The Quartic Curve and its Inscribed Configurations A DISSERTATION Submitted to the Board of University Studies of the Johns Hopkins University in Conformity with the Requirements for the Degree of Doctor of Philosophy BY H. BATEMAN May, m913 Reprinted from AMIERICAN JOURNAL OF MATHEMATICS Vol. XXXVI, No. 4 October, 1914 The Quartic Curve and its Inscribed Configurations. BY H. BATEMAN. ~ 1. Introduction. Whereas the geometry of a planar cubic curve can be regarded as fairly complete, that of the quartic is far from being so. It is true that the present knowledge of the properties of the curve is very extensive, as may be seen from the admirable article by G. Kohn and G. Loria in the "Encyklopadie der Mathematischen Wissenschaften";' but there are several important questions which have still to be answered. Some of these are mentioned in Ciani's recent report. t At present the most useful form for the equation of a general quartic is that obtained by regarding the curve as the envelope of a family of conics t 2 So + 2 S1+ S2 =0, where x is a variable parameter. This form, however, is unsuitable for a discussion of the invariants and covariants of the curve. A canonical form consisting of the sum of the fourth powers of five linear functions can not be used for this purpose, for Clebsch ~ has shown that the equation of a quartic curve can be thrown into this form only when the invariant B (the catalecticant) vanishes. The sum of six fourth powers is a possible form, II but the equation of the general quartic can be reduced to this form in Xo3 ways. It is easy to deduce from this equation that three corners of the hexagon given by the six linear forms form an apolar triad with regard to the quartic, and that the sides of two such hexagons touch a curve of the third class; but the equation is not adapted to a simple discussion of other geometrical properties of the curve. It has been found by experience that it is convenient to have a number of typical forms for the equation of the curve, each form being appropriate for the study of the properties of the curve relative to some inscribed configuration. * Bd. III2, Heft 4 (1909), pp. 517-570. f "Le curve piane di quart' ordine," Giornale di Matematiche, t. 48 (1910), pp. 259-304.: G. Salmon, "Higher Plane Curves," Dublin (1852), p. 196. ~ Crelle's Journal, Bd. 59 (1861), p. 125. II Rosanes, Crelle's Journal, Bd. 76 (1873), p. 329. Scherrer, "Progr. Frauenfeld," p. 17. Scorza Annali di Matematica (3), t. 2 (1899), p. 329. 45 358 BATEMAN: The Quartic Curve and its Inscribed Configurations. In some cases the existence of the configuration implies that the quartic curve is not general. Such cases are studied here in detail so as to prepare the way for the determination of the relations between the invariants which correspond to each particular case. A complete system of rational invariants of the general quartic curve has not yet been obtained in a form which is easy to use; but, thanks to the labors of Salmon,* Clebsch,t Maisano, f Gordan,~ Caporali, l Pascal [ and Emmy Noether,"* considerable progress has been made. In particular, Pascal has obtained criteria for certain types of degeneration, and Noether has constructed a relatively complete set of forms. Drs. Morley and Conner tt have found the relation which connects the invariants A and B when the curve contains co 1 configurations of fifteen points lying by threes on twenty lines, while Caporali has obtained some of the relations which are characteristic of certain other special types of curves. A result of considerable importance for the invariant theory has been obtained by writers on Abelian functions of genus 3.t $ It follows from the work of Riemann and Schottky that the invariants of a quartic curve can be expressed rationally in terms of six fundamental quantities which can be regarded as irrational invariants of the curve. In particular, the twenty-eight bitangents can be represented by equations in which the coefficients are rational functions of the six fundamental quantities. ~~ Cayley has shown that the quartic curve possesses a double point when the irrational invariants are connected by a certain relation. The Abelian theory has been further developed by Klein, I11 Wirtinger, Tl1 H.. Baker ** * and J. E. Wright. ttt Frobenius f ~ t has obtained similar results by algebraic methods. In his second memoir he starts with * "Higher Plane Curves." t Loc. cit. t Giornale di Matematiche, t. 19 (1881), p. 198. ~ Math. Ann., Bd. 20 (1882), p. 487. II "Memorie di Geometria," Naples (1888). f[ Napoli Atti (2), t. 12 (1905), p. 1. ** Crelle's Journal, Bd. 134 (1908). tt AMERICAN JOURNAL OF MATHEMATICS, Vol. XXXI (1909), p. 263. $: B. Riemann, "Werke," 2d edition, p. 487. H. Weber, "Theorie der Abel'schen Functionen vom Ge schlecht drei," Berlin (1876). F. Schottky, "Abriss einer Theorie der Abel'schen Functionen von drei Variabeln," Leipzig (1880); also Acta Mathematica, t. 27 (1903). Berlin Berichte (1910). ~~ See also A. Cayley, "Collected Papers," Vol. XI, p. 221; Vol. XII, p. 74. R. de Paolis, Mem. Lincei (3), 2 (1878). 1111 Math. Ann., Bd. 10 (1876). [~f "Untersuchungcn fiber Thetafunktionen," Leipzig (1895). Math. Ann., Bd. 40. *** "Multiply Periodic Functions," Cambridge (1907). ftt AMERICAN JOURNAL OF MATHEMATICS, Vol. XXXI (1909), p. 271. t$J Crelle's Journal, Bd. 99 (1886); Bd. 103 (1888). BATEMAN: The Quartic Curve and its Inscribed Configurations. 359 Hesse's method * of deriving a general quartic curve from a net of quadric surfaces touching eight associated planes. He uses the symbol fa,3s to denote the determinant formed from the homogeneous coordinates of four of these planes, and shows that the thirty-five ratios fK,: /f,,23 can be expressed rationally in terms of the sixteen ratios fxg fo01 23 (I = 4, 5, 6, 7), in which the suffixes 3, y, ~ can denote any three of the numbers 0, 1, 2, 3, and the numbers 0, 1, 2,....,7 are used to denote the eight associated planes. Now, these sixteen ratios form an orthogonal matrix, on account of the existence of relations of the type /x f /xa~,y X,,=0,6 (y =4, 5, 6,7; a =', a? ', y y'), and can consequently be expressed rationally in terms of six fundamental quantities with the aid of Cayley's formulae for the coefficients of an orthogonal linear transformation.t Frobenius remarks, however, that it is more convenient to retain the quantities fAw, in spite of the relations between them, 0 123 and to regard these as the fundamental irrational invariants of the quartic. He shows in particular that, if Xaf,A (a, O = 01, 1..., 7), are twenty-eight linear forms which, when equated to zero, give equations of the bitangents, we have relations of the type / fs Xx =0, connecting the forms belonging to four bitangents of an Aronhold system. The summation extends over the values of x which differ from a,,)y,, 8. An important consequence of this result is that, if we know the equations of seven bitangents of an Aronhold system, we can determine a set of irrational invariants. Now, the properties of a quartic curve in relation to an Aronhold system of bitangents may be discussed very conveniently with the aid of a (1, 2) transformation t in which the lines of a plane X' correspond to a net of cubics through seven fixed points in a plane X. A point P' in X' consequently corresponds to a pair of points P1, P2 in X, and when these come together, P' lies on a quartic curve~ which is called the limiting curve (Grenzkurve, Uebergangskurve) 11 of the transformation. The seven base points in the plane X correspond to seven bitangents of the limiting curve L, and these bitangents form an Aronhold system. It is important to notice that in many cases these * Crelle's Journal, Bd. 49 (1855). t Crelle's Journal, Bd. 32 (1846), p. 119. "Collected Papers," Vol. II, p. 133. + A. Clebsch, Math. Ann., Bd. 3 (1871). M. Noether, Erlangen Berichte, 10 (1878). Minchen Abh. (1889). R. de Paolis, loc. cit. G. Frobenius, loc. cit. L. Cremona and G. Battaglini, Atti R. Ace. d. Linc. (3), II, p. 152. ~ S. Aronhold, Berlin Berichte (1864). 1I Also called the synoptic curve. Miss C. A. Scott, Quarterly Journal, Vol. XXIX. 360 BATEMAN: The Quartic Curve and its Inscribed Configurations. bitangents can be derived from the seven base points by a correlation between the two planes, and so the work of calculating Frobenius's irrational invariants is much simplified. In this memoir we shall endeavor to prepare the way for a discussion of the conditions that a quartic curve may be of a particular type by looking for a (1, 2) transformation which has the species of quartic curve as limiting curve. The appropriate transformation has already been found in a number of cases. It is known, for instance, that the quartic curve L has a double point when three of the base points lie on a line or when six lie on a conic.