been defined to be a fraction expressing the ratio of the useful work to the whole work performed, which is equal to the energy expended. The limit to the efficiency of a machine is unity, denoting the efficiency of a perfect machine in which no work is lost. The object of improvements in machines is to bring their efficiency as near to unity as possible.

As to useful and lost work, see Article 12. The algebraical expression of the efficiency of a machine having uniform or periodical motion, is obtained by introducing the distinction between useful and lost work into the equations of the conservation of energy. Thus, let P denote the mean effort at the driving point, & the space described by it in a given interval of time, being a whole number of periods or revolutions, R, the mean useful resistance, s, the space through which it is overcome in the same interval, R, any one of the prejudicial resistances, s, the space through which it is overcome; then

.

2

Ꭱ 82 ; .

.(1.)

(2.)

In many cases the lost work of a machine, R. 82, consists of a constant part, and of a part bearing to the useful work a proportion depending in some definite manner on the sizes, figures, arrangement, and connection of the pieces of the train, on which also depends the constant part of the lost work. In such cases the whole energy expended and the efficiency of the machine are expressed by the equations

P8=(1+A) R1 8, + B;

1

and the first of these is the mathematical expression of what Mr. Moseley calls the "modulus" of a machine.

The useful work of an intermediate piece in a train of mechanism consists in driving the piece which follows it, and is less than the energy exerted upon it by the amount of the work lost in overcoming its own friction. Hence the efficiency of such an intermediate piece is the ratio of the work performed by it in driving the following piece, to the energy exerted on it by the preceding piece; and it is evident that the efficiency of a machine is the product of the effi ciencies of the series of moving pieces which transmit energy from the driving point to the working point.

The same principle applies to a train of successive machines, each driving that which follows it.

36. Power and Effect-Horse-Power.-The power of a machine is the energy exerted, and the effect, the useful work performed, in some interval of time of definite length, such as a second, a minute, an hour, or a day.

The unit of power called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour. The effect is equal to the power multiplied by the efficiency. The loss of power is the difference between the effect and the power. (See also Article 3.)

37. General Equation. The following general equation presents at one view the principles of the action of machines, whether moving uniformly, periodically, or otherwise:

where W is the weight of any moving piece of the machine;

h, when positive, the elevation, and when negative, the depression, which the common centre of gravity of all the moving pieces undergoes in the interval of time under consideration; v, the velocity at the beginning, and v, the velocity at the end, of the interval in question, with which a given particle of the machine of the weight W is moving;

g, the acceleration which gravity causes in a second, or 32.2 feet per second;

[R ds', the work performed in overcoming any resistance

during the interval in question;

P ds, the energy exerted during the interval in question. The second and third terms of the right hand side, when positive, are energy stored; when negative, energy restored.

The principle represented by the equation is expressed in words as follows:

The energy exerted, added to the energy restored, is equal to the energy stored added to the work performed.

SECTION 4.-Of Dynamometers.

38. Dynamometers are instruments for measuring and recording the energy exerted and work performed by machines. They may be classed as follows:

I. Instruments which merely indicate the force exerted between a driving body and a driven body leaving the distance through

which that force is exerted to be observed independently. The following are examples of this class:—

a. The weight of a solid body may be so suspended as to balance the resistance to be overcome, as in Mr. Scott Russell's experiments on the resistance of canal boats, published in the Transactions of the Royal Society of Edinburgh, vol. xiv.

b. The weight of a column of liquid may be employed to balance and measure the effort required to drag a carriage or other body, as in the mercurial dynamometer invented by Mr. John Milne of Edinburgh.

c. The available energy of a prime mover may be wholly expended in overcoming friction, the magnitude of which is measured by a weight, as in Prony's dynamometer, to be afterwards more particularly described.

d. A spring balance may be interposed between a prime mover and a body whose resistance it overcomes, so as to indicate at each instant the magnitude of that resistance.

II. Instruments which record at once the force, motion, and work of a machine, by drawing a line, straight or curved, as the case may be (such as that shown in fig. 3, Article 11) whose abscissæ represent on a suitable scale the distances moved through, its ordinates the corresponding resistances overcome, and its area the work performed.

