II9. That if we make C=log C', we shall have log at,,at log e, or = e«, and z=C'e«; therefore at the point A where =0, we have CD=C. IIIdly. That if the abscissæ are taken in arithmetical progression as x, 2, 3, &c. the ordinates will form the geometrical progression Cca C'eal, Ce a &c. IVthly. That if t∞, we have z=c', a property of the circle which, as we already know, cuts all its radii at right angles. These examples will enable the student to draw tangents to all sorts of curves whether geometrical or mechanical. Examples for Practice. 1. Let it be required to draw a tangent to the ellipse whose equation is y1= a 2. Required the expression for the subtangent to the cissoid of x3 which the equation is y α-x 3. Required the subtangent of the conchoid whose equation is ■ b+y√(aa―yy). y 4. Required the subtangent of the parabolic spiral of which the equation is y=a√px, (See art. 82 page 484) OF INVOLUTE AND EVOLUTE CURVES. IIdly. The length of MC will be equal to the line AB + the are BC; IIIdly. The indefinitely little arc Mm may be considered as a circular arc described from the centre C with the radius CM; IVly. The point C will be the point of concidence, or re-union of the two normals MN, mn, which are supposed to be indefinitely close to each other. 31. The curve BC is called the evolute of the curve AM: and reciprocally the curve AM, the involute of the curve BC; the line MC is the radius of the evolute, it is also called the radius of curvature, or of osculation, &c. This premised, we shall proceed to determine for any point M the radius MC of the evolute BC which we suppose known, Let MP, mp, be two perpendiculars to the axis AQ, indefinitely near to each other, and CO, rM two parallels to the same axis; if we call MO=u, AP=r, PM=y, Mm, or √(x2+y1)=s, we shall have Mr: Mm:: MO: MC But while AP, PM, and MO vary, MC becoming mC, does not change; therefore the fluxion of the equation MC" being taken, we shall have (us+su) x=us x and since u=mr=y, we shall find and consequently that MC = that u = *y sxy For the sake of simplifying this expression, let us suppose one of these fluxions to be constant, the elements of the curve, for example, and we shall have. MC = sy — ÿ √ (x2+y3) . if we had supposed y constant, we should have found s s xx: (s2—x3) x y x y x But if, as is usually done, we suppose a to be constant, then 32. As the curvature of circles varies in the inverse ratio of their radii, it follows that in two different points of any curve, the degrees of curvature are inversely as the radii of the evolute. Therefore in order to find in what points the curve has the greatest Hh curvature, we must determine the minimum or least value of the ra dius of curvature. If the tangent at A is perpendicular to the axis, then to determine the right line BA, or the distance from the vertex A to the origin of the evolute, we must make ro in the expression of the radius MC, and we shall have the value of BA. Lastly to find the equation of the evolute, draw CQ perpendicular to the axis, and call AB, a; BQ, t; CQ, z; in the first place supposing constant, -y. And then -xy together with the equation of the curve, enable us to find the equation of the evolute. 33. Hitherto we have supposed the ordinates to be parallel to one another. If they proceed from a fixed point or pole B, we may determine the radius MC in the following མག་་་ Taking the fluxion of this last equation (on the supposition that us that ♬ is constant,) we have -- and the fluxion of CO -xy = when y∞, or when the ordinates are parallel, as we have already found. We shall now proceed to give a few examples. p.xx 2a The equation to the ellipse and hyperbola, when we place the origin of the abscissæ at the vertex, is expressed generally by yy= pr± ; and it is evident that when a∞, we have yy=px; an equation of the parabola, which consequently is merely an ellipse or hyperbola whose transverse axis is infinite. Hence the equation uy = px ± pxx is general for all the conic sections. 2a will therefore serve to find their radius of curvature. It 34. Observing first that √(x2+y*), being equal to the nor mal (26), if we call it n, the radius of curvature, supposing a constant, will be expressed by ; and since in this example yy = --y3y we have 2yy=pr± prr; and again taking the fluxion of a this last equation we have 2ÿ ÿ+2y=±, and therefore y3ÿj 2y a = PP -y3 y conic sections the radius of curvature is equal to the cube of the nor◄ mal divided by one-fourth of the parameter. Hence in the circle where n = 4p, the radius of curvature is always equal to the nor mal, as is evident. As for the evolute of the circle, it is obviously the point which serves as the centre of the circle. 35. We haven=1⁄4 ̧ √(*a+y2) = √ { pœ±2x2x2 +?! (1±2? + pp consequently CO or PQ-NT2x+p; therefore AQ=3x+p =3x+ AB, and consequently BQ=3 x. This gives a very simple construction to determine the point C, or the centre of the osculating or equi-curve circle; take BQ=3 AP, and draw CQ perpendicular to AQ, then the point of concourse C of the two lines MC, CQ, will be the centre of the required circle. M M 2a 4 a .aa To find the equation of the evolute, let BQ=z, CQ=u; we shall have x, and p:y :: QN: CQ :: 2 x u= 4 xy P the evolute of the common parabola is the second cubical parabola*, whose parameter is 27 of that of the given parabola. All the curves represented by the general equation yx" — are called parabolas, while n and m are positive; thus if y3-a2x the curve is called the first cubical parabola, because n=1; if ya x2, the curve is the second cubical parabola, because n=2, &c. |