These results may be obtained

by processes similar to those employed in this article-but much more easily by employing the Integral Calculus:—we have therefore thought it sufficient to state the results for the information of the student.

87. We will close this chapter with the following elegant theorem-due we believe to Leibnitz.

III. If a system of forces in equilibrium acting at a point A be represented in magnitude and direction by the lines AP, AQ, AR... then will the point A be the centre of mean position of the points P, Q, R, ...; (in other words) the point A will be the centre of gravity of a system of equal particles placed at the points P, Q, R...

Take any line x'Ax passing through A and draw Pp, Qq... perpendicular to this line; then will Ap, Aq... represent the projections on x'Ax of the lines AP, AQ... i.e. of the forces P, Q... But since these forces are at equilibrium the algebraic sum of their resolved parts in any assigned direction must be zero by Art. (39).

Hence since the algebraic sum of the lines Ap, Aq... is zero,

the centre of gravity of the points P, Q... must be in the plane which passes through A at right angles to 'Ax, and since the direction of 'Ax is arbitrary, this centre of gravity must lie in every plane which can be so drawn, and must therefore coincide with the point A, the common point of intersection of these planes.

Hence, when any number of forces acting on a point are in equilibrium, this point is the centre of gravity of a series of equal particles placed at the extremities of lines which represent the forces in magnitude and direction.

And vice versa. If we consider a series of equal particles and we draw lines from each to the centre of gravity of the series, it is clear that a system of forces represented by these lines will be in equilibrium.

For as before draw the lines AP, AQ...; it is clear that A being the centre of gravity, the algebraic sum of the lines Ap, Aq... is zero; i.e. the sum of the resolved parts of the forces AP, AQ... taken in any direction x'Ax is zero, and therefore the forces are in equilibrium.

COR. 1. We see from this theorem that if three forces are in equilibrium about a point, this point is the centre of gravity of the triangle formed by joining the extremities of lines representing the forces in magnitude and direction; for the centre of gravity of a triangle is the same as that of three equal particles placed at its angular points.

Similarly, if four forces are in equilibrium about a point, this point is the centre of gravity of the pyramid whose angular points are the extremities of the straight lines

representing the forces: for the centre of gravity of a triangular pyramid is in the same position as that of four equal particles placed at the angular points.

The converse of each of these is also true.

COR. 2. More generally: If all the equal particles of a rigid body of any form are attracted to the same point by forces proportional to their distances from this point they will be in equilibrium if the point be the centre of gravity of the body; and conversely.

CHAPTER VI.

OF THE MECHANICAL POWERS.

88. THE simplest machines employed for supporting weights, communicating motion to bodies,—or speaking generally, for making a force which is applied at one point practically available at some other point, are called the Mechanical Powers; and by a combination of them all machines, however complicated, are constructed.

They are commonly reckoned as six in number:-the lever, the wheel and axle, the pully, the inclined plane, the wedge, and the screw.

In explaining and discussing these simple machines we shall suppose them to be at rest, so that the force applied at one point is balanced by the force or pressure called into action at some other point: we shall also suppose the several parts of them to be without weight and perfectly smooth except when the contrary is expressly stated.

When two forces acting on a machine balance each other, one of them is for convenience called the power and the other the weight.

89. The Lever.

A rigid rod or bar capable of turning about a fixed point of it is called a lever. The point about which it can turn is called the fulcrum, and the parts into which the rod is divided by the fulcrum are called the arms of the lever. When the arms are in a straight line, it is called a straight lever; in all other cases it is a bent lever.

We have seen in Art. 35, 36, that a body of any form capable of turning about a fixed

point may be considered as a lever, and if two forces P, Q act upon it in a plane passing through O, the lever will be in equilibrium if P. Op = Q. Oq; i. e. if the moments of P and Q which tend to turn the lever about O be equal, and tend in opposite directions.

P

In order however to render our explanation as simple as possible, we will for the present consider the arms of the lever as straight and uniform, or approximately so.

90. Levers are sometimes divided into three classes according to the relative position of the points where the power and the weight are applied with respect to the fulcrum.

Thus in levers of the first class, the power and the weight are applied on opposite sides of the fulcrum C, but act in the same direction, as in fig. 1.

In levers of the second class, the power and weight are applied on the same side of the fulcrum, but act in opposite directions (as in fig. 2, the power being applied

YP

Fig. 1.

B

Fig. 2.

at a greater distance from the fulcrum than the weight is.

In levers of the third class (fig. 3), the power and the weight act on the same side of the fulcrum

- Fig. 3.

Aw