link connecting it with B shall deviate equally to the two sides of G D during the motion; also, the length of the link.

Make D E stroke;

=

join A E; and perpendicular to it, draw E F cutting A D produced in F; A F will be the required radius. Join F B; this will be the link.

RULE XX.-Given, the data and results of Rule XIX.; also the point, G, where the middle position of a second lever connected with the same link cuts G D: to find the second lever, so that the two extreme positions of B shall lie in the same straight line, G B D, with the middle position.

=

K

L

Fig. 101.

Through G draw a straight line, L G K, perpendicular to G D; produce F B till it cuts that line in L; this point will be one end of the required second lever at mid-stroke, and F L will be the entire link. Then in D G lay off D H G B; join A H, and produce it till it cuts LK G in K; this will be the centre for the second lever. When the two extreme positions and the middle position of B lie in the straight line G D, the whole of its positions are near enough to that line for practical purposes.

RULE XXI.-Given (in fig. 102), the main centre, A, the middle position of the main lever, A F, the piston-rod-head, B, and its length of stroke; the radius, A F, of the lever, and the main link, F B, having been found by Rule XIX. Let the figure represent those parts at mid-stroke; and let it be required to construct a parallel motion consisting of a parallel

B

C

ogram, CED F (in which C E F D is called the parallel bar,

and D E

=

F C the back link), and a radius lever, or bridle, H E, jointed to the angle E of the parallelogram.

Draw the straight line A B, cutting the back link D E in G; then by Rule XX. find the lever H E, such that the middle and extreme positions of G shall lie in one straight line.

(The point G shows where a pump-rod may, if convenient, be jointed to the back link).

8. Blocks and Tackle.-RULE XXII.—The ratio of the velocity of the fall of a tackle to the velocity of the moving block is equal to the number of plies of rope by which the fixed and moving blocks are connected with each other.

9. Pistons. The area of a piston is to be measured on a plane perpendicular to its direction of motion. The stroke of a piston moving in a straight line may be measured along the line of motion of any point in the piston; when it moves in a circle the stroke is to be measured on the line described by the centre of the area. RULE XXIII. To find the volume swept by a piston per stroke; multiply the stroke by the area.

RULE XXIV. Two pistons have an invariable volume of fluid between them; to find the ratio of their velocities; take the reciprocal of the ratio of their areas.

SECTION III.-RULES RELATING TO WORK AT UNIFORM AND PERIODICAL SPEED.

1. General Principles.-In a machine moving at an uniform speed the driving and resisting forces are balanced. If the speed is varied, but in such a manner that the variations are periodic, the mean driving and resisting forces during one period, or complete revolution, are balanced. The energy exerted is equal to the whole work performed; in the former case, at all times; in the latter, during any whole number of periods or revolutions. As to units of work, see page 103.

2. Computation of Work Done.-To compute the quantity of work done:

RULE I.-When a weight is lifted to a given height:-multiply the weight by the height.

RULE II. When a body shifts through a given distance against a given force:

Case I. If the force is directly opposed to the motion (being a direct resistance), multiply the force by the distance moved;

Case II. If the force is obliquely opposed to the motion; either resolve the force into a resistance directly opposed to the motion, and a lateral force perpendicular to the motion (see page 160, Rule VIII.), and multiply the resistance by the distance moved; or otherwise:-resolve the motion into a direct component opposed

to the entire force, and a transverse component at right angles to it, and multiply the entire force by the direct component of the motion. (In symbols, let F be the force, s the distance moved, the angle of obliquity; then work done = F s cos e).

RULE III-When a rotating body turns through a given angle against a resisting couple of a given moment (see pp. 104, 161):Multiply that moment by the extent of turning in circular measure. (See page 102.)

RULE IV.—When a piston moves against a pressure of a given intensity (see p. 103):—

Multiply that intensity by the volume swept by the piston. (See page 238, Rule XXIII.)

REMARK.-The unit of volume and unit of intensity should be adapted to each other, so that the product of their numbers may express units of work. For example :—

Prism 1 ft.

x 1 in. x 1 in. li Cylinder 1 ft.

long x 1 in. diam. Ì

Cubic mètre.

Kilogrammètre.

