D

its preffure P. Let the additional pressure on the cover of the veffel be p, and the denfity of the air in the vessel bed. We fhall have P: P+p=D:d; and therefore p=Px. Then, because the preffure which expels the air is the difference between the force which compreffes the air in the veffel and the force which compresses the external air, the expelling force is p. And because the quantities of motion are as the forces which fimilarly produce them, we shall have P: PX MV: mu; where M and m exprefs the quantities of matter expelled, V expreffès the velocity with which air rushes into a void, and expreffes the velocity fought. But becaufe the quantities of aerial matter which iffue from the fame orifice in a moment are the denfities and velocities jointly, we shall have MV: mv=DVV: dvv, DV1: dv2. d--D Therefore P:p DV: dv. Hence we de

duce v=V

D

d-D

=

1332 feet in a fecond nearly. But as foon a air has come out, the density of the remain is diminished, and its elafticity is dimin therefore the expelling force is diminished. I matter to be moved is diminished in the very proportion, because the denfity and elasticit according to the fame law; therefore the ve will continue the same from the beginning end of the efflux. Hence it follows, what ap very unlikely at firft fight, that however mu air in the veffel is condenfed, it will always into a void with the fame velocity.

To find the quantity of aerial matter whic iffue during any time f, and confequentiy denfity of the remaining air at the end of this we must get the rate of efflux. In the eleme time t there iffues (by what has been said al the bulk 8/HQ i (for the velocity V is conf and therefore the quantity 8HOdt. Or other hand, the quantity of air at the begin was CD, C being the capacity of the veffel; when the air has acquired the denfity d. the q tity is Cd, and the quantity run out is CD— therefore the quantity which has run out in time t must be the fluxion of CD-C d, or Therefore we have the equation 8/HO&i=

and t__—Cả

C

T

The fluent of this is HO log. d. 7

fluent must be fo taken that may be whe D. Therefore the correct fluent will be t

C

8✓ HO

D log., for log.= log. 1, = 0. T

P

and convenient expreffion.

Hitherto we have confidered the motion of air as produced by its weight only. Let us now confider the effect of its elafticity. Let ABCD (fig. 52.) be a veffel containing air of any density D. This air is in a state of compreffion; and if the compreffing force be removed, it will expand, and its elasticity will diminish along with its denfity. Its elafticity in any state is measured by the force which keeps it in that ftate. The force which keeps common air in its ordinary density is the weight of the atmosphere, and is the fame with the weight of a column of water 33 feet high. If, therefore, we fuppofe that this air, inftead of being confined by the top of the veffel, is preffed down by a moveable piston carrying a column of water 33 feet high, its elafticity will balance this preffure as it balances the preffure of the atmo. fphere; and as it is a fluid, and propagates through every part the preffure exerted on any one part, it will prefs on any little portion of the veffel by its elafticity in the fame manner as when loaded with this column.

The confequence of this reafoning is, that if this fmall portion of the veffel be removed, and thus a paffage be made into a void, the air will begin to flow out with the fame velocity with which it would flow when impelled by its weight alone, or with the velocity acquired by falling from the top of a homogeneous atmosphere, or

Q (9-8) , and the motion will ceafe when 9(Q-D)

B

we

Some of these questions are of difficult folution, and they are not of frequent ufe in pneumatics. The cafes of greateft ufe are when the air is ex, pelled from a veffel by an external force, as when bellows are worked, whether of the ordinary form or confifting of a cylinder fitted with a moveable pifton. This laft cafe merits a particular confideration, and the investigation is extremely easy.

Let AD, fig. 52. be a piston moving downward with the uniform velocity f, and let the area of the pifton ben times the area of the hole of efflux, then the velocity of efflux ariling from the motion of the pifton will be nf. Add this to the velocity V produced by the elasticity of the air in the firft queftion, and the whole velocity will be V+nf. It will be the fame in the others. The problem is alfo freed from the confideration of the time of efflux; for this depends now on the velocity of the pifton. It is ftill, however, a very intricate problem to ascertain the relation between the time and the density, even though the piston is moving uniformly; for at the beginning of the motion the air is of common denfity. As the pifton defcends, it both expels and compreffes the air, and the denfity of the air in the vessel varies in a very intricate manner, as alfo its refiftance or reaction on the pifton. For this reason, a piston which moves uniformly by an external force will never make an uniform blaft by fucceffive ftrokes; it will always be weaker at the beginning of the ftroke. The beft way for fecuring an uniform blaft is to employ the external force only for lifting up the pifton, and then to let the pifton defcend by its own weight. In this way it will quickly fink down, compreffing the air, till its denfity and correfpond. ing elafticity exactly balance the weight of the pifton. After this the pifton will defcend equably, and the blaft will be uniform. Thefe obfervations and theorems will determine the initial velocity of the air in all important cafes of its expulfion. The philofopher will learn the rate of its efflux out of one vellel into another; the chemift will be able to calculate the quantities of the different gafes employed in the curious experiments of the ingenious but unfortunate LAVOISIER on i combustion; and the engineer will learn how to proportion the motive force of his machine to the quantity of aerial matter which his bellows muft fupply.

