Suppose we take the initial line so that C'0, then

(9) A particle, acted on by a central force varying inversely as the fifth power of the distance, is projected in any direction with the velocity from infinity; find the orbit.

Its equation is r = R cosec ẞ sin (8-0), B being the angle of projection, and the line joining the point of projection with the center being taken as the initial line.

(10) A particle acted on by a central force varying inversely as the fifth power of the distance is projected from a given point with a velocity which is to the velocity from infinity as 5 to 3, in a direction making an angle sin1 5 with the radius vector; find the orbit.

Here we have

24/6

But if V be the velocity of projection, c the initial value

of u,

Substituting and integrating we find, after the necessary reductions,

where R is the initial distance, and a a constant to be determined by the position of the initial line.

(11) A particle acted on by a force, varying partly as the inverse third, and partly as the inverse fifth, power of the distance, is projected with the velocity from infinity at an angle with the distance, the tangent of which is 2, the forces being equal at the point of projection; determine the orbit.

(12) The force tending to the center of a circle whose

radius is a being μ (r + 2),

find the velocity with which a

particle will describe the circle; and shew that if the velocity be suddenly doubled the particle will come to an apse at the distance 3a.

(13) If P=2μ

+μu3, and a particle be projected at an

angle of 45° with the initial distance (R =), with a velocity which is to the velocity in a circle at the same distance as √/2 to √3, find the curve described.

(14) If a particle be acted on by a central force varying inversely as the seventh power of the distance, and be projected from an apse with a velocity which is to the velocity in a circle at the same distance as 1 to 3; find the equation to the curve described.

p2 = R2 cos 2 (0+a).

(15) A particle, acted on by a force varying inversely as the cube of the distance, is projected from a given point with any velocity in any direction; to separate the curves according to the circumstances of projection. These curves are called Cotes' Spirals.

This resolves itself into three distinct species of curves according to the values of A and B.

SPECIES I. Let A and B have the same sign; then

The values of A and B may in these equations be expressed in terms of the initial distance, and angle of projection; but we may put the equation to the curve in a simpler form as follows. Let a be the value of corresponding to an apse,

then when = a,

du de

= 0;

which always gives a possible value of a; and therefore

As

1

Hence when 0 =a, u=2c, or is the apsidal distance.

2c

increases, u increases, or r diminishes; and when = ∞, u = = ∞o, or r=0. Hence the curve forms an infinite number of convolutions about the pole; and, as it is symmetrical on both sides of the apse, it will be as represented in the figure, where A is the apse and O the center of force.

A

μ h2

SPECIES II. Let > 1, B=0, then the equation (2) becomes

u = Acko,

the equation of the logarithmic spiral. The nature of the curve will be the same if A, instead of B, vanish.