j_ («5)2(5e)2(ca)2}. Next B1 consists of i = (ah )i an(j js absolutely complete. Hence, writing hx = H, (ƒ, Ax) = T, the fundamental system of/ is S' T, i, J.CHAPTER VI FUNDAMENTAL SYSTEMS In this chapter we shall develop, by the methods and pro- cesses of preceding chapters, typical fundamental systems of concomitants of single forms and of sets of forms. SECTION 1. SIMULTANEOUS SYSTEMS In Chapter V, Section 1, II, it has been proved that if a system of forms (A) is both finite and complete, and a sec- ond system (i?) is also both finite and complete, then the system ($) derived from (A) and (i?) by transvection is finite and complete. In view of Gordan’s theorem this proves that the simultaneous system of any two binary quantics ƒ, g is finite, and that this simultaneous system may be found from the respective systems of ƒ and g by trans- vection. Similarly for a set of n quantics. I. Linear form and quadratic. The complete system of two linear forms consists of the two forms themselves and their eliminant. For a linear form l = lx, and a quadratic ƒ, we have Then S consists of the transvectants m. Since D is an invariant S is reducible unless /8 = 0. Also B< y, and unless S = 7, (ƒ% Zy)5 is reducible by means of the product (ƒ% isyo, r~*y· Hence 7 = 8. Again, by (/*-\ i8~2y~xf, P)2, 144FUNDAMENTAL SYSTEMS 145 S is reducible if 8 > 2. Hence the fundamental system of ƒ and Z is S=\f,D, l, (ƒ, z),(/, Pf \- When expressed in terms of the actual coefficients these forms are l = a^x + afa = lx = l'x = ƒ = b = 2(8082-8?)=(a6)3 (ƒ» 0 = (Vi - Vo)2! + C^l«l - hao)x2= (ƒ> P)2 = Vl — 2 Voal + J2a0 = («0(aO· II. Linear form and cubic. If 1 = 1. and ƒ = a3 = W. = .... then (cf. Table I), (A)= W; (A) = \f, A, Q, R\, and = (ƒ aAs Z8)”· Since 22 is an invariant e = 0 for an irreducible transvectant. Also t] = 8 as in (I). If «=£ 0 then, by the product (ƒ, m.fa-WQ\ i*-y-\ S is reducible unless 8 < 3, and if S < 3 S is reducible by (ƒ, lsy(J«-lA*Q\ 1)°; unless /3 = 7 = 0, « = 1. Thus the fundamental system of ƒ and l is S = \f, A, Q, R, l, (ƒ, Z), (ƒ, Z3)3, (ƒ, Z3)3, (A, Z), (A, Z3)3, (& Z), (| = V}= ···. Then (A) = f/, DXU (B) = [<7, D2|, £ = ^2)$)*. Here /3 = 8= 0. Also 2a>e]>2a — 1, 27^e>27-l,146 THE THEORY OF INVARIANTS and consistent with these we have the fundamental system s=\fi9> A’ A’ (/’ S')' C/* 9f\· Written explicitly, these quantities are ƒ = + 2 + apl = af = <$ = d = V? + 2 *1^2 + V! = ¿1 = = ···> I)1 = 2(a0a2 — af) = (W)2, i>2 = 2(&o*2-^) = (^)2, ^ = («0*1 «A)®? C^(A ^o)^2+ C^1^2 A = (ƒ, ^)2 = «<& - 2 aA + a2&0 = 0J)2· IV. Quadratic and cubic. Consider next the simultaneous system of ƒ = a| = a'}=- ···, g = b% = bf = ···. In this case (A)= {ƒ, DU (B)= \g, A, 2 and b =*= 0, # = ( ƒ“, ^aA6Qc)v is reducible by means of the product (ƒ, A)2(/^,^A^QOY“2. Hence only three types of transvectants can be irreducible ; (ƒ, A), (ƒ, A)2, (ƒ% ga QC)Y. The first two are, in fact irreducible. Also in the third type if we take c = 0, the irreducible transvectants given by (ƒ% <7a)Y will be those determined in Chapter IV, Section 3, III, and are ƒ> 9-, (ƒ, 5')» (ƒ, (Z2, ^)8, (ƒ3, Z2)6· If 1, we may substitute in our transvectant (ƒ% gaQc')y the syzygy 1 are reducible. Tak- ing a = 0, c = 1 we note that (ƒ, $) is reducible because itFUNDAMENTAL SYSTEMS 147 is the Jacobian of a Jacobian. Then the only irreducible cases are (ƒ, Q)\ (/2, Qf- Finally if c =1, &=£ 0, the only irreducible transvectant is CA gQf- Therefore the fundamental system of a binary cubic and a binary quadratic consists of the fifteen concomitants given in Table III below. TABLE III Degree Order 0 1 2 3 1 f g 2 I) (ƒ, g)2 A CA g) 3 (ƒ, A)2 / N 09 (ƒ, A) Q 4 B (ƒ, Q)2 5 (A s'2)6 (A Qf 7 (A gQf SECTION 2. SYSTEM OF THE QUINTIC The most powerful process known for the discovery of a fundamental system of a single binary form is the process of Gordan developed in the preceding chapter. In order to summarize briefly the essential steps in this process let the form be/. Construct, then, the system A0 which is finite and complete modulo (aJ)2, i.e. a system of forms which are not expressible in terms of forms congruent to zero modulo (ab)2. Next construct Av the corresponding system modulo (ai)4, and continue this step by step process until the system which is finite and complete modulo (a6)w is reached. In order to construct the system Ak which is complete modulo (a6)2fc+2 from Ak_v complete modulo (ai)2*, a subsidiary148 THE THEORY OF INVARIANTS system Bk__x is introduced. The -system Bk_x consists of covariants of cj> = (ab')2ka%~2kb^k. If 2 w — 4Jc w, consists of <£ itself, and if 2 w — 4 k = w, consists n n n of <£ and the invariant (aJ)2(J(?)2( 2 this trans- vectant always contains a term of grade 3 and hence, by the theorem in Chapter IV, it may be transformed so that it contains a series of terms congruent to zero modulo (a&)4, and so it contains reducible terms with respect to this modu- lus. Moreover (ƒ, AT)2 is reducible for forms of all orders as was proved by Gordan’s series in Section 1 of Chapter IV. Thus Ax consists of/, H, (ƒ, AT) = T. Proceeding to construct Bt we note that i = is of order < 5. Hence Bt consists of its fundamental system : Bt = {i, B], where D is the discriminant of i. Hence A2 which is here the fundamental system of ƒ is the transvectant system given by = (f'ffPTv, The values a = /3 = y = S = r] = 0, e = 1 give JD. Since D is an invariant is reducible if and e =£ 0. Hence 6=0. If /? > 1, ^ is reducible by means of such products as (faITTv,FUNDAMENTAL SYSTEMS 149 Hence CO >8—0 (ii) « = 0, 7 = 0, /3 = 1. By Chapter IV, Section 4, IV, y2 = - *{(ƒ,ƒ)*#* - 2(/, £)2/ff+ (IT, AO2/2!· Hence 72 = - JAT8 (mod (aby). But if 7 > 1, the substitution of this in raises ¡3 above 1 and hence gives a reducible transvectant. Thus 7 = 0 or 1 (cf. Chap. V (158)). Thus we need to consider in detail the following sets only : (i) a = 1 or 2, /3 = 0, 7 = 0, (ii) « = 0, /3 = 0, 7 = 1, (iii) « = 1, 0 = 0, 7 = 1, (iv) a = 0, /3 = 1, 7 = 0. In (i) we are concerned with (ƒ“, zs)y. By the method of Section 3, Chapter IV, 2S-lg7<2£, 5« — 4<7<5«, and consistent with this pair of relations we have A/, (ƒ, *), (ƒ, iy, (ƒ, i2)3, (ƒ, (ƒ, i3)5, (Z2, (/2, i4)7, (/2, ¿4;8> (/2> O9, CA »5)10· Of these, (/2, ¿3)6 contains reducible terms from the product (ƒ, *)4C/; 02, and in similar fashion all these transvectants are reducible except the following eight: / i, (/, 0, (/, 02, (/, i2)3, (/, f*)4, (/ *3)5. ( a ^*yo■ In (ii) we have (Z% i{). But T = — (aby(bc)a3J>~.e*, and (T, i) contains the term £ = — (aby(bc)(bi')ybxc*ix. Again (bc)(bi)cjx = l\_(bc')‘Hl + (biyy - (ci)262].150 THE THEORY OF INVARIANTS Hence t involves a term having the factor/. The analysis of the remaining cases proceeds in precisely the same way as in Cases (i), (ii). In Case (ii) the irreducible transvec- tants prove to be (7, ¿)2, (7, ¿2)4, (7, ¿3)6, (7, ¿4)8, (7, ¿*)9. Case (iii) gives but one irreducible case, viz. (/7, i7)14. In Case (iv) we have (#, 0, (IT, 02, Off, i2)3, (ff, ¿2)4, (If, ¿3)5, (jH; ¿3)6. Table IV contains the complete summary. The fundamen- tal system of/consists of the 23 forms given in this table. TABLE IV De- gree Order 0 1 2 3 4 5 6 7 9 1 / 2 i jy 3 (*,/)* OM) T 4 D (*, //)* 6 (i2,/)4 (i2,/)3 (i, T)* 6 (i\ sy (i2, -S')8 7 (i·3,/)6 (¿2, r)4 — 8 (¿8, SY (l3, #)5 9 (<*, T)8 11 0’4, 5T)« 12 0’5, Z2)10 13 (i'5, T)» 18 FUNDAMENTAL SYSTEMS 151 SECTION 3. RESULTANTS IN ARONHOLD’S SYMBOLS In order to express the concomitants derived in the preced- ing section in symbolical form the standard method of transvection may be employed and gives readily any con- comitant of that section in explicit symbolical form. We leave details of this kind to be carried out by the reader. However, in this section we give a derivation, due to Clebsch, which gives the symbolical representation of the resultant of two given forms. In view of the importance of resultants in invariant theories, this derivation is of fundamental conse- quence. I. Resultant of a linear form and an n-ic. The resultant of two binary forms equated to zero is a necessary and sufficient condition for a common factor. Let ƒ= a” + «2^2= 0· Then x1: x2 = — «g : oq. Substitution in ƒ evidently gives the resultant, and. in the form R = (aa)n. II. Resultant of a quadratic and an n-ic. Let ^ = ^X = P xQx' The resultant R = 0 is evidently the condition that ƒ have either px or qx as a factor. Hence, by I, R = (^ap^)n(bq)n. Let us express R entirely in terms of a, 6, ···, and a, /3, ··· symbols. We have, since a, b are equivalent symbols, R = h\(apY(H)n + («?)"(%)"!· Let (d5p)(ig0= A6’ (fljXSp) = ^ so that152 THE THEOEY OF INVARIANTS. Theorem. -re· · p U»n 4" Vn If n is even, R = ^— is rationally and inte- grally expressible in terms of p2 = (¡i— v}2 and +(h- 1)0/-» + ff2p*-4 1 · u (h — 3)(A — 4)(A — 5) o A_fi _ 1.2.3 H ‘ But n = {0/Si + i.w-5>^"-6+..., 1·2 1*2*3 2 «.(/. + kXp·-1 +(» - 2wr’ + f>‘~2‘K”~4Vp.-,‘ + + ...,, 1 · A · o which was to be proved. Now if we write 4>=Px■ we have i>l?l = av PlVt+Ptfl = 2 «1«2’ ft?2 = «2· Then p + v= (ap)(bq) + (aq)(bp) = («1^2 - «2.PlX5l?2 - 52?l) + (al')]2= (ab)\pq)2 = — 2(a5)2(«/3)2 = — 2(a5)2D. Let the symbols of (f> be a', afr ...; /3', /3", ···, 7, ···. Then we can write for the general term of It, pn~2kak = (^fi — v)n~2kQfiv)k = ( — 2)2 Z)2 k(aK)n~2k X ^a,y(b^)\aa!r)\bff,y ... (a«w)2(5/3w)2 n . n -C-2)f i>2 A- Evidently .A*, is itself an invariant. When we substitute this in 2 R above we write the term for which k = ^ n last. This term factors. For if B == (aar)\au!fy ··· (ac/2^)2 = (6/3')W')2 ··· (¿A*. then cr2 = i?2. Thus when n is even, n n—2 n—2 »—4 R=i-Dy-2 2 A> +»(--*>) 2 2 2 A n(n-4)(n-o) ,_^ VA. 1.2-3 + ...-V?DA +B* 4 2 1 We have also, pn-2i-l(rfc(^_|_i;)==2( — 2) 2 *i) 2 *Ak, where Ak is the invariant, (159) Ak = (a6)n 1 2k(ary}(by') · (aa')2(aa")2 ··· (aaa))2 . (j/3')W')2... (6/3^>)2.FUNDAMENTAL SYSTEMS 155 In this case, B=( - 2 +(n - 2) ( - 2 By^A1 + (n - 3) (n - 4) ( _ 2 jyfT (159l) JL · A 4- ... — n A- ^ D.An_8-\-An_lt ^ 2 2 Thus we have the following: Theorem. The resultant of a form of second order with another form of even order is always reducible in terms of in- variants of lower degree, but in the case of a form of odd order this is not proved owing to the presence of the term An_v ~2~ A few special cases of such resultants will now be given; O), (ft), 00, 0*). (a) n = 1: R = A0, A0 = (a«)2. (ft) n = 2 : R= — DA0 + B2, A^ = (aft)2, B = (a<*)2. R = - (a/3)2(aft)2+ (a<*)2(ft/3)2. (c) n= 3: R =-2DA0 +Av A0 = (ab)\ay)(by). Ai=(af)(by)(aa)Xbpy. R= - 2(<*/3)2(aft)2(a7)(67) + (a7)(ft7)(aa)2(ft/3)2. (d) n=4:R=2D2A0-4DA1 + B2, A0 = (ab)i ^1=3=(aft)a(aa)2(6/8)a. J5 = (aa)2(aa')2. R = 2(a/3)2(af/3')2(aby-4(a/3)2(ab)2(aaf)2(b/3')2 + (aa)2(aa')2(b/3y2(b/3')2. SECTION 4. FUNDAMENTAL SYSTEMS FOR SPECIAL GROUPS OF TRANSFORMATIONS In the last section of Chapter I we have called attention to the fact that if the group of transformations to which a form ƒ is subjected is the special group given by the trans- formations156 THE THEORY OF INVARIANTS sin(û) — a) , , sinO» — 8) , sin a , , sin B , *¿1------: n : ’ x2 — · « : xo’ sin co sin a> sin © sin co then q — x\ + 2 a^cos to 4- af, is a universal covariant. Boole was the first to discover that a simultaneous concomitant of q and any second binary quantic ƒ is, when regarded as a function of the coefficients and variables of ƒ, a concomitant of the latter form alone under the special group. Indeed the fundamental simulta- neous system of q and ƒ taken in the ordinary way is, from the other point of view, evidently a fundamental system of ƒ under the special group. Such a system is called a Boolean system of/. We proceed to give illustrations of this type of fundamental system. I. Boolean system of a linear form. The Boolean system for a linear form, I ·—- cLqQz·^ 4" d^x2^ is obtained by particularizing the coefficients of ƒ in Paragraph I, Section 1 above by the substitution (K h ’ \1, coso), 1/ Thus this fundamental system is l = a0xx 4- axxv q = x2+2 xxx2 cos co 4- x\, a = sin2 <», b = (a0 cos co — a^)xx 4- ([a0 — at cos co^xT c =: 0%— 2 a0ax cos co 4- a\. II. Boolean system of a quadratic. In order to obtain the corresponding system for a quadratic form we make the above particularization of the b coefficients in the simulta- neous system of two quadratics (cf. Section 1, III above).FUNDAMENTAL SYSTEMS 157 Thus we find that the Boolean system of/is ƒ = a0zf + 2 axx^x2 + a2x|, q = + 2 ^2^2 cos ft) + #|, D = 2(A0a2 d = sin2 g), 0 = £*0 4- ¿*2 — 2 ax cos g), X = (a0 cos ft) — 4* («0 ~ ^2)^2 + (ai“" a2 cos ®)^i· III. Formal modular system of a linear form. If the group of transformations is the finite group formed by all transformations Tv whose coefficients are the positive residues of a prime number p then, as was mentioned in Chapter I, L = x\x2 — x\x\ is a universal covariant. Also one can prove that all other universal covariants of the group are covariants of L. Hence the simultaneous system of a linear form l and X, taken in the algebraic sense as the simultaneous system of a linear form and a form of order p 4- 1 will give formal modular invariant formations of L We derive below a fun- damental system of such concomitants for the case p = 8. Note that some forms of the system are obtained by polarization. Let ƒ = a0xx 4- axx2 ; p = 3. The algebraical system of ƒ is ƒ itself. Polarizing this, o = a*-“· (■ V = °, (a3-)B = A2 da) \ da) (mod 3), are reducible. The polar Of is also reducible. In fact, Cf = CQ —fl? (mod 3). The formal fundamental system of/modulo 3 is A, B, a i>, E, ƒ, L, Q. SECTION 5. ASSOCIATED FORMS Consider any two covariants v (¡>2 of a binary form f(xv x2) of order m. Let the first polars of these be or where >· = 4>ul4>iv, p — 24v x = xiÿi + y = HDx + HVv 2). n QXi p dx{ (1590 Let the equations (lSOj) be solved for yv y2. Then if J is the Jacobian of the two covariants (f>v 2, the result of substituting yv y2 for xv x2 in f(xv x2) is 1FUNDAMENTAL SYSTEMS 159 and the forms A0, Av ··♦, An are covariants of ƒ, as will be proved below. But the inverse of (159j) constitutes a linear transformation on the variables yv y2 in which the new variables are X, p. Hence if **> x%) = ^1’ ***’ 4*2)· Thus every covariant of ƒ is expressible rationally in terms of the m + 3 covariants of the system *^■0’ ^1’ ^2’ 4>v $2’ Such a system of covariants in terms of which all covariants of a form are rationally expressible is called a system of associated forms (Hermite). The expression for f(yv y%) above is called a typical representation of ƒ. Now we may select for in this theory the universal covariant $2 y = and then the coefficient covariants A0, Av ··· can be given in explicit symbolical form. First, however, we obtain the typical representation of ƒ as an expansion based upon a formal identity. From x = \yx + \y2, p = ptyt + p$v i.e. X = Xv, p = Py ; and ƒ = a™, we have the identity (\p)ay = (ap'yx — (aX)p.160 THE THEORY OF INVARIANTS If we raise both sides of this identity to the mth power we have at once the symbolical representation of the typical representation of/, in the form (V)m/Or */2) = BoXm ~ mB1Xm~'n+ ... + (- V)mBmnm, where -Z?o — B^ — (atiy B2 — (afl)m 2(#X)2, ···, Also (A,fx)m = Jm. Now with /i = (xy) we have J— 4" ^2^2 — 01’ by Euler’s theorem. Moreover we now have Bq = =/ A = B2 = an~\aX)2 ··., for the associated forms, and 0OO’ «i, and <#>(«0’ «1» 5 ÿr “ tj: 0C®o* Bv B2, ··· ; X, /x), 0i ; iTj, x2') = -— 0(/, “-^2’ ’ 01’ 0). 0i * Again a further simplification may be had by taking for 0j the form ƒ itself. Then we have B0 =ƒ, = (a6)a?-1J?‘1 = 0, B2 =(ab)(ac')a™-2b™-1c%-\ ... and the following theorem : Theorem. If in the leading coefficient of any covariant we make the replacements f a0, ^*2’ ^3’ *** V0l( = ƒ}’ -®lC= -®2’ -®8’ and divide by a properly chosen power of 0j( = ƒ) we have an expression for as a rational function of the set of m associated forms 0i( — ƒ)’ -®i(~ 0)’ -B2, -B3, ····FUNDAMENTAL SYSTEMS 161 ‘ For illustration let m = 3, ƒ being a binary cubic. Let $ be the invariant It. Then since B2 = ( the invariant -%R= («„«3 - «1«2>2- 4(a0a2 - ai)(ala3 - «!)* and substitutes there results That is, / aq, ct], is such a function that when/, g, A, ··· are replaced by /' = £i/+ *7i#+ ···, / =£2/+*72^ +£2^+ ···’ hf = £3ƒ + *73 (a0' av K ■■■ ■ ; *c Sv Vv D = (£*??···) = ^3’ <5 <5 CO to ?2’ £3’ (161) then is called a combinant of the set (Sylvester). We have seen that a covariant of ƒ in the ordinary sense is an invariant function under two linear groups of trans- 162COMBINANTS AND BATIONAL CURVES 163 formations. These are the group given by T and the in- duced group (23x) on the coefficients. A combinant is not only invariantive under these two groups but also under a third group given by the relations ao = £iao + Vih + + ···, ai = £iai “b Vih + -f- ···, ^0 = ^2a0 “b V2^0 "b %2C0 “b ···> (162) As an illustration of a class of combinants we may note that all transvectants of odd index off and g are combinants of these forms. Indeed %2f+V2&yr+l = £A(/,/)2r+1 + (^)(/, <02r+1+W2U 9Yr+l (163) = QvXf #yr+\ by (79) and (81). Hence (ƒ, #)2r+1 is a combinant. In- cluded in the class (163) is the Jacobian of ƒ and g, and the bilinear invariant of two forms of odd order (Chap. Ill, V). II. Theorem. Every concomitant, <£, of the set f g, A, ··· which is annihilated by each one of the complete system of Aron- hold's polar operators is a combinant of the set. Observe first that is homogeneous, and in consequence where it is the partial degree of in the coefficients a of f i2 the degree of <\> in the coefficients of g, and so forth.164 THE THEORY OF INVARIANTS Since = then ~ Thus 7 \ 5c£/ + ({,«, + ■),».+ "■+"-'»>S({l„, + „50+...+Vt) + (fl«i + ,lA + + H-------hf ^(^2an + V2^n + "· +a2en) = 0. & y «<: d'____d(&«»+ ··· +0 . (167) \ dh dVt d where i is the total degree of in all of the coefficients. In (167) we have m2 equations given by («, t = 1, ···, m). We select the following m of these and solve them for the deriv- atives^-, .. dd>f df . ,f £i-^r + 1?! a + *" +(riTLz=ih4>> f’l+<+-+^=0' 9^ M Vm dni a^_. 5(7! £m + *?m ^ + ·*· +GTm-^- = 0. (168)COMBINANTS AND RATIONAL CURVES 165 Solution of these linear equations gives dB W 8D *fx (fr»?-)1* «*x (fri?-) *Ti (£??-) But we know that #' = ¥ #1 + IT rfoV d£x 3>?x ^x Hence ,,, /52) 52) , , 52) , ¿2). ,f —s'#· Hence we can separate the variables and integrate: d' _ . dD · ** D ’ > =BitJF(a0, ·.·), (169) where J7 is the constant of integration. To determine J7, particularize the relations (162) by taking all coefficients £, 77, ··· zero except = *?2 == * * ’ = = 1· Then af0 = a0, a*, = av ··., = b{, etc., and (169) becomes = F. Hence f = 2),1<£>, which proves the theorem. It is to be noted that the set (168) may be chosen so that the differentiations are all taken with respect to %k, rjk, .·· in (168). Then we obtain in like manner (f)f =166 THE THEOEY OF INVAEIANTS That is, a combinant is such a simultaneous concomitant that its partial degrees in the coefficients of the several forms are all equal. This may be proved independently as the III. Theorem. A combinant is of equal partial degrees in the coefficients of each form of the set. We have Hence =(*! - = 0. Thus ix = iv Similarly iy = ik (/, k = 1, 2, m). IY. Theorem. The resultant of two binary forms of the same order is a combinant. Let /=/Ov xt)·, 9 = 9(xv xi)· Suppose the roots of ƒ are (rf, 4°) (* = 1, .··, «), and of g (®i’\ *2’) (* = 1, ··· w). Then the resultant may be indicated by R = g(r['\ r™)g(r?\ 4») - g(r?\ rj»>), and by R = f(s?\ O/Oi* 42») ···ƒ«, 4n)). Hence (« r) = 0, (J *242)) -/or. 4n0 = 0. Thus It is a combinant by Theorem II. Gordan has shown * that there exists a fundamental combi- nant of a set of forms. A fundamental combinant is one of a set which has the property that its fundamental system of * Mathematische Annalen, Vol. 5.COMBINANTS AND RATIONAL CURVES 167 concomitants forms a fundamental system of combinants of the set of forms. The proof of the Theorem II of this section really proves also that every combinant is a homo- geneous function of the determinants of order m, *kx bjc\ Cjci *■ ·· 4. bk2 * ' <*km bfcm ^A:m that can be formed from the coefficients of the forms of the set. This also follows from (162). For the combinant is a simultaneous invariant of the linear forms £ak + V^k + *+■ ··· + ch (Jk = 0, 1, ···, w), (170) and every such invariant is a function of the determinants of sets of m such linear forms. Indeed if we make the substitutions V = ViSr + Vtflf H-----h Vm and these are precisely the equations (162). For illustration, if the set of %-ics consists of ƒ = a0x% + 2 axxxx 2 + a2x% 9 = b0xf + 2 btxxx2 + Vi’ any combinant of the set is a function of the three second order determinants («0*1 aA)’ (^0^2 ^2^o)’ (®1^2 **2^l)* Now the Jacobian of/and <7 is J = (a/q - a^o)^ 4- (a0b2 - Vo>i^2 +0*A ~ a2hi)xl·168 THE THEORY OF INVARIANTS Hence any combinant is a concomitant of this Jacobian. In other words J is the fundamental combinant for two quadratics. The fundamental system of combinants here consists of J and its discriminant. The latter is also the resultant of ƒ and g. The fundamental system of combinants of two cubics ƒ, g, is (Gordan) * = (ƒ, g)* 0 = (ƒ, g)3, A = (tf, tf)*, 0?, #)*, (A, tf), (A, #)*. The fundamental combinants are # and 6, the fundamental system consisting of the invariant 6 and the system of the quartic & (cf. Table II). V. Bezout’s form of the resultant. Let the forms ƒ, g be quartics, f=aQx{ + a1xlx2+ ··., 9 - b2a;f + + Ml’ + atx^ 4- a^x\ _ a^xA + a4x2 % bQx\ + bxxxx2 + ¿2^2 Kxi + ^4X29 atftf 4- axx^x2 4- a^x\ + _ ^4 Vi + bxx\x2 4- b2xxx\ 4- bzx\ b± Now we clear of fractions in each equation and write We then form the éliminant of the resulting four homoge- neous cubic forms. This is the resultant, and it takes the formCOMBINANTS AND RATIONAL CURVES 169 B = Poi P02 Poz Pot Po2 P*Z+Pl2 ^04+^13 PU Poz Pot + Piz PU+P2Z Pm P04 Pu Pm Pzi Thus the resultant is exhibited as a function of the deter- minants of the type peculiar to coinbinants. This result is due to Bezout, and the method to Cauchy. SECTION 2. RATIONAL CURVES If the coordinates of the points of a plane curve are rational integral functions of a parameter the curve is called a rational curve. We may adopt a homogeneous parameter and write the parametric equations of a plane quartic curve in the form X1 =alo£i + all?l?2 + *·· + a14%2 =fl &)’ x2 =a2ofcl "b a21%V*2 “b ··· + a24^2 = «/*2 1’ £2)’ XZ = aZ0%\ “b aZ\%\%2 “b ·*' + «34^2 ~ fZ Cfl’ ?2)· We refer to this curve as the B4, and to the rational plane curve of order n as the Rn. I. Meyer’s translation principle. Let us intersect the curve B4 by two lines ux = nxxx + u%x2 -f usxs = 0, vx = vxxx -h v2x2 -f v%xz = 0. The binary forms whose roots give the two tetrads of inter- sections are uf = (a10^ + a20u2 + tf30^3)£! + (aniiA + <*21^2 + aziuz)%i%2 “b (^12^1 ”b ^22^2 “b ^32^3)?l^2“b C^13^1"b ^23^2 "b aZZ^z)£l?2 + Ca14ul + «24^2 + aZ4Uz)%V and the corresponding quartic vf. A root I2O °f uf = 0 substituted in (1702) gives one of the intersections (x{4\ x{2\ xg}) of ux = 0 and the B4.170 THE THEORY OF INVARIANTS Now uf = 0, vf = 0 will have a common root if their result- ant vanishes. Consider this resultant in the Bezout form R. We then have, by taking aiu = «1,4*1 + a>iU2 4- a3ius (i = 0, ···, 4), Pik — Q'iuQ'kv ^iv^ku' Thus Pik = (uv')1(a2iask - a2ka3i) + (uv)2(a3ialk - aua3k) + (wv)8(«w have the combinant K = («063 - asb0~) - K«A - aA~) ■ When iT= 0 the cubics are said to be apolar. The rational curve M3 has, then, the covariant curve K(x) = | xa0a3| — |· | xaxa2 | = 0. This is a straight line. It is the locus of the point (ux, vx) when the lines ux = 0, vx = 0 move so as to cut i?3 in apolar point ranges. It is, in fact, the line which contains the three inflections of i?3, and a proof of this theorem is given below. Other theorems on covariant curves may be found in W. Fr. Meyer’s Apolaritat und Rationale Curven (1883). The process of passing from a binary combinant to a ternary covariant here illustrated is called a translation principle. It is easy to demonstrate directly that all curves obtained from combinants by this principle are covariant curves. Theorem. The line K(x) = 0 passes through all of the inflexions of the rational cubic curve _S3. To prove this we first show that if g is the cube of one of the linear factors of ƒ = a(r1)«(r2)a(r3), 9 = «A + <4V%2 )3’172 THE THEORY OF INVARIANTS then the combinant K vanishes identically. In fact we then have 60 = «a>3, 51 = 3<>2<4V**> and a0 = aj1)aj2)aj8), ax = 2aj1)aj2)a^3), ···. When these are substituted in K it vanishes identically. Now assume that ux is tangent to the Rz at an inflexion and that vx passes through this inflexion. Then uf is the cube of one of the linear factors of vf, and hence K(x) vanishes, as above. Hence K(x) = 0 passes through all inflexions. The bilinear invariant