Article 34
The Tractory.


To find the equation of the curve which possesses the property that the length of the tangent from the point of contact to the axis of x is constant.


Let PT, PT be two consecutive tangents such that PT = PT = c, and let OT = t; draw TS perpendicular to PT; then if PP = ds, it is evident that ST differs from ds by an infinitesimal of a higher order. Let PT make an angle φ with OA, the axis of y; then (to the first order of infinitesimals) PTdφ = TS = TTcosφ; that is,

cdφ = cosφdt,t = cgd1φ, x = t csinφ = c(gd1φ sinφ),y = ccosφ.

This is a convenient single-parameter form, which gives all values of x, y as φ increases from 0 to 1 2π. The value of s, expressed in the same form, is found from the relation

ds = ST = dtsinφ = ctanφdφ,s = cloge secφ.

At the point A, φ = 0, x = 0, s = 0, t = 0, y = c. The Cartesian equation, obtained by eliminating φ, is

x c = gd1 cos1y c sin cos1y c = cosh1 c y 1 y2 c2 .

If u be put for t c, and be taken as independent variable, φ = gdu, x c = u tanhu, y c = sechu, s c = logcoshu.

Prob. 100.
Given t = 2c, show that φ = 7435, s = 1.3249c, y = .2658c, x = 1.0360c. At what point is t = c?
Prob. 101.
Show that the evolute of the tractory is the catenary. (See Prob. 92.)
Prob. 102.
Find the radius of curvature of the tractory in terms of φ; and derive the intrinsic equation of the involute.