other end of the base. If a, ẞ be the base angles, the angle of projection,

tan tan a + tan B.

(31) For the greatest range on an inclined plane through the point of projection the direction of motion on leaving, is at right angles to that on reaching, the plane.

(32) Particles are projected from the same point in a vertical plane: 1st, with the same vertical, 2nd, with the same horizontal, velocity; shew that in each case the locus of the foci is a parabola whose focus is at the point of projection, and axis vertical, but whose vertex is upwards in case (1) and downwards in (2).

(33) If a be the angle of projection, T the time which elapses before the projectile strikes the ground, prove that at the angle which the direction of motion

the time

T

4 sin2 a

(34) If a body describe an arc of a cycloid under the action of a force parallel to the base, shew that this force varies inversely as 2 sin - sin 20, 0 being the corresponding arc of the generating circle measured from the vertex.

(35) If the force perpendicular to a plane vary as the distance, shew that the curves described have equations of the form as the force is repulsive

y= Aa" + BaTM*,

or y= A cos (mx + B)

or attractive.

Find the circumstances of projection in the two cases that the curves may be the catenary, and the companion to the cycloid, respectively.

(36) Particles are projected in the same plane and from the same point, in such a manner that the parabolas described are equal; prove that the locus of the vertices of these parabolas will be a parabola.

CHAPTER V.

CENTRAL FORCES.

123. In this part of the subject we consider the motion of a particle under the action of a force whose direction always passes through, and whose intensity is some function of the distance from, a fixed point. The fixed point is called the Center of Force, and the force is said to be attractive or repulsive according as it is directed to or from the center. The former, as including the most important applications of the subject, we will take as our standard case; but it will be seen that a simple change of sign will adapt our general formulæ to the latter. If the center of force be itself in motion, the methods of SS 24, 28, enable us easily to treat it as fixed; but in this case the relative acceleration is not in general directed to the center, so that the problem no longer belongs to Central Forces strictly so called. It will be considered later. If the center be moving uniformly in a straight line, the results of this chapter are at once applicable to the relative motion.

124. A particle is projected in a plane, and is acted on by a force P directed to the fixed point 0 in that plane; to determine the motion.

The whole motion will clearly take place in the plane, as there is no force to withdraw the particle from it. Let Ox, Oy, any two lines through O at right angles to each other, be taken as the axes of co-ordinates. Let M be the position of the particle at the time t; and draw MN perpendicular to Ox, and join MO. Let ON= x, NM =y, OM=r, and the angle NOM=0. Then, since cos 0 sin

x

=

the com

ponents of P, parallel to the axes and in the negative di

rections, are P, P. But by the second law of motion

we may consider the accelerations in the directions of x and y separately, and we have therefore

In these, since P is a function of r, and therefore of x and y, the second members will generally contain both these variables, and the equations must be treated as simultaneous dx dy differential equations. Their integrals will give x, y, at at in terms of t; from which the position and velocity of the particle at any instant will be known, and the problem completely solved. In one case, however, viz. when P is proportional to r, the first equation will involve x and t, and the second y and t, only, and each equation may be integrated by itself. As it is the simplest example of its class, and of great importance in its applications, especially to Acoustics and to Physical Optics, we will begin by considering it.

125. A particle moves about a center of force, the force varying directly as the distance: to determine the motion.

Let μ

be the acceleration at unit of distance, usually called

the absolute force of the center, then Pμr, and equations

(4) become

A, B, A', B' being the constants introduced in the integration, to be determined by the initial circumstances of motion. Consider the particle projected from a point on the axis of a, at distance a from the center, with velocity V, and in a direction making an angle a with Ox. When t = 0, we have dy = V sin a. Hence,

dx

x = a, y = 0, dt

=

V cos a,

dt

Expanding the cosines in (1) and (2), and substituting these expressions for the constants, we obtain

which contain the complete solution of the problem. Elimi

nating t, we have

values for x, y, dt' dt'

the equation to the path of the particle; which is therefore an ellipse whose center is O. Equations (3) and (4) give periodic dx dy

such that all the circumstances of

2π

motion will be the same at the time t+ as at the time t.

result, as it is independent of the dimensions of the ellipse, and depends solely on the intensity of the force.

By taking negative in equations (B), we may apply them to the case of a repulsive force varying as the distance from 0. In the integration for this supposition the sines and cosines would be replaced by exponentials, and the curve described would be a hyperbola having O as center; but the motion would not be one of revolution, as the particle would necessarily always remain on the same branch of the hyperbola.

126. Recurring to equations (A), it will in all cases but the one we have just considered be more convenient to transform them to polar co-ordinates, especially as the general polar differential equation to the orbit described by a particle under the action of a central force can be easily formed, as follows.

127. A particle being acted on by a central force; it is required to determine the polar equation to the path.

Multiplying the second of equations (A), § 124, by x, and the first by y, and subtracting, we obtain

Changing the variables from x, y, to r, 0, where x = r cos 0,

y=r sin 0, we get as in § 22,