All the modifications of motion in water conduits take place alfo in the paffage of air through pipes and holes of all kinds. There is the fame diminution of quantity passing through a hole in a thin plate that is obferved in water. Abating the small effect of friction, water iffues with the velocity acquired by falling from the furface: and yet if we calculate by this velocity and by the area of the orifice, we find the quantity of water deficient nearly in the proportion of 63 to 100. This is owing to the water preffing towards the orifice from all fides, which occafions a contraction of the jet. The fame thing happens in the efflux of air. The motion of water is alfo greatly impeded VOL. XVII. PART I.

by all contractions of its paffage. Thefe oblige it to accelerate its velocity, and therefore require an increafe of preffure to force it through them, in proportion to the fquares of the velocities. Thus, if a machine working a pump caufes it to give a certain number of ftrokes in a minute, it will deliver a determined quantity of water in that time. Should it happen that the paffage of the water is contracted to one half in any part of the machine (which often happens at the valves), the water muft move through this con traction with twice the velocity that it has in the reft of the paffage. This will require four times the force to be exerted on the pifton. Nay, which appears very odd, and is never fufpected by engineers, if no part of the paffage is narrower than the barrel of the pump, but a part much wider, and if the conduit be again contracted to the width of the barrel, an additional force muft be applied to the pifton to drive the water through this paffage, which would not have been neceffary if the paffage had not been widened in any part. It will require a force equal to the weight of a column of water of the height necef. fary for communicating a velocity, the square of which is equal to the difference of the fquares of the velocities of the water in the wide and the narrow part of the conduit. The fame thing takes place in the motion of air, and therefore all contractions and dilatations must be carefully avoided, when we want to preferve the velocity unimpaired.

AIR alfo fuffers the fame retardation in its motion along pipes. By not attending to this, engineers of the firft reputation have been prodigiously disappointed in their expectations of the quantity of air which will be delivered by long pipes. Its extreme mobility and lightnefs hindered them from fufpecting that it would fuffer any fenfible" retardation. Dr PAPIN, a moft ingenious man. proposed this as the most effectual method of transferring the action of a moving power to a great distance. Notwithstanding his great repu tation, he could not get his fcheme patronifed in England; but in France and Germany he got fome perfons of fortune to affift him in this project; and he erected great machines in Auvergne and Weftphalia for draining mines. But, fo far from being effective, they would not even begin to move. He attributed the failure to the quantity of air in the pipe of communication, which indeed muft be condenfed before it can condenfe the air in the remote cylinder. He therefore diminished the fize of his pipe, and made his water-machine exhauft, inftead of condenfing. But he was equally difappointed, and the machines at the mines ftood still as before.

་།

Near a century after this, a very intelligent engineer attempted a much more feasible engine at an iron foundery in Wales. He erected a machine at a powerful water-fall, which worked a fet of cylinder bellows, the blow-pipe of which was conducted to the diftance of a mile and a half, where it was applied to a blast furnace. But notwithstanding every care to make the conducting pipe very air-tight, of great fize, and as smooth as poffible, it would hardly blow out a candle. The failure was afcribed to the impoffiblity of making C

the

the pipe air-tight. No very diftinct theory can be delivered on this fubject; but we may derive confiderable affiftance in understanding the causes of the obftruction to the motion of water in long pipes, by confidering what happens to air. The elafticity of the air and its great compreffibility, have given us the moft diftinct notions of fluidity in general; showing us, in a way that can hardly be controverted, that the particles of a fluid are kept at a distance from each other, and from other bodies, by the corpufcular forces. See WATER-WORKS.