Kilo. on the square mètre. 3. Computation of Energy, Power, and Efficiency.—(I.) When a given weight descends through a given height, or (II.) a given force drives a body shifting through a given distance, or (III.) a rotating body is driven by a couple of a given moment, or (IV.) a piston is driven by a pressure of a given intensity, the rules are the same as in the preceding Article; except that for resistance is to be put effort, or driving force, end for work done, energy exerted. For stored or potential energy, use the same rules, substituting possible for actual motions.

RULE V.-To find the energy which must be exerted to make a machine perform a given motion at an uniform or periodical speed against given resistances. Find, by the rules of the preceding article, the quantities of work done during the given motion against the resisting forces, and add them together; the sum will be the total work done, to which the energy to be exerted will be equal. As to Power, see page 104.

RULE VI.-To find the Efficiency of a machine; distinguish the resistances, and the work done against them, into useful and wasteful; then divide the useful work by the total work; the quotient will be the efficiency.

RULE VII.-To find the efficiency of a train of machines; multiply together the efficiencies of the elementary machines of which the train consists.

4. Computation of Driving Force.-Suppose a machine to be driven against given resistances by an effort or driving force applied at, and in the direction of motion of, the driving point; and that it is required to find the effort which will maintain an uniform speed RULE VIIL-Find the energy to be exerted, by Rule V., and divide it by the space moved through by the driving point;-of otherwise:

RULE VIII A.-Find, by the principles of mechanism (see Section I. of this part, pages 231 to 238), the ratios of the velocities of the several working points, where resistances are overcome, to the velocity of the driving point. Multiply each direct resistance by the velocity-ratio belonging to its point of application, and add together the products; the sum will be the required effort.

REMARKS. This is called "reducing the resistances to the driving point." Rule VIII. A. may be applied to a machine capable of motion, though not actually moving; it is then called the "principle of virtual velocities." When only one resistance is overcome, the effort and resistance are to each other inversely as the velocities of their points of application.

5. Friction in Machines.-RULE IX.-To calculate the resistance of friction to the sliding of two surfaces (when the pressure is not so great as to grind the surfaces, or force out the unguent), multiply the amount of the load, or direct pressure between the surfaces, by the co-efficient of friction.

Explanation of the Table.-, angle of repose; ftan ø, co-efficient of friction; 1:f=cotan o, reciprocal of that co-efficient.

In order that the load may neither grind the surfaces nor force out the unguent of the bearings of machinery, the pressure is to be limited by the following rules; in which, by area of bearing is meant the product of the length and diameter of a cylindrical bearing; although the real area on which pressure acts is much smaller.

RULE X.-Add 20 to the velocity of sliding in feet per minute, and divide 44,800 by the sum; the quotient will be the greatest proper intensity of pressure in lbs. on the square inch, with the further limitation that the intensity is in no case to exceed 1,200 lbs. on the square inch.

TULE XI. To calculate the moment of friction of an axle; multiply the resultant load by the radius of the axle, and by the sine of the angle of repose (which is sensibly equal to the co-efficient of friction).

6. Pulley and Strap.-Let T1 be the tension at the tighter side of the strap, and To the tension at the slacker side, so that T1 - To is the force to be exerted between the strap and pulley; also let c be the arc of contact between the strap and pulley, in fractions of a circumference, and f the co-efficient of friction.

RULE XII.-Given, c, f, and the force T1 - To; to find the tensions, greatest, least, and mean. Let N be the number corresponding to the common logarithm 273 fc; then

REMARK.-Whether the calculation relates to driving belts or to strap-brakes, the co-efficient, f, should be estimated on the supposition of the surfaces being oily; say 0·15 for leather on metal, and 0.08 for metal on metal.

7. Balancing of Machinery.—In a machine every piece which turns on an axis should, as far as possible, have its re-actions balanced.

RULE XIII.-In order that there may be no tendency to shift the axis, arrange the weights that turn together about it so that their common centre of gravity shall be in the axis. (This constitutes a "standing balance.")

RULE XIV.-In order that there may be no tendency to turn the axis into varying directions; multiply each of the masses that turn together about the axis by its arm or perpendicular distance from the axis. Regard the products as representing forces, each pulling the axis towards the mass to which that product belongs,

R