and therefore by § 85, the time of a small oscillation is

(1) A body is projected vertically upwards with a velocity which will carry it to a height 2g feet; shew that after three seconds it will be descending with a velocity g.

(2) Find the position of a point on the circumference of a vertical circle, in order that the time of rectilinear descent from it to the center may be the same as the time of descent to the lowest point.

(3) The straight line down which a particle will slide in the shortest time from a given point to a given circle in the same vertical plane, is the line joining the point to the upper or lower extremity of the vertical diameter, according as the point is within or without the circle.

(4) Find the locus of all points from which the time of rectilinear descent to each of two given points is the same. Shew also that in the particular case in which the given points are in the same vertical, the locus is formed by the revolution of a rectangular hyperbola.

(5) Find the line of quickest descent from the focus to a parabola whose axis is vertical and vertex upwards, and shew that its length is equal to that of the latus rectum.

(6) Find the straight line of quickest descent from the focus of a parabola to the curve when the axis is horizontal.

(7) The locus of all points in the same vertical plane for which the least time of sliding down an inclined plane to a circle is constant is another circle.

(8) Two bodies fall in the same time from two given points in space in the same vertical down two straight lines drawn to any point of a surface, shew that the surface is an equilateral hyperboloid of revolution, having the given points as vertices.

(9) Find the form of a curve in a vertical plane, such that if heavy particles be simultaneously let fall from each point of it so as to slide freely along the normal at that point, they may all reach a given horizontal straight line at the same instant.

(10) A semicycloid is placed with its axis vertical and vertex downwards, and from different points in it a number of particles are let fall at the same instant, each moving down the tangent at the point from which it sets out; prove that they will reach the involute (which passes through the vertex) all at the same instant.

th 3

(11) A particle moves in a straight line under the action of a force varying inversely as the power of the distance, shew that the velocity acquired by falling from an infinite distance to a distance a from the center is equal to the velocity which would be acquired in moving from rest at a distance a to a distance.

4

(12) A particle moves in a straight line from a distance a towards a center of force, the force varying inversely as the cube of the distance; shew that the whole time of descent

(13) A particle is placed at a given point between two centers of force of equal intensity attracting directly as the distance; to determine the motion and the time of an oscillation.

Let 2a be the distance between the centers, a the distance of the particle at any time from the middle point between them, then the equation of motion is

(14) If a particle begin to move directly towards a fixed center which repels with a force = μ (distance), and with an initial velocity (initial distance), prove that it will continually approach the fixed center, but never attain to it.

=

(15) A particle acted upon by two centers of force, each attracting with an intensity varying inversely as the square of the distance, is projected from a given point between them, to find the velocity of projection that the particle may just arrive at the neutral point of attraction and remain at rest

there.

If μ, u' be the absolute forces of the centers; a, a, the distances of the point of projection from them; and V the initial velocity; we have

(16) Supposing the Earth a homogeneous spheroid of equilibrium, the time of descent of a body let fall from any point P on the surface down a hole bored to the center Č, varies as CP, and the velocity at the center is constant.

(17) A material particle placed at a center of attraction varying as the distance, is urged from rest by a constant force which acts for one-sixth of the time of a complete oscillation about the center, ceases for the same period, and then acts as before, shew that the particle will then be retained at rest, and that the spaces moved through in the two periods are equal.

(18) A body moves from rest at a distance a towards a center of force, the force varying inversely as the distance: shew that the time of describing the space between Ba and

(19) If the time of a body's descent in a straight line towards a given center of force vary inversely as the square of the distance fallen through, determine the law of the force.

(20) Assuming the velocity of a body falling to a center of force to be as where a is the initial and x the

x

variable distance from the center, find the law of the force. (21) Find the time of falling to the center when the force ∞ (dist.) ̃3.

(22) Shew that the time of descent, to a center of force (dist.), through the first half of the initial distance, is to that through the last half as π + 2 : π − 2.

(23) A particle descends to a center of force ∞ (dist.)". Find n so that the velocity acquired from infinity to distance a, shall be equal to that acquired from distance a to distance a, from the center.

(24) A particle is placed at the extremity of the axis of a thin attracting cylinder of infinite length and of radius a, shew that its velocity after describing a space is proportional to

(25) A particle falls to an infinite homogeneous solid bounded by a plane face, find the time of descent.

(26) Every point of a fine uniform ring repels with a force (dist.), find the time of a small oscillation in its plane, about the center.

(27) Shew that a body cannot move so that the velocity shall vary as the space from the beginning of the motion. And if the velocity vary as the cube root of that space, determine the force, and the time of describing a given space.

(28) Shew that the time of quickest descent down a focal chord of a parabola whose axis is vertical is

where is the latus rectum.

(29) An ellipse is suspended with its major axis vertical, find the diameter down which a particle will fall in the least time, and the limiting value of the excentricity that this may not be the axis major itself.

(30) Particles slide down chords from a point 0 to a curved surface, under the action of a plane whose attraction is as the distance, and they reach the surface in the same time; shew that the surface is generated by the revolution (about a line whose length is a through O perpendicular to the plane) of the curve whose polar equation about O is

(31) If the particles commence their motion at the surface, and reach O after a given time, the equation to the generating curve is

p cos 0= a {sec (k cos 0)-1}.

(32) Prove that the times of falling through a given space AC towards a center of force S, under the action of two forces, one of which varies as the distance, and the other is constant and equal to the original value of the first, are as the arc (whose versed sine is AC) to the chord, in a circle whose radius is AS.

(33) The earth being supposed a thin uniform spherical shell, in the surface of which a circular aperture of given radius is made, if a particle be dropped from the center of the aperture, determine its velocity at any point of the descent.

(34) If a particle fall down a radius of a circle under the action of a force (D)3 in the center, and ascend the opposite radius under the action of the same force supposed repulsive, shew that it will acquire a velocity which is a geometric mean between radius, and the force at the circumference.