Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/36

INFINITESIMAL CALCULUS If K be a root of /() = , the corresponding value of v is ^ and the equation t/ *e - 3^-77 / ox* J represents an asymptote. jf/^K ^O, ^ C-> if _! and have a common factor y-KX, the line y = ic x is an asymptote. To each root of /(*) corresponds an asymptote, and accord - iiifly every curve of the ?ith degree has in general n asymptotes, real or imaginary. If the equation of the curve contains no terms of the degree n-l, the n asymptotes are represented by tjie equa tion u n = 0. In the case when / (/c) has a pair of roots each equal to K , then f{ K >) = Q > au( i the corresponding value of y is, in general, infinite. in such cases the corresponding asymptote is situated at infinity. The parabola is the simplest case of this, having the line at in finity for its asymptote. Branches of this class belonging to a curve are called parabolic, while a branch having an asymptote within a measurable distance is called hyperbolic. It is easy to establish an analogous method for finding asymptotes to curves whose equations are given in polar coordinates. The equations to the real asymptotes in the following curves are easily found by the above method. (1) a s y* = a*(a? s + Z 4 . Ans - =-t /=a- (2) x*if = a?( (3) x^-x i y- x-y-cQ. (4) Prove that the asymptotes to a curve of the third degree meet the curve in points which lie on a right line. (5) Show that the curve 3? - axy + aby <= has a parabolic asymptote, and find its equation. Curvature, Ewlutes, Points of Inflexion. 76. The word curvature indicates, deviation from a right line, the curvature at any point on a curve being greater or less accord ing as it deviates more or less rapidly from the tangent at the point. The curvature at any point on a curve is obtained by determining the circle which lias the same curvature as that of the curve at the point. Let ds be an indefinitely small element of the curve, and d<j> the angle between the tangents at its extremities, then represents the radius of the circle which has the same curvature. f is accordingly called the radius of curvature of the curve at the dtj> point. The circle is called the circle of curvature, and its centre the centre of curvature, corresponding to the point on the curve. Denoting the radius of the circle of curvature by p, we have ds p ~d$ Again, if x, y be the coordinates of the point, and <p be measured from the axis of x, then, since ds is the limit of the hypothenuse of a right-angled triangle of which dx, dy are the limits of the sides, we have dy d*y d<f> d<f> ds sec 3 < " 9 1T Hence p = dx ... d*y dx* This expression for the radius of curvature w r as given by John Bernoulli (Acta Eruditorum, 1701). The radius of curvature becomes infinite at a point for which ^=0. Such points are styled points of inflexion on the curve, and the tangent at a point of inflexion is called a stationary tangent (vol. vi. p. 719). Other expressions for the radius of curv"* ture can be readily obtained. For instance, since dx dy cos <b-^-. and sin <A = -^. , ds ds if the arc be taken as the independent variable, we have . cL(p ct Xi fi<p ct"ij d? Again, if p be the. length of the perpendicular drawn from the origin on the tangent at a point whose distance from the origin i r, the radius of curvature at the point is give.ii by the equation dr This value of p can be readily established from geometrical con siderations, and is frequently useful, more especially in applications f the calculus to physical astronomy. 77. If the centre of curvature for each point on a plane curve be taken, we get a new curve called its evolute. Also, with respect to he evolute, the original curve is called an involute, and may be described from its evolute by the unrolling a stretched string sup- losed wound round the evolute. In this motion each point on the string describes an involute to the curve. The curves of the system ,hus described are said to be parallel. Again, from its definition, it is plain that the evolute of a curve is the locus of the points of in tersection of the normals drawn at consecutive points on the curve. 78. Contact of Curves. Suppose two curves, represented by the equations y=f(x) and y=<p(x), to have a point (x, y} in common, then f(x) = <j>(x ). Let x + h be substituted for a; in both equations, and suppose y 1 and y 2 to be the corresponding ordinates, then y t =/(* + V =/() + (*) + nV "(*) + &c. , Now, we have h z ( ~ y * = 172 f " (x] ~ *" ( y and the curves have a common tangent. In this case the curves have a contact of the first order, and when h is small the difference between the ordinates y^ and y. 2 is a small quantity of the second order. If in addition f (x) = <f>"(x), we have . _**_ 1.2.3 In this case the difference of the ordinates is a small quantity of the third order ; and the curves are said to have a contact of the second order, and approach indefinitely nearer to each other at the point of contact than in the former case. Also, since y - y z changes its sign with that of h, the curves intersect, as well as touch, at the point of contact, If, moreover, f"(x] = $ "(x), the curves have a contact of the third order. In general, if f(x) = (f>(x), f (x) = tj> (x) , f"(x)=*<f>"(x) .... f(n)(x) = tf>( n )(x), the curves are said to have a contact of the nth order at the point. It is plain from what precedes that, if two Qtirves have a contact of the 7ith order, no curve having with either a contact of a lower order can pass between them. We shall illustrate this theory of the contact of curves by finding the circle which has a contact of the second order with the curve y=f(x) at the point (x, y). Suppose (x ~afi + (y j6) 2 = R 2 to be the equation of the circle, then, by the preceding, ^ and ^ must be the same for the circle CwC CtX and for the curve at the point. Differentiating twice successively the equation of the circle we get rf y rt and Hence _ QV - -=0 AJ / 7 .1 " ii yw v.j y t-i/ / r J2,,"2 i y d<*?) This agrees with the value for the radius of curvature found in 76, and shows, as is indeed evident, that the circle of curvature is the circle having a contact of the second order at the point in which it touches the curve. Agnin, if x, y be eliminated between the preceding differential equations and that of the curve, the resulting equation in a, p is that of the evolute of the curve. From what has been shown above, if the equation of a curve con tain n arbitrary coefficients, we can in general determine their values so that the curve shall have a contact of the order n - 1 with a given curve at any point ; for the coefficients can be determined so that y, -^ - - shall have the same values for the dx dx 2 dx n ~ l two curves at the point. The curve thus determined having a contact of the highest order with a given curve at any point is called an osculating curve. For instance, as the equation of a right line contains but two indepen dent constants, it admits in general of a contact of the first degree only. Again, the equation of a circle has three independent con-