THE TUHOKU Mathematical JOURNAL

THETUHOKU MATHEMATICAl JOURNAL

Projective Description of alone Quartic Curves, .

Joanna I. MAYER, San. Jose, Calif., U.S.A.

Ruth Gentry introduces her dissertation On the Fonrms of lane Quartic Curves as follows : " Many papers dealing with curves of the fourth order, or Q uartic Curves, are to be found in the various mathematical pert. dicals ; but these leave the actual appearance of the. curve. as as w hole so largely to the reader's imagination that it is here proposed. to give a complete enumeration off the fundamental forms of Plane Quartic Curves as they appear when projected so as to cut the line, at infinity the least possible number of times.... , together with evidence that the forms presented can exist". It is the purpose of this paper to show whether it is possible to derive, and how to construct, all of the forms proposed in Gentry's dissertation by the method described by Doctor H. P..Pettit in his " Projective Description of Some Higher Plane Curves(1)" . The accepted analysis of hgher singularities possiblein quartic curve is here given. A Tacnode is composed of two consecutive nodes, the tangents at a tacnode count as two consecutive double tangents. A NoJe-Cusp is composed.of one node and one cusp in consecu tive position and involves one real inflexion. The tangents t a Node-cusp• count as two (non-isolated) consecutive double tangents. An 0senode is composed of three consecutive nodes, either on a right line or on the • same parabola of curvature, A Tacnode-Cusp is composed of a node-cusp and a node, that is, of two nodes and one cusp in consecutive position. A Point of the _first kind is composed of three coincident nodes, formed by the crossing of three ordinary branches with real, distinct tangents. A Triple Point of the second kind is composed of one cusp and two nodes in coincidence, formed by the passage of an ordinary (1) The Tohoku Mathematical Journal, VoL 28, No8. 1, 2, June 1927. 2 J.L MAYER: branch through a cusp ; the tangents are all real, two of them consecutive, forming the cuspidal tangent. A Triple Point of the third kind is composed of two cusps and a node in coincidence ; the tangents are all, real, three of them consecutive. A triple Point of the fourth kind is composed of three coincident nodes, formed by the passage of an ordinary branch through a conjugate point (a double point with two imaginary tangents) ; one tangent is real, and two are imaginary. Classification of Quartics. I. Unicursal Rational Quartics. A. Quartics with Three real, distinct double points. i. 3 nodes ii. 2 nodes 1 conjugate point iii. 2 nodes 1 cusp iv. 3 cusps v. 2 cusps 1 conjugate point v i. 2 cusps 1 node vii. 3 conjugate points viii. 2 conjugate points 1 cusp ix. 2 conjugate points 1 node x. 1 node 1 cusp 1 conjugate point. B. Quartics with one real and Two real consecutive double points. xi. 1 Node-cusp 1 conjugate point xii: 1 Node-cusp 1 cusp xiii. 1 Node-cusp I node xiv. 1 Tacnode (opposition) 1 conjugate point xv. 1 Tacnode (opposition) 1 cusp xvi. 1 Tacnode (opposition) 1 node xvii. 1 Tacnode (embrassement) 1 conjugate point xviii. 1 Tacnode (exnbrassemermt) I cusp xix. 1 Taconrde (etnbrn.vsernent) 1 node, xx. 1 Tacriode (isolated) 1 conjugate point xxi. 1. Tacnode (isolated) 1 cusp xxii. 1 Tacnode (isolated) 1 node. C. Quartics with Three consecutive double points. xxiii. 1 Oseriode (real) PROJECTIVE DESCRIPTIONOF PLANEQUARTIC CURVES. 3

xxiv. 1 Oscnode (isolated) xxv. 1 Tacnode-cusp. D. Quartics with Three coincident double points, i.e., with a triple point. xxvi. 1. Triple point 1st kind xxvii. 1 Triple point 2nd kind xxviii. 1 Triple point 3rd kind xxix. 1 Triple point 4th kind. E. Quartics with one real, and Two imaginary double points. xxx. 1 node 2 imaginary double points xxxi. 1 cusp 2 imaginary double points xxxii. 1 conjugate point 2 imaginary double points. II. Elliptic Quartics. A. Quartics with Two real, distinct double points. i'. 2 nodes (unipartie) or bipartie( 1) ii'. 1 node 1 cusp „"" " "„ iii'. 2 cusps iv'. 2 conjugate points unipartie or bipartie v'. 1 node 1 conjugate point unipartie or bipartie vi'. 1 cusp I " B. Quartics with Two consecutive double points. vii'. 1 Node-cusp unipartie or bipartie viii'. 1 Tacnode (opposition) „ " "„ ix'. 1 Tacnode (embrasseinent) unipartie or bipartie x'. 1 Tacuode (isolated) " " "„ III. Hyperelliptic Quartics. A. Quartics with One double point i". 1 node unipartie, bipartie or tripartie ii". I cusp „ " " " "„ „ iii". 1 conjugate point unipartie, bipartie or tripartie. IV. Non-Singular Quartics. i"'. 1 closed circuit or oval ii'". 2 ovals iii'fl. 3 ovals iv"'. 4 ovals v" Annular quartic, i.e. an oval within an oval.

