which shews the curve to be a cycloid, whose base is horizontal and vertex upwards.

182. Two points being given, which are neither in a vertical nor in a horizontal line, to find the curve joining them, down which a particle sliding under the action of gravity, and starting from rest at the higher, will reach the other in the least possible time.

The curve must evidently lie in the vertical plane passing through the points. For suppose it not to lie in that plane, project it on the plane, and call corresponding elements of the curve and its projection σ and o'. Then if a particle slide down the projected curve its velocity at o' will be the same as the velocity in the other at σ. But o is never less than o', and is generally greater. Hence the time through o' is generally less than that through σ, and never greater. That is, the whole time of falling through the projected curve is less than that through the curve itself. Or the required curve lies in the vertical plane through the points.

Taking the axes of x and y, horizontal, and vertically downwards, respectively, from the starting point; if x be the abscissa of the other point, the time of descent will be

Applying the rules of the Calculus of Variations, we have,

√(1+p2)

since V or

Ny for a minimum,

the differential equation to a cycloid, the origin being a cusp and the base the axis of x.

This is a problem celebrated in the history of Dynamics. The cycloid has received on account of this property the name of Brachistochrone. Farther on we propose to investigate the nature and some of the properties of Brachistochrones for other forces besides gravity. For an investigation not involving the Calculus of Variations see Appendix.

183.

To find the curve down which if a particle, projected with a given velocity, slide under the action of gravity, it will descend equal vertical spaces in equal times.

Here we have, taking the axis of a horizontal, and that of y vertically downwards,

if the velocity is that due to a fall from the axis of x.

the semicubical parabola.

If the horizontal velocity is to be constant, we have

a parabola with its axis vertical and vertex upwards; as indeed we might have foreseen from the results of Chap. IV.

184. A particle moves on a smooth plane curve under the action of a force directed to a fixed center in the plane of the curve; to determine the motion.

Let r = f(0) be the polar equation of the constraining curve about the center of force as pole, and let P=4 (r) be the central repulsive force on a particle whose distance from the center is r.

Resolving along the tangent at any point,

Equation (2) contains the complete solution of the problem so far as the motion is concerned; since, by means of the equation to the curve, either r or s may be eliminated from it, and if the resulting differential equation be integrable, it will give s or r in terms of t.

For the pressure on the curve. Resolving along the normal at any point, p being the radius of curvature, we

have

an expression which by means of the foregoing equations will give R in terms of t or r.

Hence the solution is complete.

185. A particle, initially at rest at a point of the logarithmic spiral r = aen whose radius vector is b, moves on the curve under the action of an attracting center of force a distance, situated at the pole; to determine the motion.

At time t = 0, b=b cos ß; which gives ẞ=0,

which determines the position of the particle at any time.

When it reaches the pole r=0; the required interval is

Hence the pressure is towards the pole when the motion commences, becomes zero when the particle's distance from the pole is diminished in the ratio of and then is di

1

√/2

rected from the pole for the rest of the motion.

186. When the constraining curve is one of double cur

vature.

All we know directly about R is that it is perpendicular to the tangent line at any point.

Resolve then the given forces acting upon the particle into three, one, S, along the tangent, which in all cases in nature will be a function of x, y, z and therefore of s; another, T, in the line of intersection of the normal and osculating planes (or radius of absolute curvature); and the third, P, perpendicular to each of the other two.

Let the resolved parts of R in the directions of T and P be R1, R. Then the acceleration along the tangent is and therefore

d's

dt2

This equation together with the two to the curve is sufficient to determine the motion completely.

Now the particle at any point of its path may be considered as moving in the osculating plane. Hence, by our investigation for motion on a plane curve, § 169, if p be the radius of absolute curvature, v the velocity,