On some Osculating Figures of the Plane Curve, by Kazuhiko MAEDA(formerly Jusaku MAEDA),Seudai. 1. The parabola of index n. e shall call the curve represented by W where x and y are rectangular coordinates of a point, a parabola of index n, n being a given number positive or negative. When n is positive, the origin is a point on the curve and the tangent at this point is the x-axis or y-axis according as n is greater or less than unity. When n is negative, the origin is no longer a point on the curve, and the x-axis is an asymptote of the curve. In all cases, the origin and the x-axis have special relation to the curve. And we shall call, for the sake of convenience, the origin the initial point, while the x-axis the initial line of the curve. The parabola of index n (1.1) may be represented parametrically by (1.2) where z is the Gaussian coordinate of a point on the curve and ƒÉ a real parameter. In general, the parabola of index n can be re- presented by where ƒÄ is any point on the curve, w the initial point of the curve, a complex constant, distinct from zero and ƒÉ a real parameter. ƒ¿ Let us denote by (M) a given curve and by M an ordinary point on this curve. Let s be the arc length of (M) measured from a certain fixed point on (M) to M, and k and p be respectively the, curvature and the radius of curvature of (M) at M If we denote by z the Gauesian coordinate of the point M, we have the fundamental equation of the curve (M) as follows: (1.4), where accents denote differentiation with respect to s. We shall denote by[n] the point given by (1.5). 262 KAZUHIKO MAEDA: We shall denote by (n) the tangent circle to (M) at M, whose centre coincides with the point [n]. f the parabola of index n (n•‚0,I 1) represented by (1.3) has contact of at least the second order with the given curve (M) at the point , we have the following relations M where z', z" are the values of dz/ds, d2z/ds2 at M, and ƒÉ, ƒÉ', ƒÉ" are real numbers. Consequently we obtain, from the last equation, (1.6) Let ƒÆ be the angle which the tangent to the parabola of indt x n (1.3) makes with the initial line. Then (1.7) Putting, in general, (1.8) we obtain (1.9) Putting (1.10) we get Therefore we see that the locus of the initial point of any parabola of index n having contact of at least the second order with a curve (M) at a point M is a curve represented by (1.12) In the particular case n=-1, on writing t instead of -t-2, the locus is (1.13) which represents a circle (-1/2). Hence a known theorem: ON SOMEOSCULATING FIGURES OF THE PLANE OURVE. 263 The locus of the centre of any rectangular hyperbola having contact of at least the second order with a curve (M) at a point M is a circle (-1/2)(1). In the case n=2, by putting (1.14) where k is a numerical constant, we have (1.15) If k =1 we have (1.16) which represents a circle (1/4). Hence a known theorem: The locus of the focus of any parabola having contact of at least the second order with a "curve (M) at a point M is a circle (1/4)(2). f k=-3, on writing t instead of t-1, we obtain I (1.17) which represents an elliptic limacon of Pascal. The axis of this. limacon is the normal MN to (M), and the acnode, focus and singular focus are respectively the points [-1/4], [1/16] and [-1/2]. Hence: Let (P) be ony parabola having contact of at least the second order with a curve (M) at a point M. Let F be the focus and V the vertex of (P). If we take a point U on the axis of (P) such that Uv=3VF, the locus of the point U of the variable parabola (P) is an elliptic limacon of Pascal whoseacnode, focus and singular focus- are respectively the points [-1/4], [1/16] and [-1/2]. Let us consider the case n=3/2. The focus of the semi-cubical parabola y3=x3 is the point (-4/27, 0). The locus of the focus of any semi-cubical parabola having contact of at least the second order with a curve (M) at a point M is given by (1.18) Let us put in general (1) J. Lemaire: Etude elementaire do I'hyperbole equilatere et de quelques courbee derivdes, p. 34. G. Loria: Spezielle algebralsche tend tranazendente ebene Kurven, Theorie and Geschichte I, p. 243. (2) A. Transon: Recherches sur la courbure des lignes et des surfaces, Journal de Math. 6 (1841), 202. 264 KAZUHIKO MAEDA: x, y being the coordinates of the point ƒÄ referred to the tangent MT and normal MN to (M) as coordinate axes. Putting we obtain By elimination of ƒÆ, we get (1.19) By the translation of the coordinate axes the above equation becomes This is an acnodal circular cubic having acndde at the point [1/3], And it is nothing but the associate curve (die Begleitkurre) of the cissoid of Diocles(1) : The normal MN is the axis of symmetry of the curve, and the singular focus is the point [1/4]. The director circle is Consequently the anallagmatic centre is the point M. The real asymptote is And the middle line is Hence: The locus of the focus of any semi-cubical parabola having contact of at least the second order with a given curve (M) at a point M is an acnodal circular cubic having the normal MN to (M) as axis of symmetry and the point M as anallagmatic centre. This curve is an associate curve of a cissoid of Dioeles. Let (Pn) be any parabola of index n having contact of at least the second order. with the curve (M) at the point M. We shall denote by g(ƒÁ), a straight line passing through the initial point of (Pn) and making an angle ƒÁ with the initial line. The map-equation of g (ƒÁ) is (1) G. Loria : Ebene Kurven I, p. 37. ON SOME OSCULATING FIGURES OF THE PLANE CURVE. 265 where being a real parameter. We shall now find the envelope of g (ƒÁ). Since we have the envelope condition gives Thus the envelope of g (ƒÁ) is (1.20) where we have written T instead of t2 and Putting (1.21) the envelope is given by (1.22) which represents a deltoid with centre at the point z0. The centre z0 is given by (1.23) When ƒÁ varies, n being fixed, the centre z0 describes a circle When n varies, this circle belongs to a coaxal system with the points M and [-1/2] as limiting points. Let (Pn) be any parabola of index n having contact of at least the second order with a curve (M) at a point M. Let g (ƒÁ) be a straight line which passes through *the initial point of (Pn) and 266 KAZUHIKO MAEDA : makes a given angle y with the initial line. Then the envelope of g (ƒÁ) is a deltoid. n the I special case n = -1, the envelope is given by (1.24) Let (H) be any rectangular hyperbola having contact of at least the second order with a curve (M) at a point M. The envelope of the asymptotes of (H) is a deltoid (ƒ¢) whose director circle is the circle (-1/2) and one of whose cusps is the point [-2]. The enve- lope of two lines, each of which passes through the centre of (H) and makes a given angle ƒÁ with an asymptote, is a deltoid obtained by rotating the deltoid (ƒ¢) through angle 4ƒÁ/3 about its centre(1). In the special case n=2, we may state as follows: Let (P) be any parabola having contact of at least the second order with a curve (M) at a point M. Let g (ƒÁ) be a straight line which passes through the vertex of (P) and makes a given angle with the tangent at the vertex. The envelope of g (ƒÁ) is a ƒÁ del- toid (2). A more general result is: Let (P) be any parabola having contact of at least the second order with a curve (M) at a point M. And let l be a straight line which is in, a fixed position relative to (P). Then the envelope of l is a deltoid. In the case and only in the case where l coincides with the directrix of (P), the envelope reduces to a point [-1/2](3). In the special case n=3, we may state as follows: Let (H) be any cubical parabola having contact of at least the second order with a curve (M) at a point M. And let g (ƒÁ) be a straight line which passes throughh the point of inflexion of (II) and makes a given angle y with the inflexional tangent. The envelope of g (ƒÁ) is a deltoid. In the special case n=3/2, we may state as follows: Let (ƒ°) be any semi-cubical parabola having contact of at least the second order with a curve (M) at a point M.