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 12000 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 velocities; 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 whatever 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.

2

If we take two points A,, A, the middle point, P2, 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 A1, A, from it. Introducing a third point A,, if we join A, P2 and divide it in P3 so that Д, P=2P, P2, three times the distance of P, from any plane is equal to the sum of the distance of A, and twice that of P2 from the same plane: i. e. to the sum of the distances of A1, Д„, 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 A, AA; and A+ any

2

other point. Divide AP in P

n

n+1

so that An+P+1=nPn+1Pn•

Then from P Pa+1 An+1 draw perpen

n'

diculars to any plane, meeting it in S, T, V. Draw PQR parallel to STV. Then

n+1

QP : RAn+1:: PnPn+1: PnAn+1::1:n+1.
Hence n+1QP+1=RA2+1' Add to these

n+1Q7 and its equal nPS+RV, and we get ST

n+1(QP+1+QT)=nP2S+RV+RAn+1
i.e. n+1 P+17
T=nP12S+AntiV.

n

n+1

In words, n+1 times the distance of P, from any plane is equal to that of An+1 with n times that of P i.e. equal to the sum of the distances of A,, A2,A,+1 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 different numbers of very small parts. In any case in which the magnitudes 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 (x, 1, 21), (X2, Y2, 2), 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 approximate 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, I, from each of the points of a system is less than the mean of the squares of the distance 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 OP2=012+IP2+201·1Q, P being any one of the points and PQ perpendicular to OI. 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 IP2)=(mean of OP2)- OI2, 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, PO is its distance from the first axis, OI the distance between the two

axes.

Then, as before,

(mean of OP2)=012+(mean of IP2); 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—

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 projections 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, multipled by the cosine of the angle at which its plane is inclined to the plane of projection. This angle is acute or obtuse, according as the outside of the projected area, and the front of the plane of projection, face on the whole towards the same parts, or oppositely. Hence lines representing, as above described, moments about a point in different planes, are to be compounded as forces are. See an analogous theorem in § 107.]

201. A Couple is a pair of equal forces acting in dissimilar directions in parallel lines. The Moment of a couple is the sum of the moments of its forces about any point in their plane, and is therefore equal to the product of either force into the shortest distance between their directions. This distance is called the Arm of the couple.

The Axis of a Couple is a line drawn from any chosen point of reference perpendicular to the plane of the couple, of such magnitude and in such direction as to represent the magnitude of the moment, and to indicate the direction in which the couple tends to turn. The most convenient rule for fulfilling the latter condition is this:-Hold a watch with its centre at the point of reference, and with its plane parallel to the plane of the couple. Then, according as the motion of the hands is contrary to, or along with the direction in which the couple tends to turn, draw the axis of the couple through the face or through the back of the watch. It will be found that a couple is completely represented by its axis, and that couples are to be resolved and compounded by the same geometrical constructions performed with reference to their axes as forces or velocities, with reference to the lines directly representing them.

202. By introducing in the definition of moment of velocity (§ 46) the mass of the moving body as a factor, we have an important element of dynamical science, the Moment of Momentum. The laws of composition and resolution are the same as those already explained.

203. [If the point of application of a force be displaced through a small space, the resolved part of the displacement in the direction of the force has been called its Virtual Velocity. This is positive or