of two binary forms ƒ, g of odd order 2 n 4-1 = m is Km = a0bm - + ^a2im_2 +--------\- mam_lbl - am60, or + (^jp2m-2 -···+(- l)n(^Pnn+V where ƒ = a4- ma1x^~1x2 + ···. If two lines ux = 0, vx = 0 cut a rational curve Rm of order m = 2 n 4-1 in two ranges given by the respective binary forms uf, vf, of order m, then in order that these ranges may have the projective property Km = 0 it is necessary and sufficient that the point (ux, vx) trace the line (?) = 0. This line contains all points on the Rm where the tangent has m points in common with the curve at the point of tangency. The proof of this theorem is a direct extension of that above for the case m = 3, and is evidently accomplished with the proof of the following:COMBINANTS AND RATIONAL CURVES 173 Theorem. A binary form, f of order m is apolar to each one of the m, m-th powers of its own linear factors. Let the quantic be m ƒ = ax = «O2? + ·" = n(7f- rfx^). j=1 The condition for apolarity of ƒ with any form g = b™ is (aby = a0bm - + ···+(- l~)mamb0 = ( f, g)m = 0. But if g is the perfect m-th power, g = (r^)xl — == (xr(J))m, we have (cf. (88)) (ƒ> | — || a0axx 11 aBa^x | + £|a0a2#| | a2a^x | — ^2\aia2x\\a2asx\= 0. If iF = 0 the quartic F is said to be self-apolar, and the curve Cr(#) = 0 has the property that any tangent to it cuts the i?4 in a self-apolar range of points.CHAPTER Vili SEMINVARIANTS. MODULAR INVARIANTS SECTION 1. BINARY SEMIN VARIANTS We have already called attention, in Chapter I, Section 1, VIII, to the fact that a complete group of transformations may be built up by combination of several particular types of transformations. I. Generators of the group of binary collineations. The infinite group given by the transformations T is obtainable by combination of the following particular linear transfor- mations : t:x1 = \x, z2 = /xy, h : x = a/ + vÿ, y = t^.x' — x'v y* = 1+2a'i'kt+ ···+w<-1 ¿)/=il'I(a»’ (m) But = 0 is a necessary and sufficient condition in dv order that I(a^ *··, a^) may be free from i% i.e. in order that J«, ···) may be unaffected when we make v = 0. But when v = 0, aj. = a;· and I••*9 #in) = /(S, **> #/»)· dJ~ Hence ^- = ...)= 0 is the condition that /(a^, ···) be a seminvariant. Dropping primes, ill (a0, ···) = 0 is a nec- essary and sufficient condition that /(a0, ···) be a sem- invariant. IV. Formation of seminvariants. We may employ the operator il advantageously in order to construct the sem- invariants of given degree and weight. For illustration let the degree be 2 and the weight w. If w is even every sem- invariant must be of the form 1= a0aw + \lalaw-1+ + ··· + Then by the preceding theorem ill= (w + X^o^-i + (O — l)Xx + 2 \)axaw_2 + ··· s 0. Or w -f- Xj — 0, (w — l)Xj -b 2 X2 = 0, (w — 2)X2 -f- 3 Xg = 0, ···, ^ "b 1 -b ^X^w == 0. Solution of these linear equations for Xr X2, ··· gives "b^2^^2^-2 ***178 THE THEORY OF INVARIANTS Thus there is a single seminvariant of degree 2 for every even weight not exceeding m. For an odd weight w we would assume I— a0aw + Xjdqa^j + ··· + i)a$(w-Da%(w+i)' Then ill= 0 gives w + \1=:0, (w — l)Xx + 2 X2 = 0, ..., + 3)X^_3) + ±(w — 1)X^(M,_1) = 0, X*(IIMl) = 0. Hence Xj = \ = ... = X^(ir_1)== 0, and no seminvariant exists. Thus the complete set of seminvariants of the second degree is A = a0’ A. ^ = ^0^2 ^1’ == (IqCi^ 4 ci^cIq + 3 #2, A = aoa6 — ^ + 15 a2a4 — 10 a% == ^ + 28 $2^6 — 56 ^3+ 35 ¿?4. The same method may be employed for the seminvariants of any degree and weight. If the number of linear equa- tions obtained from ill = 0 for the determination is just sufficient for the determination of Xx, X2, X3, ··· and if these equations are consistent, then there is just one seminvariant I of the given degree and weight. If the equations are in- consistent, save for X0= X4 = X2= ... =0, there is no semin- variant. If the number of linear equations is such that one can merely express all X’s in terms of r independent ones, then the result of eliminating all possible X’s from I is an expression 1= Xj/j -f- X2J2 + ... + \rIr. In this case there are r linearly independent seminvariants of the given degree and weight. These may be chosen as l-V ^2’ ***’ -i*·SEMINVARIANTS. MODULAR INVARIANTS 179 V. Roberts’ theorem. If C0 is the leading coefficient of a covariant of ƒ= a0x™ + •••of order o>, and is its last coeffi- cient, then the covariant may be expressed in the forms M + OCq t ' t/2’ (173) 91 P»>x!» + Q—hr2 + ··· + Ojq. (174) |ft) — 1 |1 Moreover, (70 is a seminvariant and 0^ an anti-seminvariant. Let the explicit form of the covariant be K= + ... + Cjq. Then by Chapter I, Section 2, XII, Or iiCtf? + ©(11(7! - + - 2 Gi)^~24 + - +©(ii(710_1 —©— 1 C^^x^-1 + (001, — (oC^yx^ == 0. Hence the separate coefficients in the latter equation must vanish, and therefore ii o1 = O0, ii<72 = 2 Ov 12 <7„—i = (*» — 1) ii cH = » cm_v The first of these shows that C0 is a seminvariant. Combin- ing the remaining ones, beginning with the last, we have at once the determination of the coefficients indicated in (174).180 THE THEORY OF INVARIANTS In a similar manner and this leads to OC0=œCv OO^Co)-^,..., 00^= C„, 0(7,= 0; 1 0,= o)(co — !)(&> — 2) ··· (a) — i + 1) O^0(i=0, o>). This gives (173). It is evident from this remarkable theorem that a co- variant of a form ƒ is completely and uniquely determined by its leading coefficient. Thus in view of a converse theorem in the next paragraph the problem of determining covariants is really reduced to the one of determining its seminvariants, and from certain points of view the latter is a much simpler problem. To give an elementary illustration let ƒ be a cubic. Then 0 = 3 a1-+ 2 a2~^~ 4- a3 1dafi sda« o and if (70 is the seminvariant a0a2 — a\ we have O(70 = a0a3 — axa2, O2(70 = 2— «¡), O^ = 0. Then 2 if is the Hessian of ƒ, and is determined uniquely from (70. VI. Symbolical representation of seminvariants. The sym- bolical representation of the seminvariant leading coefficient (70 of any covariant Kof/, i.e. K=(aby(ac')« ... arxh*xéx ... (r + 8 + t + ··· = ©), is easily found. For, this is the coefficient of X\ in K\ and in the expansion of (o6)*(a be the seminvariant and •••)=^(ao* ···) the seminvariant relation. The transformed of by is ƒ = Oi^i + a2x2)m h · ~b ^2’ *^2 = *^2’ f = + («^ + «2)4]”· If the a0, av .·. in <£(a0, ···) are replaced by their symbolical equivalents it becomes a polynomial in cq, oq, ··· say F(av a2, fiv /32, ...). Then («7> (#y)i ·— In fact, treating SJF=0 as a linear equation with constant coefficients (ax, /3X, ··. being unaltered under tx) we have the auxiliary equations da2 _ _ dy2 _ _ dF «1 /3j 7X 0 Hence jF is a function of (a/3), (ay), ··· with constant coeffi- cients which may involve the constants ax, —. In other words, since <£(a0) = F(av ···) is rational and integral in the a’s F is a polynomial in these combinations with coefficients which are algebraical rational expressions in the ax, /3X, ···. Also every term of such an expression is invariant under tv i.e. under «'l = «r «2 = alv + «21 —1 and is of the form ro = (*/8)*(«7)* - required by the theorem. We may also prove as follows : Assume that F is a func- tion of (a/3), (ay), (aS), ··· and of any other arbitrary quantity s. Then dF a —. = a, ^«2 1 8 8 *3/8. * dF 3(«/3) dF d(a&) dQfl) , a dF d(ocy) dF ds da,2 1 d ( ay ) 1 ds da2 d(«8) , o dF d(ay) dF ds dfi2 15 (“7) d 82 1 ds d& etc, But dF aid(a8) 8 dF 1 d(<*8) d<*2 -«A dQ/3) d82 = + «A dF ¿>0/3)’ dF k«&ySEMIN V ARI ANTS. MODULAR INVARIANTS 183 Hence by summing the above equations we have 8F = ÌL + ff ** + ...') = M's* = 0 ds\ 1do, 13/3, J ds Since 8 is entirely arbitrary we can select it so that Ss =£ 0. dF Then — = 0, and Fy being free from $, is a function of the required combinations only. Theorem. Every seminvariant off of the rational integral type is the leading coefficient of a covariant off. It is only required to prove that for the terms T0 above w = p -f- a -f ··· is constant, and each index p, (a0, ···).* We have 2Γo = 2(α£)l,.(αy)l,,... — = (a0, ··.). If the degree of is i, the number of symbols involved in T0 is i and its degree in these symbols im. The number of determinant factors (ay8) ··· is, in general, w = Pi P2 “h - +Pw-D’ and this is the weight of <£. The degree in the symbols con- tributed to ro by the factors (ay8) ··· is evidently 2 w, and we have p, 2 w, that is, (ù = im — 2 w > 0. For a more comprehensive proof let d = «2-—l· *01184 THE THEORY OF INVARIANTS Then &d — dS = ax da, + ~a2'Sr~ ^-¿~ + a/?! ‘¿>«2 *d/32 Hence, since T0 is homogeneous in the symbols we have by Euler’s theorem, (&d — c?8)ro = (w + - l)<2ro, (M* - dS*)ro = *(»-* + (* = 1, 2,...), But 8T0 = 0, hence &dkT0 = &(g> — k 4- Y)dk~lT0, Also <2^ = daf~iai2 = (m—i)ai+l = Oai (i=0,1, • ••,m —1), d = ^-da o + |^ + ··· + = 0. da0 Bat Ba^ Hence ePT0 is of weight w 4- k. Then . dim~w+l T0 = 0. For this is of weight im -f 1 whereas the greatest possible weight of an expression of degree i is im, the weight of a Now assume oo to be negative. Then dim_wT0 = 0 because = (im — w + 1) [g> — (im — w + 1) 4-1 ]cfm““T0 = 0. Next dt‘m-w’~lro = 0 because 8diwi_MT0 = (im — w') [&) — (im — w') 4-1] = 0. Proceeding in this way we obtain ro = 0, contrary to hy- pothesis. Hence the theorem is proved. VII. Systems of binary seminvariants. If the binary form ƒ = -f maxx^~xx2 + ··· be transformed bySEMINVAEIANTS. MODULAR INVARIANTS 185 there will result, f = + m01x[m-%+(^Sj02x[m-2x'22 H— + (7ro4™ in which Oi = d(jV' + ia^-1 + ^ «2l'<_2 + · · · + iai-\v + ai- (176) Since i!O0 = fla0= 0,00 is a seminvariant. Under what cir- cumstances will all of the coefficients (i = 0, ···, m) be seminvariants ? If 01 is a seminvariant ilOj = il(a0 v -f aj) = a0ilv + aQ = 0. That is, flv = — 1. We proceed to show that if this condi- tion is satisfied il Oi = 0 for all values of i. Assume £lv = — 1 and operate upon Oi by il. The result is capable of simplification by ili^ = *I',_1ill' = — 8V*~\ and is D.O, = - ia/~l - ^ j J (i - l)a1i/1'-2------------------------- — i&i-i + a0vl~l + 2 + ··· + {r + l)(r + V)a^^1 + - + iai-1· But , t + 1) - ■-C--l)-C<-’- + 1Xi-r) _Q(< _ r). Hence il 0i = 0. Now one value of v for which £lv = — 1 is v = — ^. If ƒ be transformed by / (t*« / / *1 = ^1------------1 4’ ^2 = XV then 01 = 0, and all of the remaining coefficients Oi are sem- invariants. Moreover, in the result of the transformation,186 THE THEORY OF INVARIANTS T ¿= «o_1 = «o-la» (0 I«à x -iai + i l«0 3«i-2a 2 1 +c- i)^(ywi~2+c - - i)«i = S ( - l)f+ (- L'V - !)4- r=0 \ / This gives the explicit form of the seminvariants. The trans- formed form itself may now be written -v22+Ç)^|4m“343+ - + X fm 2 · Theokem. Every seminvariant of f is expressible ration- ally in terms of ro, T2, T3, ···, Tm. One obtains this expression r by replacing ax by 0, a0 by T0, and afi=f= 0, 1) by —in the o original form of the seminvariant. Except for a power of T0 = a0 in the denominator the seminvariant is rational and integral in the vfi = 0, 2, ···, m) (Cayley). In order to prove this theorem we need only note that ƒ ' is the transformed form of ƒ under a transformation of de- terminant unity and that the seminvariant, as is invarian- ti ve under this transformation. Hence s r0,0, L L ro’ H’ —— ) = S(a0 r r'V v „), (177) which proves the theorem. For illustration consider the seminvariant S = a0a4 — 4 axaz -h 3 a|. This becomes '8,=L(3ri + r4), 1 0 or S = a0aé — 4 axaz + 3 a\ = \ [3(«oa2 - ai)2+ OK - 4 «0ala3 + 6 «0aia2 “ 3 4)]· aoSE MIN V ARI ANT 8. MODULAR INVARIANTS 187 This is an identity. If the coefficients appertain to the binary quartic the equation becomes (cf. (125)) 1 a\i = 3 T| + T4. Again if we take for S the cubic invariant J of the quartic we obtain a0 0 0 -r. «0 ' — r — 1 2 a0 -r, /»2 * u0 1 Y a0 2 — r 9 1 3 a0 J_r * or laK=r2r4-r|-r|. Combining the two results for i and J we have r2r4 = 2 ^ -^2 = 6" ^2 + Now 2 T2 is the seminvariant leading coefficient of the Hessian H of the quartic ƒ, and T3 is the leader of the co- variant T. In view of Roberts’ theorem we may expect the several covariants of ƒ to satisfy the same identity as their seminvariant leaders. Substituting for T2, T for and ƒ for a0, the last equation gives H* + 2 T* - l if*H= 0, which is the known syzygy (cf. (140)). SECTION 2. TERNARY SEMINVARIANTS We treat next the seminvariants of ternary forms, the ternary quantic of order m be Ira Let f=X- m. m0 ms 0»1 + = "O’ When this is transformed by ternary collineations, Xl = + fJ-lX2 + vlxSi V: x2 = \x\ + /i2*2 + V3> x3 = X3X1 + hAl + V3> (XfiV) = °»188 THE THEORY OF INVARIANTS· it becomes/', where the new coefficients a' are of order m in the X’s, /x’s, and v’s. This form ƒ may be represented sym- bolically by ƒ = < = («1^1 + <*2X2 + <*3^3)™ The transformed form is then (cf. (76)) f = (akx i + a^2 + a,4)- (178) a^a^a^x^x^x38 (mx + m2 + m3 = m). Then we have Xr — nminm2am'. apapav - Now let ('1. d\ d d , d 1 + ^2ôx2 + ^3ô\3’ = ^ + (cf. (58)). 