* L consists of two conics when six of the base points are at the corners of a complete quadrilateral; it consists of four straight lines when six of the base points are consecutive in pairs and the lines joining the three pairs meet at the seventh base point. The conditions on the seven points usually take two or more alternative forms, for it should be noticed that the limiting curve L is unaltered when the net of cubics in the plane X is transformed into another by a quadratic Cremona transformation with base points at three of the seven points. We shall prove in this memoir that the Liroth quartic arises from a (1, 2) transformation in which the seven base points have the same polar lines with regard to a conic and a cubic. When the cubic breaks up into three straight lines, the desmic quartic is obtained; this gives a simple verification of Schur's theorem t that the desmic quartic is a particular case of Liiroth's quartic. The known fact that Liroth's quartic can be derived from eight associated points which are the poles of a plane with regard to a cubic surface, t is next used to obtain a construction for seven points which have the same polar lines with regard to a conic and a cubic. The desmic quartic arises when the plane meets the cubic surface in three lines. A (1, 2) transformation is next set up by mapping the chords of a twisted cubic in space on the points of a plane. The two points in which a chord meets a fixed quadric then correspond to the point in the plane which is associated with the chord. The (1, 2) transformation between two planes is finally obtained by mapping the quadric surface on a plane by stereographic projection, using one of the points in which the quadric is met by the twisted cubic as vertex of projection. * Also when two of the points come together. t Crelle's Journal, Bd. 95 (1883). t W. Frahm, Math. Ann,, Bd. 7 (1874). E. Toplitz, Math. Ann., Bd. 11 (1877). BATEMAN: The Quartic Curve and its Inscribed Configurations. 361 A transformation which gives rise to a desmic quartic is obtained in this way. The representation also leads to a notable property of a quadratic complex of lines which contains all the lines joining five points on a twisted cubic. It also leads to the consideration of a type of uninodal quartic containing on 1 configurations of ten points lying by threes on ten lines, and to a new proof of the theorem that a plane section of Weddle's surface contains co 1 configurations of fifteen points lying by threes on twenty lines. The method, due to E. Godt t and E. Timerding, t of deriving a quartic curve, and in fact a (1, 2) transformation, from a general Cremona quadratic transformation between the lines of a plane is next studied; and quadratic transformations are found which give rise to the Liiroth and desmic quartic respectively. Klein's quartic is next obtained as the limiting curve of a (1, 2) transformation; and a set of eight associated points in space, from which the curve can be derived, is deduced from this result. The known equations of the twenty-eight bitangents of Klein's quartic are obtained by a simple method. Dr. Coble ~ has extended the known reduction of the equations of a point conic and line conic to the sums of three squares. His extension relates to a point quartic and a line quartic. Two sets of six conics take the places of the sides and vertices of the common self-polar triangle, and each quartic is consequently represented as the sum of the squares of six quadratic forms. Now, it seems natural to try to extend other simplified forms of the equation of a conic to a quartic curve. An extension of the equation of a conic referred to a circumscribed triangle leads to the known form VX 1X2 + VY1 Y2 + VZ1 2 - of the equation of the general quartic curve. The corresponding generalization of the equation of a conic referred to an inscribed triangle leads to the problem of reducing the equation of the general quartic to the form Y1 Y2 Z2 + Z Z2X1X2 + X1X2 Y1Y2= 0. This problem is shown to be equivalent to that of finding a triangle which is inscribed in a given cubic and circumscribed to a given conic; and, in general, this problem possesses a limited number of solutions. In a special case, however, o 1 quadrilaterals can be circumscribed to the conic and completely in* H. Bateman, Proc. London Math. Soc., Ser. 2, Vol. III (1905). Morley and Conner, loc. cit. t "Dissertation," Gottingen (1873). Clebsch-Lindemann, "Vorlesungen," p. 1007. t Math. Ann., Bd. 53 (1900), p. 193. ~ Trans. Am. Math. Soc., Vol. IV (1903). 362 BATEMAN: The Quartic Curve and its Inscribed Configurations. scribed in the cubic. I have shown that in this case the corresponding quartic curve is a desmic quartic. It is known that Caporali's quartic possesses ao 1 sets of twenty-four points with the property that each set can be divided into a group of three quartets such that any two quartets belonging to different groups lie on a pair of lines. Now, I have discovered a second type of quartic curve containing o 1 sets of twenty-four points possessing the same property, but these configurations of twenty-four points have other properties which differ from those possessed by the configurations inscribed in Caporali's quartic. The parts of the thesis which have been reserved for later publication are devoted to the following topics: 1) The analytical study of Hesse's configuration. 2) Discussion of the case "D" with the aid of elliptic functions. 3) A set of eleven points in a space of four dimensions with the property that each point is the vertex of a quadric cone passing through the other ten. 4) A new property of the four-nodal cubic surface. 5) A diagram showing the twenty-eight bitangents of a quartic curve, with their symbols in Hesse's notation. ~ 2. Liiroth's Quartic. Liiroth* has shown that a quartic curve which passes through all the intersections of five lines is not general, for it is the covariant S of a quartic of Clebsch's type, i. e., a quartic whose equation can be expressed as the sum of five fourth powers; and since this type of quartic depends on only thirteen constants, it follows that the equation of Liiroth's quartic involves only thirteen independent constants. Liiroth has shown, moreover, that when one pentagon t is known to be completely inscribed in a given quartic curve, there are oo 1 pentagons with the same property, and their sides all touch a conic. * A particular (1,2) transformation which has the Liiroth quartic as limiting curve may be obtained as follows: Let a point P correspond to the two points Q, Q' in which the polar of P with regard to a conic C, meets the polar conic of P with respect to a cubic C3. There is evidently a (1, 2) correspondence between P and Q; for when Q is given, P is the point of intersection of the polar lines of Q with regard to the * Math. Ann., Bd. 1 (1869), Bd. 13 (1878). t I use the word "pentagon" here to mean the figure formed by five lines; it has ten vertices. t See also Darboux, "Sur une classe remarquable de courbes et de surfaces algebriques," Paris (1896), p. 186. W. K. Clifford, "Math. Papers," p. 205. BATEMAN: The Quartic Curve and its Inscribed Configurations. 363 conic and the cubic. To find the locus of P when Q and Q' come together, we take the equations of the cubic and the conic in the forms * (ax3) -0, (bx2) -0, (x) 0, where the symbol (a x3) is used to denote the sum of four terms with different suffixes. The four lines x1, x2, x3, x =0 are the common tangents of the oo 1 conics apolar to C2 and C3; they form a quadrilateral whose six corners lie on the Hessian of C3. Let (Y1, Y2, ys,, y4) be the coordinates of P. Its polar with regard to C2 is (b y x) = 0, and its polar conic with regard to C3 is (a y 2) = 0. When Q and Q' come together, the line Q Q' touches this polar conic at the point Q (z,, Z23, Z3 4), and so its equation must be equivalent to (ayzx) = 0. Since (x) = 0, we must have a set of relations of the type aryrzr = bry, + y, (r = 1, 2, 3, 4). Using the relations (z-) =0, (b y z) = 0, we find that X (0) + ) ( ) = X ( a + I ( 0. \a/ \ay/ a a Hence, b2Y) ( 1) (b)2 a \ay/ \a Putting ( ) + -) a5y5 0, and using the symbol [c] to denote the sum of \ a / \a/ five terms with different suffixes, we see that the locus of P, when Q and Q' come together, is the Liiroth quartic [- =0. Since the equation of the Hessian of C3 is (1) =0, it appears that the Hessian H3 touches the Liiroth quartic L at the six corners of the quadrilateral. This indicates a method by which a suitable C3 can be determined when L is given. Since there are cc 1 pentagons completely inscribed in L, it appears that there are Go 1 suitable C3's. Returning to the transformation, we notice that, if P describes a straight line (b y) = 0, the two corresponding points Q describe a cubic curve whose equation is obtained by eliminating y1, Y2, Y3, y4 from the four equations (y) 0, (ky) 0, (byx)= 0, (ayx2) 0. It is easy to see that the equation of the cubic can be derived from that of the line by writing * A reduction to these forms is given by R. A. Roberts, Proc. London Math. Soc., Ser. 1, Vol. XXI (1889), p. 62. 