Then, by (2), we infer

(Zx) = 0, and (Zy) = 0........ (7).

And from (2) and (5),

(Yx - Xy) = 0, and (Xx+Yy) = 0......(8). And when (6), (7), and (8) are true, (3), (4), and (5) are true for all values of 0.

106. It appears from the preceding article that when forces act in one plane on a rigid body and maintain equilibrium, the necessary and sufficient condition in order that equilibrium may subsist after the body has been turned round an axis perpendicular to the plane while the forces remain parallel to their original directions, is

Σ(Xx+Yy) = 0.

107. We have remarked in Art. (10) that the property of the divisibility of matter leads us to the supposition that every body consists of an assemblage of material particles or molecules which are held together by their mutual attraction. Now we are totally unacquainted with the nature of these molecular forces; if, however, we assume the two hypotheses that the action of any two molecules on each other is the same, and also that its direction is the line joining them, then we shall be able to deduce the conditions of equilibrium of a rigid body from those of a single particle.

PROP. To find the conditions of equilibrium of a rigid body from those of a single molecule.

Let the body be referred to three rectangular axes; and let x, y, z, be the coordinates of one of its constituent particles; X1, Y, Z, the resolved parts, parallel to the axes, of

19

27

the forces which act upon this particle exclusive of the molecular forces; P, P, P, ...... the molecular forces acting on this particle; a,, B1, Y; a B2, Va...... the angles their respective directions make with the three axes of coordinates. Then, since this particle is held in equilibrium by the above forces, we have, by Art. (27),

We shall have a similar system of equations for each particle in the body; if there be n particles there will be 3n equations. These 3n equations will be connected one with another, since any molecular force which enters into one system of equations must enter into a second system; this is in consequence of the mutual action of the particles.

There are two considerations which will enable us to deduce from these 3n equations six equations of condition, independent of the molecular forces. These will be the equations which the other forces must satisfy, in order that equilibrium may be maintained.

The first condition is this, that the molecular actions are mutual; and that, consequently, if P, cosa, represent the resolved part parallel to the axis of x of any one of the molecular forces involved in the 3n equations, we shall likewise meet with the term - P, cosa, in another of those equations which have reference to the axis of x. Consequently, if we add all those equations together which have reference to the same axis, we have the three following equations of condition independent of the molecular forces

ΣΧ = 0, ΣΥ= 0, ΣΖ = 0.

The second consideration is this: that the straight lines joining the different particles are the directions in which the molecular forces act.

Thus, let P, be the molecular action between the particles whose coordinates are (x, y, z,) and (×2, Y2, ≈2),

the corresponding resolved parts of P, for the two particles.

Then

1

·P, cosy

and x1

These enable us to obtain three more equations free from molecular forces; for if we multiply (1) and (2) by y1 respectively, and then subtract, we have

Y®

Y1x, X11 + ...P {x, cos B,-y, cosa,} + ... =

0...(4).

1

By the same process we obtain from the system of equations which refer to the particle (x, y, z),

2

Y ̧¤ ̧ - X ̧31⁄2 + ... - P(x, cosẞ, -y, cosa,) + ... = 0... (5).

2 2

1

But the values of cosa, and cosß1, given above, lead to the condition

(xx) cosß,- (y2-y1) cosa, = 0.

will not involve P1, the molecular action between the particles whose coordinates are (x, y, z,) and (x, y, z) respectively.

It follows readily from what we have shewn, that if we form all the equations similar to (4) and (5), and add them

independent of the molecular forces.

In like manner we should obtain

Σ(Zy - Y2) = 0, Σ(Xz - Zx) = 0.

Moreover we can shew that these six equations are the only equations free from the molecular forces, supposing the body to be rigid, and consequently the molecules to retain their mutual distances invariable. For if a body consist of three molecules, there must evidently be three independent molecular forces to keep them invariable; if to these a fourth be added, we must introduce three new forces to hold it to the others; if we add a fifth, we must introduce three forces to hold this invariably to any three of those which are already rigidly connected; and so on; from which we see that there must be at least 3+3 (n−3) or 3n-6 forces. Hence the 3n equations resembling (1), (2), and (3) contain at least 36 independent quantities to be eliminated; and therefore there cannot be more than six equations of condition connecting the external forces and the coordinates of their points of application.

CHAPTER VIII.

CENTRE OF GRAVITY.

108. Weight is measured like other quantities, by means of an arbitrary unit. If a certain upward force be necessary to prevent a body from falling, then another body which requires an equal force to sustain it, is said to have a weight equal to that of the first. When two weights have been recognised to be equal, a body which requires to sustain it a force equal to the sum of the two equal forces which would sustain the two equal weights, is said to have a weight double that of either of the two equal weights; and so on.

It appears from experiment that the weight of a given. body is invariable so long as the body remains at the same place on the earth's surface, but changes when the body is taken to a different place. We shall suppose therefore, when we speak of the weight of a body, that the body remains at one place.

When a body is such that the weight of any portion of it is proportional to the volume of that portion, it is said to be of uniform density; the density of such a body is measured by the ratio which the weight of any volume of it bears to the weight of an equal volume of some arbitrarily chosen body of uniform density.

The product of the density of a body into its volume is called its mass.