On the Quartic Curve having two double points. By K. OGURA.( READ JANUARY 18, 1908.)

Transcription

1 230 K. OGURA: [VOL. W. On the Quartic Curve having two double points. By K. OGURA.( READ JANUARY 18, 1908.) 1. Any point (x1 x2 x3) on a quartic curve having two double points P, P' can be expressed by the elliptic functions of a single parameter.* Let us use Loria's notatiou õ ƒë being the parameter, and ƒï, c1, c2, c3,ƒì, ƒð, ƒâ2, ƒâ3,ƒã being some con stants, and Among the intersections ƒë1, ƒë2, cof this curve with a curve Cm of the mth order, there exists a relation : when Cm passes through neither P nor P', when Cm passes through either P or P', when Cm passes through both P and P', * Clebsch-Lindemann: Vorlesungen uber Geometrie. Bd. I. S. 903, &c. (1876) õ Loria: Spezielle algebraische und transeendente Kurven der Ebene. S (1902)

2 . No. 12.] ON THE QUARTIC CURVE HAVING TWO DOUBLE POINTS First of all we shall shew the -residual theorem ö on the quartic curve. Let the intersections of with be where m1+m2=4ƒê1-4; then [mod.2k, 2K'i]. Next, let the intersections of with be where n1+m2=4ƒë-4; then [mod. 2K, 2K'i]. Lastly, let the intersections of with be where m1+n2= 4ƒÊ2-4; then [mod. 2K, 2K'i]. From these relations we have [mod. 2K, 2K'i], which shows the existence of the curve passing through the where This result is nothing but the co-residual theorem on the curve ö Stahl: Theorie der Abel'schen Funktionen. S. 77. (1896)

3 232 K. OGURA: [V OL. Y. 3. Take two points A and B on the curve. If we consider a conic passing through P,P' and A, and touching the curve at A and again at another point Ai, tnen we have Hence [ mod. 2K, 2K'i]. Similarly, for B and Bi, But [mod. 2K, 2K'i]. Therefore the six points P, P', A, B, Ai, Bi are on a conic. This result is similar to that which Hesse obtained for a cubic.* Fig. 1. In the particular case of a bicircular quartic, every above-stated conic becomes a circle, and the figure takes a simple but õ (Fig. 1). More particularly in the case where the bicircular quartic degenerates into two circles, we obtain a well-known theorem in the elementary geometry (Fig. 2). * Clebsch-Lindemann. S Durege: Die ebenen Kurven dritter Ordnung. S (1871) õ This result is also true, when the bicircular quartic has the third double point. In this case, by the transformation of reciprocal radii vectores, we can get a simple theorem on the conic.

4 No. 12.] ON THE QUARTIC CURVE HAVING TW0 DOUBLE POINTS. 233 Fig Let be the points of contact of the four tangents to the quartic curve through the double point P; and similarly let be those for the other double point P'. Since [mod. 2K, 2K'i], there exists a conic through the six points P, P', A1, A2, A1', A2'. De note the conic by In the same way, we can prove the existence of the other eleven conies. Thus we obtain the following twelve conics in all. (I) (II) (III) (IV) (V) (VI)

5 ( Z) ( [) ( \) ( ]) (XI) (XII) Hence there are twelve conics through the double points of the quartic curve and the four points of contact of the tangents to the curve through the double points. Remark i. Two tangents PAi and P'Ai' may be considered as the degenerated conic which passes through P, P' and touches the quartic at Ai and Ai'. ii. This theorem is a particular case of the double tangent theorem on the quartic curve having - no singular point. Evidently we have 5. the following congruence: Hence the ten points P, P', A1, A2, A3, A4, A1', A2', A3', A4', are on a cubic curve. Denote it by [P A1 A2 A3 A4 P' A1' A2' A3' A4']. Now, since the conics (I) and (XII) have four common points, P', A1', A2' on a cubic, two straight lines A1 A2 and A3 A4 interest P at a point on the cubic, by the co-residual theorem; the same is true for A1 A3 and A2 A4, A1 A4 and A2 A3, A1' A2' and A3' A4', A1' A3' and 4', A1' A4' and A2' A3'; further for A1 A1' and A2 A2', A1 A2' and Therefore 1 A1' and A3 A3', &c.

6 No. 12.] ON THE QUARTTC CURVE HAVING TWO DOUBLE POINTS. 235 These four points of concurrence, A1", A2", A3", A4" say, are on the cubic [P A1 A2 A3 A4 P' A having the six points of interection of 1' A2' A3' A4'] These twelve points and sixteen lines form the following configuration (Fig. 3): Fig. 3. Groups of three collinear points. Lines of collinearity.