364 BATEMAN: The Quartic Curve and its Inscribed Configurations. Y1 - a b2 x x - a2 b3 x2 x + a2 b4 x2 4 - a4 b x2 X + a4 b3 x3 x -3 a3 b4 x 4, Y2 - a b3 x3 x2 -a b3 b 1 + a3 b4 x 3 - a4 b3 X3 42 + a4 b1 x x - al b4 xi x4, a2 b1 x x2 - a b2 x2 + a b4 x1 4 -4 bl Ix + a4 b2 x2 x4 a2b4 2 x4, Y4 -= a23 x 2X3 -- a b2 x2 x + as b1 x 1 -- aI b3 x2 + aa b2 x x2 -- a 2 b1 2 a x2 and these are the equations of the transformation. The cubic curve corresponding to a given line t can be generated by the points of intersection of corresponding members of a pencil of conies and a pencil of lines. The conics are the polar conics of points on t with regard to C3, and so pass through the four poles of t with regard to C3. The lines are the polars of points on t with respect to C2; they all pass through T, the pole of t with respect to C2. This point T evidently lies on the cubic 3 corresponding to t, and the four tangents from T to r3 are the polars with respect to C, of the four points in which t meets L. Hence, the invariant of r, i. e., the cross ratio of the four tangents which can be drawn to it from any point of the curve, is equal to the cross ratio of the four points in which r meets L. The four cubics y, - 0, Y2 0, 8 - 0, y4 - 0 all pass through the seven points which have the same polar lines with regard to C2 and C3. These seven points are the base points of the (1, 2) transformation; and their corresponding lines, i. e., their polars with regard to C2, are by a well-known theorem bitangents of the limiting curve L. Hence, we have the following theorem: The seven points which have the same polar lines with regard to a conic C, and a cubic C3 are such that these polar lines form an Aronhold set of bitangents of a Liiroth quartic. It appears from this result that the seven points can not be chosen arbitrarily; they depend on thirteen independent constants instead of fourteen. This would not be expected a priori, because an arbitrary cubic and conic give fourteen arbitrary constants. It is not easy to find the relation between the seven base points. At Prof. Morley's suggestion I have considered the cubic C3 and two points as given, and found the locus of the other five. Take the two given points and the intersection of their polar lines with regard to C3 as corners of the triangle of reference. The equations of C3 and C2 then take the forms ax3 + b y3 + cz1S + 3 f y2z + 3 gyz2 + 3 k z X2 + 31X2 y + 6nxyz- 0, -Zx2 + bgy2 + cfz2 + 2fgyz 0, where x is arbitrary. The possible conics C2 all have double contact at two points on the line x 0. BATEMAN: The Quartic Curve and its Inscribed Configurations. 365 A point (x, y, z) has the same polar line with regard to C3 and C2, if the three derivatives of qp are proportional to the three derivatives of 4A; hence, it lies on the two cubic curves f (gy + cz) (lx2 + by2 + gz2 + 2fyz + 2nzx) g(by + fz) (kx22 + fy2 + cz2 + 2gyz + 2nxy), Ax (lx2 + by2 + gz2 + 2f yz + 2nzx) g (by + f z) (ax2 + 2nyz + 2kzx + 21xy). The first equation does not contain AX, and so represents the required locus when two points and C3 are given. The locus is a cubic curve which circumscribes the triangle of reference and passes through the four poles of x = 0 with regard to C3. It also passes through the two points in which the polar conic of one of the given points meets its polar line. Since the term in xyz is absent from the equation, it appears that the corners of the triangle of reference form a conjugate triad; i. e., the mixed polar of two points passes through the third. It is clear that this cubic locus is the cubic t3 corresponding to the line x= 0 in the transformation. The associated point T is the point y 0, z 0, and this point is coresidual to the seven base points of the transformation. If in the previous notation the equation of C3 is (ax3) -=0, where (x) =-0, and (Y1, Y2, y, DY), (Z1, 2 2 Z3, 4) are the two given points, the equation of the locus is (axy2) (ax2z) (ayz2) - (axz2) (ax2y) (ay2z). This equation was obtained by Professor Morley. Returning to the previous work, we notice that as X varies the second equation represents a pencil of cubics which pass through four fixed points on er, viz., the two given points and the two points in which the polar conic of one of these points meets its polar line. It appears, then, that the different possible sets of five points associated with the two given ones and C3 are cut out by a pencil of cubics through four fixed points on r,, and so form an involution on r3. ~ 3. Derivation of the Desmic Quartic from a (1, 2) Transformation. Let us now consider the case when the cubic C3 consists of three straight lines. Taking these lines as sides of the reference triangle and using the equation of the conic in the form F - (a, b, c, f, g, h$x, y, z)2 = 0, we find that the polar conic of a point P (X, Y, Z) with regard to the cubic is Xyz + YzX + Zxy 0, 46 366 BATEMAN: The Quartic Curve and its Inscribed Configurations. while the polar line of P with regard to the conic is U x + Vy + Wz - 0, where U=aX+hY+gZ, V=hX+bY+fZ, W=gX+fY +cZ. The line touches the conic if /XU+ VrYV+ V ZW=0. (1) This, however, is Humbert's equation for a desmic quartic curve.* It evidently represents a quartic curve having the lines X 0, Y = 0, Z 0, U- =O V= 0, W- O as bitangents. Now, when X =0, we have bY2 CZ2; hence, it appears that the points of contact of the three bitangents X = 0, Y = 0, Z = 0 are the corners of a complete quadrilateral whose sides are the lines x Va + Y Vb ~ Z Vc= 0. Humbert has shown that there are altogether six triads of bitangents of the curve which possess this property. The lines U = 0, V =0, W = form another of these triads. It should be mentioned that since a, b, c, f, g, h are quite arbitrary, the equation we have obtained represents the general desmic quartic, and so we have the result that the desmic quartic may be derived from a transformation associated with a conic C2 and a cubic C3 which consists of three straight lines. This gives a simple verification of Schur's theorem that the desmic quartic is a particular case of Liiroth's quartic. This result may also be deduced from Humbert's remark that the sides of the quadrilateral formed by the lines (1) meet the quartic again in collinear points. This remark also enables us to prove that there are more than one system of pentagons completely inscribed in the desmic quartic; for, if we can show that the quadrilaterals belonging to two triads of bitangents do not touch a conic, we can be sure that the pentagons derived from these quadrilaterals by adding an associated line, belong to different systems. Now, in the case of the two triads of bitangents whose equations have already been obtained, the sides of the two quadrilaterals are given by XV a~YVb+~ZV/c 0, UVbc- f2 ~ V7ca- g2 W/ab -h2 =0. Conics touching the sides of these quadrilaterals have equations of the types 1X2 + mY2 + jnZ2=0, f U2 + t V2 + v W2 0, respectively. These equations are the same only if g h + Ib b f + vf c = ga + y f h + v cg = a h + y b + v gf = 0; i. e., if g2 h2(f2- b c) +abfg (ch -fg) + cahf (bg- hf) = O. * Liouville's Journal, t. 6 (1890), p. 423. BATEMAN: The Quartic Curve and its Inscribed Configurations. 367 Since this equation is not generally satisfied, we must infer that there are different systems of pentagons associated with the two triads of bitangents. Now, Humbert has shown that there is an equation of the form (1) associated with any two of the six triads of bitangents; hence, we are led to the conclusion that there are at least six different systems of pentagons completely inscribed in the desmic quartic. To determine the nature of the involution on the conic touching the sides of all the pentagons of one system we use Darboux's method. Let y2 = xz be the equation of the conic; then the equations of the five sides of a circumscribed pentagon may be written in the form x + 2 ay + 0, (s 0,1,...,4). If now we put ax+2a,+A Xs- _ SS y'2 _y + z (s- 4., 4.) fX(), ), where f(a) = (a - a) (a -- a2) (a - a3) (a - a,), the condition (x) = O is satisfied. The equation of a desmic quartic derived from a cubic C3 consisting of the three lines (x3) = 0 and the conic C2 whose equation is (bx2) =0, is easily found to be (b2) () (b)2. Taking (b2x) = 0 to be the line ao, and noticing that (b) is arbitrary since any multiple of (x) can be added to (b2x) without altering the equation of the quartic, we find that this equation takes the form 4 f' (as) z ~ ~~ 2 + =0. a2x + 2 ay + z a2x + 2 a y + z + 2 a, +^ oilx+2o yIntroducing Darboux's coordinates x: 2 y z = 1: 0 + p: 0 q, we find that the equation may be written in the form f'4() x f'_ (as) 1 as a 0 a(-s a- 1P Hence, the pencil of quintics giving the parameters of sets of five tangents which intersect on a desmic quartic is of the form — 0) (0) f ' (as) + ( -) —O ao —O 0,+]=, where y is a variable parameter. The desmic quartic is also obtained when the cubic C3 and the conic C2 are represented by the equations x3 + y3 + z3 + 6 mxyz 0, ax2 + by2 + cz2 = 0 respectively. 