cient; it has been aptly compared by Professor Leslie, to "a reservoir which collects the intermitting currents, and sends forth a regular stream."* To equalize a motion which is subject to variation at each reciprocation in the steam engine, the fly is used. Its heavy mass of matter must be so shaped, as to balance itself in any position on an axis connected with the machinery, and turning round with a part of it.

The proportions of the fly wheel must be derived from the laws of rotary motion. They are not often stated very clearly, and rather in too comparative a form for the purpose of application; Dr. Jackson's equation+ is derived most in unison with my own methods, and adding the time, the radius corresponding to the angular velocity of the exterior ring of the wheel, and comparing with the force of gravity to obtain the co-efficient, it is

In this equation P is the mean quantity the moving force varies in its intensity in excess above the resistance, and t the time in which that variation takes place; v the velocity, and n v the greatest variation of velocity; d the leverage the force P acts with, and r the radius corresponding to the velocity v; and b the weight of the fly acting at the distance x from the axis.

It is obvious that the mass of the fly must be sufficient to receive the excess of force during the time it acts, and afford it again to the machine in an equal lapse of time; and so that the velocity shall not vary more than the nth part. The only point therefore, which depends on practical experience, is what variation of velocity may be allowed. On this point however there is no difficulty, as the practice of different makers is so different, as to shew that it may be taken with considerable latitude.

The weight of the rim may always be considered to be collected at the extremity of the radius; and then a=r, and the equation becomes

The effect of the arms of the wheel may be neglected, as it is a problem which neither requires nor admits of a very refined solution, in consequence of the uncertainty regarding the precise variation of the intensity of the moving force; hence, it ought not to be rendered complicated.

-537.-From this equation it appears, that when the weight or the diameter of the rim is considerable, and still more when both are so, it may acquire a great momentum with

Natural Philosophy, Vol. I. p. 152.

+ Theoretical Mechanics, art. 400-403.

but little increase of angular velocity, or lose a considerable momentum with little diminution of that velocity. It thus becomes a receptacle for the surplus energy of the power, when it acts with most intensity, or when the resistance is least, and preserves it for future demand.

By either a diminution of resistance, or an increase of power, the machine would otherwise be considerably accelerated; the excess of motive force is however, in a great measure, expended upon the fly, in which it generates a proportional momentum with little increase of velocity again, when the resistance is increased, or the moving power diminished, the machinery would be very sensibly retarded, if the momentum accumulated in the fly did not continue the motion with little diminution of its own velocity; and other things being the same, the shorter the interval of reciprocation, or of unequal resistance, the less will be the change of velocity.

The greater the angular velocity of the axis of the fly is, the greater will be its dominion or equalizing power, all other things being equal, for the variation of velocity is inversely as the velocity of the rim.

Every part of a machine which has either a continuous or pendulous motion, particularly when it is massive, will obviously act as a fly in equalizing the motion of the machine.

The greater part of these remarks have been made in a less general form by Dr. Robison,* and Dr. Jackson;† but they also state that when a more perfect equalizer is wanted, we should increase the power of the fly wheel by enlarging the diameter rather than the mass, because we thus produce the same effect with less weight, consequently with less transverse strain upon the axle and supports, and less friction.

This must however be carried only to small extent, for a mass of matter with an immense velocity, sustained by arms which must be completely incapable of resisting its impulse, becomes a very dangerous appendage to a machine. Arms of cast iron could not resist a sudden check with a rim moving at the velocity of eighteen feet per second, and equal to the weight of the arms, consequently, such wheels should be of limited diameter.

538.-When it is necessary to exceed a velocity of twelve feet per second at the rim, malleable iron arms should always be used, and a velocity of thirty-three feet per second at the rim is about the extreme limit for a fly, even where the ring is of malleable iron. For cast iron rims, with arms of malleable iron, I should not think a velocity exceeding eighteen feet per second safe.

* Mechanical Philosophy, Vol. II. p. 250.

Essay on the Strength of Cast Iron, art. 261.

+ Theoretical Mechanics, p. 227.

