On the Curve of Constant Inclination, by TONOSUKE IWYATUKI in Sendai. In the following note it is designed to give a general method of finding the curve of constant inclination. Let us consider in general the curve of constant inclination on a given surface, formed by the rotation. of a curve z=g(as) lying in. the plane of rx, about the axis of z, where g(m) is a finite, one valued, con- tinuous, and differentiable function in an interval. If a particle moves with uniform velocity along the curve of constant inclinationon the surface, the orthogonalprojections of the particle,on the plane of my and the axis of z will move with uniform velocities Vain a and Vcos a respectively, where V and a axe the actual velocity of the particle and the. inclination of its direction to the axis of z. In the time-interval t the particle moves from the starting point A to the point P on the surface and its projectionson the plane of xy and the axis of r come to S and B, respectively; P lies on a circular section by a plane pass- ing through B and parallel to the plane of x, and S lies on the projection of the section on the plane of say. Let Q be the point of intersection of the circular section and the curve z=g(x) in the plane of zx, T a cor- responding point of Q in the plane of say; then T and S lie on the circle of centre 0 and of radius BQ. Put RQ:f(0R)f where f(x) is the inverse function of g(as). On the other hand, OR=Vcosa-t so that BQ=f(Vcos a-t)=OT. ON THE URVE OF CONSTANT INCLINATION 61 Then the equation of the circle (centre 0, radius T) is X+Y=ifWoosat)12. For. the sake of brevity we put Voos a=1, Vsin a-v, then OTfi(t), OR=t. if, in the small time interval M, T moves to T and S to S, then T, S lie on the circle (centre rad. T). Let us put T=x, so that x=f(t), TZ=ux, SS=88=v dt, and put SW=8a, W being the point of intersection of the axe and the radius OS. I n, the triangle SS'W, we have; neglectibg the tesirnals of higher orders than the second, Sae-I-8x2=aas2=e(by)2. In the. limit (da)2=v2(dt)2-(dx)2, whence (da)a=l v2 Jlt)ate; therefor WOO=do=T02 L.f(t)2I. and therefore 9=I.1 v2 V(01111 and moreover OS=rf(t). The polar equation to our curve is, therefore, given by the follow- ing relations (i) or (ii) 62 TORANANOS IWATSUKI: The sign of is taken positive or negative according as the ourvero=Ax) has the differentialcoefficient negative or positive, prom the result (ii), we see that the coordinatesare expressedonly in the terms of a, x, and the initial constant. Therefore the projection of the particle on. the plane, of my which moves along a curve of constant inclination on the given surface of revolution,describes the same locus whatever the velocity of the moving particle may be. In the case of right circular cone: x=nz, tga=m, whence 1/mv-nSd-1og= and r=nx, 11)=nrro; therefore ytlTti-n, leer. This representsa logarithmicspiral In order to find the intrinsicequation of the orthogonalprojection of the curve of constant inclination we proceed as follows: Since H=11V2-f (t) 2 as et=ft, the radius of curvature becomes and we have t- These give us the required intrinsic equation, the double,sign being so. determined that p is positive. This may be obtained by a direct geometrical consideration. If a particle moves. with a constant velocity along the curve in consideration, the orthogonal projection of the particle on the plane of xy moves equal distance in equal time, so that Os=coast. if 3t=coast. (1) This formula has, already been obtained by Bnschke, in a different manner, by using a formula in natural geometry. ON THE CURVE OF CONSTANT INCLINATION. 63 Thus the projections of the particle on the plane of y at and t-opt, lie on the circumference of the circle with centre S, radius S being the position of the moving particle, on our curve at the instant t On the other hand they lie also on the two circles (centre radius 01 and Centre radius OT) respectively, where T and T are the two positions on the axis of X at t+81 and t-31, respectively. Thus the two points S and S of the four points of intersection of the preceding circle and, the last two consecutive circles, must be the two consecutive positions of S on our curve. Now take OS as the axis of X and a stinaight line perpendicular to OS as that, of then the equations of the circle (centre S rad.) and the circles (Centre rad. T and centre rad. T) are respectively (X-OS)2+Y2=Ss2 (1) 12+Y2=(0S-TIT), (2) XZ+Ya=(OS+TT), (3) which become IX-f(4124-Y2=882, X2-Y2 f(t-8t)Z, X2+Y2=t+δt)2. If from these equations, we calculate the abseissae of S and S i.e. OL and OM; then, remembering SL=OL-x and MS=x-OM, we get finally EL=SL-MS 64 TORANOSUKE WATSUK: f(t+2)+r(94-e0 +r2 1f(t)8t12+1f(t+e2) f(t+61) 1f(t)8-28121(t) -8X, say, El, s2 being evanescent with t. Then we have f(t)f11(t)+f(t)12-(Ifi(t) On the other band 8X28Y2=881=v28t, (4) where ax and cSY denote MS' and MS" respectively; from which and and therefore which means lim LS GE-lim LOSE, O being the point of intersection of S"S and the circle. For ES=SM=SX, EO=S"M=aY, EL=SH=X; therefore and therefore lim L GSE=lim LS'GE=say. ON THE CURVE OF CONSTANT IAOLINATION. 65 Thus the radius of curvature is But from (4) we have and as eX differs from by a small qmntity of higher order than the second, we have Hence the required formulae are or where the double sign is to be taken as before. Ini the case of sphere 2+0=a2 , tga=m. Differentiating with respect to r. (1) Examples (1), (2), (3) and (4) have been already obtained by Enneper in the Mathematisohe Annalen, 19. The papers mentioned below may also be consulted. Pirondini: Sur quelquos lignes liees l'helice oylindrique, Crelle J. 121. Huth: Kurven von konstanter Steilheit, Jahrosberioht der Realschule zu Stollberg 1893. Appell: Sur une proprlet caractristique (lea h6lices, Archiv d. Math. 64. Jonas: Kurven konstanter Steilheit auf der Kugelflttoho, Archiv d. Math. 3. Beihe,Bd. 8. B1 as ehke: Allgemeine Sohraubenlinien, Monatshefto fair Math. and Physilc, 19. Jahrgang. Naito: Curve of constant Inclination, Tokyo Sitgaku-Butpurigakkwai Kizi IV. Nugel: Die Sechraubenlinien, Diss. Ialle, 1912. 86 TORANOSUKEIWATSUKI: whence 10=71 n2iCL7-(ms+11x2 Mr=8: therefore Therefore the curve is an fepioycloid. Ex. 2. In the case of spheroid, Hence we find p w nV, n2b4a-(ib2-a2)az(b2m2+a2), mz=s. Therefore Therefore the equation represents an epicycloid. Ex. 3. In the case of hyperboloid of revolution, The required equation may be obtained by putting -b2 in plat of b2 in the previous result. Therefore we have If bm2-a2>O, or I tg a I>a/b, the curve becomes a pseudocycloid; if b2m2-a2<O or I tg a (<a/b, thebecomescurve a hypocycloid; if b2m2-a20, the curve becomes a straight line. Hence according as the inclination of the curve is greater than, ON THE CXRVEOF CONSTANTINCLINATION. 67 equal to, or less than that of the generatingline of the asymptoticcone, the required curve is a pseudooyeloid,a straight line, or a hypocycloid. Ex. 4. In the case of f(z)=v/4z, xa=4dz, tga=m. We have p=V4dmaz-41m, rnz=s, map2=4d(ms-d), and the required curve is an involute of a circle..