29\2 ‘ "*d\s d\J 1d\1 Then, evidently (cf. (75) and (23x)) \m ( ô W d\m» Vsk) < («H+ ». + »»—»)· (179) This shows that the leading coefficient of the transformed form is a™, i.e. the form ƒ itself with (#) replaced by (X), and that the general coefficient results from the double ternary polarization of a™ as indicated by (179). Definition. Let be a rational, integral, homogeneous function of the coefficients of/, and r the same function of the coefficients of ƒ'. Then if for any operator ^/x^-^, (xi) ···, say for the relation is true, is called a seminvariant of/.SEMINVARIANTS. MODULAR INVARIANTS 189 The reader should compare this definition with the ana- lytical definition of an invariant, of Chapter I, Section 2, XI. I. Annihilators. A consequence of this definition is that a seminvariant satisfies a linear partial differential equation, or annihilator, analogous to il in the binary theory. For, (*AV... ... \ dfij darm00\ dfi J da'm \ dfi J = (*■ = m2a^+1a^~la^ = i»2«m1+i Mi_x ^ and dsj/t X W dp Hence = S wlaa“.+» ».-i — = 0(m1 + m2 + ms = m) ' ^ mi ^WjWgWig (1 Now since the operator 5) m2a%+l ro2-l msT-j rm 'JU/mxmjn9 annihilates then the following operator, which is ordinarily indicated by is an annihilator of . (180) ^m2am1+l m2-l w3 (m2 4- m2 + m3 = m) (181) "mxintfiiz % The explicit form of a ternary cubic is ƒ= «300*1 + 3 «210*1*2 + 3 «120*1*1 + «030*2 + 3 «201*1*3 4- 6 a^x^x3 4- 3 «021^2^3 “b 3 ^102^1^3 + 3 «012^2^3 “b ^003^3* In this particular case ^*2*1 — ^300 3 , 0 3 q 3 , 3 + J a2io 7“---r O am —— 4- «201; 3a2io } da. 120 }dat 0 3 3 + 2am^T + iil02^ '030 3a in '021 5 3a, 012 (182)190 THE THEORY OF INVARIANTS This operator is the one which is analogous to fl in the /, by like processes, one obtains the analogue of 0, e.g. ilw Similarly ilXA, iiVi, flw may all be derived. An independent set of these six opera- tors characterize full invariants in the ternary theory, in the same sense that il, 0 characterize binary invariants. For such we may choose the cyclic set ilw Now let the ternary m-ic form ƒ = amoox? + rnam_ llf/^r~1x2 + — + a0m0x^1 + “ l)«m-211^r“2:r2 H----^ aOr*-nX™~1')XZ +........................................................? binary theory. From r d_ d\ be transformed by the following substitutions of determinant unity : x1 = x\ - - 5s=iai^, arrM ^rnOO x2 = *^2’ X 3 = X r 3* (183) Then the transformed form ƒ' lacks the terms x^^x^. The coefficients of the remaining terms are seminvariants. We shall illustrate this merely. Let m = 2, ƒ = «200*1 + 2 «110*1*2 + «020*1 + 2 «101*1*3 + 2 «011*2*3 + «002*3' Then a20off ^ioo^i2 (^020^200 allo)X2 ^ (^011^200 a101^11o)^2^3 + (a002a200 ~ ai01 )^32* It is easy to show that all coefficients of ƒ ' are annihilated by il . w Likewise if the ternary cubic be transformed by a2l0 rJ _ ^201 rJ ^2’ x300 *300SEMINV ARI ANTS. MODULAR INVARIANTS 191 and the result indicated by «|û0ƒ'r = Amxf + 3 Amxffar2 + ···, we have ^-300 = «300’ (I®4) ^210 = ^120 = «30o(«300a120 “ a21o)’ ^■030 = ^ «210 — ^ ^210«120a300 d" «030«300’ ■^201 “ 0’ ^111 = «30o(«300«lll “ «210«20l)’ "^■021 = «300«021 «300«201a120 ^ «210«111«300 d~ ^ «210«201’ *^■102 = «30o(a300«102 «20l)’ -^•012 = «300^012 «300«102«210 ^ «300«201«111 d" ^ a20la210' ^■003 = ^ «201 ^ «300«201«102 d" «003«300* These are all seminvariants of the cubic. It will be noted that the vanishing of a complete set of seminvariants of this type gives a (redundant) set of sufficient conditions that the form be a perfect mth power. All seminvariants of ƒ are expressible rationally in terms of the As, since f is the transformed of/by a transformation of determinant unity. II. Symmetric functions of groups of letters. If we mul- tiply together the three linear factors of ƒ = + a<%2 + aÿ)xB)(afx1 + 4% + 4%) the result is a ternary cubic form (a 3-line), ƒ = a300^ d- ···· The coefficients of this quantic are a300 = 2otJ1)aj2)a^3) = 41)ai2)ai3)’ «210 = 2aj1)aj2)aÿ) = 41)42)<43) + 41)42M3) -f 41)42)«i3)’ «120 = 2<}42)<43) = <}42)43) + 41}42)43) + 41}42)43), «030 = 241}42)43) = 41)42)43), «201 = 2«j1)ai2)a33) = 41)ai2)43) d" «i1)42)«i3) d- 41}42)43), am = 241)42)43) = 41}42)43) -i- 41)42)43) + 41)42)43) + 4l}42)43) + 41}42)43) + 41)42)43),192 THE THEORY OF INVARIANTS «021 = = a^afaf + aa)«(2)«(3) + «u>a<2>a<3>, am = 2aj1)«|j2)aij3) = aj1)a^2>a^3) + «^1)aJ2,ci^3) + a^1)a|2)«J3), a0i2 = 2<4»<4»<43) = «f«'3\ aoo3 = 2<41)a82)as8) = «¿M2*®»8’· These functions 2 are all unaltered by those interchanges of letters which have the effect of permuting the linear factors of ƒ among themselves. Any function of the a[j) having this property is called a symmetric function of the three groups of three homogeneous letters, «>, ««>, «), of, «f, «f), Of, af, ). In general, a symmetric function of m groups of three homo- geneous letters, ccj, a2, a3, i-e. of the groups YiOf, «f, of), 72 («f, «f, 42)), 7“2m)’ «ro» is such a function as is left unaltered by all of the permuta- tions of the letters a which have the effect of permuting the groups yv 72’ Vm among themselves: at least by such per- mutations. This is evidently such a function as is left un- changed by all permutations of the superscripts of the a’s. A symmetric function of m groups of the three letters av a2’ a3’ every term of which involves as a factor one each of the symbols a(1), a(2), ···, a(m) is called an elementary sym- metric function. Thus the set of functions a310, a210, ··· above is the complete set of elementary symmetric functions of three groups of three homogeneous variables. The non- homogeneous elementary symmetric functions are obtained from these by replacing the symbols <43) each by unity.SEMINVARIANTS. MODULAR INVARIANTS 193 The number TV of elementary symmetric functions of m groups of two non-homogeneous variables «m,0,o’ am-i,i,o* ··· is, by the analogy with the coefficients of a linearly factorable ternary form of order m, JVr=m + m-h(m — l) + (m — 2)+ ··· + 2 + 1 = J m(m + 3). The jV equations = 2, regarded as equations in the 2 m unknowns «jr), <45) (r, $= 1, ···, ra), can, theoretically, be com- bined so as to eliminate these 2 m unknowns. The result of this elimination will be a set of \ m(m -f 3) — 2 m = \ m(m — 1) equations of condition connecting the quantities am_no, ··· only. If these a’s are considered to be coefficients of the general ternary form f of order m, whose leading co- efficient a003 is unity, the \m(m — 1) equations of condition constitute a set of necessary and sufficient conditions in order that ƒ may be linearly factorable. Analogously to the circumstances in the binary case, it is true as a theorem that any symmetric function of m groups of two non-homogeneous variables is rationally and integrally expressible in terms of the elementary symmetric functions. Tables giving these expressions for all functions of weights 1 to 6 inclusive were published by Junker* in 1897. III. Semi-discriminants. We shall now derive a class of seminvariants whose vanishing gives a set of conditions in order that the ternary form ƒ of order m may be the product of m linear forms. The present method leads to a set of conditional relations containing the exact minimum number \ m(m — 1) ; that is, it leads to a set of \m(m — 1) independent seminvariants of the form, whose simultaneous vanishing gives necessary and sufficient conditions for the factorability. We shall call these seminvariants semi-discriminants of the form. They * Wiener Denkschriften for 1897.194 THE THEORY OF INVARIANTS are all of the same degree 2 m — 1 ; and are readily formed for any order m as simultaneous invariants of a certain set of binary quantics related to the original ternary form. If a polynomial, fSm, of order m, and homogeneous in three variables (xv x2, #3) is factorable into linear factors, its terms in (xv x%) must furnish the (xv x2) terms of those factors. Call these terms collectively afx,, and the terms linear in xs collectively xzaf~x. Then if the factors of the former were known, and were distinct, say m m < = «00II (4'X - 4%) II (4°)» i=l i= 1 the second would give by rational means the terms in xs re- quired to complete the several factors. For we could find rationally the numerators of the partial fractions in the decomposition of a^~l/a^x, viz. Ih .(*) nm-\ ------ alx — i=1 u0x u00 i=l ^*2 X1 7 i *^2 and the factors of the complete form will be, of course, <{i) rp __ A»(0 rY'Xo. rfxi — 4% + aixs O' = 2» —> «»)· Further, the coefficients of all other terms in fim are rational integral functions of the r(i) on the one hand, and of the on the other, symmetrical in the sets (r|°, —r^\ a*). We shall show in general that all these coefficients in the case of any linearly factorable form are rationally expressible in terms of those occurring in a<£., af~l. Hence will follow the important theorem, Theorem. If a ternary form f3m is decomposable into linear factors, all its coefficients, after certain 2 m, are expressible rationally in terms of those 2 m coefficients. That is, in the space whose coordinates are all the coefficients of ternary forms of order m, the forms composed of linear factors fill a rational spread of 2 m dimensions.SEMINV ARI ANTS. MODULAR INVARIANTS 195 We shall thus obtain the explicit form of the general ternary quantic which is factorable into linear factors. Moreover, in case fSm is not factorable a similar development will give the theorem, Theorem. Every ternary form /3m, for which the discrimi- nant D of afx does not vanish, can be expressed as the sum of the product of m distinct linear forms, plus the square of an arbitrarily chosen linear form, multiplied by a u satellite ” form of order m — 2 whose coefficients are, except for the factor D~\ integral rational seminvariants of the original form fZm. A Class oe Ternary Seminvariants Let us write the general ternary quantic in homogeneous variables as follows: where fsm — aZc “h aTx -f- ag 2#§ _|_ ... _|_ afp — aioz™ 1 + aaxf- ^ + anx™-' 2^2 ... _|_ iXm-i (i = 0,1, 2, ···, m). Then write ^ ^______________ ^ ak II (4W ¿=1 (% = 41>r22) — r2m)); and we have in consequence, assuming that 2) =£ 0, and writing Jm ______ a0r<*'> - the results Hence also 'd< dxl -K=r<*>, x2^r»-i 3a,. (190) -«55*,, A2a»(i> = =-^Ja$*„ and from these relations we deduce the following : A,ff a* = ----__________= „ y a”a)0) — <(m-D m.. f-i ( —----------------------------------±----’ <191> „ ^or, L a0r(DV2) ”· a0r(»»-l) or, from (185) Ai/£„ (192) (_ ^ — 2... In (191) the symmetric function 2 is to be read with refer- ence to the r s, the superscripts of the r’s replacing the sub- scripts usual in asymmetric function. Let us now operate with Aa on both members of (191). This gives A-| A>2 Rn &£& - 2 aMDa£li) ··· r^-1} rf">y —-· #aa2------— 00 Vw ~~ rim~1) ~ rFvSEMINVARIANTS. MODULAR INVARIANTS 197 Let 2^ represent an elementary symmetric function of the two groups of homogeneous variables rv r2 which involves h distinct letters of each group, viz. = 1, 2, ..., K). Then we have SC(- ··· <‘®>> We are now in position to prove by induction the follow- ing fundamental formula: (- |m- 3- t\tD (194) = 2[(— ··· ... ri*+t)r] (s = 0, 1, ···, m; t = 0, 1, ··., m — $), where the outer summation covers all subscripts from 1 to m, superscripts of the r’s counting as subscripts in the sym- metric function. Representing by Jm-s-.t,t the left-hand member of this equality we have from (190) Vm_,_M=2 (-1) t+1 airMairW a] m—1 '1 r^"1) ntm n1m /»·($) x HM2) ··· rp q-2. nfm aQr(s~V (,S) m—s 1 r(*+l) This equals 2 2(- l)**1^ ... where 8 is a symmetric function each term of which involves t + 1 letters ^ and m — s — t letters r2. The number of terms in an elementary symmetric function of any number of groups of homogeneous variables equals the number of permutations of the letters occurring in any one term when the subscripts (here superscripts) are removed. Hence the number of terms in 2m_4 is \m — s 1 m — s — t\t* and the number of terms in S is (m — s -f 1)| m — s/\t\m — s — t.198 THE THEORY OF INVARIANTS But the number of terms in - ^+^r<+i) - rjr>) is \m — s + 1/1 m — s — t\t + 1. Hence S ~(t -h l)Sm_5+1, and so t “T 1 This result, with (193), completes the inductive proof of formula (194). Now the functions are evidently simultaneous in- variants of the binary forms afx, a!