It appears from this expression, that the weights of the pulleys diminish the advantage of this system.

204.

If all the pulleys are equal, then

W = 2" P - A1 (2′′-1 + 2"-2 + ...... + 1)

Hence, if we suppose both the power and weight diminished by the weight of a pulley, we may then neglect the consideration of the heaviness of the pulleys.

205.

In the system (fig. 47) where each string is attached to the weight, let T1, T... be the tensions of the first, second... strings; then if the weights of the pulleys are inconsiderable, we have

Now W is supported by the tensions of the n strings fastened to the block B, and

206. In the system (fig. 48), let T1, T... be the tensions strings; then T1 = P; and T2 has

of the first, second... strings; then Ti

to support three tensions equal to P; therefore

and if there be (n) different strings, the tension of the last is

T= 32-1 P.

Now the weight W is supported by two strings whose tensions are T1, two of which the tensions are T2, &c.;

REMARK.

1

2

+ 2 Tn

If the weights of the pulleys and blocks are not inconsiderable, they may be taken into account, in this and every other system, by adding each to the tension of that string which supports it, as in Art. 203.

207. In the system, fig. 49, the weight W is supported by the tensions of all the strings at the lower block, and as it is the same string which passes round all the pulleys, the tension of every part = P; wherefore, if there ben pulleys in the lower block, there are 2n strings supporting the weight, and therefore

III. On the Wheel and Axle.

208. The wheel and axle consists of a cylinder and a wheel firmly attached to each other, and being moveable about a fixed axis coinciding with the axis of the cylinder, and passing through the centre of the wheel at right angles to its plane, as in fig. 50.

The power P acts by means of a cord wrapped round the circumference of the wheel C, and the weight W is fastened to a cord which is wound upon the cylinder AB as P turns the machine round its axis; and thus W is raised.

209. To find the condition of equilibrium on the wheel and axle.

We may consider P and W as forces acting upon a rigid body with a fixed axis, and therefore their moments. about that axis must be equal;

. Px (perpendicular upon its direction from the axis), = W. (perpendicular upon its direction from the axis).

Now these perpendiculars are respectively the radii of the wheel and of the cylinder ;

.. P. (radius of the wheel) = W. (radius of the axle).

210. If the thickness of the rope be considerable, it must be taken into account.

We may suppose the actions of P and W to be transmitted along the middle or axis of the rope, and then the perpendiculars upon the directions of P and W will be respectively equal to

radius of wheel + radius of rope,

and radius of axle + radius of rope,

and the condition of equilibrium is

P. (rad. wheel + rad. of rope) = W (rad. axle + rad. of rope). This diminishes the advantage of the machine.

211. The pressure on the axis of this machine may be found by transposing P and W in their own planes, parallel to themselves, to the axis.

212.

IV. On the Inclined Plane.

This machine is nothing more than a plane inclined to the horizon. The condition of equilibrium may be thus found.

Let AB (fig. 51.) be the plane; AC parallel and BC perpendicular to the horizon; W the weight, P the power. Draw WR perpendicular to the plane, WG perpendicular to the horizon. P is supposed to act in the plane RWB. The weight W is kept at rest by three forces, viz. P in the direction WP; gravity (= W) in the direction WG, and reaction R of the plane in the direction WR.

Denote the angle PWB by 0, and the inclination BAC of the plane to the horizon by i; and resolve the three forces, acting on the point W, in a direction parallel to the plane, the sum will be

P cos PWB

W cos AWG + R cos RWB

= P cos 0 W. sin i.

But since there is an equilibrium, this sum must be equal to zero,

.. P cos 0 = W sin i,

which is the condition of equilibrium.

213. If P's direction should happen to be parallel to the plane, 0 and cos 0 = 1;

.. P = W sin i.

But if P's direction should happen to be parallel to the horizon, i and cos (-i) = cos i ;

=

214. To find the reaction of the plane.

Resolve the forces in a direction at right angles to that in which P acts;

215. This mechanical power is a combination of the lever and inclined plane; it may be conceived to be thus generated.

Let ABCD (fig. 52) be a cylinder; BEFC a rectangle whose base BE is equal to the circumference of the cylinder. Divide this rectangle into any convenient number of equal rectangles GE, IH, CK; and draw their diagonals BH, GK, IF. Then, if this rectangle CE be wrapped upon the cylinder, so that BE coincides with the circumference of the base, E, H, K, F will respectively fall upon the points B, G, I, C of the cylinder, and the lines BH, GK, IF will trace out upon its surface a continuous spiral thread BLGMINC winding uniformly up the cylinder. The cylinder is usually made protuberant where the spiral line BLGMINC falls upon it so that the thread becomes a winding inclined plane, projecting from the cylinder as in fig. 53, and differing from the inclined plane BH in no

* The following illustration renders this very clear :—

"When a road directly ascends the side of a hill, it is to be considered as an inclined plane; but it will not lose this mechanical character, if, instead of directly ascending towards the top of the hill, it winds successively round it, and gradually ascends so as after revolutions to reach the top. In the same manner a path may be conceived to surround a pillar by which the ascent may be facilitated upon the principle of the inclined plane. Winding stairs constructed in the interior of great columns partake of this character; for although the ascent be produced by successive steps, yet if a floor could be made sufficiently rough to prevent the feet from slipping, the ascent would be accomplished with equal facility. In such a case the winding path would be equivalent to an inclined plane, bent into such a form as to accommodate it to the peculiar circumstances in which it would be

required