shows us that, of matter of uniform density, the mass or quantity is proportional to the volume or space it occupies.

Let M be the mass, p the density, and V the volume, of a homogeneous body. Then

M = Vp;

if we so take our units that unit of mass is that.of unit volume of a body of unit density.

If the density be not uniform, the equation

gives the Average (§ 26) density; or, as it is usually called, the Mean density, of the body.

It is worthy of particular notice that, in this definition, Newton says, if there be anything which freely pervades the interstices of all bodies, this is not taken account of in estimating their Mass or Density.

175. Newton further states, that a practical measure of the mass of a body is its Weight. His experiments on pendulums, by which he establishes this most important remark, will be described later, in our chapter on Properties of Matter.

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

176. The Quantity of Motion, or the Momentum, of a rigid body moving without rotation is proportional to its mass and velocity conjointly. The whole motion is the sum of the motions of its several parts. Thus a doubled mass, or a doubled velocity, would correspond to a double quantity of motion; and so on.

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

177. 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 § 31. Thus, in the figure 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.

178. Rate of Change of Momentum, or Acceleration of Momentum, is proportional to the mass moving and the acceleration of its velocity conjointly. Thus (§ 44) the rate of change of momentum of a falling body is constant, and in the vertical direction. Again (§ 36) the rate of change of momentum of a mass M, describing a circle of MV2 radius R, with uniform velocity V, is R

and is directed to the

centre of the circle; that is to say, it depends upon a change of direction, not a change of speed, of the motion.

179. 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 and the square of its velocity.

180. 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.

Suppose the velocity of a mass M to be changed from v to v, any time ; the rate at which the kinetic energy has changed is

Now

T

T

in

M (v, — v) is the rate of change of momentum in the direction of motion, and (v, + v) is equal to v, if τ be infinitely small. Hence the above statement. It is often convenient to use Newton's Fluxional notation for the rate of change of any quantity per unit of

1

T

time. In this notation (§ 28) v stands for (v,-v); so that the rate of change of Mv2, the kinetic energy, is Mv.v. (See also §§ 229, 241.) 181. It is to be observed that, in what precedes, with the exception of the definition of density, we have taken no account of the dimensions of the moving body. This is of no consequence so long as it does not rotate, and so long as its parts preserve the same relative positions amongst one another. In this case we may suppose the whole of the matter in it to be condensed in one point or particle. We thus speak of a material particle, as distinguished from a geometrical point. If the body rotate, or if its parts change their relative positions, then we cannot choose any one point by whose motions alone we may determine those of the other points. In such cases the momentum and change of momentum of the whole body in any direction are, the sums of the momenta, and of the changes of momentum, of its parts, in these directions; while the kinetic energy of the whole, being nondirectional, is simply the sum of the kinetic energies of the several parts or particles.

182. Matter has an innate power of resisting external influences, so that every body, so far as it can, remains at rest, or moves uniformly 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.

183. 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. Force is wholly expended in the Action it produces; and the body, after the force ceases to act, retains by its inertia the direction of motion and the velocity which were given to it. Force may be of divers kind, as pressure, or gravity, or friction, or any of the attractive or repulsive actions of electricity, magnetism, etc.

184. 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.

(a) The place of application of a force. The first case to be considered is that in which the place of application is a point. It has been shown already in what sense the term 'point' is to be taken, and, therefore, in what way a force may be imagined as acting at a point. In reality, however, the place of application of a force is always either a surface or a space of three dimensions occupied by matter. The point of the finest needle, or the edge of the sharpest knife, is still a surface, and acts as such on the bodies to which it may be applied. Even the most rigid substances, when brought together, do not touch at a point merely, but mould each other so as to produce a surface of application. On the other hand, gravity is a force of which the place of application is the whole matter of the body whose weight is considered; and the smallest particle of matter that has weight occupies some finite portion of space. Thus it is to be remarked, that there are two kinds of force, distinguishable by their place of application-force whose place of application is a surface, and force whose place of application is a solid. When a heavy body rests on the ground, or on a table, force of the second character, acting downwards, is balanced by force of the first character acting upwards.

(b) The second element in the specification of a force is its direction. The direction of a force is the line in which it acts. If the place of application of a force be regarded as a point, a line through that point, in the direction in which the force tends to move the body, is the direction of the force. In the case of a force distributed over a surface, it is frequently possible and convenient to assume a single point and a single line, such that a certain force acting at that point in that line would produce the same effect as is really produced.