^, agT1· We shall show in the next paragraph that the expressions 1-m-s—t, t = Dast DJm_s_t t (s = 2, 3, ·· ·, w; t = 0, 1, ···, w s) are, in reality, seminvariants of the form /3m as a whole. Structure of a Ternary Form The structure of the right-hand member of the equality (194) shows at once that the general (factorable or non-fac- torable) quantic /3m(Z) =£ 0) can be reduced to the following form : fjfl nl· in—o ƒ*» = fl(rfxt-r{k)x2 + «*) + ]£ X(«·<~ t)· k= 1 (195) This gives explicitly the “satellite” form of /3w, with coeffi- cients expressed rationally in terms of the coefficients olfZm. It may be written m m—s s ^**-2 = X X(Ba>' - r i=2 N)^ v ■ !)*>»(»»-» I m-s-t\tj'1'1 m in — (196) 5=2 t=0SEMIN Y ARI ANTS. MODULAR INVARIANTS 199 Now the coefficients are seminvariants of fZm. To fix ideas let m=3 and write the usual set of ternary operators, — Æfti „ + ^^02 ^ 4~ f^ft3 ^ 4” ^ 4“ ^2-^01^1 ■ —MQa ' '-™da oaQ0 aoi oa02 da Iff &da - 4“ 21 11 da, '20 0x^=3 a™--------\-2a01-----ha™--------1" 2alft---l· #n~----ha, dat 01 Ô«,2 ’ “a«Ò8 da il da„ ^20 da« * 12 ^21 w v w u Ô Ô ■+2aio^+3aoo^;+«ii^;+2Si^;+«o2^< Jòa, '30 *10 ^21 ni 12 etc. Then J10 is annihilated by fl^ but not by fl X\X21 ¿01 is anni- hilated by f2#1& but not by fl*^, and Im is annihilated by but not by ilXjfCl. In general Im_s_ut fails of annihilation when operated upon by a general operator VLXix. which con- tains a partial derivative with respect to a8t. We have now proved the second theorem. The Semi-discriminants A necessary and sufficient condition that fZm should de- generate into the product of m distinct linear factors is that fim_2 should vanish identically. Hence, since the number of coefficients in fim_2 is \ m(m — 1), these equated to zero give a minimum set of conditions in order that fZm should be fac- torable in the manner stated. As previously indicated we refer to these seminvariants as a set of semi-discriminants of the form fZm. They are m—s—t, t D&gf (- Ar^A {Rm m_ ■8 — t 's=2, 3, t=0,1, ,m; } (197) ·, m — sj They are obviously independent since each one contains a coefficient (a8*) not contained in any other. They are free from adventitious factors, and each one is of degree 2m — 1.200 THE THEORY OF INVARIANTS In the case where m = 2 we have ^00“·“ a20 2 a{ '00 ^01 *01 2 a, 02 aoo «01 «02 + aio «11 0 0 «10 «Il This is also the ordinary discriminant of the ternary quadratic. The three semi-discriminants of the ternary cubic are given in Table V. In this table we have adopted the following simpler notation for the coefficients of ƒ : ƒ = a0a^ + axx\x2 + a2xxx\ 4- + b0x\xz 4- b]X]X2x§ 4- b2xpz 4- c0x+ cxx2: + doxl TABLE V ho 4 aja2a3b$ — a\bl -9a|ò* -j· 3 — d\b\ -(- 3 d2b2 — af&f 4* 6 did360^2 — 2 4- Gridai)0&1 4* 3 d2dsb0b1 — 4 a?a3&0&i 4“ did2bib2 — 9 d3b]b2 — afafco — 18 djd2d3CQ 4" 4 a2c0 4- 4 a?a3c0 4- 27 a|c0 —/oi a|a36§ — 3 &o 4- a?a3&i — 3 d2dÿb^ 4- dfb2 — 4 dYd2bI 4- 9 a36| 4- 9 — did2d3bQbi 4- 2 a{asb0b2 —■ 6 d2d^)()b2 4- 4 d2bib2 — 3 didsb1b2 — aia2ò1ò2 4" d[d\<\ 4-18 a1a2«3c1 — 4 a|cj — 4 -27 d\cY -loo albi 4- d^bçpl 4- — 2 ü2b(fil — d^sbfài 4· a%b\b2 — 2 dxdzblb2 4- 3 dj)1b2 — d]dçjbçf)-ÿb2 -a3b\ — d^b-Jy^ 4- Q>2^lp2 4- b2 4- 4* 18 did2d3diQ — 4 a2d0 — 4 a\d3dQ — 27 a|(?oSEMINVARI ANTS. MODULAR INVARIANTS 201 In the notation of (197) the seminvariants in this table are 100 = “h -®8» I\o = -®^20 "h 101 ~ -®«21 4" ^2^3’ where 2) is the discriminant of a = a^xf + «01^2 H----+ «ob2!» and ii3 the resultant of a and /3 = a10x f + anxtx2 + anx\. Corresponding results for the case m = 4 are the following: ^00 = 2T «4o(l Z*1 “ Jf> ■” where *1 — a02 ^ a01Æ03 "h 12 a00a04i ’ Jl = = 27 a^aM + 27 aooao3 4- 2 a§2 — 72 «00«02^04 9 «oi«02«03 ’ «10 «11 «12 » H-i CO 0 0 0 «10 *11 «12 «18 0 II 0 0 %«10 «00«11 ^02«10 "" «00«12 «10 «03«10 «00«13 «11 «04«10 «12 0 CO e ° «00 «01 «02 CO o « «04 0 0 «00 «oi «02 «03 «04 the other members of the set being obtained by operating upon i?4 with powers of Av A2: A, = 4 «, oo da ■ -f- 3 (X/ '01 10 5a H" 2 a02~ l· «03^ » n da12 ' mda13 A2 = 4 «04^ + 3 «03^ + 2 «02^ + a01£ X1S wu>12 according to the formula 10 4—s—t, t = a,tD |4 — s — t\t (8 = 2^ 3, 4 ; t = 05 1, ··♦, 4-*).202 THE THEORY OF INVARIANTS IV. Invariants of m-lines. The factors of a™x being assumed distinct we can always solve Im_s_ut = 0 for asi, the result be- ing obviously rational in the coefficients occurring in a%~1. This proves the first theorem of III as far as the case D =£ 0 is concerned. Moreover by carrying the resulting values of asi (* = 2, 3, ···, m; t = 0, 1, ···, m — s') back into fZm we get the general form of a ternary quantic which is factorable into linear forms. In the result agj., af~l are perfectly general (the former, however, subject to the negative condition D =£ 0), whereas (- )Da%-* = A ™-Œr \ m-j Am—j 1A 7? LI] | m—j — 111 + Am—j 7? 2 1 l’m \m — j x m—j 2. (j = 2, 3, ···, m). Thus the ternary form representing a group of m straight lines in the plane, or in other words the form representing an m-line is, explicitly, f=aZ + xzaf-1 m m—j A m—i—j A i T> + D~ 1( - iy™(™-l) Vx{ V 1----- 2 (198) j=2 i=0 1-----— I— This form, regarded as a linearly factorable form, possesses an invariant theory, closely analogous to the theory of binary invariants in terms of the roots. If we write <& = al = (aoo = O’ and assume that the roots of l0£ = 0 are — rv — r2, — r3, then the factored form of the three-line will be, by the partial fraction method of III (185), 3 /=n(*l + r‘Z2 ~ i=I Hence the invariant representing the condition that the 3-line ƒ should be a pencil of lines is 1 r2 Zi-r2/^_ra . 1 ^3 h-rjl^r^ Q =SEMINVARIANTS. MODULAR INVARIANTS 203 This will be symmetric in the quantities rv r2, r3, after it is divided by V i2, where It = (rl - r2)2(r2 ~ rs)\r3 - rlf is the discriminant of the binary cubic a$x. Expressing the symmetric function = Q/^fR in terms of the coefficients of aft# we have Q1 = 2 00!#!2 — ttQiaQ2an + 9 ^00^03^11 ® a01aQSal0 + ^ ^02^10 — 6 #00002#12· This is the simplest full invariant of an m-line ƒ. SECTION 3. MODULAR INVARIANTS AND COVARIANTS Heretofore, in connection with illustrations of invariants and covariants under the finite modular linear group repre- sented by Tp, we have assumed that the coefficients of the forms were arbitrary variables. We may, however, in con- nection with the formal modular concomitants of the linear form given in Chapter VI, or of any form ƒ taken simulta- neously with L and Q, regard the coefficients of ƒ to be them- selves parameters which represent positive residues of the prime number p. Let ƒ be such a modular form, and quadratic, ƒ = a0z* + 2 #!^2 + #¿4 Letjt>=3. In a fundamental system of formal invariants and covariants modulo 3 of ƒ we may now reduce all expo- nents of the coefficients below 3 by Fermat’s theorem, #? = 0t· (mod 3) (i = 0, 1, 2). The number of individuals in a fundamental system of ƒ is, on account of these reductions, less than the number in the case where the 0’s are arbitrary variables. We call the in- variants and covariants of ƒ, where the a s are integral, modular concomitants (Dickson). The theory of modular invariants and covariants has been extensively developed.204 THE THEORY OF INVARIANTS In particular the finiteness of the totality of this type of con- comitants for any form or system of forms has been proved. The proof that the concomitants of a quantic, of the formal modular type, constitute a finite, complete system has, on the contrary, not been accomplished up to the present (December, 1914). The most advantageous method for evolving funda- mental systems of modular invariants is one discovered by Dickson depending essentially upon the separation of the totality of forms ƒ with particular integral coefficients modulo p into classes such that all forms in a class are permuted among themselves by the transformations of the modular group given by Tp.* The presentation of the elements of this modern theory is beyond the scope of this book. We shall, however, derive by the transvection process the funda- mental system of modular concomitants of the quadratic form ƒ, modulo.3. We have by transvection the following results (cf. Appendix, 48, p. 241): TABLE VI Notation Trans- VECTANT Concomitant (Mod 3) A (AfY2 a\ — a0a2 1 (JY QY a%a2 + a0a\ + a0a\ + a\a2— a% — a% L xfx2 — x±x\ Q ((L,Z)*L) * x\ + x\x\ + xfxt + %2 f %x\ + 2 aYxxx + a&\ A c/; QY «0*1 + aYx%x2 + afaxl + a&\ Cl (/8, QY {alaY — a\)x\ + (a0 - a2) («! + a0a2)x1x2 + (of - aYa\)x2 c2 if\ QY («o + «i - «o«2)^1 + «i(«o + «2)^2 + («? + «!— a0a2)xI Also in q and 01 we may make the reductions a\ = a{ (mod 3) (¿ = 0, 1, 2). We now give a proof due to Dickson, that these eight forms constitute a fundamental system of modular invariants and covariants of/. * Transactions American Math. Society, Vol. 10 (1909), p. 123.SEMINVARIANTS. MODULAR INVARIANTS 205 Much use will be made, in this proof, of the reducible in- variant 1= (a2 - 1) Of - 1) (a\ - 1) = ?2 + A2 - 1 (mod g). In fact the linearly independent invariants of ƒ are 1, A, J, q, A2. (i) Proceeding to the proposed proof, we require the semin- variants of/. These are the invariants under x1 = xr1-\- xf2, x2 == xf2 (mod 3). These transformations replace ƒ by ƒ', where ar0 = a0, a[ = a0 + av a2 = a0 — a1 + a2 (mod 3). (t) Hence, as may be verified easily, the following functions are all seminvariants : «0’ a% ^o^2’ = K2 — ($) Theorem. Am/ modular seminvariant is a linear homo- geneous function of the eleven linearly independent seminvari- ants (if (s'). For, after subtracting constant multiples of these eleven, it remains only to consider a seminvariant S = axaxa\ + a^a^a^ + azax + a^a\a\ + aha\a2 + aQa\ + $a\ + 7 a2, in which av a2 are linear expressions in a%, a0, 1 ; and a3, ···, a6 are linear expressions in a0,1 ; while the coefficients of these linear functions and /3, 7 are constants independent of a0, av av In the increment to S under the above induced transformations (t) on the a’s the coefficient of axa\ is -r a0a4, whence a4 = 0. Then that of a\a2 is at = 0 ; then that of axa2 is /3 — a0a5, whence j3 = oc5 = 0; then that of a\ is — a2 = 0 ; then that of ax is — 7 — a0a6, whence 7 = aQ = 0. Now S=a3av whose increment is a3a0, whence a3 = 0 Hence the theorem is proved. Any polynomial in A, J, q, a0, B is congruent to a linear206 THE THEORY OF INVARIANTS function of the eleven seminvariants (i), ($) by means of the relations J2=-l =6£-|-2. For t odd, the covariants fu+\ Q% cr\fl0v CfCv have as coefficients of x\ a& ^0? A2 #2A 4- cf^i ¿fyA 4- B, respectively. Linear combinations of products of these by invariants give the seminvariants (s) and A, A2. Hence, by the lemma, C~P^LQ\ where P is a polynomial in the208 THE THEORY OF INVARIANTS covariants of the table omitting/4. For t even the co- variants o2q\ cxq< have a0, a%, A + 0%, B as coefficients of zx. Taking up next covariants O of order 2 1 0 - x\ + A + xlX2 "l· X\x\ To show that L and Q are not functions of the remaining concomitants we use case (1). For /4, use case (4). No linear relation holds between ƒ, Cv 02 in which Ox is present, since Ot is of index 1, while ƒ, C2 are absolute covariants. Now f^lcC2 by case (4); Q2^kf by case (5). Next A) by (2) and (3) ; A =£ F(q) by (4) and (5).CHAPTER IX INVARIANTS OF TERNARY FORMS In this chapter we shall discuss the invariant theory of the general ternary form f=a^ = b?= .... Contrary to what is a leading characteristic of binary forms, the ternary ƒ is not linearly factorable, unless indeed it is the quantic (198) of the preceding chapter. Thus ƒ repre- sents a plane curve and not a collection of linear forms. This fact adds both richness and complexity to the invariant theory of ƒ. The symbolical theory is in some ways less adequate for the ternary case. Nevertheless this method has enabled investigators to develop an extensive theory of plane curves with remarkable freedom from formal difficulties.* SECTION 1. SYMBOLICAL THEORY As in Section 2 of Chapter VIII, let f(x) =a™ = (axxx + «2*2 + «3%r =&*?=···. Then the transformed of ƒ under the collineations V (Chap. VIII) is f=(a>,x’i + + aAT- (199) I. Polars and transvectants. If (yx- j/2> y%) is a set co' gredient to the set (xx, xv xs), then the (y) polars of ƒ are (cf. (61)) fv* = a?"