The equation to the locus of the extremity of the subtangent is evidently being measured from a line 90° distant from the original axis as is at right angles to r. If in a similar way we find the locus of the extremity of the subtangent of the curve ra0, and so on in succession, we shall have a series of spirals, the equations to which are the angle in each case being measured from a line 90° distant from that in the preceding curve. (2) The equation to the hyperbolic spiral is The locus of the extremity of the subtangent is evidently a circle, the radius of which is a: and as = 0 makes r = ∞ while the subtangent remains finite and equal to a, it appears that a line drawn parallel to the axis at a distance a is an asymptote. (3) The equation to the lituus is and the subtangent = 0, it appears is measured is an asymptote to the Also since r2 = a2 it appears that if a circle be described with radius r, the sector between the axis and the radius r is of constant area. (4) The equation to the Lemniscate is r2 = a2 cos 20. (5) The equation to the logarithmic spiral is The locus of the extremity of the subtangent is the involute of the curve, the equation to it being r1 = ar = ace", and therefore a similar spiral. Also if r be the subnormal, that is, the portion of a perpendicular to the radius vector at the origin cut off by the normal, the locus of the extremity of r is the evolute of the spiral, its equation being If r' be a radius in the direction of r produced backwards, r' = a {1 − cos (0 + π)} · = a (1 + cos 0). Therefore r+r' 2a, or the chords passing through the pole are of constant length. If, be the value of o corresponding to an angle + π; that is, to a tangent at the other extremity of the chord passing through the origin, 1 = n (0+) and 1 = nπ. 4, There fore the angle between two tangents at the extremities of any chord passing through the origin is constant. (8) Let the equation to a spiral be 0 (2ar - r2) = 1. Then when = ∞, (2ar — r2)1 = 0 and r = 0, r = 2a. Therefore the circle, the radius of which is 2a, is an asymptote to the spiral. The pole also, for which r = 0, may also be considered as an asymptotic circle the radius of which is zero, as the curve makes an infinite number of revolutions before it reaches it. The same remark applies to the logarithmic spiral, and many other curves for which r is zero when is infinite. d Ꮎ a 03 the sub dr 2 tangent corresponding to 1 is a, and there are therefore two rectilinear asymptotes inclined at angles + 1 and -1 to the axis. circle whose radius is a is asymptotic to the spiral. The form of the curve is given in fig. 28. CHAPTER X. SINGULAR POINTS OF CURVES. By the Singular Points of Curves are usually meant those for which any of the differential coefficients of the one variable 0 with respect to the other take the values 0, ∞ or We 0 shall confine our attention to the first and second differential coefficients only; and of these the first is the more important. When dy dx = 0, the curve is at that point parallel to the axis of a, and if the first differential coefficient which does not vanish along with be of an even order, the ordinate is at dy dx that point a maximum or a minimum. We shall not here consider any examples of such points, as the subject has been already sufficiently illustrated in Chap. VII. When d'y da2 = 0, the curve coincides at that point with a straight line; for as y = ax + b is the equation to a straight dy dx2 = 0. The same result line, it follows that for such a line d2y = da2 that point the radius of curvature is infinite, or the line at has no curvature, or is straight. If the first differential coefficient which does not vanish along with dy dx odd order, then dy is a maximum or minimum, and the curve has a point of contrary flexure. Instead of finding what differential coefficient vanishes, it is generally more dy dx2 convenient to try whether change sign on substituting in it values of x a little greater and a little less than that which makes it vanish. If it do change sign, the point is one of contrary flexure, otherwise not. If day = 8, there may be a point of contrary flexure provided that it change sign for values of a a little greater or a little less than that which makes it infinite. dy 0 make y it is an indica = dx 0 If any values of x and tion generally that the point in question is a multiple point, or that several branches of the curve pass through it. The dy multiplicity may be of different kinds. 1st. If is found dx by the usual method of evaluating vanishing fractions, to have several different possible values there are as many branches of the curve cutting each other in one point. 2nd. dy If is found to have two or more equal and possible dx values, there are two or more branches of the curve touching each other in one point, which is called a point of osculation. 3rd. If all the values of dy the point in question is an isolated or conjugate point, that dy equal values for a given value a of one of the variables, we find that for a value a +h the other variable is possible, |