8 = 4a (sin y-sin yo). (87) If : begins at the highest point (the vertex) of the cycloid, Yo = 0, so that in this case s = 4a sin y. Ex. 5. The intrinsic equation of the catenary whose equation is and if : begins at the lowest point of the curve, s = a coty. In this case p= -a (cosec y)2. Ex. 6. In the curve whose equation is w+y} = aš, if : begins when y = 0, (88) Ex. 9. Find the intrinsic equation to the Epicycloid whose equations are X = (a + b) cos 0–6 cos (92) y = (a + b) sin 6-b sin From the preceding equations and (79) we have sina 720-sino cos 0—cosabo sin v dô= cos vodo = nini :. sin (47b0 + x) = sin (0+ ); :: 0 = a +27 (5—24). Now let us suppose s to begin at the cusp, where 0 = 0, and v=r; so that y decreases as s increases; then we have a 6=-sin a # 26 (5 –v) dll; (a + b) {1-cos a 26 (5 –v)} = 80(+ D) { sin 120 (1 - %)}; which is the required intrinsic equation. If b=a, the epicycloid becomes the cardioide, and (94) becomes 8 = 16a (sin - 724). (95) (93) (94) ion And in this case if y=-, s=8a, which is half the perimeter of the cardioide. Ex. 10. The intrinsic equation to the involute of the circle is (96) SECTION 5.- Involutes of Plane Curves. 169.] Another geometrical problem of considerable interest, which requires single integration in its solution, is that of the discovery of an involute of a curve, when the curve is given. Our investigation will comprise the determination of certain general formulæ which express reciprocal properties of evolutes and involutes, and the application of these formulæ to certain examples. Let an, fig. 11, be a part of the curve whose involute is to be found. Let on = $, no = n, and let the equation to an be n = f(E); (97) let the length-element of this curve be do, and let po be the tangent at n, whose extremity p is the generating point of the involute; so that pn is the radius of curvature of the involute. Let Pn=p, then, by Vol. I, Art. 292, equation (40), do = dp; (98) c being a constant, the value of which depends on the position of the point on an, at which o begins. Thus in fig. 11, if An= , and np = An, then o =p, and c=0; and if pn be longer than an, then c is the excess of length; that is, if a string of the length pn is wrapped round an, and ultimately becomes a tangent at a, c is the length of the string that remains when the wrapping is complete. Let om = X, MP=y; then, since = tan ntn, (99) an = d 4 = do' and from the geometry of the figure, I dE I x = ON-NM = $-pha (100) y = xn-nr = n-potong in which equations p must be expressed in terms of § and n; and & and n having been eliminated from them and (98), the resulting equation will contain x and y only, and be that to the required involute. 170.] From (100) by differentiation we have But by Art. 285, Vol. I, (21) and (22), the numerator and denominator of the right-hand side of (102) are proportional to the direction-cosines of the tangent of the evolute; and therefore we conclude that it is perpendicular to the tangent of the involute. Again, squaring and adding the two equations (101), we have d.x2 + dyo = pe {(1.*+ (d. anm)}. But if s' is the radius of curvature of the evolute at the point (€, n), then, by Vol. I, Art. 285, (19), 171.] Examples of involutes. Ex. 1. To find the involute to the catenary, the generating point being in contact with it at its lowest point. By equations (19) and (20), Art. 155, Y the equation to the tractory, the form of which is evident from fig. 7. Ex. 2. To find the equation to the involute of the cycloid, the generating point being in contact with it at its vertex. Let the cycloid be placed as in fig. 12; and let on=Ě, NO=n; 01 = X, MP = y; then the equation to the cycloid is { = a versin-17 +(2an-n2); |