of the point at which it leaves the curve. Then the value of x will be a root of the cubic equation

(a3 — b2) (x3 — 3ax13) — 3a2b3x + a3 (b2 + 2ah) = 0.

Fontana; lb. p. 175.

(13) A particle descends from rest down the convex side of the Cissoid of Diocles, the asymptote of the Cissoid being vertical; the initial place of the particle being known, to find the point at which it will leave the curve.

Let P (fig. 146) be the initial position of the particle, and Q its place on leaving the curve: draw PS, QN, at right angles to Ox, Oy; let PS=h, QN=x. Then, a being the radius of the generating circle, the value of x will be a root of the cubic equation

If the motion commence at the cusp 0, h=0, and therefore

8
x= a.

Fontana; lb. p. 181.

(14) A particle is projected with a given velocity along the convex side of a parabola from a given point of the curve: at the focus of the parabola there is a centre of attractive force which varies inversely as the square of the distance; to determine the reaction of the curve on the particle at any point of its path.

Let S (fig. 147) be the focus of the parabola; B the point from which the particle is projected; BT the tangent at B; P the position of the particle after any time; let SP=r, SB = a, SA=m, B= the velocity of projection, μ the absolute force towards S. Then, at P,

=

(15) There is a centre of force at one extremity of the diameter of a semi-circle, the force being repulsive and varying as

W. S.

21

the distance: to find the pressure exerted upon the curve by a particle which moves from rest from the centre of force along its concave side, and the time which elapses before it reaches the other extremity of the diameter.

=

If a the radius of the circle, m = the mass of the particle, μ the absolute force, and R= the pressure when the particle is at a distance r from the centre of force,

=

(16) A particle is attached to the end of a fine thread which just winds round the circumference of a circle, at the centre of which there is a repulsive force varying as the distance: to find the time of unwinding, and the tension of the string at any time.

2π

If μ = the absolute force, and a = the radius of the circle, the time of unwinding is equal to and the tension at any time t is equal to 2μ3. a. t.

(17) The major axis of an ellipse is vertical: to find the velocity with which a particle must be projected vertically upwards from the extremity of the minor axis along the interior of the elliptic arc, so that after quitting the curve it may pass through the centre.

If a, b, denote the semi-axes major and minor, the required velocity will be equal to

(8a2 + b2) g)
За. 34

(18) A particle moves in a parabolic tube under the action of a repulsive force from the focus and a force parallel to the axis, each force varying as the focal distance of the particle to find the pressure on the tube.

Let m be the focal distance of the vertex, μ and v the abso

lute forces, V the velocity at the vertex: then, when the particle's focal distance is r, the pressure is equal to

COR. If v = 3μ and V2 = (μ+v) m2, the pressure is always zero and the particle would describe its path without constraint.

=

(19) A molecule moves in a narrow tube in the form of the lemniscate a cos 20 and is attracted towards the node by a force varying inversely as the seventh power of the distance: to find the law of the pressure exerted by the molecule on the tube.

The pressure varies directly as the distance of the molecule from the node.

(20) A particle moves along the convex side of an ellipse, under the action of two forces tending to the foci and varying inversely as the square of the distance, and a third force tending to the centre and varying as the distance; to find the reaction of the curve at any point.

Let R denote the reaction of the curve on the particle at any point, p the radius of curvature; f, f', the initial focal distances, and μ, μ, the corresponding absolute forces; " the absolute force to the centre, 2a the axis major of the ellipse, and ẞ the initial velocity. Then

If B', B", "", denote the velocities which the particle ought to have initially, in order to revolve freely round the three centres of force taken separately,

and therefore, when the forces are taken conjointly, it will revolve about them freely when

SECT. 3. Motion of a Particle on Rough Plane Curves.

(1) A heavy particle slides on a rough cycloid, the base of which is horizontal and vertex downwards, starting from instantaneous rest at the highest point: to determine the coefficient of friction in order that the particle may come to rest at the vertex.

Let the tangent at the vertex be the axis of y, the axis of the curve being that of x. Let μ be the coefficient of friction, v the velocity at any point, R the normal reaction of the curve. Then for the motion we have, p denoting the radius of curvature, and m the mass of the particle,

x = a (1 − cos 0), y=a (0+ sin 0), s = 4a sin

2'

= 4a cos 2: p=4

hence 2v dv - μv2d0 = — 2ag sin Ode + 2μag (1 + cos 0) dė,

==

d (v2. e-μ0)=-2age-μ sin odo

+2μage-μ0 (1 + cos 0) do:

integrating from 0= 0 to 0=π, and bearing in mind that v=0 at both limits, we have

and therefore μe=1, a relation by which the value of μ is determined.

(2) A heavy body is placed on a rough inclined plane, the inclination of which is greater than the angle of indifference, and is connected with a fine elastic string parallel to the plane and attached to a fixed point: if the body be initially at rest and the string of its natural length, to determine the circumstances of the resulting motion.

Let a be the natural length of the string, x its length at the end of any time t, a the angle of the plane, m the mass of the body, and the tension requisite to double the length of the string: then

SECT. 4. Inverse Problems on the Motion of a Particle along immoveable Plane Curves.

(1) To find a curve EPF (fig. 148) such that, A and B being two given points in the same horizontal line, the sum of the times in which a particle will descend by the action of gravity down the straight lines AP, BP, may be the same whatever point of the curve P may be.

Bisect AB in 0; let Ox, a vertical line, be the axis of x, and OAy, which is horizontal, the axis of y; let AB = 2a. Then, x, y, being the co-ordinates of P, the times down AP, BP, will be respectively equal to

Hence, k denoting the sum of the times,

(}k*gx)1 = {x2 + (a − y)*}* + {x2 + (a + y)3}* ....................................... (1).