368 BATEMAN: The Quartic Curve and its Inscribed Configurations. The polar conic and polar line of (X, Y, Z) are now X (x2 + 2 myz) + Y (y2 + 2 mzx) + Z (2 + 2 mxy) 0, aXx+ b Yy + c Yz 0, respectively. The conic touches the line if a2 X2 (YZ - m X2) + b2 Y2 (z X - m2 2) + c2 Z2 (X y _ m2 Z2) + 2bcYZ (nm2YZ —mX2) + 2caZX (m2ZX —mY2) + 2abXY (m2XY -mZ2) 0, or [(a2 — 2mbc)X + (b2- 2 mca)Y+ (c2 - 2 mab)Z] XYZ = m2[a2X4 + b2Y4 + C2Z4 2bcY2Z2- 2caZ2X2- 2abX2Y2]. This, however, is one of Humbert's equations for a desmic quartic. The lines X _0, Y -0. Z=0 are bitangents whose points of contacts lie at the corners of the quadrilateral formed by the four lines V/aX ~ 7,D Y V~ \cZ = 0. It appears, then, that if we want the (1, 2) transformation to give rise to a desmic quartic, it is not necessary for the cubic C3 to break up into three lines. ~ 4. Derivation of the (1, 2) Transformation from Eight Associated Points and vice versa.* Let a point P whose coordinates are (X, Y, Z) relative to a fixed triangle in a plane E be made to correspond to a quadric surface whose equation is XS + YS, + ZS 0, (1) where So, S1, S2 are of the second degree in the homogeneous coordinates x, y, z, t. As P varies, the quadric always passes through a set of eight associated points Ao, A,...., A7. Let the two generators through Ao of the quadric corresponding to P meet a fixed plane M in two points Q, Q' and consider the (1, 2) transformation by which P is derived from Q. As P describes a straight line, Q and Q' describe a cubic curve; viz., the projection of the biquadratic common to a pencil of quadrics through Ao, A,...., A7. If al,...., a7 are the projections of A1...., A7 on the plane M when Ao is taken as vertex of projection, the net of cubics through a,...., a7 is the image of the net of lines in the plane E. The points Q and Q' come together when the quadric corresponding to P is a cone, and then by Hesse's theorem the locus of P is a quartic curve which is the limiting curve L of the (1, 2) transformation. * The leading ideas of this section are due to Hesse, Clebsch and Frobenius. See the memoirs cited above. BATEMAN: The Quartic Curve and its Inscribed Configurations. 369 If the quadric (1) is always the polar quadric of P with regard to a cubic surface S3, the eight associated points are the poles of the plane E with regard to S., and L is a plane section of the Hessian H4 of the cubic surface. Now, when the equation of S8 is written in Sylvester's canonical form (ax)3- 0 of the sum of five cubes, the equation of H4 is (1/a x) = 0, and it appears that any plane section is a Liiroth quartic. The Liiroth quartic can be derived by Hesse's method from eight associated points which are the poles of a plane with regard to a cubic surface.* The condition to be satisfied in order that three quadrics through eight associated points may be polar quadrics of three points with regard to a cubic surface is discussed by T6plitz.t The invariant A must vanish. $ It appears that neither the cubic surface nor the three poles are uniquely determined when A = 0. The plane of the three poles and the five planes associated with the canonical form of S3 osculate a twisted cubic. ~ Dr. Coble has shown, with the aid of T6plitz's invariant, that when six of the eight points are given, the other two lie on a quartic surface. To deduce an equation for Liiroth's quartic from Frahm's result, we take four poles of the plane E as vertices of the tetrahedron of reference. If, now, the equation of the cubic surface is (a, b, c, d, f, g, h, k, 1, m, u,, A3, w, y, p, q, r, six, y, z, t)3 - 0, the polar planes of the corners of the reference tetrahedron coincide with the plane X + Y + Z + T 0, if a = I k =u, m = b = f -v, h = g c- w, a = y = d. The equation of the polar quadric of an arbitrary point (X, Y, Z, T7) of this plane is now found to be X[ (s-p)yz + (a-q)zx+ (a-r)xy-+ (a-d)xt+ (r-d)yt + (q-d)zt] + Y [ (b —p)yz+ (s-q)zx+ (b-r)xy+ (r —d)xt+ (b-d)yt+ (p —d)zt] + Z [ (c-p)yz+ (c-q)zx + (s-r)xy+ (q-d)xt + (p-d)yt+ (c-d)zt] = 0. This represents a cone if [X (s-p) + Y (b — p) + Z (c-p) ] [X(a-d) + Y (r-d) +Z(q-d) ] ~ [X(a-q)+Y(s-q)+Z(c-q) ] [X(r-d)+Y(b —d)+Z(p —d) ] ~ [X(a-r)+Y(b-r)+Z(s-r)] I[X(q-d)+Y(p-d) +Z(c-d)]- 0. * W. Frahm, Math. Ann., Bd. 7 (1874). t Math. Ann., Bd. 11. See also H. S. White and Miss K. G. Miller, Bull. Amer. Math. Soc., Vol. XV (1909), p. 347. $ Cf. Salmon's "Geometry of Three Dimensions," 4th edition, pp. 209-210. ~ A. C. Dixon, Proc. London Math. Soc., Ser. 2, Vol. VII (1909), p. 150. Toplitz, loc. cit. 370 BATEMAN: The Quartic Curve and its Inscribed Configurations. Hence, this is an equation for a Liiroth quartic. The six lines obtained by equating the square brackets to zero are bitangents of the curve. The tangent plane to the quadric (2) at the point X Y Z 0 or Ao is easily found to be X F+ Y -F + ZF = 0, where F_ (a-d) x2+ (b-d)y2 ax y o + (c-d)z2 + 2(p-d)yz+2(q-d)zx + 2 (r-d)xy=O; and this is the polar plane of (X, Y, Z, T) with regard to the quadric cone F- 0. Hence, if we make the plane M the same as E, it appears that the point P corresponds to the two points Q, Q' in which its polar plane with regard to the cone meets the section of its polar quadric with regard to S3. Let C2 be the section of the quadric cone and C3 that of the cubic surface S3 by the plane E; then the tranformation is clearly derived from C. and C3 by the method of ~ 2. The seven base points al, a2,...., a7 consequently have the same polar lines with regard to C2 and C3; hence, we have the following theorem: If we project from one of the eight poles of a plane with regard to a cubic surface, the other seven poles project into seven points which have the same polar lines with regard to a conic and a cubic. Taking each of the eight poles in turn, we find that there are eight Aronhold systems of bitangents of a Liiroth quartic which possess the property of being the polar lines of seven points with regard to a conic and a cubic. We are not justified yet in asserting that every Aronhold system of bitangents possesses this property. To obtain a set of eight associated points Ao,...., A7 when a,...., a7 are given, we proceed as follows: It is known, that eight associated points are the vertices of two tetrahedra which are self-polar with regard to a quadric.* Consequently, if one of these tetrahedra be chosen as tetrahedron of reference, the coordinates of the vertices of the other can be represented by (X, Y1, Z 1, t1), (x2, Y2 Z2, t2), (X3, y3, z3, t3)' (X4, Y4, Z4, t4), where the relations of the type ux, X+ vyrys + W zs+ ktr ts =0, (r*s=), are satisfied. By using the known properties of an orthogonal matrix, we may deduce from these six relations of the type 4 4 4 4 4 4 arxt, =O a yr t = r arZ tr - CrYrzr y - Ear czx= ar- y, = 0. 1 1 1 1 1 1 Now let t = 0 be the plane containing the seven points al,...., a, and let three of these points be at the corners of the reference tetrahedron. The coordinates of the other four may then be represented by * Hesse, "Analytische Geometrie des Raumes" (1861), 3d edition, Leipzig (1876), p. 214. BATEMAN: The Quartic Curve and its Inscribed Configurations. 371 (X,, y1 a, 0), (, ), (3,, y, z2, 0), (X4, y,, 4, 0) When these are given, our six equations determine the ratios of (al, a,, 4) and of (t1, t2, t3, t4) uniquely. Hence, when al,...., a are given, we can calculate Frobenius' irrational invariants for the associated quartic curve, and if we know the relation between a1,...., a7 for a particular type of quartic, we can (theoretically) find the corresponding relation between the eight associated points. ~ 5. Derivation of a (1, 2) Transformation by Mapping the Chords of a Twisted Cubic on a Plane. Let the polar planes of a point P with regard to a pencil of quadrics meet in a line 1. This line meets a given quadric S in two points Q, Q' which may be said to correspond to P. If, moreover, P is restricted to lie in a certain plane X, there is generally only one position of P for a given position of Q, for the polar planes of Q with regard to the quadrics of the pencil intersect in a line I which meets X in P. There is consequently a (1, 2) transformation between the points of the plane X and the points of the quadric S. If P moves along a straight line, the corresponding line I generates a quadric surface which meets S in a biquadratic C4. This curve C4 passes through six fixed points A,...., A6 on S, viz., the points in which S is met by the twisted cubic K3 which is the locus of the poles of X with regard to the quadrics of the pencil. If now we project the points of S on a plane Y, taking A1 as vertex of projection, the biquadratics C4 project into cubics through seven fixed points, viz., the projections of A2...., A6 and the two points where the generators through A, meet Y. The two points Q, Q', and consequently their projections on the plane Y, come together when the line I touches S; and then the locus of P is a quartic curve L, the limiting curve of the (1, 2) transformation between the two planes X and Y, or between X and S. The twenty-eight bitangents of L are derived from the six points A,...., A6, the twelve generators through these points and the ten pairs of plane sections of S which pass through the six points. The curve on S which corresponds to L is the intersection of S with the Weddle surface which is the locus of the vertex of a quadric cone through A,...., A; for Geiser* has shown that this surface is also the locus of a point Q such that Al,...., A6, Q, Q' form a set of eight associated points. Now, this is what happens in the present case; for when Q and Q' coincide, the ca 1 lines through P correspond * Crelle's Journal, Bd. 67 (1867). 