pose such forces to produce any change in the state of the body, whether from rest to motion, or vice versa, or in the velocity of the motion which the body may have previously had, would be equivalent to a supposition that the forces applied to the body being in equilibrium were capable of producing a dynamical effect, which would be a contradiction in terms. This, though not always clearly understood by mere practical men, or by persons superficially informed, is, in fact, among the fundamental principles of mechanical science.

(89.) When the piston is at the top of the cylinder, and about to commence its motion downwards, the steam acting upon it will have not only to overcome the resistance arising from the friction of the various parts of the engine, but will also have to put in motion the whole mass of matter of the piston pump rods, pump pistons, and the column of water in the pump barrels. Besides imparting to this mass the momentum corresponding to the velocity with which it will be moved, it will also have to encounter the resistance due to the preponderance of the weight of the water and pump rods over that of the steam piston. The pressure of steam, therefore, upon the piston at the commencement of the stroke must, in accordance with the mechanical principles just explained, have a greater force than is equal to all the resistances which it would have to overcome, supposing the mass to be moving at a uniform velocity. The moving force, therefore, being greater than the resistance, the mass, when put in motion, will necessarily move with a gradually augmented speed, and the piston of the engine which has been described in the last chapter would necessarily move from the top to the bottom of the cylinder with an accelerated motion, having at the moment of its arrival at the bottom a greater velocity than at any other part of the stroke. As the piston and all the matter which it has put in motion must at this point come to rest, the momentum of the moving mass must necessarily expend itself on some part of the machinery, and would be so much mechanical force lost. It is evident, therefore, independently of any consideration of the expansive principle, to which we shall presently refer, that the action of the

moving power in the descent of the piston ought to be suspended before the arrival of the piston at the bottom of the cylinder, in order to allow the momentum of the mass which is in motion to expend itself, and to allow the piston to come gradually to rest at the termination of the stroke.

Thus, if we were to suppose that after the piston had descended through three-fourths of the whole length of the cylinder, and had acquired a certain velocity, the steam above it were suddenly condensed, so as to leave a vacuum both above and below it, the piston, being then subject to no impelling force, would still move downwards, in virtue of the momentum it had acquired, until the resistance would deprive it of that momentum, and bring it to rest; and if the remaining fourth part of the cylinder were necessary for the accomplishment of this, then it is evident that that part of the stroke would be accomplished without further expenditure of the moving power.

In fact, this part of the stroke would be made by the expenditure of that excess of moving power, which, at the commencement of the stroke, had been employed in putting the machinery and its load in motion, and in subsequently accelerating that motion.

Although under such circumstances the resistance, during the operation of the moving power, shall not have been at any time equal to the moving power, since while the motion was accelerated it was less, and while retarded greater than that power, yet as the whole moving power has been expended upon the resistance, the mechanical effect which the moving power has produced under such circumstances will be equal to the actual amount of that power. If in an engine of this kind the steam was not cut off till the conclusion of the stroke, a part of the moving power would be lost upon those fixed points in the machinery which would sustain the shock produced by the instantaneous cessation of motion at the end of the stroke.

Independently, therefore, of any consideration of the expansive principle, it appears that, in an engine of this kind, the steam ought to be cut off before the completion of the stroke.

(90.) To render the expansive action of steam intelligible, let A B (fig. 28.) represent a cylinder whose area we will suppose,

Fig. 28.

