ordinates of the point at which the final resultant acts, we shall at length obtain

(F1 + F2 + F ̧ + ... + F„) ≈ = F11 + F2≈2 + F53 + ... + Fn&u

2

or, more concisely, ZF.Z= (F≈).

By similar reasoning we shall obtain

The last three equations determine the values of xyz; and since those values do not contain any terms depending on the inclinations (to the co-ordinate axes) of the lines in which the forces act, those forces may be turned about the points on which they act without affecting the position of the point whose co-ordinates are xyz. On this account this point is

called the centre of parallel forces.

122. DEF. The product of a force into the distance of the point on which it acts from a plane, is called the moment of the force with respect to the plane. Hence Σ(Fx), Σ(Fy), Σ(Fx) are the sums of the moments of the forces with respect to the planes of yx, xx, xy: and ΣF.x, ΣF.J, ZF.z are the moments of their resultant with respect to the same planes. Hence, remembering that the co-ordinate planes were taken in any position, it follows, that the sum of the moments of any parallel forces with respect to a plane is equal to the moment of their resultant_with_respect to the same plane.

123. If the proposed plane be drawn through the centre of parallel forces, the moment of the resultant with respect to it will be zero; consequently, the sum of the moments of any parallel forces with respect to any plane passing through their centre is equal to zero.

124. If ΣF be equal to zero, there is then no centre of parallel forces, as we likewise know from Art. 73.

125. The formulæ of (121) are true if the co-ordinates are oblique and in that case Σ(Fx), Σ(Fy), Σ(Fs) are called the oblique moments of the forces with respect to the co-ordinate planes of yx, xx, xy.

THE CENTRE OF GRAVITY.

126. It has been found by experiment, that under the exhausted receiver of an air pump bodies of unequal magnitudes, and differing altogether in their nature and form (such as a piece of lead, a shilling, a feather, &c.) fall from the top to the bottom of the receiver exactly in the same time: from which it has been inferred, that the Earth exerts an equal force on all equal portions of matter; and that the weight of a body at a given place, measured according to the principles laid down in Arts. 7-10, is proportional to the quantity of matter in the body: that is, if M be the quantity of matter in a body whose weight is W at a given place, then

W ∞ M.

But we have stated in Art. 8, that the weight of a body, measured by a standard spring, is not the same at all places of the Earth's surface; it is in fact (as is shewn in Dynamics) proportional to the accelerating force of gravity, at the respective places. This force is generally denoted by g; and hence we have for a given body

Wag.

Consequently, for different bodies at different places W∞ Mg. For reasons stated in Dynamics we assume that

W = Mg.

127. The size or bulk of a body is called its volume and is denoted by V: but it is necessary to explain, both with regard to V and M, that they are expressed in numbers on the following principle. A known body, composed of matter uniformly diffused through all its parts, is taken as a standard to which all others are referred. The volume and mass of this body are called the units of volume and of mass. If a body be V times the size, and contain M times the quantity of matter, of the standard body; V and M are taken as the measures of the volume and mass of that body. Also, supposing the matter of the second body to

ρ

be uniformly diffused through its parts, if a portion of it of the same size as the unit of volume contains times as much matter, p is called the density of the body; and it is evident that

M = pV.

128. The direction in which a body descends when let fall is called the vertical direction; it may be discovered by suspending a heavy body by a thread, or by drawing a line perpendicular to the surface of still water. A plane at right angles to the vertical is called a horizontal plane; and it is evident, since the Earth is spherical, that the horizontal plane changes its position in passing from place to place: but since the distances of the bodies of systems usually treated of in STATICS are exceedingly small compared with the radius of the Earth (4,000 miles, nearly) we may consider the surface of still water as a horizontal plane to a small extent, and consequently the verticals as parallel.

129. Hence, in every body, and in every rigid system of bodies, there is a certain point through which the resultant of the forces which the Earth exerts on the different parts always passes in every position of the body or system. This point is called the centre of gravity of the body or system: it is sometimes also called the centre of mass.

130. One property of the centre of gravity, particularly worthy of remark, is, that it does not depend at all upon the intensity of the force of gravity. For divide the whole system into very small equal molecules, the quantity of matter in each being m, and their number n, and denote the force exerted upon a unit of matter by g; then the force exerted on each molecule And if x1 y1X1, X2 Y2 Z29••• be the co-ordinates of the molecules, and xyz those of the centre of gravity, we have, by Art. 121,

= mg.

It appears then, that the co-ordinates of the centre of gravity are the means* of the co-ordinates of the molecules, and consequently its position is independent of the intensity of gravity. Hence the centre of gravity of any body is a certain point within it, the place of which depends only on the relative disposition of its equal molecules. The investigation of its place is therefore purely geometrical, and may be applied to any body whatever; and for this reason we often speak of the centre of gravity of bodies far removed from the influence of the Earth, and when, in fact, no reference is intended to be made either to the Earth or to gravity; the point alluded to, being no other than the one determined from the geometrical principles just laid down, viz. that its co-ordinates are the respective means of the co-ordinates of all the equal molecules of which the body is composed.

131.

Since the resultant of the forces which act on the particles of a body passes through the centre of gravity, if that point be supported the body will be in equilibrium in every position. For instead of the forces themselves, we may substitute their resultant, which will be counteracted by the point of support, and this will be the case if the body be turned round that point into any position what

soever.

132. And since the resultant may be applied at any point in the line of its direction (Art. 21), if the point of support be not in the centre of gravity, but in any point of a vertical passing through it, the body will be in equilibrium. And conversely, if a body be suspended from any point in it, it will not be at rest till the centre of gravity

Hence the centre of gravity of two equal bodies is the middle point between them.

and the point of suspension are situated in the same vertical.

This property may sometimes be employed in finding the centre of gravity. For if the body be successively suspended from two points in it, and the corresponding verticals be drawn upon or through the body, their common point of intersection will be the centre of gravity.

133. It follows at once, from Art. 131, that if all the particles which are situated in a line passing through the centre of gravity be supported, the body will rest in equilibrium on that line in all positions. And the converse is true, viz.—that if a body rest in equilibrium, in all positions, on a fixed line, the centre of gravity must be in that line; for, unless the centre of gravity were in that line, a position might be found in which the vertical through the centre of gravity did not pass through a point of support, and consequently the body would not be in equilibrium in all positions, which is contrary to the hypothesis.

Hence, if we can find two lines on which a body will rest in all positions, the centre of gravity will be in their common point of intersection.

134. Since the resultant of all the forces of gravity, which act on the particles of a body, may be supposed to act at the centre of gravity, and is equal to their sum (Art. 121), we may, in any investigation in which this resultant is required, suppose the whole mass united at the centre of gravity; and hence it becomes important to know the situation of this point in bodies of different figures.

135. It is not always convenient to divide a proposed body into equal molecules, as was done in Art. 130, it therefore becomes necessary, in that case, to use other formulæ for the determination of the centre of gravity.