On the of a and Some Applications of the Method of. Moving Trihedral,

by

Tsunami jIHAYAsx, Sendai,

In this note, I will give some propertiesof the involutes of a curve. Most of them are got by using the method of moving trihedral. Some are new, but some are well known. They may be adoptedas illustrative examplesof the method. 1. Tangents to the involutes of a twisted curve at the correspond- ing points (lying on the same tangent to the twisted curve) are all parallel to the principal normal to the curve (at the point of contact of the tangent). 2. Principal normalsto the involutes at the correspondingpoints are parallel to each other and all lie on the rectifying plane of the original curve. 3. Binormals to the involutes at the corresponding points are parallel to each other and all lie on the rectifying plane of the original curve. 4. Centres of of the involutes at the corresponding points lie on a straight line passing through the point on the original curve and parallel to the binormals to the involutes. 5. All the involutes have the same polar line at the corresponding points. 6. All the involutes have the same polar developable. 7. The rectifying developable of the original curve is nothing but the polar developable of the involutes. 8. The necessary and sufficient condition for a twisted curve, that its involutes are plane , is that the curve is a cylindrical helix. 9. The change of direction of an to a cylindrical helix is proportional to that of the original curve, measured from the direc- tions at the corresponding points. ONTHE INVOLUTES OFA CURVE. 237

10. Let us treat the ruled , that is the totality of straight lines, each of which passesthrough a point on a curve and has constant direction-cosinesa, b, c with respect to the tangent,principal normal and binomial at the point on the curve. Then the orthogonal trajectoriesof the generatinglines of the surface are-got by measuringk-as units of length along the generating lines from the points on the curve, k being a constant acid s being the arc-length of the original curve. It is note- worthy-that the length is independentof b and o. When a=1, the generating line is nothing but the tangent line of the original curve, and so the surface is the tangent surfaceiof the original curve and the orthogonal trajectories are the involutes of the curve. When a=0, the generating lines come in the normal plane of the original curve. So the orthogonal trajectoriesof the straight lines on the normal planes of a curve making constant angles with the principal normals are got by measuringoff constant lengths along the straight lines from the curve. 11. Taking an arc of a space curve and the correspondingarcs of its involutes, the points which would be the mean-centresof the arcs of the involutes,if the densitiesat the points of the arcs be supposed to be proportional to the reciprocals of k-s, k being the parameter, all lie in a straight line in general, or else are one and the same point. 12. For the involutes of a cylindrical helix, the at correspondingpoints of tie involutes are proportionalto the reciprocals of k-8(1 Hence for such involutes, the Steinerian mean-centres, namely the "Kriimmungs-Schwerpunkte"after Stei ner(2), of the cor- respondingarcs of the involutes, lie in a straight line in general, or else are one and the same point. 13, For the involutes of an arc of a plane curve, the Fameis true. Especiallyfor a closedconvex plane curve, of which the iiioving tangent sweepsout an even multiple of 7r(=2mirsay) when it revolvesalong the curve, the Steinerian mean-centres of the correspondingarcs of the involutes of the curve are one and the same point, which lies on the straight line perpendicularto the initial direction of the moving tangent and passing through the Steinerian mean-centreof the original curve,

(1) This is a ehnraoteristio property of the cylindrical helix. (2) Werke II, pp. 97-169 [p. 122], or orelle's Journal XXI, pp. 33-63 and pp. 101-133. 238 TSULUJIOHT HAYASHI: from which the point is at the distance 1/2mnr,t being the perimeter of the original curve. If the angle swept out by the moving tangent be an odd multiple of x, the Steinerian mean-centre of the involute traces the straight line perpendicular to the initial direction ofthe moving tangent and passing through the. Stoinerian mean-centre of the original curve, the distance between the two points being 2k-1 divided by the multiple of r. It can be easily proved that the Stein a ria n mean-centres of a closed plane curve and its (closed) are one and the same point. So the Steinerian mean-centres of a system of parallel convex closed plane curves coincide with the Steinerian mean-centre of their commonevolute and therefore are one and the same point.(1)

14. If we treat the family of planes passing through tangents and making constant.angles with osculating planes, all the characteristicsof the family intersect the given curve. If the angle made by the plane with the osculatingplane be equal to 4b or 135, the characteristiclies in the normal plane and bisectsthe angles between the principal normal and binormal. If the angle be considered as parameter, the charact- eristics of such families of planes at one point on the curve form the cone

referred to the tangent, principal normal and binormal as and C axes, p and r being the radii of curvature and torsion of the curve at that point respectively. Hence such cones at all the points of a cylind- rical helix are congruent. 15. For the family of planes passing through principal normals and making constant angles with normal planes, all the characteristics of the

(1) It is informed by Prof. Kubota that (luring his study about the locus of the ordinary mean- centres of parallel conbex closed plane curves, he has substantially obtained, but overlooked, the latter part of this statement, and that Prof. F.Schilling has proved the first part of this statement for the curves of constant breadth in the Zeitsehrift ftir Mathematik and Physik. Bd. 63 (1914), pp. 67-136 [p. 124]. Prof. Kubo- ta also adds that he has still substantially obbined the theorem: The Steinerian mean-centres of a system of parallel convex closed surfaces are one and the same point, the curvature being that of Gauss, i.e. total cnlrvature. For Prof. Kubota's study on the locus of the ordinary mean-centres of parallel curve see his paper Uber die Schwerpunkte der konvexen gesohlosseiienKurven and Tlttohon," which 3 to be published in Vol. 14 of this Journal. ON THE INVOLUTESOF A CURVE. 239 family are parallel to rectifying planes and intersect the principal normals. If the original curve is a B e r t r a n d curve, the angle made by the plane with the normal plane can be so chosen that the distance between the characteristic and rectifying plane is constant for all the points on the curve, and then the relative direction of the characteristic with respect to the tangent, principal normal and binormal is also constant. If the angle made by the planes with the normal planes be considered as parameter, the characteristics of such families of planes at one point on the given curve form the quadric

16. For the family of planes passing through binormals and mak- ing constant angles with normal planes, the characteristics of the family, are not so peculiarly situated. If the angle made by the planes with the normal planes be considered as parameter, the characteristics of such families of planes at one point on the given curve form the quadric

October 1917.