372 BATEMAN: The Quartic Curve and its Inscribed Configurations. to co1 quadrics through A,...., A6, Q. Q', and since S also passes through these points, it follows that the eight points form an associated set. To find the equation of L, we shall suppose that the pencil of quadrics is given by the equation (a + -) X2 + (b +;) y2 + (c + -) Z2 + (d + 2) t2 = 0, where x is a variable parameter. If (X, Y, Z, T) are the coordinates of P, the line common to the polar planes of P has coordinates proportional to (b-c)YZ, (c-a)ZX, (a-b)XY, (a-d)XT, (b —d)YT, (c-d)ZT and generates a tetrahedral complex. The condition that this line should touch the quadric S is of the fourth degree in X, Y, Z, T. If, in particular, the equation of S is 1x2 + my2 + n 2 + pt2 =0, the condition is lp(b-c)2 Y2Z2 + m p(c-a)2Z2X2+ np(a-b)2X2Y2 + mn(a-d)2X2T2 + nl(b-d)2Y2T2 + lm(c-d)Z2 T2 = 0. If I = (a-b) (a-c) (a-d), m — (b-a) (b —c) (b-d), n -(c-a) (c-b) (c-d), p (d-a) (d-b) (d-c), the equation becomes (b-c) (a-d) (Y2Z2 + X2T2) + (c-a) (b-d) (Z2X2 + Y2 2) + (a-b) (c-d) (X2 y2 + Z2T2) _ 0, and represents a desmic quartic surface. The curve L, being a plane section of this surface, is a destmic quartic curve. If in the general case the equation of the plane X is x + r y + z + rt =0, the equations of the twisted cubic K3 are *'y'z't _ - __. K. ~. x~yz~ ^: Y: ': j a + b + x c + d + -' where 2 is a variable parameter. Taking two points on the cubic with parameters x and u, we see that the coordinates of the line joining them are proportional to six quantities of the type (d-a),r 2 - (a +>) (a + ) (d + P) (d +-) Comparing these with the coordinates of the line 1, we see that this chord of K3 is the line I corresponding to a point P with coordinates (X, Y, Z, T) of type BATEMAN: The Quartic Curve and its Inscribed Configurations. 373 X (a+X) (a +,) i(a- b)(a -c)(a-d) ' and this point lies in the plane X. Hence, a point P in the plane X corresponds to a chord of the twisted cubic K3. A tangent of the twisted cubic corresponds to a point PO of a certain conic C2 obtained by putting X = in the above equations. If a point P moves along a tangent to C2, the corresponding chord I always passes through a fixed point on K3. Hence, a point on K3 is associated with a tangent to C2, the chord joining two points on K3 with the point of intersection of the two associated tangents. It is now evident that the six bitangents of L which correspond to the points A A, A2...., Ao are tangents to the conic C2; they therefore form a Brianchon set. It should be noticed that the polar planes of a point on K,3 with regard to the pencil of quadrics meet in a line which lies in the plane X and touches the conic C2, this is why Ar corresponds to a line and not to a single point. An interesting theorem may be obtained by considering the case when the chords joining five points on K3 all belong to a quadratic complex R2. It is clear from the expressions for the coordinates of a line I that the points in the plane X corresponding to these chords all lie on a quartic curve C4. Now, these points are the ten points of intersection of five tangents to C2; hence, it follows that the curve C4 is a Liiroth quartic. Making use of the fact that there are cc 1 pentagons circumscribed to C2 and inscribed in C4, we obtain the following theorem: If a quadratic complex R2 and a twisted cubic K3 are such that a set of five points can be found on K3 whose joining lines all belong to R2, then c 1 such sets of five points on K' can be found. There is a similar theorem for a complex of degree n and sets of 2 ni + 1 points on a twisted cubic. It should be remarked that the quadratic complex R2 can not degenerate into the complex of tangents to a quadric surface; for a quadric surface can not touch the ten lines joining five distinct points. We have no reason to believe, however, that the complex is of a special type, for I have not yet succeeded in finding a relation between its invariants.* It is clear that, if one set of five distinct points can be found whose joins belong to a quadratic complex, there are at least oo3 others; for we may take any one of the o 2 twisted cubics on the five points, and there will be c 1 sets of five points situated thereon. We can expect that there are co 5 such sets of five points. * In the dissertation I suggested that the quadratic complex may be of a special type and contain Go 6 sets of five points whose joins belong to the complex. 47 374 BATEMAN: The Quartic Curve and its Inscribed Configurations. If we reciprocate with regard to one of the fundamental linear complexes of R2, we find that when sets of five points exist whose joins belong to R2, there are also sets of five planes whose lines of intersection belong to R2, and the sets of points and planes are equally numerous. It is evident, from what has gone before, that if we reciprocate with regard to one of the quadrics of our pencil, the points on the Liiroth quartic C4 correspond to the planes joining some point on K3 to the lines joining the different pentads of points on K3. Now, these lines generate a ruled surface R8 having K3 as a fourfold curve, and our theorem tells us that the tangent cone to R8 from any point of K3 is the reciprocal of the Liiroth quartic C4 with regard to some quadric of the pencil. It can also be shown that any plane section of R8 can be derived from C4 by a suitable quadratic transformation. Let Y be the plane of the section; then the transformation under consideration is obtained by finding where the line I corresponding to a point P in X meets Y. Now, if S, S' are two quadrics of the pencil, there is a correlation between P and the line p in which its polar plane with regard to S meets Y; similarly, there is a correlation between P and the line p' in which the polar plane with regard to S' meets Y. If p and p' meet in Q, there is a quadratic transformation which sends P into Q. The base points in the plane Y are the points where this plane meets K3. We shall now make further use of this quadratic tranformation. Let A,, A2,...., A6 be six arbitrary points on K3. The lines joining them correspond in X to the points of intersection of six tangents to the conic C2, while they meet the plane Y in fifteen points lying on the section of the Weddle surface having A1,...., AC as nodes. This section is a quartic curve which also passes through the three base points in which K3 meets Y. The quadratic transformation consequently sends it into a quintic curve C5 with three nodes, and this quintic passes through the fifteen intersections of the six tangents to C2. Now, Darboux has shown that when a quintic C5 and a conic C2 are such that a set of six tangents to C2 intersect in fifteen points lying on C, there are ao 1 such sets. Transforming this theorem, we are led to the conclusion that there are oo 1 sets of points B1,...., B2 on K3 whose joining lines cut out a configuration of fifteen points on a plane section of the Weddle surface having A1,....,,A as nodes. This result has been known for some time.t It should be noticed that the correspondence between a point P of the plane X and a chord I of the twisted cubic K3 enables us to map the Weddle * T. Reye, Zcitschr. filr Math. u. Phys., Bd. 11. "Geometrie der Lage," 3d edition, Part II, ~ 25. t H. Bateman, Proc. London Math. Soc., Ser. 2, Vol. III (1905), p. 288. F. Morley and J. R. Conner, AMERICAN JOURNAL OF MATHEMATICS, Vol. XXXI (1909), p. 2G3. BATEMAN: The Quartic Curve and its Inscribed Configurations. 