the given forces; let, therefore, the axis of X coincide with the direction of

Hence X = p1+P2 cos 15° + P3 cos 45° + p, cos 105°

Y = p2 sin 15°+p, sin 45°+p, sin 105°

= 3√6-2√2+6√2+ 5 √6+ 5 √⁄2 = 7√2+4√/6.

2

2

... R = √/X2+Y2 = {(4+10√√/2−√/6)2 + (7√/2+4√6)°}*
=25·184297 = magnitude of the resultant.

Also, tan RPX =

Y 19.697454

= 1.2552028; and hence

X 15.6926463

angle RPX = 51°27′ 22′′ = angle included by force p, and the resultant.. Ex. 2. Two forces, represented by 7 and 5, act at an angle of 60°; find their resultant, and the angle it makes with the less force.

Ans. R 10-4403065, and p = 35° 30′. Ex. 3. The resultant of two forces is 24, and the angles it makes with them are 30° and 45°; find the component forces.

Ex. 4. Resolve a given force into two others, such that

(1.) Their sum shall be given, and act at a given angle.

(2.) Their difference shall be given, and act at a given angle.

Ex. 5. If a stream flows at the rate of two miles an hour, find the course which a boat, rowed at the rate of four miles an hour, must pursue, that it may pass directly across the stream.

Ex. 6. Two chords AB, AC of a circle, represent two forces; one of them, AB, is given; find the position of the other, when the resultant is a maximum. Ex. 7. Three forces represented by 13, 14, 15, acting at a point, keep each other in equilibrium; find the angles which their directions make with each other. Ans. 112° 38', 120° 30′, and 126° 52'.

Ex. 8. Three forces P1, P2, P3, act upon a given point and keep it at rest; given the magnitude and direction of p1, the magnitude of p1, and the direction of p2, to find the magnitude of p2, and the direction of p3.

Ex. 9. A string 15 inches in length is attached at its extremities to two tacks, in the same horizontal line, at the distance of 10 inches from each other; a weight of 12lbs. is suspended between the tacks, by means of a string attached to the first, at the distance of 7 inches from one of its extremities; find the strain upon each tack.

Ex. 10. A cord PABQ passes over two small pulleys A, B, whose distance AB is 6 feet, and two weights of 4 and 3lbs., suspended at the extremities P and Q respectively, support a third weight W of 5lbs. ; find the position of the point C to which the weight W is attached, when AB is inclined to the horizon at an angle of 30°.

Ex. 11. P and Q are two equal and given weights suspended by a string passing over three fixed points, A, B, C, given in position; find the actual pressure, and also the horizontal and vertical pressures on each of the three points A, B, C. Also, compare the pressures on A, B, C, when the angles at A, B, C are 150°, 90°, 120° respectively.

ON THE MECHANICAL POWERS.

17. WEIGHT and Power, when opposed to each other, signify the body to be moved, and the body that moves it; or the patient and agent. The power is the agent, which moves, or endeavours to move, the patient or weight.

18. A Machine, or Engine, is any mechanical instrument contrived to move bodies; and it is composed of the mechanical powers.

19. Mechanical Powers are certain simple machines, which are commonly employed for raising greater weights, or overcoming greater resistances, than could be effected by the natural strength without them. These are usually accounted six in number; namely, the Lever, the Pulley, the Wheel and Axle, the Wedge, the Inclined Plane, and the Screw.

20. Centre of Motion, is the fixed point about which a body moves. And the Axis of Motion, is the fixed line about which it moves.

21. Centre of Gravity, is a certain point, upon which a body being freely suspended, it will rest in any position.

OF THE LEVER.

22. A Lever is any inflexible rod, bar, or beam which serves to raise weights, while it is supported at a point by a fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds of levers.

26. A Fourth Kind is sometimes added, called the Bended Lever. As a hammer

drawing a nail.

W

27. In all these machines, the power may be represented by a weight, which is its most natural measure, acting downwards; but having its direction changed, when necessary, by means of a fixed pulley.

PROP. V.

