Properties of Curvature in the Ellipse and Hyperbola. [From Runkle's Mathematical Monthly 2.6 (March 1860): 198-204.] ⁸

By Chauncey Wright, Nautical Almanac Office, Cambridge, Moss.

The equation of curvature, (which, as furnishing a direct and simple means of constructing points m the evolute of the ellipse or hyperbola by the method of the fourth proportional, was published in No. X. Vol. I. of this Journal, among the Prize Problems,) can itself be easily deduced from a simple geometrical construction.

1. In this equation r and r' denote the radii vectores from the foci to any point of the curve; p denotes the perpendicular line from the centre to the tangent of this point, and g the radius of curvature.

Further, let p1 and p2 denote the perpendicular lines from the foci to the tangent, and let ε denote the angle formed by both the radii with the tangent; A and B the semi-axes of the curve, and φ the angle formed by the radius r with the axis A.

If to two points, M and N, of an ellipse or an hyperbola, distant from each other by the element of arc d s, radii be drawn from both foci, and if the angle included by the radii from the first focus F be

denoted by d φ, and if a tangent be drawn through one of these points M, and the radius from the first focus F to the other point N be extended to this tangent (an infinitesimal distance of the second order), and from the point of intersection the line N L be drawn, forming the same angle with the tangent as the intersecting radius; then the angle included between this line N L and the radius r' from the second focus F' to the first point M, will be d φ, and the distance apart of these two lines at the second focus F' will be in the ellipse, (r + r' d φ, and in the hyperbola (r' - r) d φ, or in both 2 A d φ.

The angle which the line NL forms with the other radius N F' from the second focus is obviously equal to twice the angle which the tangents of the two points M and N form with each other, and we may express it by 2 d τ. The distance apart of these lines at the second focus is therefore 2r' d τ; hence the distances 2Adφ and 2 r' d τ are equal or

The construction of the centre of curvature O is given in the figures.

which is easily proved as follows.

The angle included by the radii vectores to any point is in the ellipse the supplement of 2 ε and in the hyperbola it is equal to 2 ε. If now the distance between the foci be denoted by 2 C we have by trigonometry

3. The mechanical properties of the ellipse and hyperbola may be easily deduced from the equations

in the two cases of central forces for which these curves are the paths described.

First, if the centre of attraction or repulsion be the centre of the ellipse or hyperbola, and if v denote the velocity with which the material point is at any time, t, moving, and re the radius vector of this point from the centre, εe, the angle which it forms with the tangent

and φe the angle which it forms with the axis A; the principle of equal areas may be expressed by the equation,

p v = a = a constant,

If the area and time be reckoned from the same origin, and if T denote the time of one complete revolution, a is equal to twice the


or the force is proportional to the distance from the centre.

hence the time of revolution is independent of the magnitude or form of the ellipse. The small oscillations of a free pendulum are therefore isochronous. .

Secondly, when one of the foci F of the ellipse or hyperbola is the centre of force, we have the mechanical equations

and the equations of curvature

The time of revolution is therefore independent of the minor axis B, and its square is proportional to the third power of the major axis, according to Kepler’s third law.

The general equation

includes the two cases above, and all cases in which rx and px are the radius and perpendicular to the tangent from any centre of force whatever.

If we substitute in it the radius and perpendicular from the centre, re, and p = A sin ε, or the radius and perpendicular from the

focus, r and p1 = r sin ε, the force R becomes independent of direction, and is a function of the radius only.

If we resume the mechanical equations

and the equation of curvature

we have in general

hence

The square of the velocity is therefore proportional in the first case to the product of the radii from the foci; and in the second case to their ratio. Moreover, it is obvious that in the second case the product of any two velocities in two positions at equal and opposite distances from the minor axis is the constant m/A; that is,

These values of v2 may also be deduced, though not so readily, by integration, and the determination of constants from the fundamental equation d(v2) = 2 R d rx.