(c) The third element in the specification of a force is its magnitude. This involves a consideration of the method followed in dynamics for measuring forces. Before measuring anything it is necessary to have a unit of measurement, or a standard to which to refer, and a principle of numerical specification, or a mode of referring to the standard. These will be supplied presently. See also § 224, below.

185. The Measure of a Force is the quantity of motion which it produces in unit of time.

The reader, who has been accustomed to speak of a force of so many pounds, or so many tons, may be reasonably startled when he finds that Newton gives no countenance to such expressions. The method is not correct unless it be specified at what part of the earth's surface the pound, or other definite quantity of matter named, is to be weighed; for the weight of a given quantity of matter differs in different latitudes.

It is often, however, convenient to use instead of the absolute unit (§ 188), the gravitation unit—which is simply the weight of unit mass. It must, of course, be specified in what latitude the observation is made. Thus, let W be the mass of a body in pounds; g the velocity it would acquire in falling for a second under the influence of its weight, or the earth's attraction diminished by centrifugal force; and Pits weight measured in kinetic or absolute units. We have P = Wg.

The force of gravity on the body, in gravitation units, is W.

186. According to the common system followed in modern mathematical treatises on dynamics, the unit of mass is g times the mass of the standard or unit weight. This definition, giving a varying and a very unnatural unit of mass, is exceedingly inconvenient: and its clumsiness is in great contrast to the clear and simple accuracy of the absolute method as stated above, to which we shall uniformly adhere, except when we wish, in describing results, to state forces in terms of the gravitation unit, as the vernacular of engineers in any locality. In reality, standards of weight are masses, not forces. It is better, though less usual, to call them standard masses than standard weights; as weight properly means force, and ambiguity is the worst fault of language. 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.

Whereas a merchant, with a balance and a set of standard masses, would give his customers the same quantity of matter however the earth's attraction might vary, depending as he does upon masses for his measurement; another, using a spring balance, would defraud his customers in high latitudes, and himself in low, if his instrument (which depends on forces and not on masses) were correctly adjusted in London.

It is a secondary application of our standards of mass to employ them for the measurement of forces, such as steam pressures, muscular power, etc. In all cases where great accuracy is required, the results obtained by such a method have to be reduced to what they would have been if the measurements of force had been made by means of a perfect spring-balance, graduated so as to indicate the forces of gravity on the standard masses in some conventional locality.

It is therefore very much simpler and better to take the imperial pound, or other national or international standard mass, as, for instance, the gramme (see Chapter IV.), as the unit of mass, and to derive from it, according to Newton's definition above, the unit of force.

187. The formula, deduced by Clairault from observation, and a certain theory regarding the figure and density of the earth, may be employed to calculate the most probable value of the apparent force of gravity, being the resultant of true gravitation and centrifugal force,

in any locality where no pendulum observation of sufficient accuracy has been made. This formula, with the two co-efficients which it involves, corrected according to modern pendulum observations, is as follows:

Let G be the apparent force of gravity on a unit mass at the equator, and g that in any latitude λ; then

g= G (1 + 00513 sin2 λ).

The value of G, in terms of the absolute unit, to be explained immediately, is

32.088.

According to this formula, therefore, polar gravity will be

g = 32.088 × 1·00513 = 32.252.

188. As gravity does not furnish a definite standard, independent of locality, recourse must be had to something else. The principle of measurement indicated as above by Newton, but first introduced practically by Gauss in connection with national standard masses, furnishes us with what we want. According to this principle, the standard or unit force is that force which, acting on a national standard unit of matter during the unit of time, generates the unit of velocity.

This is known as Gauss' absolute unit; absolute, because it furnishes a standard force independent of the differing amounts of gravity at different localities.

189. The absolute unit depends on the unit of matter, the unit of time, and the unit of velocity; and as the unit of velocity depends on the unit of space and the unit of time, there is, in the definition, a single reference to mass and space, but a double reference to time; and this is a point that must be particularly attended to.

190. The unit of mass may be the British imperial pound or, better, the gramme; the unit of space the British standard foot or, better, the centimetre; and the unit of time the mean solar second.

We accordingly define the British absolute unit force as the force which, acting on one pound of matter for one second, generates a velocity of one foot per second.'

191. 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 the force of gravity on a given mass. 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 preceding formula gives exactly 32.2, for the latitude 55° 35', which is approximately that of Edinburgh. The variation in the force of gravity for one degree of difference of latitude about the latitude of Edinburgh is only 0000832 of its own amount. It is nearly the same, though somewhat more, for every degree of latitude southwards, as far as the southern limits of the British Isles. On the