(1) A plane elliptic quartic can consist of one closed circuit or two closed circuits, with both double points on the same or separate circuits. Hence there is it possibility of 12 types of elliptic quartics with two distinct real double points. 4 J.I. MAYER:

The method described by Doctor H. P. Pettit in his " Projective Description of Some Higher Plane Curves, is as follows : " Consider two curves in the plane , C, of order n, and U2 of order m. On a point A1 chosen at random in the plane let a line 1, be drawn intersecting the curve C1, in n points P1,P2, ...... Pn. Let these n points determine n lines g1,g2 ... gn through a second point A2. Each of the lines cuts the curve C2 in m points R1 R2, .... , Rm which determine in lines k1, k2...... km through a third fixed point A3. Thus we have through A3 mn lines corresponding to the one line 1, through A, and which cut 1, in mm points. The locus of these points 'for all possible choices of the line through A, is the curve generated. Thus we have a higher type of projectivity. ` The product of two pencils of rays projectively related by means of two base curves of order m and n respectively (as described above) is a curve of order 2mn having mn-fold points at the vertices of the two pencils(1).' " When C1 is of order 2, C2 of order 2 and A0, 0, 0), A2(0,1,0) and A3(0, 0, 1) are the vertices of the triangle of reference in the plane (homogeneous coordinates) the curve generated is an octic. It is here proposed to show how this octic can be made to degenerate into different types of plane quartic curves. Let the three fixed points A1, A2,A3 be the vertices of the triangle of reference in the plane (homogeneous coordinates). Then the pencil on A1,will have the equation (1) The curve C, of order 2 will have the equation (2) -cohere i= 1, 2, 3 and ,j =1, 2, 3. The equation of the connecting pencil on A will be (3) The curve C2 of order 2 will have the equation (4) the pullcii On A3 Will have the equation (5) Eliminating x1,x2•> from equations (1), (2) and (3) there results the relation (1 ) Tohoku Mathematical Journal Vol. 28, No. 1, 2 June 1927. PROJECTIVE DESCRIPTION OF PLANE QUARTIC CURVES. 5

(6) which expresses in in terms of k. Similarly by the elimination of X2,x3 from equations (3), (4) and (5) there results the relation (7) or

expresses n as a function of m, or m• as a function of n. Equation (8) is of order 4 in m and vi. Hence by the elimination of in from (6) and (8) there results an equation of degree 8 between k and n. Then if the values of k and n from equations (1) and (5) be introduced in this last equation there results the explicit equation of the generated curve(1) in the following determinant form

Part I. Unioursal Plane Quarties. The three double points may be all real, or two be conjugate imaginaries. The case when all three are real will be first con- sidered : in this case the double points may be all distinct ; twe consecutive ; all three consecutive ; or all .three coincident. A. Quartics with Three real, distinct double points. The general appearance of the curve depends so largely upon the nature and relative position of the tangents at the double points and those from the, double points that it is here proposed to consider the curve in . subdivisions according s the tangents at the double points are real and distinct, consecutive, or imaginary, that is, ac- em-di-noras the double points are nodes cnsps or Conjugate points. Theorem I. The curve generated is a plane quartic curve having .three real distinct dps(2) at the vertices of the triangle of reference, if A1,and A2 are on c1; and A1A3,A2A3 are tangent (1) Tohoku Mathematical Journal Vol. 23, No. 1, 2 June 1927, pp. 73-74. (2) dps=double points, dp=double point.

