(6) The velocity of a point parallel to each of three rectangular axes is proportional to the product of the other two co-ordinates; what are the equations to the path, and what is the time of describing a given portion when the curve passes through the origin?

(7) A point moves in a plane, its velocities parallel to the axes of x and y are

u +ey and v + ex respectively,

shew that it moves in a conic section.

(8) Two points are moving with uniform velocity in two straight lines, 1st in a plane, 2nd in space; given the initial circumstances, find when they are nearest to each other. Shew also that in both cases the relative path is a straight line, described with uniform velocity.

(9) A number of points are moving with uniform velocity in straight lines in space; determine the motion of their common center of inertia. (§ 53.)

(10) A cannon-ball is moving in a direction making an acute angle with a line drawn from the ball to an observer; if V be the velocity of sound, and nV that of the ball, prove that the whizzing of the ball at different points of its course will be heard in the order in which it is produced, or in the reverse order, according as n <> sec 0.

(11) A particle projected with a velocity u, is acted on by a force, which produces a constant acceleration f, in the plane of motion, inclined at a constant angle a to the direction of motion. Obtain the intrinsic equation to the curve described, and shew that the particle will be moving in the opposite direction to that of projection at the time

(12) Shew that any infinitely small motion given to a plane figure in its own plane is equivalent to a rotation through an infinitely small angle about some point in the figure.

(13) The highest point of the wheel of a carriage rolling on a road moves twice as fast as each of two points in the rim whose distance from the ground is half the radius of the wheel.

(14) A rod of given length moves with its extremities in two given lines which intersect; shew how to draw a tangent to the path described by any point of the rod.

(15) Investigate the position of the instantaneous center about which the rod is turning, and apply this also to solve the preceding question.

(16) One circle rolls on another whose center is fixed. From the initial and final positions of a diameter in each determine how much of their circumferences have been in contact.

(17) One point describes the diameter AB of a circle with uniform velocity, and another the semi-circumference AB from rest with uniform tangential acceleration, they start together from A and arrive together at B, shew that the velocities at B are as π: 1.

(18) In the example of § 30 find in the case of e= 1 the ultimate distance of the particles, and for e<1 the length of time occupied in the pursuit.

(19) In the example of § 31 find the greatest distance the boat is carried down the stream, and shew that when it is in that position its velocity is √ (u2 — v2).

When uv, shew directly that the curve described is a parabola.

(20) Shew that if p be the radius of curvature of the curve of pursuit, we have in the figure of § 30,

(21) In the case of a boat propelled with velocity u relatively to the water in a stream running with velocity v; shew that the boat passes from one given point to another in the least possible time when its actual path is a straight line.

(22) The velocity of a stream varies as the distance from the nearest bank; shew that a man attempting to swim directly across will describe two semiparabolas. (Shew that the sub-normal is constant.) Find by how much the mean velocity is increased.

(23) A point moves uniformly in a circle; find an expression for its angular velocity about any point in the plane of the circle.

(24) If the velocity of a point moving in a plane curve vary as the radius of curvature, shew that the direction of motion revolves with uniform angular velocity.

(25) Two bevilled wheels roll together; having given the angular velocity of the first wheel and the inclinations of the axes of the cones, find their vertical angles that the second may revolve with given angular velocity.

(26) Supposing the Earth and Venus to describe in the same plane circles about the Sun as center; investigate an expression for the angular velocity of the Earth about Venus in any position, the actual velocities being inversely as the square roots of their distances from the Sun.

(27) A particle moving uniformly round the circular base of an oblique cone is projected by generating lines on a subcontrary section; find its angular velocity about the center of the latter.

η

(28) If, n denote the co-ordinates of a moving point referred to two axes, one of which is fixed and the other rotates with uniform angular velocity w, prove that its component accelerations parallel to these axes are

CHAPTER II.

LAWS OF MOTION.

39. HAVING, in the preceding chapter, considered the purely geometrical properties of the motion of a point or particle, we must now treat of the causes which produce various circumstances of motion; and of the experimental laws, on the assumed truth of which all our succeeding investigations are founded. And it is obvious that we now introduce for the first time the idea of Matter.

We commence with a few definitions and explanations, necessary to the full enunciation of Newton's Laws and their consequences.

40. The Quantity of Matter in a body, or the Mass of a body, is proportional to the Volume and the Density conjointly. The Density may therefore be defined as the quantity of matter in unit volume.

If M be the mass, p the density, and V the volume, of a homogeneous body, we have at once

M=Vp;

if we so take our units that unit of mass is the mass of unit volume of a body of unit density. If the density vary from point to point, we have, of course,

M=SSSpd V.

As will be presently explained, the most convenient unit mass is an imperial pound of matter.

41. A Particle of matter is supposed to be so small that, though retaining its material properties, it may be treated so

T. D.

3

far as its co-ordinates, &c. are concerned, as a geometrical point.

42. The Quantity of Motion, or the Momentum, of a moving body is proportional to its mass and velocity conjointly.

Hence, if we take as unit of momentum the momentum of a unit of mass moving with unit velocity, the momentum of a mass M moving with velocity v is Mv.

43. Change of Quantity of Motion, or Change of Momentum, is proportional to the mass moving and the change of its velocity conjointly.

Change of velocity is to be understood in the general sense of 10. Thus, with the notation of that section, if a velocity represented by OA be changed to another represented by OC, the change of velocity is represented in magnitude and direction by AC.

44. Rate of Change of Momentum, or Acceleration of Momentum, is proportional to the mass moving and the acceleration of its velocity conjointly. Thus (§ 16) the acceleration d's of momentum of a particle moving in a curve is M along dt

the tangent, and M in the radius of absolute curvature.

P

45. The Vis Viva, or Kinetic Energy, of a moving body is proportional to the mass and the square of the velocity, conjointly. If we adopt the same units of mass and velocity as before, there is particular advantage in defining kinetic energy as half the product of the mass into the square of its velocity.

46. Rate of Change of Kinetic Energy (when defined as above) is the product of the velocity into the component of acceleration of momentum in the direction of motion.