(1) A particle describes a path in one plane under the action of a force the direction of which is perpendicular to a given straight line in the plane: to determine the nature of the hodograph.

Letv be the velocity of the particle at any time, ẞ the component of v parallel to the given line, the inclination of the tangent of the particle's path to the given line. Then v cos B, which, since B is a constant quantity, is the equation to the hodograph, which is consequently a straight line at right angles to the given straight line.

=

(2) A particle describes an equiangular spiral about a centre of force at the pole: to find the form of the hodograph. The hodograph is also an equiangular spiral.

(3) Two particles are describing free paths in one plane, which are hodographs to one another; if the particles be always at corresponding points, to determine the forms of their paths and the nature of the forces acting on the particles.

The paths are conic sections: the forces are centric, varying as the distance from a common centre.

(4) A particle moves in a semicircle, under the action of gravity, in a medium where the resistance varies as the density and the square of the velocity; to find the hodograph.

If a be the radius of the semicircle, the polar equation to the hodograph, the prime radius vector being horizontal, is r2 = ag cos 0.

(5) To prove that the whole accelerating force, which acts on a particle at any instant, is represented, both in direction and magnitude, by the element of the hodograph divided by the element of the time.

Hamilton: Proceedings of the Royal Irish Academy, 1846-7, No. 58, p. 345. (6) To prove that the force is to the velocity, in any varied motion of a particle, as the element of the hodograph is to the corresponding element of the orbit.

Hamilton: Ib. p. 345.

(7) If a particle be describing an orbit about a fixed centre of force, to prove that half the chord of curvature of the hodograph (passing through or tending towards the fixed centre of force) is to the radius vector of the orbit as the element of the hodograph is to the element of the orbit.

Hamilton: Ib. p. 346.

(8) Under the circumstances of the preceding theorem, to prove that the radius of curvature of the hodograph is to the radius vector of the orbit, called the vector of position, as the rectangle under the same radius vector and the force is to the parallelogram under the vectors of position and velocity.

Hamilton: Ib. p. 347.

(9) If a particle describe an orbit about a fixed centre of force, which varies inversely as the square of the distance, to prove that the hodograph is a circle.

Hamilton: 76. p. 347.

(10) If the hodograph of a particle, describing an orbit about a fixed centre of force, be a circle, to prove that the force must vary inversely as the square of the distance.

Hamilton: Ib. p. 347.

(11) If two circular hodographs, having a common chord, which passes through or tends towards a common centre of force, be cut perpendicularly by a third circle, to prove that the times of hodographically describing the intercepted arcs will be equal.

Hamilton: Ib. No. 63, p. 417.

CHAPTER IV.

CONSTRAINED MOTION OF A PARTICLE.

SECT. 1. Constrained motion of particles without friction. LET a particle, under the action of any force in one plane, move within an indefinitely thin curvilinear tube APB (fig. 133). Let x, y, be the co-ordinates of the place P of the particle, after a time t from the commencement of the motion; and let AP=s, where A is some assigned point in the tube. Let X, Y, represent the resolved parts of the accelerating force acting on the particle parallel to the axes Ox, Oy, and S the resolved part along the tangent to the curve APB at P. Then the equation for the motion of the particle will be

or, by integration, v denoting the velocity of the particle at the point P,

ds

v2 or

dt2

2 [(Xdx+ Ydy) + C = 2 [Sds + C......(B),

where C is an arbitrary constant, introduced by the integration, which may be determined if we know the initial velocity and the initial position of the particle.

If the force acting on the particle be a central force; then, P representing its intensity at a distance r, we have, taking the centre of force as the origin of x, y,

(1) A particle, acted on by gravity, descends from rest down a given circular arc, the tangent to which at the lowest point is horizontal; to compare the initial accelerating force estimated along the curve with that which would correspond to motion down the chord, when the arc is indefinitely diminished.

Let A (fig. 134) be the lowest point of the arc, P the initial position of the particle, T the intersection of the tangent at P with the tangent at A. Let F= the accelerating force at P down the arc PA, f= that down the chord PA, ‹ PAT=0 = < APT, g = the force of gravity.

and therefore, when the arc is indefinitely diminished, F= 2ƒ. Saurin; Mémoires de l'Académie des Sciences de Paris, 1724, p. 70. Liouville; Ib. p. 128. Courtivron, Ib. 1744, p. 384.

(2) The highest point of the circumference of a circle in a vertical plane is connected by means of a chord with some other point in the curve; to determine the time in which a particle, under the action of gravity, will fall down this chord.

Let AB (fig. 135) be the diameter through the highest point A of the circle; AC the chord down which the particle descends. Join BC, and let P be the position of the particle after a time t from the commencement of its motion. Let AP=s, = AP 8, AB 2a, BAC = a, AC-l. Then, the resolved part of g, the force of gravity, along AC, being g cos a, we have, for the motion of the particle,

=

= gt cos a + C;

0, t = 0, simultaneously; hence C=0; and therefore

Integrating again, and observing that s=0 when t=0, we have

8 = gť2 cos a.

Let T denote the whole time of descent down AC; then

This result, being independent of the inclination of the chord to the vertical, shews that the descents down all such chords are performed in the same time; a proposition established by Galileo.

Wolff; Elementa Matheseos Universe, Tom. II. p. 58.

(3) From one extremity of the horizontal diameter of a circle in a vertical plane, two chords are drawn subtending angles a, 2a, at the centre; given that the time down the latter chord is n times as great as that down the former, to find the value of a.

Let 1, 7, be the lengths of the two chords, and t, t', their times of description. Then, as in the preceding problem, it will be found that

But, r denoting the radius, it is evident from the geometry

that

α

1 = 2r sin

l' = 2r sin x :