CHAPTER II. LAWS OF MOTION. 44. HAVING, in the preceding chapter, very briefly considered the purely geometrical properties of the motion of a point, we must now treat of the causes which produce various circumstances of motion of a Particle; 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 ideas of Matter, and of Force. We commence with a few definitions and explanations, necessary to the full enunciation of Newton's Laws and their consequences. 45. 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. As will be presently explained, the most convenient unit mass is an Imperial Pound of matter. 46. A Particle of matter is supposed to be so small that, though retaining its material properties, it may be treated, so far as its co-ordinates, &c. are concerned, as a geometrical point. 47. 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. 48. 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. 49. Rate of Change of Momentum, or Acceleration of Momentum, is proportional to the mass moving and the acceleration of its velocity conjointly. Thus (§ 17) 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. v2 P 50. 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. 51. 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. 52. Matter has the innate property of resisting external influences, so that every body, as far as it can, remains at rest, or moves with constant velocity in a straight line. This, the Inertia of matter, is proportional to the quantity of matter in the body. And it follows that some cause is requisite to disturb a body's uniformity of motion, or to change its direction from the natural rectilinear path. 53. Impressed Force, or Force simply, is any cause which tends to alter a body's natural state of rest, or of uniform motion in a straight line. The three elements specifying a force, or the three elements which must be known, before a clear notion of the force under consideration can be formed, are, its place of application, its direction, and its magnitude. 54. The Measure of a Force is the quantity of motion which it produces in unit of time. According to this method of measurement, the standard or unit force is that force which, acting on the unit of matter during the unit of time, generates the unit of velocity. Hence the British absolute unit force is the force which, acting on one pound of matter for one second, generates a velocity of one foot per second. [According to the common system followed till lately in mathematical treatises on dynamics, the unit of mass is g times the mass of the standard or unit weight; g being the numerical value of the acceleration produced (in some particular locality) by the earth's attraction on falling bodies. This definition, giving a varying and a very unnatural unit of mass, is exceedingly inconvenient. In reality, standards of weight are masses, not forces. They are employed primarily in commerce for the purpose of measuring out a definite quantity of matter; not an amount of matter which shall be attracted by the earth with a given force.] 55. To render this standard intelligible, all that has to be done is to find how many absolute units will produce, in any particular locality, the same effect as gravity. The way to do this is to measure the effect of gravity in producing acceleration on a body unresisted in any way. The most accurate method is indirect, by means of the pendulum. The result of pendulum experiments made at Leith Fort, by Captain Kater, is, that the velocity acquired by a body falling unresisted for one second is at that place 32-207 feet per second. The variation in gravity for one degree of difference of latitude about the latitude of Leith is only 0000832 of its own amount. The average value for the whole of Great Britain differs but little from 32.2; that is, the attraction of gravity on a pound of matter in this country is 32.2 times the force which, acting on a pound for a second, would generate a velocity of one foot per second; in other words, 32.2 is the number of absolute units which measures the weight of a pound. Thus, speaking very roughly, the British absolute unit of force is equal to the weight of about half an ounce. 56. Forces (since they involve only direction and magnitude) may be represented, as velocities are, by vectors, that is, by straight lines drawn in their directions, and of lengths proportional to their magnitudes, respectively. Also the laws of composition and resolution of any number of forces acting at the same point, are, as we shall presently shew, § 67, the same as those which we have already proved to hold for velocities; so that, with the substitution of force for velocity, § 10 is still true. 57. The Component of a force in any direction, sometimes called the Effective Component in that direction, is therefore found by multiplying the magnitude of the force by the cosine. of the angle between the directions of the force and the component. The remaining component in this case is perpendicular to the other. It is very generally convenient to resolve forces into components parallel to three lines at right angles to each other; each such resolution being effected by multiplying by the cosine of the angle concerned. The magnitude of the resultant of two, or of three, forces in directions at right angles to each other, is the square root of the sum of their squares. 58. The Centre of Inertia or Mass of any system of material points whatever (whether rigidly connected with one another, or connected in any way, or quite detached), is a point whose distance from any plane is equal to the sum of the products of each mass into its distance from the same plane divided by the sum of the masses. The distance from the plane of yz, of the centre of inertia of masses m1, m,, etc., whose distances from the plane are x1, x2, etc., is therefore ין is And, similarly, for the other co-ordinates. Hence its distance from the plane S=rữ tuy+vz-a=0, D=λπ + μÿ + vz-a, Σ {m (λx + μy + vz − a)} ___ Σ (md) Ση = as stated above. And its velocity perpendicular to that plane is from which, by multiplying by Σm, and noting that & is the distance of x, y, z from 8=0, we see that the sum of the momenta of the parts of the system in any direction is equal to the momentum in that direction of the whole mass collected at the centre of mass. 59. By introducing, in the definition of moment of velo |