therefore s' + 2a (a + a)2 = 1. За Suppose we measures from the point for which x'= 2a, that is from the point which corresponds to the vertex of the parabola; then we see that s' increases with x', so that we must take the lower sign in the last equation; also by supposing = 2a and s=0 we find 7=-2a; thus This value of s' may also be obtained by the application of the ordinary method of integration. 99. When the length of the arc of a curve is known in terms of the co-ordinates of its variable extremity, the equation to the involute can be found by ordinary processes of elimination. For we have (Dif. Cal. Art. 331), where the accented letters refer to a point in a curve, and the unaccented letters to the corresponding point in the involute. Thus If then s' is known in terms of x, or of y', or of both, by means of this relation and the known equation to the curve dx we may find and ds' dy; and p is known from the equation ds s' pl. It only remains then to eliminate x and y' from (1) and (2) and the known equation to the curve; we obtain thus an equation between x and y, which is the required equation to the involute. supposing measured from the point for which x=0 and y=c; we shall now find the equation to that involute to the catenary which begins at the point of the curve just specified. and ps', no constant being required, because by supposition p vanishes with s. Hence equations (1) and (2) of the preceding article become thus x = x′ — √√ (c2 — y2); therefore x = √√ (c2 — y2) +x. We have then to substitute these values of x and y' in the equation to the catenary, and thus obtain the required relation between x and y. The substitution may be conveniently performed thus This curve is called the tractory; on account of the radical, there are two values of x for every value of У less than c, these two values being numerically equal, but of opposite signs. There is a cusp at the point for which x=0 and y= c; and the axis of x is an asymptote. 101. The polar formulæ may also be used in like manner to determine the involute when the length of an arc of the evolute can be expressed in terms of the polar co-ordinates of its variable extremity. We have (Dif. Cal. Art. 332), Here, as before, the accented letters belong to the known curve, that is, to the evolute, and the unaccented letters to the required involute; thus since the evolute is known, there is a known relation between p' and r'. And s+p=l, so that if s' can be expressed in terms of p' and ' we may eliminate p' and by means of (1), (2), and the known equation to the evolute. Thus we obtain an equation connecting p and r, which serves to determine the involute. 102. Application to the Equiangular Spiral. = In this curve p' r' sin a, where a is the constant angle of the spiral. If we suppose the involute to begin from the pole of the spiral, and s to be measured from that point, we have pr' sec a, (Art. 84). Thus (1) of the preceding article becomes 22 = p2 sec2 a+r2 — 2r'p sec a = r2 sec2 a+ r22 sin2 a +p2 - 2r'p sec a, by (2). From this quadratic for p we obtain If we take the upper sign we find p= r2. But this solution must be rejected, because from it we should find P or dr 1+ 3 cos2 a dp cos a (1+cos* a) equation pr' sec a. r', which is inconsistent with the involute is an equiangular spiral with the same constant angle as the evolute has. curve; Intrinsic Equation to a Curve. 103. Let s denote the length of an arc of a curve measured from some fixed point, & the inclination of the tangent at the variable extremity to the tangent at some fixed point of the then the equation which determines the relation between 8 and is called the intrinsic equation to the curve. In some investigations, especially those relating to involutes and evolutes, this method of determining a curve is simpler than the ordinary method of referring the curve to rectangular axes which are extrinsic lines. 104. We will first shew how the intrinsic equation may be obtained from the ordinary equation. Suppose y = f(x) the equation to a curve, the origin being a point on the curve, and the axis of y a tangent at that point; from the given equation we have thus x is known in terms of tan o, say x = F(tan 4); then from this equation s may be found in terms of p by integration. A similar result will be obtained if at the origin the axis of x be the axis which we suppose to coincide with a tangent. 105. Application to the Cycloid. By Dif. Cal. Art. 358, we have The constant will be zero if we suppose s measured from the fixed point where the first tangent is drawn, that is, from the vertex of the curve. |