itself exists. The wave-length for that particular ray, i.e. the space through which light is propagated in vacuo during the time of one complete vibration of this period, gives a perfectly invariable unit of length; and it is possible that at some not very distant day the mass of such a sodium particle may be employed as a natural standard for the remaining fundamental unit. This, the latest improvement made upon our original suggestion of a Perennial Spring, is due to Clerk Maxwell.

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 other hand, the variation per degree would be sensibly less, as far north as the Orkney and Shetland Isles. Hence the augmentation of gravity per degree from south to north throughout the British Isles is at most about of its whole amount in any locality. The average for the whole of Great Britain and Ireland differs certainly but little from 32 2. Our present application is, that the force of gravity at Edinburgh 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 in this latitude. Thus, speaking very roughly, the British absolute unit of force is equal to the weight of about half an ounce.

192. Forces (since they involve only direction and magnitude) may be represented, as velocities are, by straight lines 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 show later (§ 221), the same as those which we have already proved to hold for velo cities; so that with the substitution of force for velocity, §§ 30, 31 are still true.

193. 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.

194. [If any number of points be placed in any positions in space, another can be found, such that its distance from any plane what ever is the mean of their distances from that plane; and if one or more of the given points be in motion, the velocity of the mean point perpendicular to the plane is the mean of the velocities of the others in the same direction.

If we take two points 41, 42, the middle point, P1, of the line joining them is obviously distant from any plane whatever by a quantity equal to the mean (in this case the half sum or difference as they are on the same or on opposite sides) of their distances from that plane. Hence twice the distance of P, from any plane is equal to the (algebraic) sum of the distances of 41, 4 from it. Introducing a third point 4,, if we join A,P, and divide it in P, so that A,P,= 2P,P,, three times the distance of P, from any plane is equal to the sum of the distance of A, and twice that of P, from the same plane: i. e. to the sum of the distances of A1, Æ ̧, and A, from it; or its distance is the mean of theirs. And so on for any number of points. The proof is exceedingly simple. Thus suppose P to be the mean of the first n points A1, A,...A; and A., any other point. Divide AP in P., so that A.Pa+z = nPP

་

Then from P, Patil Anti draw perpen

diculars to any plane, meeting it in S, T, V

Draw PQR parallel to SV. Then

Hence n+1QP+= RA. Add to these

+1QT and its equal nPS+ RV, and we get $

ST

n+1(QP+1 + QT)=nP2S+ RV + RA

i.e. n+1P11T=nP2S+ A2+, V.

n+1

In words, n + 1 times the distance of Pt, from any plane is equal to that of Ạ, with n times that of P, i.e. equal to the sum of the

distances of A,, A,,...A+, from the plane. Thus if the proposition be true for any number of points, it is true for one more-and so on -but it is obviously true for two, hence for three, and therefore generally. And it is obvious that the order in which the points are taken is immaterial.

As the distance of this point from any plane is the mean of the distances of the given ones, the rate of increase of that distance, i.e. the velocity perpendicular to the plane, must be the mean of the rates of increase of their distances-i. e. the mean of their velocities perpendicular to the plane.]

195. The Centre of Inertia or Mass of a system of equal material points (whether connected with one another or not) is the point whose distance is equal to their average distance from any plane whatever (§ 194).

A group of material points of unequal masses may always be imagined as composed of a greater number of equal material points, because we may imagine the given material points divided into dif ferent numbers of very small parts. In any case in which the magni. tudes of the given masses are incommensurable, we may approach as near as we please to a rigorous fulfilment of the preceding statement, by making the parts into which we divide them sufficiently small.

On this understanding the preceding definition may be applied to define the centre of inertia of a system of material points, whether given equal or not. The result is equivalent to this :

The centre of inertia 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.

We also see, from the proposition stated above, that a point whose distance from three rectangular planes fulfils this condition, must fulfil this condition also for every other plane.

The co-ordinates of the centre of inertia, of masses w1, w2, etc., at points (x1, 1, 1), (X2, y2, 22), etc., are given by the following formulae :

These formulae are perfectly general, and can easily be put into the particular shape required for any given case.

The Centre of Inertia or Mass is thus a perfectly definite point in every body, or group of bodies. The term Centre of Gravity is often very inconveniently used for it. The theory of the resultant action of gravity, which will be given under Abstract Dynamics, shows that, except in a definite class of distributions of matter, there is no fixed point which can properly be called the Centre of Gravity of a rigid body. In ordinary cases of terrestrial gravitation, however, an ap proximate solution is available, according to which, in common parlance, the term Centre of Gravity may be used as equivalent to

Centre of Inertia; but it must be carefully remembered that the fundamental ideas involved in the two definitions are essentially different.