*»* (& = 0, 1, m). (200) * Clebsch, Lindemann, Vorlesungen liber Geometrie.210 THE THEORY OF INVARIANTS If the point (y) is on the curve ƒ = 0, the equation of the tangent at (j/) is ' = 0. (201) The expression d _d_ M — 1 |w — 1 — 1 dx1 dx2 dx3 d ƒ<»£(# WO) (202) | m | n | p tyl d _d_ Ô dzt dz2 dz8 y=z~x is sometimes called the first transvectant of ƒ(#), (x), ^(V), and is abbreviated (ƒ, <£, i/r). If /O) = C = = ■» 'i> 'X — then, as is easily verified, (ƒ, 4>, •^) = (a5c)a™-1ir1cr1· This is the Jacobian of the three forms. The rth trans- vectant is (ƒ, , W = (abeyaf-^-rcrr (r = 0, 1,...). (203) For r = 2 and = ^ this is called the Hessian curve. Thus (ƒ, ƒ ƒ )2 = {abcya^-2by2c^-2 = 0 is the equation of the Hessian. It was proved in Chapter I that Jacobians are concomitants. A repetition of that proof under the present notation shows that transvectants are like- wise concomitants. In fact the determinant A in (202) is itself an invariant operator, and A' = A. Illustration. As an example of the brevity of proof which the symbolical notation affords for some theorems we may prove that the Hessian curve of ƒ = 0 is the locus of all points whose polar conics are degenerate into two straight lines.INVARIANTS OF TERNARY FORMS 211 If g = al = /32 = ··· = amx\ H---is a conic, its second trans· vectant is its discriminant, and equals O 0 consisting of fac- tors of the two types (abc), ax is a concomitant. In fact if cj) = (abc)p(abcr)q ··· arxb% ···, then k Ox V ax k dK ’ = a^ k C!x a^ k d. av k ov av k dv since, by virtue of the equations of transformation a!x = ax^ ···. Hence by the formula for the product of two determinants, or by (14), we have at once f = (\/iv)p+q+ '''(abc)p(abcT)9 ··· arxb% ··· =(\/ap)p+«+-0. The ternary polar of the product of two ternary forms is given by the same formula as that for the polar of a product in the binary case. That is, formula (77) holds when the forms and operators are ternary. Thus, the formula for the rth transvectant of three quan- tics, e.g. t= (ƒ, <#>, fy = (abcya™-rbrrc»-r,212 THE THEORY OF INVARIANTS may be obtained by polarization : That is, by a process analo- gous to that employed in the standard method of transvec- tion in the binary case. Let (^Ol = ^2^3 ^3^2' (^^2 = ^3^1 ^1^3’ (^^3 = ^1^2 ^2^1* (205) ThCn a(M = (abc). (206) Hence T may be obtained by polarizing af r times, changing y{ into (6*?)* and multiplying the result by bn~rc%~r. Thus (<#*, 4, o2=: a)(î)“.‘,A+(2)(J)^„<'. = § (acd)(bcd)axcx + %(acd)2bxcx. Before proceeding to further illustrations we need to show that there exists for all ternary collineations a universal co- variant. It will follow from this that a complete fundamental system for a single ternary form is in reality a simultaneous system of the form itself and a definite universal covariant. We introduce these facts in the next paragraph. II. Contragrediency. Two sets of variables (xv a?2, #3), (uv uv u%) are said to be contragredient when they are sub- ject to the following schemes of transformation respectively : x1 = \lX[ + fjLlXf2 + Vlxf3 V: + v2xf3 x3 = \sx[ + /¿8x'2 + v3x'3 = XjWj "f· ^*2^2 “f” \'j^3 A: u'2 = ^1w1 + ^u2 + fi3u3 u's = v-jU! + v2u2 + v3u3. Theorem. A necessary and sufficient condition in order that (x) may be contragredient to (u) is that ux = utxx -f u2x2 -f u3x3 should be a universal covariant.INVARIANTS OF TERNARY FORMS 213 If we transform ux by V and use A this theorem is at once evident. It follows, as stated above, that the fundamental system of a form ƒ under V, A is a simultaneous system of ƒ and ux (cf. Chap. VI, § 4). The reason that ux = uxxx 4- u2x2 does not figure in the cor- responding way in the binary theory is that cogrediency is equivalent to contragrediency in the binary case and ux is equivalent to (xy) = xxy2 — x2yv which does figure very prominently in the binary theory. To show that cogredi- ency and contragrediency are here equivalent we may solve + \2u2 uf2 = 4- p2u2, we find - (\/*>! = X2u2 + - ui)> = \y2 + /n^- w'), which proves that yx = 4- u2, y2 = — u1 are cogredient to xv x2. Then ux becomes (yx)(cf. Chap. 1, § 3, V). We now prove the principal theorem of the symbolic theory which shows that the present symbolical notation is sufficient to represent completely the totality of ternary concomitants. III. Theorem. Every invariant formation of the ordinary rational integral type, of a ternary quantic f = a™ =..· = (2% = TO), can be represented symbolically by three types of factors, viz. (aie), (abu), ax, together with the universal covariant ux. We first prove two lemmas.214 THE THEORY OF INVARIANTS Lemma 1. The following formula is true : d d d d\8 n *1 *3 AnDn= d dfix 1 co =1 1 d d _d_ d dv1 dv2 1 CO 1^0 V1 v% vz (207) where 0^= 0 is a numerical constant. In proof of this we note that j9n, expanded by the multi- nomial theorem, gives D" = X WÙKWi ~ /Vs)*2 0V2 “ /Vi)*3· ÇZij = n). Also the expansion of An is given by the same formula where now (\r/Aai/t) is replaced by (-^—— We may \d\r dfi8 dvtJ call the term given by a definite set iv i2·, i8 of the exponents in 2)n, the correspondent of the term given by the same set of exponents in An. Then, in An2>w, the only term of D" which gives a non-zero result when operated upon by a definite term of An is the correspondent of that definite term. But Dn may be written . I n Dn = X X1X2 X3 1 (^'*'>2 (^•*'>3 * An easy differentiation gives J98(/W,>i(MI;) 2 = h(h + *2 + *3 + 1) OOÎ (^v)2(^)t\ and two corresponding formulas may be written from symme- try. These formulas hold true for zero exponents. Employ- ing them as recursion formulas we have immediately for A nDn,INVARIANTS OF TERNARY FORMS 215 A= V( - ) (\h\h\h)2\h + h+ «3+1 = X (|w)> + 1 = ^(|w)3(w + 1)2Q + 2). (208) V=° This is evidently a numerical constant (7 =£ 0, which was to be proved (cf. (91)). Lemma 2. If P is a product of m factors of type n of type 0^ and p of type 7v, then AhP is a sum of a number of monomials, each monomial of which contains k factors of type (a^y), m — k factors of type aKtn — k of type 0^ and P~k of type 7v. This is easily proved. Let P = ABO\ where A =«A1)«i2) ··· a^m), B = 0?0? 0?\ (7 = 7(i)7(2) ... 7(a, x), and216 THE THEOEY OF INVAEIANTS suppose that it does not contain the variables (w), and that the corresponding invariant relation is <£(«', a?', ·.·) = (\fiv)w <£(a, a?, ···). (210) The inverse of the transformation J^is x\ = [ 0*01^1 + etc. Or, if we consider (x) to be the point of intersection of two lines V* = + v2x2 + vzxz, wx = wxxx + W2X2 -I- wzxz, we have xx:x2:xz = (vw\ : (vw\ : (w)3. Substitution with these in xrv ··· and rearrangement of the terms gives for the inverse of V F-i: 1 (X*«0 ’ , _ Vvwx - V\Wv 2 (Xa"0 3 cx^) ’ We now proceed as if we were verifying the invariancy of <£, substituting from V~x for xfv x2, xz on the left-hand side of (210), and replacing afmm2ms by its symbolical equivalent (cf. (199)). Suppose that the order of is ©. Then after performing these substitutions and multiplying both sides of (210) by (\/jlv)w we have vKwM ···) = (\pvy+* (a, as, ···)> and every term of the left-hand member of this must contain + factors with each suffix, since the terms of the right- hand member do. Now operate on both sides by A. Each term of the result on the left contains one determinant factor by lemma 2, and in addition w + co — 1 factors with eachINVARIANTS OF TERNARY FORMS 217 suffix. There will be three types of these determinant fac- tors e.g. (abe), (aw) = a*, (abv). The first two of these are of the form required by the theorem. The determinant (a6r) must have resulted by operating A upon a term containing a^b^Vy and evidently such a term will also contain the factor w^ or else W\. Let the term in question be Then the left-hand side of the equation must also contain the term — Raxh^v^Wy, and operation of A upon this gives — R(^abw)v^ and upon the sum gives R^abvywp — (abw^v^. Now the first identity of (212) gives (abv^Wfj. — (abw)vil = (bvw)a/lt — (aw)JM = bxa^ — b^a*. Hence the sum of the two terms under consideration is -B(Mm - and this contains in addition to factors with a suffix fi only factors of the required type ax. Thus only the two required types of symbolical factors occur in the result of operating by A. Suppose now that we operate by Aw+Ui upon both members of the invariant equation. The result upon the right-hand side is a constant times the concomitant (a, x) by lemma 1. On the left there will be no terms with X, /a, v suffixes, since there are none on the right. Hence by dividing through by a constant we have <£(a, x) expressed as a sum of terms each of which consists of symbolical factors of only two types viz. (■abe), ax,218 THE THEORY OF INVARIANTS which was to be proved. Also evidently there are precisely co factors ax in each term, and w of type (abc), and co = 0 if is an invariant. The complete theorem now follows from the fact that any invariant formation of ƒ is a simultaneous concomitant of ƒ and ux. That is, the only new type of factor which can be introduced by adjoining ux is the third required type (abu). IV. Reduction identities. We now give a set of identi- ties which may be used in performing reductions. These may all be derived from ax dy az K K K ^x Cy Cz = (abc')(xyz), (211) as a fundamental identity (cf. Chap. Ill, § 3, II). We let uv w2, m3 be the coordinates of the line joining the points O) = (xv xv xa), O) = (yv yv ya). Then «1 : m2 : m3 = (xy\ : (xy\ : (xy\. Elementary changes in (211) give (bcd)ax — (cda)bx 4- ( dab)cx — (abc)dx = 0, (bcu)ax — (cua)bx -f- (uab)cx — (abc)ux = 0, (212) (abc)(def) — (dab^(eef) + (cda)(bef) — (bcd)(aef) =0. Also we have axby-aybx=(abu), vawb — vbwa = (abx). ( ’ In the latter case (a?) is the intersection of the lines v, w. To illustrate the use of these we can show that if ƒ= a%= ··· is a quadratic, and D its discriminant, then (a be) 0abd) cxdx = £ Df. In fact, by squaring the first identity of (212) and inter- changing the symbols, which are now all equivalent, this result follows immediately since (abc)2 = D.INVARIANTS OF TERNARY FORMS 219 SECTION 2. TRANSVECTANT SYSTEMS I. Transvectants from polars. We now develop a stand- ard transvection process for ternary forms. Theorem. Every monomial ternary concomitant of ƒ= = (abc)p(abd)q ··· (bcd)r ··· (abu)8(bcuy ··· a* ···, is a term of a generalized transvectant obtained by polarization from a concomitant j of lower degree than (f>. Let us delete from the factor a%, and in the result change a into v, where v is cogredient to u. This result will contain factors of the three types (bcv), (bed), (buv), together with factors of type bx. But (uv) is cogredient to x. Hence the operation of changing (uv) into x is invari- antive and (buv) becomes bx. Next change v into u. Then we have a product ^ of three and only three types, i.e. (bcu), (bed), bx, (f>1 = (bcd)a ··· (bcuy ··· byxchz ···. Now does not contain the symbol a. Hence it is of lower degree than . Let the order of <£ be co, and its class fi. Suppose that in there are i determinant factors con- taining both a and u, and k which contain a but not u. Then <7 + i + k = m. Also the order of <\>x is and its class (ol = & + 2 i + k — m, = H — i'+k. We now polarize <\>t by operating upon it and dividing out the appropriate constants. If in the resulting220 THE THEORY OF INVARIANTS polar we substitute v = a, y = (au) and multiply by a,™~i~k we obtain the transvectant (generalized) r =(<#>r a?, t4)M· (214) The concomitant is a term of t. For the transvectant r thus defined k + i is called the index. In any ternary concomitant of order a> and class p the number o> + is called the grade. Definition. The mechanical rule by which one obtains from a concomitant (7= Aalxa2x ··· ···««*? any one of the three types of concomitants Ci = *** “* ^2 ==: *** *“ C3 = ^1^2*" ^rx^iu is called convolution. In this aiai indicates the expression allall + a12a12 + a13a13· Note the possibility that one a might be #, or one a might be u. II. Theorem. The difference between any two terms of a transvectant r equals reducible terms whose factors are concom- itants of lower grade than t, plus a sum of terms each term of which is a term of a transvectant r of index 1 is of lower grade than and is obtainable from the latter by convolution. Let t be the concomitant O above, where A involves neither u nor x. Then, with \ numerical, we have the polar = ^I^axya^y ··· ··· arxctiv ··· a-kv^k+iu *** *fm· (215)INVARIANTS OF TERNARY FORMS 221 Now in