375 surface W on a plane by making P correspond to the pair of points in which I meets W. The o 3 plane sections of W are then represented by trinodal quintics which pass through the fifteen intersections of six tangents to the conic C2. The result that there are o 3 trinodal quintics through the fifteen points is unexpected, as the method of counting constants suggests only c 2. The result may, however, be verified as follows: Let x: y: z 1 0: 02 be the parametric representation of the conic C,; then, if a,, a..., a, are the parameters of -the points of contact of the six tangents, and 1 2, 1, 93 are three arbitrary parameters, the equation 6 6 (0- a) n ( —a 1 1 (0_- _1)2 (0 -_ 2)2 (0__- 33)2 () - 1)2 (<) -- 2)2 ( -" 3)2 in Darboux coordinates (0, p) represents a quintic curve* which passes through the fifteen intersections of the tangents to C2 at the points al,...., a. and has double points at the points of intersection of the tangents at the points 1, /, /3.t The theorem can evidently be generalized as follows: There are Coo curves of degree 2n-1 which pass through the n (2 n — 1) intersections of 2n tangents to a conic, and have double points at the jn(n-1) intersections of n tangents to the conic. Let us next consider a set of five points B1, B2...., B3 on the curve K3. If I is any chord of K3, the twisted cubics that pass through B1, B2i X.. X, B5 and meet I generate a surface of the fifth degree S5 having K3 as double curve. This surface has been discussed by Clebsch,$ who shows that it can be represented on a plane in such a way that the images of the plane sections are Liiroth quartics passing through a fixed point T and the ten intersections of five lines. The surface S5 meets a plane Y in a quintic with double points at the three base points in which K3 meets Y. This quintic rF contains a configuration of ten points which lie by threes on ten lines, viz., the points where the joins of B,...., B5 meet Y; for, since B1,...., B5 are triple points of S5, the lines joining them lie entirely on the surface. The quintic F5 is transformed by our quadratic transformation into a Liiroth quartic C4 passing through the intersections of five tangents~ to the conic C2. * Cf. Darboux, loc. cit. t Cf. A. C. Dixon, Quarterly Journal, Vol. XXVI (1893), p. 212. t Math. Ann., Bd. 1 (1869), p. 253; Bd. 4, p. 249. See also Sturm, Geometrischen Verwandtschaften, Bd. 4, p. 315. ~ The fact that the ten points in which the joins of five points in space meet an arbitrary plane can be transformed by a quadratic transformation into the ten intersections of five lines, has been known to Dr. Morley for some time. 376 BATEMAN: The Quartic Curve and its Inscribed Configurations. Now, since there are I configurations of ten points on C4 which lie by fours on five tangents to C2, it follows that there are likewise xc configurations of ten points on Pr lying by threes on ten lines. Hence, any plane section of S5 contains co 1 configurations of ten points lying by threes on ten lines. If the plane Y passes through the line I, the curve P5 breaks up into the line I and a quartic with a double point. Hence, we are led to the conclusion that a quartic with a double point can sometimes contain o 1 configurations of ten points lying by threes on ten lines. The corresponding Liiroth quartic has a double point. The skew cubic which generates S, cuts out a series of triads of points on r5 which, together with the three base points cut out by K3, are all conjugate triads with regard to a certain conic r,. Hence, any such triad of points lies on a conic through the three base points, and so transforms into a set of three collinear points on C4. These points are, moreover, collinear with a fixed point on C4, viz., the point corresponding to the line I. To see this, we have only to remark that any generating cubic of S5 lies on a quadric through K3, and this quadric contains the line 1, since it is met by it in three points. The conic through the base points which contains the triad of points cut out by this generating cubic consequently passes through the point in which I intersects Y, and so transforms into a line passing through a fixed point T of the Liiroth quartic C4. When the generating cubic touches the plane Y, the associated line will touch the Liiroth quartic. Hence, the ten tangents from T to C4 are derived from the ten generating cubics which touch the plane Y. Now it is clear that the ten points of contact of these cubics lie on the conic r2, and are in fact the ten points in which r2 meets rP. Transforming this result, we find that the points of contact of the ten tangents from T to C4 lie on a quartic curve with three biflecnodes which lie on C4. It follows from the plane representation of the surface S5 that the point T can be regarded as an arbitrary point on the Liiroth quartic C4; hence, we have the following theorem: If tangents are drawn to the curve from an arbitrary point T on a Liiroth quartic, the ten points of contact lie on a quartic curve with three biflecnodes situated on the Liiroth quartic. The lines joining these nodes touch the conic C2, and form with the two tangents from T to C2 a set of five lines intersecting in ten points on the Liiroth quartic. * T. Reye, "Geometric der Lage," Vol. I (1886), p. 225. A. C. Dixon, Quarterly Journal, Vol. XXIII (1889). G. Humbert, Journal de l'Ecole Polytechnique, Cah. 64 (1894). BATEMAN: The Quartic Curve and its Inscribed Configurations. 377 It is clear that the three biflecnodes can be found at once when T, C2 and C4 are known. The second part of the theorem follows at once when we consider the set of five points on K. consisting of the two points in which it is met by 1 and the three points in which it is met by Y. ~ 6. Derivation of a Quartic Curve from a Quadratic Transformation between the Lines of a Plane. E. Godt * and E. Timerding t have shown that the general quartic curve can be derived from a quadratic transformation between the lines of a plane by considering the locus of points which lie on their corresponding conics. t Let A, B, C; A', B', C' be the two sets of fundamental points of the transformation, (f, v,,), (E', a', 7 ') the coordinates of a line referred to these triangles; then a point whose equation is x + y Y + z - 0 corresponds to a conic whose tangential equation is p e f t. The point equation of this conic is Vxx' + Vyy' + V zz- 0. Let x' - a1x + b1y + c1z, y' = a2x -+ b2y + c2z, z' - a3x + b3sy + c3z be the relations connecting the two systems of point coordinates; then the point (x, y, z) lies on its corresponding conic if Vx (alx + bly + c1z) + V/y (a2x + b2y + c2z) + V'z (a x + b3y + c3z) = 0. Comparing this equation with the equations obtained in ~~ 3 and 4 for the quartics of Liroth and Humbert, we can easily find the equations of quadratic transformations which lead to these quartics. ~ 7. Klein's Quartic Considered as the Limiting Curve of a (1, 2) Transformation. The general equation of a cubic through the seven points Py with coordinates v 4v E, (V-1 2,...., 7), where e7 - 1, is (X3 - y Z2) + (y- Z X2) + 3 (z3 x y) 0. * "Dissertation," Gittingen (1873). Clebsch-Lindemann, "Vorlesungen," p. 1007. t Math. Ann., Bd. 53 (1900), p. 193. t This method of generation and a number of others are included in a general scheme studied by Caporali, Memorie di Geometria (1888), and Segre, Annali di Mat. (2), t. 20 (1892). 378 BATEMAN: The Quartic Curve and its Inscribed Configurations. Consider the transformation X Y Z X -y 2 y-3 __ X2 Z3 z y2 which makes a line ~ X + Y -+ Z 0 correspond to a cubic curve through the seven points P,. It is evidently a (1, 2) transformation whose Jacobian is the sextic curve J = x5 y + y5 z + z5 x - 3 x2 y2 z2 = 0. To find the limiting curve of the transformation, we must eliminate (x, y, z) from this last equation and the equations (1). We easily find that Xz + Yx + Zy, (2) and J - 0 is equivalent to X 2 y + Y y2 z + Z z2x = 0; consequently X Y Z X_ Y Z or (x2z- y3 ) (xy2 -z) y (Z2- 3) o z- y Hence, X2 y2 Z2 2+ +Y,0 or X3Z + Y3 X+ZY 0, the equation of Klein's quartic.* HIence, the limiting curve L is a Klein quartic. The line joining the two points (x, y, z) corresponding to a given point (X, Y, Z) is represented by the equation Xz+Yx+Zy 0. There is evidently a correlation connecting this line and the point (X, Y, Z), which will be denoted hereafter by Q. If Q lies on L, the two corresponding points P are consecutive and the line joining them envelops the curve whose tangential equation is t3 + r3 + 3,5 - 0. If P, P' are the two consecutive points, P lies on the curve J, and the line P P' may be called the principal direction at P. A curve which crosses J at P in a direction different from the principal one, corresponds to a curve which touches the limiting curve L at Q.t For instance, a line which meets J in six distinct points generally corresponds to a rational cubic touching L in six distinct points. To find the equation of this cubic, we take the equation of the line in the form $+ y+ - =0. (3) * Klein-Fricke, "Vorlesungen fiber die Theorie der elliptischen Modulfunktionen," Leipzig (1892), pp. 675, 678, 701. Math. Ann., Bd. 14 (1879), p. 428. Haskell, AMERICAN JOURNAL OF MATHEMATICS, Vol. XIII (1890). H. F. Baker, I"Multiply Periodic Functions," p. 269. E. M. Radford, Quarterly Journal, Vol. XXX, p. 263. t De Paolis, loc. cit. BATEBAN: The Quartic Curve and its Inscribed Configurations. 