C

A

B

for the sake of illustration, to be a square foot, and whose length, A B, shall also be a foot. If steam of a pressure equal to the atmosphere be supplied to this cylinder, it will exert a pressure of about one ton on the piston; and if such steam be uniformly supplied from the boiler, the piston will be moved from A to B with the force of one ton, and that motion will be uniform if the piston be opposed throughout the same space by a resistance equal to a ton. When the piston has arrived at B, let us suppose that the further supply of steam from the boiler is stopped by closing the upper steam valve, and let us also suppose the cylinder to be continued downwards so that BC shall be equal to A B, and suppose that B C has been previously in communication with the condenser, and is therefore a vacuum. The piston at B will then be urged with a force of one ton downwards, and as it descends the steam above it will be diffused through an increased volume, and will consequently acquire a diminished pressure. We shall, for the present, assume that this diminution of pressure follows the law of elastic fluids in general; that it will be decreased in the same proportion as the volume of the steam is augmented. While the piston, therefore, moves from B downwards it will be urged by a continually decreasing force. Let us suppose, that by some expedient, it is also subject to a continually decreasing resistance, and that this resistance decreases in the same proportion as the force which urges the piston. In that case the motion of the piston would continue uniform. When the piston would arrive at P', the middle of the second cylinder, then the space occupied by the steam being increased in the proportion of 2 to 3, the pressure on the piston would be diminished in the proportion of 3 to 2, and the pressure at B being one ton, it would be two-thirds of a ton at P'. In like manner when the piston would arrive at c, the space occupied by the steam being double that which

M

it occupied when the piston was at B, the pressure of the steam would be half its pressure at B, and therefore at the termination of the stroke, the pressure on the piston would be half a ton.

If the space from в to c, through which the steam is here supposed to act expansively, be divided into ten equal parts, the pressure on the piston at the moment of passing each of those divisions would be calculated upon the same principle as in the cases now mentioned. After moving through the first division, the volume of the steam would be increased in the proportion of 10 to 11, and therefore its pressure would be diminished in the proportion of 11 to 10. The pressure, therefore, driving the piston at the end of the first of these ten divisions would be 19ths of a ton. In like manner, its pressure at the second of the divisions would be 40ths of a ton, and the third 19ths of a ton; and so on, as indicated in the figure.

Now if the pressure of the steam through each of these divisions were to continue uniform, and, instead of gradually diminishing, to suffer a sudden change in passing from one division to another, then the mechanical effect produced from B to c would be obtained by taking a mean or average of the several pressures throughout each of the ten divisions. In the present case it has been supposed that the force on the piston at в was 2240 pounds. To obtain the pressure in pounds corresponding to each of the successive divisions, it will therefore only be necessary to multiply 2240 by 10, and to divide it successively by 11, 12, 13, &c. The pressures, therefore, in pounds, at each of the ten divisions, will be as follows:

If the mean of these be taken by adding them together

and dividing by 10, it will be found to be 1498 pounds. It appears, therefore, that the pressures through each of the ten divisions being supposed to be uniform (which however, strictly, they are not,) the mechanical effect of the steam from B to c would be the same as if it acted uniformly throughout that space upon the piston with a force of about 1500 pounds, being rather less than three-fourths of its whole effect from A to B.

But it is evident that this principle will be equally applicable if the second cylinder had any other proportion to the first. Thus it might be twice the length of the first; and in that case, a further mechanical effect would be obtained from the expansion of the steam.

The more accurate method of calculating the effect of the expansion from в to c, would involve more advanced mathematical principles than could properly be introduced here; but the result of such a computation would be that the actual average effect of the steam from в to c would be equal to a uniform pressure through that space, amounting to one thousand five hundred and forty-five pounds, being greater than the result of the above computation, the difference being due to the expansive action through each of the ten divisions, which was omitted in the above computation.

(91.) It is evident that the expansive principle, as here explained, involves the condition of a variation in the intensity of the moving power. Thus, if the steam act with a uniform energy on the piston so long as its supply from the boiler continues, the moment that supply is stopped, by closing the steam valve, the steam contained in the cylinder will fill a gradually increasing volume by the motion of the piston, and therefore will act above the piston with a gradually decreasing energy. If the resistance to the moving power produced by the load, friction, &c. be not subject to a variation corresponding precisely to such variation in the moving power, then the consequence must be that the motion imparted to the load will cease to be uniform. If the energy of the moving power at any part of the stroke be greater than the resistance, the motion produced will be accelerated; if it be less, the motion. will be retarded; and if it be at one time greater, and another