the ¿th polar of a simple product like p = 7i*7ar — two terms are said to be adjacent when they differ only in that one has a factor of type 7^7^ whereas in the other this factor is replaced by 7^7^. Consider two terms, tv t2 of P. Suppose that these differ only in that a^a^a^a^ in tx is re- placed in ^ by a>m^Kvahxajy · Then tx —12 is of the form ¿1 ^2 = *®( ^vv^tcu^hy^jx tX'iiuP’KvQ'hxtyy)· We now add and subtract a term and obtain t] ^2 = [^yv^KU(Q’hyQjx Q'hxQ'jy) ”b Q’h,xQ'jy(jt’r\vaKu ^rfu^Kv) J · (216) Each parenthesis in (216) represents the difference between two adjacent terms of a polar of a simple product, and we have by (213) h - t2 = B(px(ahaj))avvaKU + B(aKav(uv))a hxtyy · (217) The corresponding terms in t are obtained by the replace- ments v = a, y = (au). They are the terms of S=- B'((au)(ahaj)x)aailaKU -IP((m)aKan')(aJau)ahx, or, since ((au)(ahaj)x) = (aahaj)ux - (ahasu)ax, of S = Bf (ahajU)amaKUax - Br(ahaJà)a1laateuux + B^a^au) ) (ajaii)ahx, where B becomes P' under the replacements v = a, y = (aw). The middle term of this form of P is evidently reducible, and each factor is of lower grade than r. By the method given under Theorem I the first and last terms of S are re- spectively terms of the transvectants Tj == (Pj(^ö^'M')ccTjttttKW, ax, ux ) , T2 = (-®l(a <*r,x)ajxahx' ax’ x = Oux by convolution. Also each is of lower grade than r Again if the terms in the parentheses in form (216) of any difference tx — t2 are not adjacent, we can by adding and subtracting terms reduce these parentheses each to the form * [(Tl - T2> + (T2 - T3> + - (T/-1 - Ti)], (218) where every difference is a difference between adjacent terms, of a simple polar. Applying the results above to these dif- ferences — ri+i the complete theorem follows. As a corollary it follows that the difference between the whole transvectant r and any one of its terms equals a sum of terms each of which is a term of a transvectant of af with a form ^ of lower grade than v obtained by convolution from the latter. For if T = vlTl + v2T2 + ... + iyrr + ... where the ?’s are numerical, then rr is a term of t. Also since our transvectant r is obtained by polarization, Si/* = 1. Hence t - rr = i/jCtj - Tr) 4- ^2(t2 “ Tr) H-* and each parenthesis is a difference between two terms of t. The corollary is therefore proved. Since the power of ux entering t is determinate from the indices A, i we may write r in the shorter form r= (*rOM. The theorem and corollary just proved furnish a method of deriving the fundamental system of invariant formations of a single form ƒ = a™ by passing from the full set of a given degree i — 1, assumed known, to all those of the fundamental *Isserlis. On the ordering of terms of polars etc. Proc. London Math. Society, ser. 2, Vol. 6 (1908).INVARIANTS OF TERNARY FORMS 223 system, of degree i. For suppose t^at all of those members of the fundamental system of degrees < i — 1 have been previously determined. Then by forming products of their powers we can build all invariant formations of degree i — 1. Let the latter be arranged in an ordered succession <*>', <#>", - in order of ascending grade. Form the transvectants of these with a”*, Ty=(^w), a™)*·*. If t;- contains a single term which is reducible in terms of forms of lower degree or in terms of transvectants rpj1 4 are therefore of the form (/^/)*’U*‘ + *<3), and hence are reducible. Thus the fundamental system of/ is u*, ƒ, £, D. The explicit form of D was given in § 1. A symmetrical form of L in terms of the actual coefficients of the conic is the bordered discriminant a200 a110 aioi «1 ano tt020 aoii «2 aioi a011 a002 «3 Ul u2 uz 0 To verify that L equals this determinant we may expand (abu)2 and express the symbols in terms of the coefficients. We next give a table showing Gordan’s fundamental system for the ternary cubic. There are thirty-four in- dividuals in this system. In the table, i indicates the degree. The reader will find it instructive to derive by theINVARIANTS OF TERNARY FORMS 225 methods just shown in the case of the quadratic, the forms in this table of the first three or four degrees. TABLE VII i Invariant Formation 0 1 al 2 (abufaxbx 3 (abu)2(bcu)axc«£ = (abc)2axbxcx, sj = (abc)(abu)(acu)(bcu) 4 (aau)alal, ats\a\, S = a®, = (abu)2(cdu)2(bcu)(adu) 5 a,sl(abu)axbl, o,6ss„o^, a,(abu)2slbx, t® = atb,su(_abu)2 6 «A8u(bcu)albxcl, a8s2(abu)2(bcu)cj, T=af 7 «ui>d(sp»), a,6tiua|6f, a,tl(abu)\ 8 Qx = atbtctaÿ)^ att2u{abu)2(bcu)c% s2(2(stx) 9 {aqu)alql, pfrl(ptx), ats2utual{stx) 10 atsltu(abu)2bz(stx) 11 {a.qu)alql 12 0wq)alalql p£«ÿ2(p*t) IV. Fundamental system of two ternary quadrics. We shall next define a ternary transvectant operation which will include as special cases all of the operations of trans- vection which have been employed in this chapter. It will have been observed that a large class of the invariant for- mations of ternary qualities, namely the mixed concomitants, involve both the (x) and the (u) variables. We now assume, quite arbitrarily, two forms involving both sets of variables e.g. = Aalxa2x · · · O'rxUlvSh.u · · · am' 'f = BhiJ)2x ·.. bpx/3iu02u ··· Ami226 THE THEORY OF INVARIANTS in which A, B are free from (x) and (u). A transvectant of <£, and yfr of four indices, the most general possible, may be defined as follows: Polarize by the following operator, wherein et·, , An example is (ct\xct'2x(*'u·) ^lx^2x^u)\] 0 = “h + lM)· If, now, we introduce in place of (/> successively products of forms of the fundamental system of a conic, i.e. of ƒ = a% L= a* = (a'a"u)2, D = (aa’a")2, and for products of forms of the fundamental system of a second conic, g = % V = ft = (6'j"*02, = (WW we will obtain all concomitants of ƒ and g. The fundamental simultaneous system of ƒ, g will be included in the set of transvectant of and yjr of index Gcfl)INVARIANTS OF TERNARY FORMS 227 transvectants which contain no reducible terms, and these we may readily select by inspection. They are 17 in num- ber and are as follows : *=(<£ b^l = (abu)\ , T; eight mixed con- comitants Ci(i = 1, ···, 8). SECTION 3. CLEBSCH’S TRANSLATION PRINCIPLE Suppose that (y), (z) are any two points on an arbitrary line which intersects the curve ƒ = a™ = 0. Then Uiiu2: uz = (yz\ : (jjz\ : (gz\ are contragredient to the #’s. If (x) is an arbitrary point on the line we may write x! = Wi + H = Vtf* + H = 'niVz + Vs> and then (7jv' ?;2) may be regarded as the coordinates of a representative point (V) on the line with (^), (z) as the two reference points. Then ax becomes dx — d]X·^ + d2X2 ^3*^3 = Vl&y 4" Q'z') and the (77) coordinates of the m points in which the line intersects the curve ƒ = 0 are the m roots of 9 = 97 = (V?l + a*’?2>m= (Ml + Kv 2)“= .... Now this is a binary form in symbolical notation, and the notation differs from the notation of a binary form h = a™ = (a1xi + d2x2)™ = ··· only in this, that av a2 are replaced by dy, a9 respectively. Any invariant, Ix = ^k(cfoy(ac)q .··, of h has corresponding to it an invariant I of g, 7= 2k(dybz - azbyy(aycz - dzcvy ....INVARIANTS OF TERNARY FORMS 229 If I = 0 then the line cuts the curve ƒ = ax = 0 in m points which have the projective property given by It = 0. But (cf. (213)), (aybz — ajby) = (abu). Hence, Theorem. If in any invariant Ix = ^k{ab)p(ac)q ··· of a binary form h = ax = (a1x1 + a2x2)m= •••we replace each second order determinant (aV) by the third order determinant (aSw), and so on, the resulting line equation represents the envelope of the line ux when it moves so as to intersect the curve f=af=· (a+ a^2 + azxz)m = 0 m points having the projective property = 0. By making the corresponding changes in the symbolical form of a simultaneous invariant I of any number of binary forms we obtain the envelope of ux when the latter moves so as to cut the corresponding number of curves in a point range which constantly possesses the projective property J=0. Also this translation principle is applicable in the same way to covariants of the binary forms. For illustration the discriminant of a binary quadratic h = a\ = b\ = ··· is Z> = (a&)2. Hence the line equation of the conic ƒ = a2 = + a2x2 + a3^3)2= ··· = 0 is L = (abu)2 — 0. For this is the envelope of ux when the latter moves so as to touch ƒ = 0, i.e. so that D= 0 for the range in which ux cuts ƒ= 0. The discriminant of the binary cubic h = (a+ a2x2)3 = b% = ··· is R = (a6)2(ai?)(6c?)(i?c?)2. Hence the line equation of the general cubic curve ƒ= ··· is (cf. Table VII) Pu = L= {abuY(fLcu)(bdu){cdu)2==: 0.230 THE THEORY OF INVARIANTS We have shown in Chapter I that the degree i of the dis- criminant of a binary form of order m is 2(m — 1). Hence its index, and so the number of symbolical determinants of type (ai) in each term of its symbolical representation, is k = | im = m(m — 1). It follows immediately that the degree of the line equation, i.e. the class of a plane curve of order m is, in general, m(m — 1). Two binary forms hx = a% = a!™ = ···, h2 = b™ = ···, of the same order have the bilinear invariant 1= (a5)m. If JT=0 the forms are said to be apolar (cf. Chap. Ill, (71)); in the case 2, harmonic. Hence (aJw)m = 0 is the envelope of ux == 0 when the latter moves so as to inter- sect two curves ƒ = a% = 0, g = bf = 0, in apolar point ranges.APPENDIX EXERCISES AND THEOREMS 1. Verify that I = — 4 axa3 + 3 a2 is an invariant of the binary quartic ƒ = a$[ 4- 4 axx^x2 4- 6 a&fa% 4- 4 + a4x2> for which r=(M*L 2. Show the invariancy of + axx2) — Oo(«i^i + 7i -f p2V2> V2 = y2>7l + 82?72> in the extended sense indicated by the invariant relation all ali «1 A «2 @2 a ii a2i «12 ak yi 8, 72 ai2 0*22 11. Verify the invariancy of the bilinear expression Hfg = ^11^22 “H a22^11 ^12^21 — tt21&12> for the transformation by r of the two bilinear forms ƒ = %aikXiyk, g = '2tbikxiyk- 12. As the most general empirical’definition of a concomitant of a single binary form ƒ we may enunciate the following: Any rational, integral function of the coefficients and variables of ƒAPPENDIX 233 which needs, at most, to be multiplied by a function \j/ of the coefficients in the transformations T, in order to be made equal ’ to the same function of the coefficients and variables of ƒ', is a concomitant of ƒ. Show in the case where is homogeneous that ip must reduce to a power of the modulus, and hence the above definition is equivalent to the one of Chap. I, § 2. (A proof of this theorem is given in Grace and Young, Algebra of Invariants, Chapter II.) 13. Prove by means of a particular case of the general linear transformation on p variables that any p-ary form of order m, whose term in is lacking, can always have this term restored by a suitably chosen linear transformation. 14. An invariant of a set of binary quantics /i = cwT H------,/2 = b^ + /3 = Co#f + ···, satisfies the differential equations 2D<£ = ^ ai5a" *" —b &o-5r H” ^ da. udbi 36* + -+^+ ·■· ]* = 0, 20 —fmctL-—(m — 1 )a2-—h ··· + —b w&i-J- + (n —+ ··· 4-pCi-—h •••>] = 0. OCq J The covariants of the set satisfy fsa- x2 J-]d> = o, V (s°— 15. Verify the fact of annihilation of the invariant «0 «i a2 J= 6 «1 «2 a3 U2 «3 a4 of the binary quartic, by the operators O and O.234 THE THEORY OF INVARIANTS 16. Prove by the annihilators that every invariant of degree 3 of the binary quartic is a constant times J. (Suggestion. Assume the invariant with literal coefficients and operate by Q and O.) 17. Show that the covariant J^ A of Chap. II, § 3 is annihilated by the operators so-» ^ so _ .a Ofl/j OX2 18. Find an invariant of respective partial degrees 1 and 2, in * the coefficients of a binary quadratic and a binary cubic. The result is I = tt0(^i^3 — &D — — ^1^2) "h ^2(^0^ — bf). 19. Determine the index of I in the preceding exercise. State the circumstances concerning the symmetry of a simultaneous invariant. 20. No covariant of degree 2 has a leading coefficient of odd weight. 21. Find the third polar of the product ƒ · g, where ƒ is a binary quadratic and g is a cubic. The result is (ZsOvs = ^¡{fgyz + 6/^2 + 3 f^gv). 22. Compute the fourth transvectant of the binary quintic ƒ with itself. The result is (ƒ, ƒ)4 = 2(a0a4 — 4 aYaz + 3 a|)^ + 2(a0a5 — 3 + 2 a2a3)a^2 +2(a1a5 — 4 a2a4 + 3 a|)a|. 