379 This equation and (2) give x y z X- Z Y - X - z - Y' Now X y2 + Y z2 + Zx2 - 0; therefore X (<Y-_X)2+ Y ($Z-,Y)2 + Z ( X —Z)2 0 is the equation of this cubic. The coordinates of the double point are evidently X -=, Y =, Z =-; this point is evidently associated with the line (3) in the correlation already mentioned. A line (3) which passes through one of the base points P, corresponds to a rational cubic which breaks up into factors. One of the factors is the line E4^y + EvZ + E2X =0, (4) associated with the base point Pv; this line is a bitangent of L. The other factor is a conic e5^ X2 + E3 ^ 7Y2 + E6v 2 Z2 _ (2 + 2 3 ev) YZ - (2 + 2 Y E,)ZX - (42 + 2 5 v) X Y 0, which touches the quartic L in four points. If the line passes through a second base point PI, the conic breaks up into a bitangent e4 y + EA Z + e2/ X - 0, corresponding to P, and a second bitangent 5v-2A,2 X + E3v-4 r2 Y + e66v- t2 Z = 0. Since E4 t v + Ev + = 0, E4 + +- E 0 the equation of this last bitangent may be written in the form (E - EV)2X + (E2- e2)2 Y + E E (E- 3_ )2Z = 0. (5) The quartic curve L has seven bitangents with equations of type (4), and twenty-one with equations of type (5). Other equations for the bitangents and quadritangent conics have been obtained by Klein. It should be mentioned that many writers have used a different system of coordinates in studying the properties of the curve.* A set of eight associated points in space from which Klein's quartic may be derived by Hesse's method, is easily deduced from our set of seven points. If we take four of the points at the corners of the tetrahedron of reference, the other four points may be represented by (1,-0,1 +0, 20+1), (1+0, 1,1, —), (-0,-1, 0, 0+1), (0, 1+0, 1, 0), where 0 = e3 + 5 + e6 and consequently 02 + 0 + 2 = 0. * See, for instance, A. Wiran, Stockholm Bihang till Handlingar 21 (1895), N. 3. A. B. Coble, AMERICAN JOURN.AL OF MATHEMATICS, Vol. XXVIII (1906), p. 333. 380 BATEMAN: The Quartic Curve and its Inscribed Configurations. 8. Reduction of the Equation of a General Quartic to a Special Form. Salmon has remarked that the general quartic curve can be obtained as the locus of the poles of the tangents to a conic C2 with regard to a cubic C3, or as the envelope of the polar conics of points on C2 with regard to C3. Let x = 0, y 2 0, z 1 be the parametric equations of C2; then the equation of the polar conic with regard to C2 of a point on C2 with parameter 0 is represented by A f (0) 02So+ 20S + S2 = 0, and the envelope of the conic is the quartic curve Q = SoS -S = 0. If we use Darboux coordinates (0, p), where x = -0, y- 0+ -, z-1 are the coordinates of an arbitrary point P and (0, <q) the parameters of the points of contact of the two tangents from P to C2, it follows that the polar conic of P, viz., N -- f (0, ) =- 0 So + (6 + ) S1 + S, = 0, passes through the poles of the two tangents through P to C2 with regard to C3, and therefore the points of contact of the two conics f (0) = O, f (p) = O with their envelope Q. The conic N consists of two straight lines if P lies on the Hessian H of C, and then the two lines intersect in the correspoding point P' on H. The condition for this is that H (x, y, z) - 0, or H [60, 0 + 4, 1] = 0, where H is homogeneous and of the third degree. We are now led to ask the following question: Can we find three councs A f() =0-, B =f (() =0, C= f () 0, such that the points of contact with Q of each of the three pairs BC, CA, AB lie on two straight lines? The question is evidently equivalent to the following: Can we find a triangle which is inscribed in H and circumscribed to C? The answer seems to be that we can generally find two proper triangles of the required type. For, if we eliminate 4 from the equations H[06, 0 +', 1] =0, H[ p 4, + ', 1] =0, by means of a determinant, using Sylvester's method, we are led to an equation of the ninth degree in 0 and of the ninth degree in p. The first three rows of the determinant contain only 0; the last three rows contain only p. It is clear, then, that ( -- )3 is a factor of the determinant. Rejecting this factor, we are left with an equation of the sixth degree in 0, 6 + <p, 1, which, when written in the form F (x, y, z) _ 0, represents a curve of the sixth degree. * "Higher Plane Curves." See also Gerbaldi, Rend. Palermo (1893); Ciani, Rend. Lombardo (1895). BATEMAN: The Quartic Curve and its Inscribed Configurations. 381 The curves F -0, H 0 intersect in eighteen points, but these do not all give rise to proper triangles. If RS is a common tangent of F and C2, R being on F, and T is a point in which the other tangent from R to C2 meets C3, RRT must be regarded as a degenerate triangle fulfilling the conditions,* and so R is one of the points of intersection of F and H. Since C2 and H generally have twelve common tangents, there are generally twelve points of type R. The points of intersection of C, and H do not give rise to degenerate solutions in this case (cf. Clifford, loc. cit.). Since we have only accounted for twelve intersections by means of degenerate cases, we must conclude that there are six other intersections which give rise to two proper triangles fulfilling the conditions. This number agrees with the well-known result for the case when the cubic H consists of three straight lines.t Our argument is not quite conclusive; it would be more satisfactory if the algebraic work could be carried out in detail. There is an important exceptional case in which a 1 complete quadrilaterals can be inscribed in H and circumscribed to C2. Darboux t and Clifford ~ have shown, in fact, that if one complete quadrilateral is circumscribed to C2 and completely inscribed in H, then x 1 such quadrilaterals can be found. This is to be expected, because a quadrilateral inscribed in H and circumscribed to C2 gives four triangles fulfilling the conditions, and this is more than the proper number; consequently there must be an infinity of triangles. It will be convenient to call this exceptional case the case " D," and to refer to the special type of quartic which arises from it as a "D" quartic. It will be shown presently that this quartic is a desmic quartic. In the general case the result we have just obtained may be used to reduce the equation of the general quartic curve to the form 1+ n + n 0. X1 2 Y1 Y2 Z1Z2 Let us write x1x2- =f (, 4), yy2 =f(4 (, 0), z x2 =f(0, P); we can then find constants a, #/, y such that a Yl Y Zl Z2+ P/ Zx1 2 x1 2 + y Xl X2 Yl Y2 S0 2 --. To see this, let us put S, O 1, 1 -t, 82 t2; then the above identity holds if * Cf. Clifford's proof of Poncelet's theorem, "Math. Papers," p. 17. Clifford makes a few remarks on the present problem, but arrives at a slightly different result. t Salmon's "Conic Sections," p. 273. The reciprocal theorem is given.: Loc. cit. ~ "Math. Papers," p. 205. 48 382 BATEMAN: The Quartic Curve and its Inscribed Configurations. L(t+G)2 (t+cp) (t+4') +~3(t+(p)2 (t+4~) (t-f-) +2'(t+4~)2 (t+O) (t+qp) =: 0. Now, this equation can be satisfied by putting and it is easy to verify that we have identically ((p-4) Y Yy2 ZIZ2 + (4-) Z, Z2XI X2 + (O -cP) XI X2 YlY2 Hence, the equation of the general quartic curve can be expressed in the form (q-4') Yly2 1 Z2 +( — ) Zl 2XI X2-V (O - ()XI X2 Yly2 - If -we change the arbitrary constants, in x2$.' 2 the equation may be written YYZ Z+ZlZXXN+XlX2Y~Y2 It is clear that the quartic curve passes through the four points of intersection of two pairs of lines such as XI X2 - 0, Yly2- 0, and that the three pairs of lines give rise to twelve points. The conics A, B, C now have the simple equations Y1Y2+Z1Z22z0, Z1Z2-FXlX2-0, X1X2+-Y1Y2=07 and it is easy to verify that each of these conics touches Q in four points. It should be noticed that the four points irn which A cuts X1 X2 z 0, the four points in which B cuts Y1 Y2 - 0 and the f our points in which C cuts Z1 Z2= 0, all lie on the conic xi1X2 + Y1Y12 -F Z1 Z2 0.O This may be written in the alternative forms Xix2_+ YIY2 + zlz2 - where a1_O+q-K'k, a2_(4'+4 +O02 a3O00/4 ~9. Configurations of Sixteen Points Inscribed in a Quartic Curve of Type D. if,7 in the case D, 0, cp, 4', vX are the parameters of the points of contact with C2 of the sides of one of the quadrilaterals, we have six pairs of straight lines: The equation of the 4uartic curve may be thrown into the forms BATEMAN: The Quartic Curve and its Inscribed Con figurationis. 