The writers on hydrodynamics have always confidered the obstruction to the motion of fluids along canals of any kind, as owing to fomething like the friction by which the motion of folid bodies on each fide is obftructed; but we cannot form any diftinct notion of refemblance, or even analogy between them. The fact is, that a fluid running along the canal has its motion obstructed; and that this obftruction is greatest in the immediate vicinity of the folid canal, and gradually diminishes to the middle of the ftream. It appears, therefore, that the parts of fluids can no more move among each other than among folid bodies, without fuffering a diminution of their motion. The parts in phyfical contact with the fides and bottom are retarded by thefe immoveable bodies. The particles of the next ftratum of fluid cannot preferve their initial velocities without overpaffing the particles of the firft ftratum; and it appears from the fact that they are by these means retarded. They retard in the fame manner the particles of the third ftratum, and fo on to the middle ftratum or thread of fluid. The fact, therefore, is, that this fort of friction is not a confequence of rigidity alone, but that it is equally competent to fluids. Nay, fince it is a matter of fact in air, and is even more remarkable there than in any other fluid, and as our experiments on the compreffion of air fhow us the particles of air ten times nearer to each other in fome cafes than in others, and 1000 times denfer, and thus force us to acknowledge that they are not in contact, it is plain that this obftruction has no analogy to friction, which fuppofes inequality of furface. No fuch inequality can be fuppofed in the furface of an aerial particle; nor would it be of any service in explaining the obftruction, fince the particles do not rub on each other, but pass each other at fome fmall imper. ceptible distance.

We must therefore have recourfe to fome other mode of explication. We fhall apply this to air only in this place; and, fince it is proved by the incontrovertible experiments of CANTON, ZIMMERMAN, and others, that water, mercury, oil, &c. are alfo compreffible and perfectly elaftic, the argument from this principle, which is conclufive in air, muft equally explain the fimilar phenomenon in hydraulics.

The moft highly polished body which we know must be conceived as having an uneven surface, when we compare it with the small spaces in which the corpufcular forces are exerted; a quantity of air moving in a polished pipe may be compared to a quantity of small fhot fliding down a channel with undulated fides and bottom. The row of particles immediately contiguous to the

5

fides will therefore have an undulated motion; but this undulation of the contiguous particles of air will not be fo great as that of the surface along which they glide; for not only every motion requires force to produce it, but alfo every change of motion. The particles of air refift this change from rectilineal to an undulating motion; and, being claftic, that is repelling each other and other bodies, they keep a little nearer to the surface as they are paffing over an eminence, and their path is lefs incurvated than the furface. The undulation of the next row of particles will be less than that of the firft; that of the third row less than that of the fecond, and so on, as reprefented in fig. 54. pl. CCLXXXI. And thus, while the mafs of air has a progreffive motion along the pipe or canal, each particle is defcribing a waving line, of which a line parallel to the direction of the canal is the axis, cutting all these undulations. This axis of each undulated path will be straight or curved as the canal is, and the excurfions of the path on each fide of its axis will be lefs and lefs as the axis of the path is nearer to the axis of the canal.

Let us now fee what fenfible effect this will have; for all the motion which we here fpeak of is imperceptible. It is demonftrated in mechanics, that if a body moving with any velocity be deflected from its rectilineal path by a curved and perfectly fmooth channel, to which the rectilineal path is a tangent, it will proceed along this channel with undiminished velocity. Now, the path in the prefent cafe, may be confidered as perfectly smooth, fince the particles do not touch it. There should not, therefore, be any diminution of the velocity. Let us grant this of the abfolute velocity of the particle; but what we obferve is the velocity of the mafs. Let us fuppofe a fingle atom to be a fenfible object, and let us attend to two fuch particles, one at the fide, and the other in the middle; although we cannot perceive the undulations of thefe particles during their progreffive motions, we fee the progreffive motions themselves. Let us fuppofe then that the middle particle has moved without any undulation whatever, and that it has advanced ten feet. The lateral particle will alfo have moved ten feet; but this has not been in a ftraight line. It will not be so far advanced, therefore, in the direction of the canal; it will be left behind, and will appear to have been retarded in its motion; and in like manner each thread of particles will be more and more retarded (apparently only), as it recedes farther from the axis of the canal, or what is ufually called the thread of the stream. And thus the observed fact is a neceffary confequence of the nature of a compreffible or elaftic fluid; and without fuppofing any diminution in the real velocity of each particle, there will be a diminution of the velocity of the fenfible threads of the general ftream, and a diminution of the whole quantity of air which paffes along it during a given time.

Let us now fuppofe a parcel of air impelled along a pipe, which is perfectly smooth, out of a larger veffel, and iffuing from this pipe with a certain velocity. It requires a certain force to change its velocity in the veffel to the greater velocity which it has in the pipe. This is clearly demonftrated. How long foever we fuppofe this pipe,

there