(8) 6 J. I. MAYER : respectively to CI at A1 and A2. If A1 and •A2 are on C1,and A1A3 and A2A3 are tangent to C, at A1 and A2respectively, then a11=a22=a13=0(1 ). Substituting these values in determinant I, we have when expanded : a233x 32(b11„2x21+2b12x1x2+b22x22) +4a122b33x12x22- 4a12anx1x2x3•((b13x1+b23x2) =0. The highest powers occuring in are of the second degree; therefore A1,A2, A3 are dps(2 ). The tangents at the dps(3) are :

being the terms multiplying x12,x22, x32 respectively equated to zero. The conditions that A1,A2, A3 be nodes, cusps, conjugate points respectively is that which is also the conditions that A1A3(x2=0),A2A3(x,= 0), A,A3(x3=0) intersect G2 in two real distinct, coincident or imaginary points. Hence the combination of these three conditions on the sides of the triangle A1A2A3with the conditions given in Th. I taken three at. a time gives 10 plane rational quartics with three real distinct double points at the points A1, A2 and. A3 respectively. A. Quartics with three real, distinct dps( 3). A1 and A2 on G1, A1A3 and A2A3 tangent to G1 at A, and A2 respectively. C2 intersects C1 in two real and distinct pts. Type i, 3 nodes : A1A2, A2A3 and A1A3 intersect C2 in 2 real distinct pts. „ ii. 2 nodes 1 conjugate pt : AlA2 intersects C2 in 2 imaginary pts ; A1A3, and A2A3 intersect 6 in 2 real distinct pts. „ iii. 2 nodes 1 cusp : A2A3 tangent to C2; A1A2 and A1A3 intersect Ca in 2 real distinct pts. „ iv. 3 cusps : A1A2, A2A3and A1A3 intersect C2 in 2 coincident pts. V. 2 cusps conjugate pt A2A3_ intersect C2. in 2 coincident pts ; AlA2 intersects C2 in 2 imagin-

(1) Veblen and Young : Projective Geometry, Vol. I, p. 289. (3) Hilton, H.: Plaice Algebraic Curves, p. 23, pt=points, pts=points. PROJECTIVEDESCRIPTION OF PLANEQUARTIC CURVES. 7

ary pts, Type vi. 2 cusps 1 node : AlA3 and A2A3 tangent to C2, AlA2 intersects C2 in 2 real distinct pts, „ "vii. 3 conjugate pts : A1A2,A2A3 and A1A3 intersect C2 in 2 imaginary ptg. " viii. 2 conjugate pts 1 cusp : A1A2 and A2A3 intersect C2 in 2 imaginary pts : AlA3 tangent to 02. ix. 2 conjugate pts 1 node : A1A3 and A2A3 intersect C2 in 2 imaginary pts : AlA2 intersects C2 in 2 real distinct pts. „ "x. 1 node 1 cusp 1 conjugate pt : A1A3 intersects C, in 2 real distinct pts ; A2A3 tangent; to (C2 A1A2 intersects 02 in 2 imaginary pts. B. Quartics with one distinct and two consecutive dps. " Tacrlodes formed by the contact of . real branches are here distinguished as cmbrassement, in which both branches lie on the same side of the tangent, and opposition, in which the branches lie on opposite sides of the tangent (Cramer, Analyses des Lignes Courbes 1750 calls this , latter variety osculation, but, this term is perhaps open to objections. owing to. its ordinary use as indicating, contact of an order higher than the first). When the branches are imaginary the point is called ann isolated tacnode(1)." Theorem II. The curve generated is a plane quartic with two consecutive dps and one real dap ,if the triangle A1 A2A3is ingeribed

If A1A2A3 are on C1, then an=a22=a33= 0, and the equation of the quartic is :

that is, according as A1A3 intersects C2 in 2 real distinct, coincident, or imaginary pts .

(1) Gentry, Ruth: On the Forms of Plane Quartic Curves, Art. 35. 8 J. I. MAYER:

The expansion, the tangents (x22=0), and the construction show that A3 is a tacnode (embrassement), Node-cusp,-tacnode (isolated) or tacnodo (opposition) according as C2 lies inside C1,and A2A3 intersects C2 in two real distinct, coincident or imaginary pts ; or C1 and A2A3 intersect C2 in two real distinct pts. Hence by the combination of these seven conditions taken two at a time 12 different types of plane rational quartics with two consecutive dps and one real dp can be generated. B. Quartics with one and two consecutive dps. Triangle A1A2A3 inscribed in C1. For type xiv-xvi C2 intersects C1 in 2 real distinct pts ; for types xi-xiii, xvii-xdii , C2lies inside C1A1 is a dp and A3 is 2 consecutive dps. Type xi. A1A3tangent to C2; A1A3intersects C2 in 2 imaginary pts.

