of given capacity, the density will be doubled, and so on. But it also 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 the volume, of a homo geneous 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

M = Vp

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 MVs 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 di rection, 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, in any time; the rate at which the kinetic energy has changed is

¦ ¦ . } M (v,; − v') = — M (v, − v) . § (v, +v).

Now M (v,-v) is the rate of change of momentum in the direction of motion, and (v,+v) is equal to v, if r 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 time. In this notation (§ 28) ₺ stands for(v,-v); so that the rate of change of Mu, the kinetic energy, is Mv. v. (See also §§ 229, 241.)

T

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. Force 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 kinds, 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 P its 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 system commonly followed in mathe matical treatises on dynamics till fourteen years ago, when a small instalment of the first edition of the present work was issued for the use of our students, the unit of mass was g times the mass of the standard or unit weight. This definition, giving a varying and a very unnatural unit of mass, was exceedingly inconvenient. By taking the gravity of a constant mass for the unit of force it makes the unit of force greater in high than in low latitudes. 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.

A merchant, with a balance and a set of standard weights, would give his customers the same quantity of the same kind of matter however the earth's attraction might vary, depending as he does upon weights 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 constant forces and not on the gravity of constant masses) were correctly adjusted in London.

It is a secondary application of our standards of weight to employ them for the measurement of forces, such as steam pressures, mus cular 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 weights in some conventional locality.

It is therefore very much simpler and better to take the imperial pound, or other national or international standard weight, as, for instance, the gramme (see the chapter on Measures and Instruments), as the unit of mass, and to derive from it, according to Newton's definition above, the unit of force. This is the method which Gauss has adopted in his great improvement of the system of measurement of forces.

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 coefficients 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

8=G(1+00513 sin 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 x 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 connexion 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. It is however terrestrial and inconstant if the unit of time depends on the earth's rotation, as it does in our present system of chronometry. The period of vibration of a piece of quartz crystal of specified shape and size and at a stated temperature (a tuning-fork, or bar, as one of the bars of glass used in the musical glasses') gives us a unit of time which is constant through all space and all time, and independent of the earth. A unit of force founded on such a unit of time would be better entitled to the designation absolute than is the 'absolute unit' now generally adopted, which is founded on the mean solar second. But this depends essentially on one particular piece of matter, and is therefore liable to all the accidents, etc. which affect so-called National Standards however carefully they may be preserved, as well as to the almost insuperable practical difficulties which are experienced when we attempt to make exact copies of them. Still, in the present state of science, we are really confined to such approximations. The recent discoveries due to the Kinetic theory of gases and to Spectrum analysis (especially when it is applied to the light of the heavenly bodies) indicate to us natural standard pieces of matter such as atoms of hydrogen, or sodium, ready made in infinite numbers, all absolutely alike in every physical property. The time of vibration of a sodium particle corresponding to any one of its modes of vibration, is known to be absolutely independent of its position in the universe, and it will probably remain the same so long as the particle