7 Groups of four concurrent lines. Points of concurrence This configuration is similar to Hesse's* on the cubic curve, and also similar to that which can be usually found under the name of Poncelet or Reye. õ 6. Let 1, 3 and 5 be the straight lines which pass through the other intersections of the conies (I), (III) and (V) respectively with any conic through P and P'. Since the conics (I), (III) and (V) have the four points P, P', A1, A1' common, the three straight lines 1, 3 and 5 are concurrent. Denote the point of concurrence by a11'. In this way, we get the theorem: There are twelve straight lines passing through every pair of the two points at which any conic through P, P' intersets the twelve conics; and these twelve straight lines form sixteen groups of three concurrent lines (Fig. 4). * Durege, S õ Emch: Introduction to peojective geometry and its applications. p. 86. (1905)

8 Fig. 4. Groups of three concurrent lines. Points of concurrence.

9 These sixteen points of concurrence form twelve groups of four It is note-worthy that this configuration is reciprocal to that in the last article. 7, There are nine conics passing through P, P' and a point (q) on the quartic curve and having a ccatact of the second order with the quartic. then Hence These nine points, three by three, are on a conic through P, P' and Q. If we denote the point by (m, n), we get the following arrangement.* * Weber: Lehrbuch der Algebra. Bd. II. S (1899). Netto-Cole: Theory of substitutions. p (1892). Clebsch-Lindemann. S. 609, &c.

10 No. 8.] ON THE QUARTIC OURVE IIAVING TWO DOUBLE POINTS These nine points form a triple system, and may be written as These results show us a great resemblance between the cubic and the quartic having two double points. 8. There are sixteen conics passing through the two double points P, P' and having a contact of the third order with the quartic curve. The sixteen points of contact are Now, if the cubic curve through the eight points P, P', A1, A2 A3, A1', A2', A3' touch the quartic curve at the point v, then we have [mod. 2K, 2K'i]. Hence Therefore, there are four cubic curves passing through P, A1, A2, A3, P' A1', A2', A3' and touching the quartic curve at the poin's X1, Y1, Z1, W1 respectively. Denote these cubic curves by &c. Repeating the same process, we get the follow ing theorem. There are sixty four cubic curves passing through the two double

11 points and six points of contact of the tangents to the quartic curve through the double points, and touching the quatic at the points of contact of the conics which pass through the two double points and have a contact of the third order with the quartic curse.

12 No. 12.] ON THE QUARTG CURVE HAVING TWO DOUBLE POINTS The quartic curve having two double points has the same deficiency 1 as the cubic curve having no singular point, and they are transformable to each other by a certain birational transformation. Take two points.p and P' on a cubic curve having no singular point, and the third point P" not on the curve. Then by the Cremona quadratic transformation whose principal points are P, P', P", the cubic curve can be transformed into a quartic curve having two double points P and P'. In general, if the original curve Cn1 passes through the point P, a1 times; P', a1' times; and P", a1" times; then the curve can be transformed into a curve Cn2 which passes through P, a2 times; P', a2' times; and P", a2t times; and there exist the following fundamental relations:* and By these relations, we are able to shew the resemblance of some properties of the two curves, as in Salmon's theorem,** Hart's theorem, õ Steiner's theorem of closure, õ õ &c. Fig. 5. Again, the theorem in Art. 7 is nothing but the resnlt of transforma tion of the well-known theorem on the cubic. Next, by the transformation of the bicircular quartic, we can shew * Salmon-Fiedler: Analytische Geometrie der hoheren ebenen kurven. S (1882)

13 K. O GUA: ON THE QUARTC OURVE ETC. Fig. 6. [Degenerated case] that the circular cubic has the same properties as the bicircular quartic, as shown in Art. 3 (Figs. 5-6). Lastly, the theorem in Art. 4 are transformable to the following theorem: There are twelve conics through two ints on a cubic curve and the four points of contact of the tangents to the curve through the two points. We shall add a direct proof of this theorem briefly.* From any point P (u) on the cubic, four tangents can be'drawn to the curve, and their points of contact are and similarly, for another point But [mod, ƒö,ƒö']. Therefore P, A1, A2, P', A1', A2' are on a conic, &c. From this result, we can obtain configurations similar to those in Art. 5-6; one of them is, of course Hesse's configuration itself. * For the notations, see Clebsch-Lindemann. S. 607.