D. Quarties with Three coincident double points, i.e. Triple Point. Theorem III. The curve generated is a quartic with a Triple Point if A3 is on C1 and C2 ; A2 on C1; A1A2 tangent to C1, A1A3 tangent to C2 or AM, tangent to C1 and C2. if A2 is on C1, A3 on C1, and C2; A1A2 tangent to C1; A1A3 PROJECTIVE DESCRIPTION OF PLANE QUARTIC CURVES. 9 tangent to C2 then a33=L33=a22 =a12 b13= 0. The equation of the quartic is

The tangents at A3 are 2b33 X1(a11b23x1x2- a33b11x21-2b12a13x1x2•--b22a13 x22=0 being the terms multiplying x3•3 equated to zero, all three real and distinct, two real and coincident or two imaginary according as

These tangent will be real and distinct or two coincident according as all A13•‚O or a13=0, that is according as A1A3 is not or is tangent to

C1. According as A2 lies inside or outside C2, A3 is a triple point of the 1st or 4th kind. If A2 lies inside C2 and A1A3 is tangent to

Ct then A3 is a triple point of the 2nd kind.

Hence a plane quartic with a triple point of the 1st, 2nd and

4th kind can be generated by this projective method.

U. Quartics with Three coirncident dps.

The following unproved theorems are given in order to show the variety of choices possible for the positions of the base points and curves. It might be noted here that one is not limited to c uves of order two for nine base curves. A cubic and a straight line might have been used but conics presented a greater possibility in the case of plane quartic curves. Theorem IV. The curve generated is a. rational quartic with 2 imaginary dps if 10 J. I. MAYER :

(1) (2) In (1) for construction A3 is on both C1 and C2, A1 is the center of C1; C1 is tangent to A2A3 at A3. A2A3 is perpendicular to A1A3 at A3. In(2) A3is „,C2_and C2 ; A1 is the center of C2; C1is tangent to A2A3at A3. A2A3is perpendicular to A1A3at A3. In this last case A3 is a cusp in the other it is a• node. If triangle AlA2A3is a right triangle with the right angle at A3 and C1 is moved along A2A3we see that A3 can be made a node, cusp or a conjugate pt. Theorem V. The curve generated is an elliptic plane quartic with 2 real distinct dps if : (1) A1 and A3 are on C1,and C2. It is unipartie or bipartie according as A2is inside or outside C1 and C2. A1 and A3 are nodes. (2) A1 and A3On C1,and C2 ; A1A2tangent to C1. A3 is a node . A1 is a cusp. (3) Al and A3 on C1,and C2; A1A2 tangent to C, ; A2A3 tangent to C2. A1 and A3 tire cusps. (4) Triangle AlA2A3 is self-polar with respect to C1 and C2. A1 and A3 are conjugate points if A2 is inside C1 and C2; Al is a conjugate point, and A3 is a node if A3 is inside C, and C2. (5) A2 is on C1 and C2; A3 is on C1,; A1A2is tangent to C1,and C2 at A2; A1A3 is tangent to C1. A3 is a cusp and Al is a node, cusp or a conjugate point according as A1A3 intersects C2 in 2 real distinct, coincident or imaginary points.

Part II. Theorem I. If the three base points are taken on a straight line in the plane and the base curves are conics, the generated curve is always a plane quartic curve. Let the three fixed points A1A2', A3 be the points A, (1, 0, 0), A2'(1, 0, 1), and A3(0, 0, 1) in the plane (homogeneous coordinates). Then the pencil on A, will have the equation (1) The curve C, of order 2 will have the equation ' (2) where i=1., 2, 3 and j=1, 2, 3. The equation of the connecting pencil on. A2' will be PROJECTIVE DESCRIPTION OF PLANE QUARTIC CURVES. 11

(3)