With these explanations we may now proceed to form rules from the equation. The equation is

but the dimensions of the rim will be more convenient than the weight; and the weight b = 2 × 3·1416 ra x 32 pounds for cast iron = 20 r a, consequently,

the area of the section of the rim in inches. If the fly wheel shaft makes N revolutions per minute, then in the time t it will make

539.-The next point to be considered, is the degree of equalization a machine requires. Its own parts have much effect, and the species of parts which act as flies, are most numerous in machines which require the equalizing power of the fly the most. At a mean, perhaps a variation of one-tenth is nearly corresponding with practice, and with this condition the rule is

540.—Case I. A double engine with a crank. In this case the variation is from the full force of the steam to nothing at each quarter of the stroke; hence, the mean excess is one-fourth of the greatest force P on the piston, and the rule in the nearest simple expression is

RULE. Multiply forty times the pressure on the piston in pounds, by the radius of the crank in feet, and divide this product by the cube of the radius of the fly wheel in feet, and by the number of its revolutions per minute, the result is the area of the rim of the fly in inches.

The number of horses' power, multiplied by 200, will be the greatest pressure on the piston, nearly.

Example. The pressure on the piston of an engine being 4000 pounds, the radius of the crank 2.5 feet, and the revolutions per minute twenty-two, required the section of the rim of a fly of nine feet radius. In this case

twenty-five inches nearly, for the area of the section of the rim.

541.-Case II. In a single engine with a crank the mean excess is half the moving force; hence,

or the rim of the fly wheel should be double that required for a double engine with the same sized cylinder, or of twice the power.**

542.-Counter weights. If the beam of a single engine be balanced when at rest, that weight which it is necessary to add or subtract, to cause the piston to rise at the proper speed, is called the counter weight. The excess of force of the steam overcomes the friction of the parts, and the additional weight ought to be sufficient to cause it to rise and acquire double the velocity of the engine, if it freely accelerated during the whole stroke. If W be the whole weight of matter moved, w = the counter weight, and 7 = the length of the stroke, then

But (art. 342,) v2 = 2·66 1; v being here in seconds: hence, w = 0·2 W; consequently, the counter weight should with these proportions be one-fifth of the mass of matter it has

* In single acting atmospheric engines a weight has been applied to the fly wheel, such that its force to turn the shaft should be exactly half the force of the steam to turn it, and placed so as to rise while the piston was descending, and descend during the rise of the piston. To find the weight, we have

when w is the weight and P the mean pressure on the piston. It is supposed to be applied to the rim of the fly, and the section of the continued rim should be the same as for a double engine of the same power. This mode is described in Fenwick's Essays on Practical Mechanics, p. 39. Woolf proposed to equalize the motion of an engine by a piston working in a cylinder; this however has no other effect than a weight, while the friction and expense of construction are considerably increased. See Nich. Journal, Vol. VI. p. 218, and Vol. VII. p. 134.

to move, supposing the whole to be collected at the ends of the beam, and it is most easily found by trial. The resistance of the water in the pumps will reduce the accelerated to an uniform motion of half the final velocity it would have acquired with no such resistance.*

Of regulating the Power of Engines.

543.—An engine is frequently applied where the work to be done is not constantly the same; and when the machinery of a part of it is suddenly stopped, or suddenly set on, if the moving power were to remain the same, an alteration of the velocity must take place, it must move faster or slower. This change of velocity would in most cases be very hurtful to the work, and cause considerable loss; besides, there is always a velocity at which a machine will act with greater advantage than at any other; therefore the change of velocity arising from the above cause, is in all cases a disadvantage, and in all delicate operations exceedingly injurious. In a cotton mill, for example, where the power moved the spindles with a given speed, if so much of the work were at once thrown off as to increase the velocity in a considerable degree, a loss of work would immediately take place, and an increase of waste from the breaking of the threads; on the other hand, there would be much loss of the time of the attendants, if the machinery moved too slow.

An equally bad effect is observed in raising water, and other species of work.

;

544.-The throttle valve. The power of a steam engine is usually regulated by increasing or diminishing the steam passage, and this is generally performed by admitting the steam into the cylinder, more or less freely, by means of what is called a throttle valve this valve is formed of a circular plate of metal, a, Fig. 1, Plate VIII. having a spindle fixed across its diameter. The plate is accurately fitted to an aperture in a metal ring of some thickness, through which the spindle is fitted steam-tight, and the ring is fixed between the flanches of that joint of the steam pipe which is next to the cylinder. A square part is formed on one end of the spindle to receive an arm or lever b, by which the valve may be turned in either direction.

Smeaton arranged his engines to make the returning stroke in less time than the acting one, (Reports, Vol. II. p. 360.) Watt states it to be generally agreed that the reverse should be the case. (Robison's Mech. Phil. Vol. 11. p. 99.) The reasons for making them equal are stated in (art. 340.)