The second proposition in § 194 may now evidently be stated thus:-The sum of the momenta of the parts of the system in any direction is equal to the momentum in the same direction of a mass equal to the sum of the masses moving with a velocity equal to the velocity of the centre of inertia.

196. The mean of the squares of the distances of the centre of inertia, 7, from each of the points of a system is less than the mean of the squares of the dis. tance of any other point, O, from them by the square of OI. Hence the centre of inertia is the point the sum of the squares of whose distances from any given points is a minimum. For OP=OI' + IP2 +2OI IQ, P being any one of the points and PQ perpendicular to OZ. But IQ is the distance of P from a plane through I perpendicular to OQ. Hence the mean of all distances, IQ, is zero. Hence.

(mean of IP') = (mean of OP1) – Or, which is the proposition.

197. Again, the mean of the squares of the distances of the points of the system from any line, exceeds the corresponding quantity for a parallel line through the centre of inertia, by the square of the distance between these lines.

For in the above figure, let the plane of the paper represent a plane through I perpendicular to these lines, O the point in which the first line meets it, P the point in which it is met by a parallel line through any one of the points of the system. Draw, as before, PQ perpendicular to OI. Then PI is the perpendicular distance, from the axis through I, of the point of the system considered, PÓ is its distance from the first axis, OI the distance between the two

axes.

Then, as before,

(mean of QP) = OI3 + (mean of IP"); since the mean of IQ is still zero, IQ being the distance of a point of the system from the plane through I perpendicular to OI.

198. If the masses of the points be unequal, it is easy to see (as in § 195) that the first of these theorems becomes

The sum of the squares of the distances of the parts of a system from any point, each multiplied by the mass of that part, exceeds the corresponding quantity for the centre of inertia by the product of the square of the distance of the point from the centre of inertia, by the whole mass of the system.

Also, the sum of the products of the mass of each part of a system by the square of its distance from any axis is called the Moment of Inertia of the system about this axis; and the second proposition above is equivalent to➡

Vol. 23-4

The moment of inertia of a system about any axis is equal to the moment of inertia about a parallel axis through the centre of inertia, I, together with the moment of inertia, about the first axis, of the whole mass supposed condensed at I.

199. The Moment of any physical agency is the numerical measure of its importance. Thus, the moment of inertia of a body round an axis (§ 198) means the importance of its inertia relatively to rotation round that axis. Again, the moment of a force round a point or round a line (§ 46), signifies the measure of its importance as regards producing or balancing rotation round that point or round that line.

It is often convenient to represent the moment of a force by a line numerically equal to it, drawn through the vertex of the triangle representing its magnitude, perpendicular to its plane, through the front of a watch held in the plane with its centre at the point, and facing so that the force tends to turn round this point in a direction opposite to the hands. The moment of a force round any axis is the moment of its component in any plane perpendicular to the axis, round the point in which the plane is cut by the axis. Here we imagine the force resolved into two components, one parallel to the axis, which is ineffective so far as rotation round the axis is concerned; the other perpendicular to the axis (that is to say, having its line in any plane perpendicular to the axis). This latter component may be called the effective component of the force, with reference to rotation round the axis. And its moment round the axis may be defined as its moment round the nearest point of the axis, which is equivalent to the preceding definition.. It is clear that the moment of a force round any axis, is equal to the area of the projection on any plane perpendicular to the axis, of the figure representing its moment round any point of the axis.

200. [The projection of an area, plane or curved, on any plane, is the area included in the projection of its bounding line.

If we imagine an area divided into any number of parts, the projections of these parts on any plane make up the projection of the whole. But in this statement it must be understood that the areas of partial projections are to be reckoned as positive if particular sides, which, for brevity, we may call the outside of the projected area and the front of the plane of projection, face the same way, and negative if they face oppositely.

Of course if the projected surface, or any part of it, be a plane area at right angles to the plane of projection, the projection vanishes. The projections of any two shells having a common edge, on any plane, are equal. The projection of a closed surface (or a shell with evanescent edge), on any plane, is nothing.

Equal areas in one plane, or in parallel planes, have equal projec tions on any plane, whatever may be their figures.

Hence the projection of any plane figure, or of any shell edged by a plane figure, on another plane, is equal to its area, multiplied