The curve C1 of order 2 will have the equation (4) where i = 1, 2, 3 ; j =1, 2, 3. The pencil on A3 have the equation (5) Eliminating x1,x2 from equations (1), (2), (3) and (5) there results the relation (6) f (l + kn, k, 1) = 0, which expresses n in terms of k. Similarly by the elimination of x1x3 from equations (3), (4) and (5) we have (7) which expresses n as a function of m, or m as a function of n. Equation (7) is of order 4 if m and n . Hence by the elimination of n from (6) and (7), by using Sylvestey's Eliminant-Method, there results an equation of degree 8 between m and k. then if the valueq of m and n from equations (1) and (5) be introduced in this, last equation there results the following equation

which when expanded has a factor.x2= 0 throughout and hence is always of the fourth order and hence is always a plane quartic. N. B. The 1st and 2nd and the 3rd and 4th rows are the same except for a multiple of x2 in the 2nd and 3rd columns. This 12 J. I. MAYER : writing is to be omitted. We shall show how many of the different types of plane quartic curves given in the classificationPart I can be derived by this.special method,that is, the base points lying on a straight line in the plane and the base curves are conics. In order to simplify the work we shall begin with a Non-singularplane quartic with 2 ovals and then proceedto demonstratehow many possibletypes of plane quartics can be derived from a Non-singular plane quartic with 2 ovals, either by causing an oval to disappear; by shrinking an oval to a point causing a conjugate point ; joining a closed circuit to itself or joining two closed circuits causing a node; or causing a 'loop-node' to evanescentinto a cusp ; and etc ; or a combinationof these conditions(1). Theorem II, If C1 and C2 intersect in 2 real and distinct points, A2'R1 and A2R2(tangents to C2 from A2') intersect C1,in 2 real distinct points p1p2 and p1'p2' respectively (A2' must be chosen such that the arcs P1 Q P2 and F1' Q' P2' do not overlap one another, see Fig. 1) ; the points of tangency M1 and M2 (tangents to C1 from A1,) lie outside the arcs P1 Q P2, and P1'Q'p2' the generated Curve is a Non-singular quartic consisting of 2 ovale. See Fig. 1. (1) One of these ovals disappears when P1 and P2 are imaginary, or shrinks to a conjugate point when .P1 and P2 coincide. (2) One of these ovals joins to itself forming a 'loop-node' when M, lies within the arc P1QP2and M2 without the arcs P1QP2 and P1'Q'P2'. Similar results follow if M1, is replaced by M2. (3) This ` loop-node' evanesces into a cusp when M1 or M2 coincides with P1,

(1) Gentry, Ruth : On the Forms of Plane Quartic Curves, Art. 70. PROJECTIVEDESCRIPTION OF PLANE QUARTICCURVES. 13

That this condition for a cusp is correct will be shown later. The construction is sufficient proof for the 'loop-node' and the conjugate point. Hence these three conditions for a conjugate point,'loop-node, and a cusp taken one at a time combined with the above conditions for an oval or a non-oval gives a method of 9 Hyper-elliptic quartics given in the classification Part I. (4) If M1 and M2 both lie within P1QP2 or P1'Q'P2, 2 ' loop- nodes ' are formed on the same closed circuit and (5) 2 cusps will be formed if M1 and M2 coincide respectively with P1 and P2 or P1' and P2'. These 'loop-nodes' or cusps will lie on the same closed circuits according as M1, and M2 lie both within P1QP2 or P1'Q'P2' or one within each of the arcs. Hence combinating these conditions with the preceeding inform- ation we can generate 12 of the 18 Elliptic quartics with 2 real and distinct double points.

Summary. A1A2'A3 is a straight line and C1, and C2 are conies. A2' must be chosen such that the arcs P1QP2 and P1'Q'P2' do not overlap one another see Fig. I. 14 J. I. MAYTER :

(P1 and P2 are real and distinct and M2 lies within P1'Q'P2' the curve will be bipartie with a . 'loop - nodg' on each circuit.

Theorem III. As A2' approaches closer to C1 the 2 closed circuits in Fig. I. approach each other. In the limiting position, that is A2 on C1 they join forming a 'loop-node.'

If A2' is on C1=‡”a1x1xj=0 then a11„+x22+2a13x1x3+a33x32=0 must be identically satisfied, by the point A2'(1, 0, 1) and this is satisfied if (a11 + 2a13 + a33) = 0. Making use of this above condition in Deter- ininant I Part II we have when expaiided :

The highest power occuring in a is of the second degree and hence according to Hilton : Plane Algebraic Curves pp. 23-24 A3„0(0,0,1) is a double point. A3 is a 'loop-mode', cusp or a conjugate -point according as 8 (the point of intersection of A2'R1with C1,)lies below A2', that is A2' lies within SR1 ; S coincides with A2' ; or S lies within A2'Ri. These three conditions combined with the three conditions for different kinds of double points given in Theorem II taken three at a time give 6 of the 10 plane rational quartics with P ROJEONVE DESCRIPTIONOF PLANE QUARTIC CURVES. 15

3 real and distinct double points.