A dynamometer of this class consists essentially of two principal parts: a spring whose deflection indicates the force exerted between a driving body and a driven body, and a band of paper, or a card, moving at right angles to the direction of deflection of the spring with a velocity bearing a known constant proportion to the velocity with which the resistance is overcome. The spring carries a pen or pencil, which marks on the paper or card the required line. The following examples of this class of instruments will be described in the sequel:

a. Morin's Traction Dynamometer.

b. Morin's Rotatory Dynamometer.

c. Watt and M'Naught's Steam Engine Indicator.

III. Instruments which record the work performed, but not the resistance and motion separately.

39. Prony's Friction Dynamometer measures the useful work performed by a prime mover, by causing the whole of that work to be expended in overcoming the friction of a brake. In fig. 12, A represents a cylindrical drum, driven by the

BF F

H

EVE

Fig. 12.

K

prime mover. The block D, attached to the lever B C, and the

smaller blocks with which the chain E is shod, form a brake which embraces the drum, and which is tightened by means of the screws F, F, until its friction is sufficient to cause the drum to rotate at an uniform speed. The end C of the lever carries a scale G, in which weights are placed to an amount just sufficient to balance the friction, and keep the lever horizontal. The lever ought to be so loaded at B that when there are no weights in the scale, it shall be balanced upon the axis. The lever is prevented from deviating to any inconvenient extent from a horizontal position by means of safety stops or guards H, K.

The weight of the load in the scale which balances the friction being multiplied into the horizontal distance of the point of suspension C from the axis, gives the moment of friction, which being multiplied into the angular velocity of the drum, gives the rate of useful work or effective power of the prime mover.

As the whole of that power is expended in overcoming the friction between the drum and the brake, the heat produced is in general considerable; and a stream of water must be directed on the rubbing surfaces to abstract that heat.

The friction dynamometer is simple and easily made; but it is ill adapted to measure a variable effort; and it requires that when the power of a prime mover is measured, its ordinary work should be interrupted, which is inconvenient and sometimes impracticable.

40. Morin's Traction Dynamometer. The descriptions of this and some other dynamometers invented by General Morin are abridged from his works entitled Sur quelques Appareils dynamométriques and Notions fondamentales de Mécanique.

Fig. 13 is a plan and fig. 13 a an elevation of a dynamometer for recording by a diagram the work of dragging a load horizontally. aa, bb, are a pair of steel springs, through which the tractive force is transmitted, and which serve by their deflection to measure that force. They are connected together at the ends by the steel links ff. The effort of the prime mover is applied, through the link r, to the gland d, which is fixed on the middle of the foremost spring; the equal and opposite resistance of the vehicle is applied to the gland c, which is fixed on the middle of the aftermost spring. When no tractive force is exerted, the inward faces of the springs are straight and parallel; when a force is exerted, the springs are bent, and are drawn apart, through a distance proportional to the force. The springs are protected against being bent so far as to injure them by means of the safety bridles i, i, with their bolts e, e. Those bridles are carried by the after-gland, and their bolts serve to stop the foremost spring when it is drawn forward as far as is consistent with the preservation of elasticity and strength.

The frame of the apparatus for giving motion to the

paper band

Fig. 13.

#

Fig. 13 a.

is carried by the after-gland. The principal parts of that apparatus are the following:

1, store drum on which the paper band is rolled before the commencement of the experiment, and off which it is drawn as the experiment proceeds;

9, taking-up drum, to which one end of the paper band is glued, and which draws along and rolls up the paper band with a velocity proportional to that of the vehicle. Fixed on the axis of this drum is a fusee having a spiral groove round it, whose radius gradually increases at the same rate as that at which the effective radius of the drum g is increased during its motion by the rolling of successive coils of paper upon it. The object of this is to prevent that increase of the effective radius of the drum from accelerating the speed of the paper band;

n is a drum which receives through a train of wheelwork and endless screws, a velocity proportional to that of the wheels of the vehicle, and which, by means of a cord, drives the fusee. The mechanism is usually so designed that the paper moves at onefiftieth of the speed of the vehicle.

Between the drums / and g, there are three small rollers to support the paper band and keep it steady.

One of the safety bridles carries a pencil k, which, being at rest