23. If F = aj&jjc,, prove \aValbybxCuAPPENDIX 235 24. Express the covariant Q=(ab)\cbyxax of the binary cubic in terms of the coefficients of the cubic by ex- panding the symbolical Q and expressing the symbol combina- tions in terms of the actual coefficients. (Cf. Table I.) 25. Express the covariant +j = ((ƒ,ƒ)4, ƒ )2 of a binary quintic in terms of the symbols. The result is — j =(ab)\bc)\ca)2axbxcx = — (aby(ac)(bc)âx. 26. Let be any symbolical concomitant of a single form ƒ, of degree i in the coefficients and therefore involving i equivalent symbols. To fix ideas, let <£ be a monomial. Suppose that the i symbols are temporarily assumed non-equivalent. Then <£, when expressed in terms of the coefficients, will become a simul- taneous concomitant x of i forms of the same degree as ƒ, e.g. Also will be linear in the coefficients of each ƒ, and will reduce to again when we set bj = — = lj = a,·, that is, when the symbols are again made equivalent. Let us consider the result of operat- ing with upon <£. This will equal the result of operating upon 1? the equivalent of 8, and then making the changes Now owing to the law for differentiating a product the result ƒ = QqxT + + · · ·, fi = b0x? + mb&T-'xz H---, fi-1 = io*r + rnl^-% + ···. b, = - = lj = a, (j = 0, -, m).236 THE THEORY OF INVARIANTS upon i and then making the changes b = ■■■ — 1 = a. Hence the operator which is equivalent to 8 in the above sense is When is operated upon fa it produces i concomitants the first of which is fa with the a’s replaced by the p’s, the second is fa with the 6’s replaced by the p’s, and so on. It follows that if we write < = + mp^-% -f . and =(ab)r(ac)· — a?Jfx ···, we have for Scf> the sum of i symbolical concomitants in the first of which the symbol a is replaced by 71-, in the second the symbol b by tr and so forth. For illustration if is the covariant Q of the cubic, Q = (aby(cb)clax, then SQ =(7rb)2(cb)c^7rx +(a7r)2(c7r)cZax 4- (ab)\irb)^xax. Again the operator 8 and the transvectant operator Í1 are evidently permutable. Let g, h be two covariants of ƒ and show from this fact that S(g, h)r =($g, h)r + (g, ^)r. 27. Assume ƒ =«2, A = (/,/)2 = (a6)2aA = Ax2, Q = (ƒ, (/, ƒ )*) = (cA)cIAx = (ab)2(cb)c*ax = B = (A, A)2 _ ((ab)%cdy(ac)(bd), and write Q = Q2 — Qofii 4* 3Q^x2 4* 3 Q&iofi 4“ Qs%2* Then from the results in the last paragraph (26) and those in Table I of Chapter III, prove the following for the Aronhold polar operator 8 = :APPENDIX 237 8/=Q, SA = 2(aQ)2axQ, = 2(f, Q)> = 0, 8Q = 2(ƒ, (ƒ, Q)1) +(Q, A)=-1 i?/, Si? = 4(A, (ƒ Q)2)2 = 0. 28. Demonstrate by means of Hermite’s reciprocity theorem that there is a single invariant or no invariant of degree 3 of a binary quantic of order m according as m is or is not a multiple of 4 (Cayley). 29. If ƒ is a quartic, prove by Gordan’s series that the Hessian of the Hessian of the Hessian is reducible as follows: m H)\ (H, Hyy = - rfai*Jf+ Adduce general conclusions concerning the reducibility of the Hessian of the Hessian of a form of order m. 30. Prove by Gordan’s series, ((/,02i/)2=F + *W04/i where i = (ƒ, ƒ)4, and ƒ is a sextic. Deduce corresponding facts for other values of the order m. 31. If ƒ is the binary quartic ƒ = < = cl = · · ·, show by means of the elementary symbolical identities alone that (aby(acyblc* = $f. (aby. (Suggestion. Square the identity 2(ab) ( = a” 37. Prove that* Q =(ab)(bc)(ca)aj)xcx and all covariants of Q are combinants of the three cubics al, b% cj (Gordan). 38. Let ƒ and g be two binary forms of order m. Suppose that is any invariant of degree i of a quantic of order m. Then the invariant constructed for the form vxf + v2g will be a binary form of order i in the variables vx, v2. Prove that any invariant of Ft is a combinant of ƒ, g. (Cf. Salmon, Lessons Intro- ductory to Modern Higher Algebra, Fourth edition, p. 211.) 39. Prove that the Cartesian equation of the rational plane cubic curve is — &ioil “1“ "b ··· H" at3^l (i— I> 2, 3), Oi, x2, aj,) = |OoOi*| \a0a2x\ \a0asx\ \aQa2x\ \a0a3x\ + \axazx\ \a0a3x\ lojo^l \a2azx\ = 0. 40. Show that a binary quintic has two and only two linearly independent seminvariants of degree five and weight five. The result, obtained by the annihilator theory, is X(oJa5 — 5 alaxaA + 10 aottfa3 — 10 a0ai<*2 + 4 of) + Kaoa2 — — 3 a0«i«2 + 2 of). 41. Demonstrate that the number of linearly independent seminvariants of weight w and degree i of a binary form of order m is equal to (w; i, m) — (w — 1; i, m),APPENDIX 289 where (w; i9 m) denotes the number of different partitions of the number w into i or fewer numbers, none exceeding m. (A proof of this theorem is given in Chapter VII of Elliotts’ Algebra of Quantics.) 42. If ƒ = ax = b™ = ··· is a ternary form of order m, show that Prove also (ƒ, /)°’2*= (abuyka”rH: 1 2m — 4:1c m — 2 &V m — 2 k' s — i ]k\abcy X (abu)2k~r(bcu)(acuya^-i-2kb^8+i-2kc^-r-8. 43. Derive all of the invariant formations of degrees 1, 2, 3, 4 of the ternary cubic, as given in Table VII, by the process of pass- ing by transvection from those of one degree to those of the next higher degree. · 44. We have shown that the seminvariant leading coefficient of the binary covariant of ƒ= a™, 0 = (ab)p(ac)« ··· aÿ>l -, is 0O = (ab)p(ac)q ·-· af&f —. If we replace ax by ax, bx by bx, etc. in 0O and leave a2, b2, — unchanged, the factor (ab) becomes (a&i + a2x2)b2 — (bxxx + b2x2)a2 = (ab)xx. At the same time the actual coefficient ar = ax~Ta^ of ƒ becomes a™~ra\ 1 m — rdrf I m dx2 Hence, except for a multiplier which is a power of xl9 a binary covariant may be derived from its leading coefficient 0O by re- placing in 0O, a0, ax, —, am respectively by ƒ 1 ay _ \m-rdrf 1 dmf 9 mdx29 m(m — l)da%’ 9 |m dxr2 91m dx2 Illustrate this by the covariant Hessian of a quartic.240 THE THEOEY OF INVAEIANTS 45. Prove that any ternary concomitant of ƒ = a” can be de- duced from its leading coefficient (save for a power of ux) by re- placing, in the coefficient, apqr by Le/ |m (Cf. Forsyth, Amer. Journal of Math., 1889.) 46. Derive a syzygy between the simultaneous concomitants of two binary quadratic forms ƒ, g (Chap. YI). The result is — 2e/i2 = D!02 + A/2 “ 2 hfg, where J12 is the Jacobian of the two forms, h their bilinear in- variant, and A? A the respective discriminants of ƒ and g. 47. Compute the transvectant (f, f)°·2 = (abufaX of the ternary cubic f=ai = bl = ^ [j Letelr in terms of its coefficients aMr (p + q + r = 3). The result for ■£(ƒ, /)0,2 is given in the table below. Note that this mixed concomitant may also be obtained by applying Clebsch’s translation principle to the Hessian of a binary cubic. 44 *1^1% 44 4ulu3 *?«2W3 44 01200102 “«111 2 «1110201 — 2 «2100102 01020300 — «201 2 02100111 — 2 «1200201 2 «2010210 — 2 01110300 03000120 — «210 «i*2 w? *1*2w1% *!*2W2 ^*2^3 *1*2*4 01200012 — 2 «m«021 + «1020030 2 «hi — 2 «2100012 — 2 «1020120 4- 2 «2010021 01020210 — 2 «2010111 4* 03000012 2 «2100021 — 2 «2010030 2 «2010120 — 2 «3000021 03000030 — 02100120APPENDIX 241 44 2 2 x\uxuz 4U2UZ 0*1^3 #0300012 — #021 2 #021#111 — 2 012O#O12 #012#210 -«ill 2 #120#021 2 ao30#m 2 #111#120 — 2 ao2i#2io #210#030 am x\xz4 a'*la*3wlw3 a5la?3M2w3 #120 #003 — 2 #lll#012 + #102#021 2 «2010012 — 2 «2100003 #300#003 — #201#102 2 «111 — 2 «2010021 — 2 «1200102 + 2 «2100012 2 «210#102 — 2 «3000012 #120#201 — 2 «1110210 + #300#021 *2*3*4 *2*3^2 *2*3*4 *2*3*i2**3 *2*3*4 #0300003 — 0O21«O12 2 «0210102 — 2 «1200003 «0120201 — 2 «1110102 -f- 021O«OO3 2 «1200012 — 2 «0300102 2 afu — 2 «0210201 — 2 «2100012 4* 2 «120#102 #210#021 — 2 «1200111 + «0300201 *1*4 *3*h«* 44 *3**1«3 a|w2ti3 *|*4 #0210003 — a2 — #012 2 aoi2#io2 — 2 «1110003 #0030201 2 ~' 0102 2 #1110012 — 2 «0210102 2 «1020111 — 2 «0120201 #2010021 “ #m 48. Prove that a modular binary form of even order, the modulus being p> 2, has no covariant of odd order. (Suggestion. Compare Chap. II, § 2, II. If X is chosen as a primitive root, equation (48) becomes a congruence modulo p — 1.)ERRATA PAGE LINE PAGE LINE 8 In (12); for u read ^ 30 119 Delete Kx, · · · . . . . 6 25 For d/dx2 read d/dx2 . . 18 122 For fx ^ § n read y, 2: § n . 21 28 In the subscript of the 137 For C| read C2Y2 . . . 24 second element of the 137 For read 0 25 first row read x2 for x2 17 142 For read ... . 5 29 For J nn (n — 1)” read 145 For 2a>i read 2a^e . 28 |mn(m-l)n .... 5 158 For £~ 2 read and 31 For J.0, Am read a0, am 16, 17 in line 26 read Am for An 18 33 For (2, 2) read (2, 3) . . 12 160 In Bx and B2 read m for n 13 37 For u — w read w ^ w . . 15 184 Read — dkd for — ddk . . 6, 8 39 Read (xv x2)“ for (aj1? x2) . 16 187 For (\fxv) = 0 read (Xfiv) ^ 0 26 45 For a'=f0fJLm read a'm=f^n 5 225 The form of degree 11 52 For (— l)r read (— 1)· . . 22 should read (aqu)a2^ . 16INDEX Absolute covariants, 2, 42 Algebraically complete systems, see fundamental systems Anharmonic ratio, 3 Annihilators: binary, 25 ternary, 189 Anti-seminvariants, 176, 179 Apolarity, 51, 173 Arithmetical invariants, 12, 32, 48, 157 Aronhold’s polar operators, 46 Associated forms, 158 Bezout’s resultant, 168 Bilinear invariants, 51 Boolean concomitants: of a linear form, 156 of a quadratic, 157 Canonical forms: binary cubic, 108 binary quartic, 111 binary sextic, 112 ternary cubic, 111 Class of ternary form, 230 Classes in modular theory, 204 Cogrediency, 20 Combinants, 162 Complete systems: absolutely, 129 relatively, 130 Conic, system of, 224 Contragrediency, 212 Contra variants, 228 Conversion operators, 70 Convolution, 93, 220 Coordinates, 15 Covariant curves, 171 Covariants : definition, 23 systems, 144-161 universal, 32 Cubic, binary : fundamental system, 68,100,141 Cubic, binary: canonical form, 108 syzygy, 107, 110, 161 Cubic, ternary: fundamental system, 225 canonical form, 111 semi-discriminants, 193 Degree, 20 Determinant, symbolical, 55, 170 Differential equation: satisfied by combinants, 163 (see also annihilators) Differential invariant, 9 Diophantine equations, 116 Discriminant, 4, 31 Eliminant, 30 End coefficients, 179 Euler’s theorem, 44 Existence theorem, 40 Factors of forms, 69, 191 Fermat’s theorem, 14, 21 Finiteness: algebraical concomitants, 66 formal-modular concomitants, 204 modular concomitants, 204 Formal modular concomitants, 12, 157 Fundamental systems, 144, 161, 204, 223, 225 Geometry of point ranges, 78 Gordan’s proof of Hilbert’s theorem, 112 Gordan’s series, 83 Gordan’s theorem, 128 Grade: of binary concomitant, 123 of ternary concomitant, 220 Group: of transformations, 18 the induced group, 19 243244 THE THEOEY OF INVARIANTS Harmonically conjugate, 6 Hermite’s reciprocity theorem, 76 Hesse’s canonical form, 111 Hessians, 28 Hilbert’s theorem, 112 Identities, fundamental: binary, 66 ternary, 218 Index, 34 Induced group, 19 Inflexion points, 171 Intermediate concomitants, 47 Invariant area, 1 Invariants: fundamental systems, 144-161 modular, 203 formal modular, 157, 204 Involution, 78 Irreducible systems, see fundamental systems Isobarism, 35 Jacobians, 27 Jordan’s lemma, 119 Line equation: of conic, 223 of cubic, 229 of form of order m, 230 Linear transformations, 15 Linearly independent seminvariants, 178, 205. Mixed concomitants, 228 Modular: concomitants, 203 forms, 203 transformation, 12 Operators (see annihilators) conversion, 70 Aronhold, 46 Parametric representation, 169 Partitions, 238 Polars, 42 Projective properties, 78 Quadratic, 65 Quadric, 225 Quartic, 89 Quaternary form, 33 Quintic, 147 Range of points, 78 Rational curves, 169 Reciprocity, Hermite’s law, 76 Reduction, 64, 83 Representation, typical, 159 Resultants, 29, 166, 168 Resultants in Aronhold’s symbols, 151 Robert’s theorem, 179 Roots, concomitants in terms of, 69 Semi-discriminants, 193 Seminvariants : algebraic, 175 modular, 205 Sextic, canonical form of, 112 Simultaneous concomitants, 23 Skew concomitants, 39 Standard method of transvection: binary, 57 ternary, 219 Stroh’s series, 89 Symbolical theory: binary, 53 ternary, 209 Symmetric functions: binary, 69 ternary, 191 Symmetry, 39 Syzygies: algebraic, 104 modular, 208 Tables: I. Concomitants of binary cubic, 68 II. Concomitants of binary quar- tic, 89 III. System of quadratic and cubic, 147 IV. System of quintic, 150 V. Semi-discriminants of ter- nary cubic, 200 VI. Modular system of quad- ratic, 204 VII. System of ternary cubic, 225 Ternary quantics: symbolical theory, 209 transvection, 219 fundamental systems, 223,225INDEX 245 Transformations, non-linear, 9 (see linear transformations) Transformed form: binary, 16 ternary, 187 Translation principle: Clebsch’s, 228 Meyer’s, 169 Transvectants, binary: Definition, 51 Theorems on, 92 Transvectants, ternary: Definition, 209, 219 Theorems on, 220 Types, 48 Typical representation, 159 Uniqueness of canonical reduction, 109, 112 Universal covariants, 32, 212 Weight, 34