Summary.

Theorem IV. As Al approaches closer to C1and M1 and M2 lie within P1'QP2' (see Fig. I) the 2 'loop-nodes' as in type i' approach each other and in the limiting position, that is Al on C1, they actually coincide forming two consecutive double points at A1. When Al is on C1,that is a11=0, the equation of the plane quartic is as follows :

The equation of the tangents, (a12X2+ a13x3)2= 0, the expansion at Al and the construction show that Al is 2 consecutive double points. Al is a tacnode (embrasseinent), Node-cusp, or a tacnode (isolated) according as A1,lies within P1'Q'P2' (see Fig. 1) ; Al coincides with P2' ; or P2' lies within A1Q'P1'. The generated curve will be bipartie or unipartie according as P1 andP2 are real distinct or imaginary (See Fig. I.) Hence 6 of the 8 Elliptic plane quartics with 2 consecutive double points can be generated. (A method of generating a plane quartic. with a tacnode (opposition) will be given later). Since A1 coinciding with P2' (see Fig. I.) (i.e. M1 and M2 coin- 16 J.I. MAYER: ciding with P2' involves the condition for a cusp) causes a Node- cusp the condition imposed for a cusp in Theorem II is necessarily correct. If A2' is on C1 then (a11+2a13 + a33)= 0 and making use of this in the preceeding equation we see that A3 is then a double point. Henco combining these three conditions placed on A1 with the three conditions placed on A3 in Theorem III taken 2 at a time 9 of the 12 rational plane quartics with 2 consecutive double points can be generated. It might be here noted that A1A2'tangent to C1 at A2', that is an„==a33=-a13, causes 2 consecutive double points at A3. A3 is a tacnode (embrassement), Node-cusp, or a tacnode (isolated) according as S (point of intersection of A2R1, where A2'R1 is the tangent to . C2 with C1,)below A2R1 ; A2', i.e. A2' within SR1,; A2' and S coincides ; S2 lies within A2R1.The generated quartic will be rational or elliptic depending upon the position. of A1 that is depending upon the position of M1 and M2 of Fig. I. Summary. PROJEOTIVE DESCRIPTION OF PLANE QUARTIC CURVRES. 17

From Theorem IV we see that A1 On C1 causes two consecutive double points, and from Theorem II we see that A2'R1 tangent to C1 causes one double point ; if we combine these two results we derive a plane rational quartic with three consecutive double points.

Theorem V. if A1A2'A3 is tangent to C1 and C2 at A1, the curve generated has an Osenode at A1. If A1A2'A3 is tangent C1 to C1 andC2and C2 at A1, then and the equation of the plane quartic is as follws :

The expansion at A1, the tangents x22= 0 and the construction show that the two tangent branches at A1,cross and therefore A1 is an Osenode. Al is a real or isolated Osenode according as C1 and C2 are tangent internally or externally at A1. Theorem VI. If A3 is on C2 and A2' is on C1 the generated curve is a plane rational quartic with a Triple point at A3. If A3 is on C2 and A2' is on C1then b33=0 and (an„+ 2a13+a33)=0 and the equation of the plane quartic is :

The highest power occuring in x3, equated to zero gives the equation of the tangents at A3, which is of the first degree and therefore A3 is a triple point. If A1A2'A3 is tangent to C1, that is a11•==a33= -a13 two tangents coincide; if A1A2A3 is tangent to C1 and C2, that is b13=0 and a11= 18 J.1 MAYER;