23. When the Weight and Power keep the Lever in equilibrio, they are to each other reciprocally as the Distances of their Lines of Direction from the Prop. That is, P: W:: CD: CE; where CD and CE are perpendicular to WO and AO, which are the Directions of the two Weights, or the Weight and Power W and P

For, draw CF parallel to AO, and CB parallel to WO: Also, join CO, which will be the direction of the pressure on the prop C; for there cannot be an equili- P brium unless the directions of the three forces all meet in, or tend to, the same point as O. Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle CBO or CFO, which are drawn in the direction of those forces; therefore, P: W:: CF: FO or CB.

But, because of the parallels, the two triangles CDF, CEB are equiangular, therefore

Hence, by equality,

·

CD: CE:: CF : CB.

P: W:: CD: CE.

E

B

That is, each force is reciprocally proportional to the distance of its direction from the fulcrum.

And it will be found that this demonstration will serve for all the other kinds of levers, by drawing the lines as directed.

Corollary. 1. When the two forces act perpendicularly on the lever, as two weights, &c.; then, in case of an equilibrium, D coincides with W, and E with P; consequently then the above proportion becomes P: W:: CW: CP, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces.

Corollary. 2. If any force P be applied to a lever at A; its effect on the lever, to turn it about the centre of motion C, is as the length of the lever CA, and the sine of the angle of direction CAE. For the perp. CE is as CA × sine of angle at A.

Corollary. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power by the distance of its direction, is equal to the product of the weight by the distance of its direction.

That is, P X CE W x CD.

Corollary. 4. If the lever, with the weight and power fixed to it, be made to move about the centre C; the momentum of the power will be equal to the momentum of the weight; and their velocities will be in reciprocal proportion

to each other. For the weight and power will describe circles whose radii are the distances CD), CE; and since the circumferences, or spaces described, are as the radii, and also as the velocities, therefore the velocities are as the radii CD, CE; and the momenta, which are as the masses and velocities, are as the momenta and radii; that is, as P × CE and W × CD, which are equal by corol 3.

Corollary 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two.

Corollary 6. If several weights,

P, Q, R, S, act on a straight lever, and keep it in equilibrio, then the sum of the products on one side of the prop, will be equal to the sum on the other, made by multiplying

each weight by its distance; namely,

A

P × AC + Q × BC = R × DC + S × EC.

For, the effect of each weight to turn the lever, is as the weight multiplied by its distance; and in the case of an equilibrium, the sums of the effects, or of the products on both sides, are equal.

That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.

29. SCHOLIUM. Upon the foregoing principles depends the nature of scales and beams, for weighing all sorts of commodities. For, if the weights be equal, then will the distances be equal also, which gives the construction of the common scales, which ought to have these properties:

1st, The points of suspension of the scales and the centre of motion of the

D

E

beam, ABC, must be in a straight line; 2d, The arms AB, BC must be of an equal length: 3d, That the centre of gravity be in the centre of motion B: 4th, That they be in equilibrio when empty : : 5th, That there be as little friction as possible at the centre B. A defect in any of these properties, makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibro when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal; but the de

ceit will be shown by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.

30. To find the true weight of any body by such a false balance :—First, weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required For, if any body b weigh W pounds or ounces in the scale D, and only w pounds or ounces in the scale E; then we have these two equations;

the mean proportional, which is the true weight of the body b.

31. The Roman Statera, or Steelyard, is also a lever, but of unequal brachia or arms, so contrived that one weight only may serve to weigh a great many, by sliding it backwards and forwards to different distances on the longer arm of the lever; and it is thus constructed:

make a notch in the beam, marking it with a cypher 0. Then hang on at A a weight W equal to I, and slide I back towards B till they balance each other; there notch the beam, and mark it with 1. Then make the weight W double of I, and sliding I back to balance it, and there mark it with 2. Do the same at 3, 4, 5, &c., by making W equal to 3, 4, 5, &c. times I; and the beam is finished. Then, to find the weight of any body b by the steelyard; take off the weight W, and hang on the body b at A; then slide the weight I backwards and forwards till it just balance the body b, which suppose to be at the number 5; then is b equal to 5 times the weight of I. So, if I be 1 pound, then b is 5 pounds; but if I be 2 pounds, then b is 10 pounds; and so on.

OF THE WHEEL AND AXLE.

PROP. VI.

32. In the Wheel and Axle; the Weight and Power will be in equilibrio, when the Power P is to the Weight W, reciprocally as the Radii of the Circles