383 (-)Y1 y2 ZI Z + (4-) zi z2 xix2 + (O - p) X1 X2Y1 Y2 -O, (O - k) u Iu 2VIv2 -F ((P- X) V1 v2z<WI2 -F (x-O) z1z2u1 u2 0O, or into the f orm %xi X2fl1 U2 + Y Y1 Y2 V1 V2 + 1 Z1 Z2 W1 W2 = 0, where 2X, yi, v are subject to the relation 2X + ~t+ v - 0, but are otherwise arbitrary. It should be noticed that ((p-4' (O-X)x XI 2 U1 lt + (4~-0) ((P-X)y Y. 2 V1 v2 -I (O-qP) (4-X) Z, Z2 WI 2 -0.O lt is clear that the quartic curve passes through the four tetrads of points in which the six pairs of lines U1 U2,7 VI V2,7 W1 W2, Ixx2, Y1y2, ZI Z2 intersect. Prom what has gone bef ore it appears that the three diagonal points of each of these tetrads lie on the Hessian H of the cubic curve C.3, while the six pairs of lines touch the Cayleyan r3 of this cubic. It should be noticed that we have the 'identities (4.-O() vI v2 - (4'- X) z1zX2 + (O- X) x1 X2 0,O Consequently we may write (4~ —C0) VI v2 + ((P-zX) Y1 Y2 (-X) ZI Z2 - (O-(P) WI-w2 - 7 and the equation of the quartic curve takes the form (cp-4)(O-~)(M2-N2) + i ~ O )+ (L2-M2) -0, or aL 2 + pM2 +TN2-=07 where + r 0 The conics L, MY1 N are so related that the equations M~ —N =0, N L L 07 L'-i-M -0 all represent pairs of straight lines; their equations must consequently be of the f orm L==BC+AD-0,' M-CA.+BD=0, N==AB+CD-0, where A, B, C, D a~re linear functions of the coordinates. 384 BATEMAN: The Quartic Curve and its Inscribed Configurations. The equation of the quartic curve may now be written in the form a (B2C2 + A2D2) + p (C2A2 + B2D2) + t (A2B2 + C2D2) 0, which shows that it is a desmic quartic, i. e., a plane section of the desmic surface studied by Stephanos, * Humbert, t Schroeter I and others. It should be noticed that an equation of the form p XI X2 U, U2 + q y y2 Vl 2 + rz z2 w w2 0 always represents a desmic quartic, for with the aid of the identical relation (1) we may reduce it to the form (2); we may also reduce it to the form F x1 x2 u1 u2 GYlY2 vv2, which indicates that the curve passes through the sixteen points of the configuration. Hence, there are o 1 desmic quartics through these sixteen points. ~ 10. A Quartic Curve with co1 Inscribed Configurations of Twenty-four Points. When the configuration of twelve lines touching the curve r is discussed with the aid of the parametric representation of this curve in terms of elliptic functions,~ it appears that the twelve lines can be divided into three tetrads such that the points of contact of the four lines of a tetrad lie on another tangent to r. The three new tangents obtained in this way meet in a point T, and so it appears that the configuration of points and lines is identical with one whose reciprocal has been studied by Hesse. ii As the point T moves along a straight line, the configuration of sixteen points derived from it with the aid of the curve r, describes a desmic quartic. Caporali has shown that Hesse's configuration of twelve points and sixteen lines is associated with a second configuration of the same type and that each configuration is derived from the other by the same construction. He has shown, moreover, that the two sets of twelve points lie on a quartic curve T which contains o 1 configurations of a similar type. This quartic is usually known as Caporali's quartic and depends on eleven arbitrary constants. * Darboux's Bulletin, ser. 2, t. 3 (1879), p. 424. f Liouville's Journal, ser. 4, t. 7 (1891), p. 353. t Crelle's Journal, Bd. 109 (1892), p. 341. ~ Humbert, loc. cit. 11 Crelle's Journal, Bd. 36 (1848), p. 153. See also Durege, "Die ebene Curven dritter Ordnung," Leipzig (1871); Caporali, "Memorie di Geometria," p. 338; Schroeter, Crelle's Journal, Bd. 108 (1891), p. 269; J. de Vries, Acta Math., t. 12 (1888), p. 63; Martinetti, Atti dell' Accad. Catania, ser. 4a, t. 3 (1891) p. 20; S. Nakagawa, Proc. Tokyo Math. Phys. Soc., April, 1907. f Ace. di Napoli Rend., December, 1888, "Memorie di Geometria," pp. 336, 340, 349. Ann. di Mate. matica (2), t. 20 (1892), p. 274. See also Ciani, Ace. di Napoli Rend. (3), t. 2 (1896), p. 126. The curve was first studied by Hesse as the Jacobian of a line, a cubic curve and its Hessian. BATEMAN: The Quartic Curve and its Inscribed Configurations. 385 The two conjugate sets of twelve points are such that any quartet of one set and any quartet of the other set lie on a pair of lines. Two sets of twelve points which possess this property generally depend on twelve arbitrary constants. In Hesse's configuration some other condition is satisfied, and the nine pairs of lines through the quartets of points do not generally belong to a net of conics. We shall now obtain a type of quartic which contains o 1 conjugate pairs of sets of twelve points with the above property and for which the nine pairs of lines do belong to a net. Going back to the leading ideas of ~ 8, we now consider the analogue of Darboux's theorem that if the sides of a triangle A intersect the sides of a second triangle A' in nine points lying on a given cubic curve H, and both triangles circumscribe a conic C2, there are oo1 pairs of triangles circumscribing C2 whose sides intersect on H. The quartic curve generated by the poles of tangents to C2 with regard to a cubic curve C3 having H as Hessian, will possess co 1 inscribed configurations of twenty-four points, viz., the poles of the sides of two triangles A, A', and these twenty-four points can be arranged into two sets of three quartets possessing the property mentioned above.* The lines containing quartets of different sets touch the Cayleyan r3 of the cubic and belong to the net of conics which includes all the polar conics of C3. To obtain an equation for our quartic curve, let us consider the case when one vertex of the triangle A' lies on the conic; then two of the sides coalesce, and the sides of the triangle A touch H at points on a line. Now let (ax3) 0 be the equation of the cubic curve, (1) 0 the equation of its Hessian, the notation being the same as in ~ 2. The line x4 0 meets the Hessian at points on the lines x1, x2, X3 = 0, and the equations of the tangents at these points are respectively al X + a4x4 0, a2x2 + a4x4, a3x3+ + a4x4 = 0. Now, these tangents meet the Hessian again at points on the line (ax) = 0, and so may be taken as the sides of the triangle A. The four lines whose equations have just been given and the line x4 0 all touch the conic C2; also if (y1, y 2 y 3 y 4) are the coordinates of any point P on the quartic, the polar line * If we take two triangles whose sides intersect on H but do not touch a conic, the poles of their sides with regard to C3 will form two sets of twelve points possessing the same property, but the twentyfour points do not generally lie on a quartic curve. This configuration clearly depends on twelve constants. I am inclined to think that there are two distinct types of configurations of twenty-four points with the above property. 386 BATEMAN: The Quartic Curve and its Inscribed Configurations. of P with regard to the cubic, viz., (ay2 ) = 0, must also touch C2. Making use of the relation (x) =0 to get rid of x1, we find that the six lines touch a conic if a1a4 (a2- a3) (a2a3a4+ ala2a3-a3aa4-aa2a4) y1+ a2a4(a3a) (a3ala + aa2a - a1 a2- a 23 aa4) Y2 a a4(a l - a2) (al a2 a4 al a 3 aa3 4- a3 a14)Y3 + a2 a3 a (a2 - a3) (a2 a3 4 + - a al, a4 a+ a -a 2 a a3- 2 a1 a) y2 y3 + a a2 a3a (a3a- al) (a2 a3 a4 + a3 a, a4 + a aa a4-al a a- 2 a2 a2) y2 y2 +a a2a3 (a1- a2) (a2a3a4+ a3a, a4 +a a a -a a a- 2 aa2a 2a) y y -=a4 Y [a2y2 (a2-a3) (a2a3 a4 a2a3 - al a2a4 -- a a4) + a y2 (a3 -- a) (a3 a1 a4 + a a2a3 -- a a4 - a2 a3 a4) + a3L y (a1 - a2) (al a2 a4 + a1 a2 a - a2 a3 a4 - a3 a a4)]. It is clear from this equation that the curve passes through the vertices of the triangle ylY2 y3- =0, through the four points given by a, y2 = a2 y a y2 and through a second set of four points given by equations of the type c, y2 = C y2 = c y2, the two sets of four points being situated on a conic whose equation is obtained by equating to zero the expression within the square brackets. It appears from the equation that a quartic of the present type depends on eleven arbitrary constants. In conclusion I wish to thank Professors Morley and Coble for the encouragement and helpful suggestions they have given me during the preparation of this thesis. BIOGRAPHY. I, Harry Bateman, was born at Manchester, England, on May 29, 1882, son of Samuel and Marnie Elizabeth Bateman. I was educated at Manchester Grammar School, and proceeded from there to Trinity College, Cambridge (1900-1905), where I took the degrees of B. A. and M. A. After my election to a fellowship at Trinity College, I spent six months at Gottingen University and two months at the Sorbonne, Paris (1905-1906). I was Lecturer in Mathematics at Liverpool University (1906-1907) and Reader in Mathematical Physics at Manchester University (1907-1910). From 1910 to 1912, I was Lecturer in Mathematics at Bryn Mawr College; and since September, 1912, I have held the James Buchanan Johnston Scholarship at Johns Hopkins University.