a33=- a18three tangents coincide. The tangents are all three real and distinct or 2 imaginary depending upon the position of S (S is the point of intersection of A2'R1,with C1). Hence As is a triple point with 3 real distinct, 2 coincident, 3 coincident, or 2 imaginary tangents according as- A2' lies within SR1; A1A2'Aa'is tangent to C1; A1A2'Asis tangent to both C1 and C2; or s lies within A2'R1. It might be noted here that if A1 is on C1,and A2' on C2the generated quartic has a triple point at A1 of the 1st, 3rd, or 4th kind according as A1 is on C2 and A2' inside C1,and A3 within A1A2'; A3 outside C1and A3 in between A1A2';or A2'outside C1and A1 within A2'A3. C1 and C2,intersect symmetrically for all three types. In conclusion, of this special method that was described in Theorem II we can say that most of the plane quartics given. in the classification Part 1 have been found ; for the construction for some if not all of the quartics with deficiency (the deficiency of a curve is the number .D by which its number . of double points is short of the maximum which is for an nth degree curve =1/2(n-1) (n -• 2) Sa1mon : higher Plane Curves p. 30) zero, one, two and three have been given. Since this special method at the most gave a plane quartic Non-singular with 2 ovals it was not to be expected to find plane quartics with three or four ovals. A few types that were not found by this special method have been found by different positions of U, and C2 arid. of the points A1,A2' and A3. A1,A2', and A3 will still be taken on a straight line in the plane and C1, and C2,will still be conics. Under Theorem IV two methods were given for the construction of plane quartics with 2 consecutive double points, rational or elliptic. Neither of these methods gave a Tacnode (opposition). It is proposed here to give a method which includes all four types of quartics with 2 consecutive double points. Theorem VII. If A3 is on C2, the generated quartic has 2 consecutive double points at A3. If A3 is on C2,then b33=0 find the equation of the quartic is: PROJECTIVE DESCRIPTION OF PLANE QUARTIC CURVES. 19

The tangents, (b13x1,+ b23x2)2=0, the expansion at A3 and the construction show that A3 is 2 consecutive double points. If C1 and C2 intersect symmetrically ; A1A2'A3is the line of symmetry ; the tangents formfrom• A A1 intersect C1 in M1 and and according as 2'M2intersect C2 in 2 real distinct or coincident point the curve generated will be bipartie or unipartie with A3 a tacnode (embras- ement). If C1 is moved up along the other line of symmetry until it becomes tangent to A1A2'A3then A3 becomes a Node-cusp. If C1, is moved still farther up A3 becomes a tacnode (isolated). Or if A1, is the center, of C1 and C2 (A1 within A2'A3 and A3 is on C2) then the curve generated is bipartie with a tacuode (opposition) at A3, with an oval surrounding A1. If A2' is moved over on Cs (outside Cl) A, becomes a conjugate point, and if C1,becomes tangent to C2 at A3,then A1,becomes a node. Or if C1,and C2 intersect symmetrically A1A2'A3is the diagonal of the rectangle formed by the joins of the intersections of C1 and C2 and A3 is on both C1,and C2 the curve generated consis of 2 ovals tangent externally at A3. The following method has been selected to generate an annlar quartic. The points A1A2' and A3,are taken on a straight line and C1 and C2 are conics and the triangle A1A2A3is self-polar with respect to C1 and C2 then if A1,is inside C1,and C2 and A2' outside C1 and C2,but within A1A3 the generated curve is annular ; if A2' is moved until it is inside Cl and C2 the two ovals intersect in 2 real distinct points. Or if A3 is inside C1 and C2, A2' outside C1 and C2 but within A1A3the ovals do not intersect (not annular). Or if A2' is inside C1 and C2 and A1 within A2'A3,the 2 ovals join to them- selves forming 2 'loop-nodes'. It might perhaps be of interest to know that if when the two base curves are conics and the three base points are A1(1, 0, 0), A2(0 ,1,0), A.3(0,0, 1), the two conics coincide then the generated curve is always a plane quartie curve with double points at A1 and

A3. (Fig. 2.) In conclusion, the following piano quartics, enumerated in the classification Part I, can be generated (using the method described in Dr. H. P. Pettit's " Projective Description of Some Higher Plane Curves ") by either the general method (the three base points form- 20 J. I. MAYER :

ing a triangle) or the particular method (the three base points lying on a straight line) (1) all rational quartics except a quartic with a tacnode-cusp ; (2) all elliptic quartics except those with 2 imaginary double points ; 2 conjugate points bipartie ; Lode and a conjugate point bipartie ; cusp and a conjugate point bipartie ; and a taenode (opposition) unipartie ; (3) all quartics with deficiency 2 except those with 3 parts ; and a conjugate point with one oval ; (4) all non PROJECTIVEDESCRIPTION OF PLANEQUARTIC CURVES . 21 singular quartics except those which involve 3 or 4 ovals. (It must be remembered that a full classification wqs not given for the Non- singular plane quartics). Whether or not a method of construction can be found, using for the base curves two conies, for those types not included in this paper, has not as yet been determined.