The denomination Machine is now vulgarly given to a great variety of subjects, which have very little analogy by which they can be clasped with propriety under any one name. We say a travelling machine, a bathing machine, a copying machine, a threshing machine, an electrical machine, &c. &c. The only circumstance in which all these agree seem to be, that their construction is more complex and artificial than the utensils, tools, or instruments which offer themselves to the first thoughts of uncultivated people. They are more artificial than the common cart, the bathing tub, or the flail. In the language of ancient Athens and Rome, the term was applied to every tool by which hard labour of any kind was performed; but in the language of modern Europe, it seems restricted either to such tools or instruments as are employed for executing some philosophical purpose, or of which the construction employs the simple mechanical powers in a conspicuous manner, in which their operation and energy engage the attention. An electrical machine, a centrifugal machine, are of the first class; a threshing machine, a fire machine, are of the other class. It is nearly synonymous, in our language, with engine; a term altogether modern, and in some measure honourable, being bestowed only, or chiefly, on contrivances for executing work in which ingenuity and mechanical skill are manifest. Perhaps, indeed, the term engine is limited, by careful writers, to machines of considerable magnitude, or at least of considerable art and contrivance. We say, with propriety, steam engine, fire-engine, plating-engine, boring-engine; and a dividing machine, a copying machine, &c. Either of these terms, machine or engine, are applied with impropriety to contrivances in which some piece of work is not executed on materials which are then said to be manufactured. A travelling or bathing machine is surely a vulgarism. A machine or engine is therefore a tool; but of complicated construction, peculiarly fitted for expediting labour, or for performing it according to certain invariable principles: And we should add, that the dependence of its efficacy on mechanical principles must be apparent, and even conspicuous. The contrivance and erection of such works constitute the profession of the engineer; a profession which ought by no means to be confounded with that of the mechanic, the artisan, or manufacturer. It is one of the artes liberales; as deserving of the title as medicine, surgery, architecture, painting, or sculpture. Nay, whether we consider the importance of it to this flourishing nation, or the science that is necessary for giving eminence to the professor, it is very doubtful whether it should not take place of the three last named, and go pari passu with surgery and medicine. The inconsiderate reader, who peruses Cicero de Oratore with satisfaction, is apt to smile at Vitruvius, who requires in his architect nearly the same accomplishments which Cicero requires in his orator. He has not recollected, or perhaps did not know, that the profession of an architect in the Augustan age was the most respectable of all those which were not essentially connected with the management of state affairs. It appears that the architects were all Greeks, or the pupils of Greeks, altogether different from the members of the Collegium Murarius, the corporation of builders and masons. The architecture of temples, stadiums, circuses, amphitheatres, seems to have been monopolised, by flatter authority, by a society which had long subsisted in Asia, connected by certain mysterious bonds, both civil and religious. We find it in Syria; and we learn that it was brought thither from Persia in very ancient times. From thence it spread into Ionia, where it became a very eminent and powerful association, under the particular protection of Bacchus, to whom the members had erected a magnificent temple at Teos, with a vast establishment of priests and priestesses, consisting of persons of the first rank in the state. They were the sole builders of temples and stadiums throughout all Greece and the Lesser Asia; and the contractors for the machinery that was employed in the theatres, and in the great temples, for the celebration of the high mysteries of paganism. By the imperfect accounts which remain of the Eleusinian and other mysteries, it appears, that this machinery must have been immense and wonderful, and must have required a great deal of mechanical skill. This indeed appears, in the most convincing manner, to any person who reflects on the magnificent structures which they erected, which excite to this day the wonder of the world, not only on account of their magnificence and incomparable elegance, but also on account of the mechanical knowledge that seems indispensably necessary for their erection. This will ever remain a mystery. There are no traces of such knowledge to be found in the writings of antiquity. Even Vitruvius, writing expressly on the subject, has given us nothing but what is in the lowest degree of elementary knowledge.

This association of the Dionysiacs undoubtedly kept their mechanical science a profound secret from the uninitiated, the profane. They were the engineers of antiquity, and Vitruvius was perhaps not one of the initiated. He speaks of Myro and other Greek architects in terms of respect which border on veneration. Perhaps the modern association of free masons is a remain of this ancient fraternity, continued to our times by the company of builders, who erected the cathedrals and great conventual churches. No one who considers their works with scientific attention, can doubt of their being deeply versed in the principles of mechanics, and even its more refined branches. They appear to have carried the art of vault-roofing almost to its acme of perfection; far outstripping their Grecian instructors in their knowledge of this most delicate branch of their art. It were greatly to be wished that some such institution did yet exist, where men might be induced by the most powerful motives to accomplish themselves in the knowledge necessary for attaining eminence in their profession.

We have been informed (and we thought our authority good), that our gracious Sovereign has signified his intention of patronizing an institution of this kind. We heard, that it was proposed to institute degrees similar to our university degrees, and proceeding on similar conditions of a regular education or standing, which would ensure the opportunities of information, and also on an examination of the proficiency of the candidate. This examination, being conducted by persons eminent in the profession, perhaps still exercising it, would probably be serious, because the successful candidate would immediately become a rival practitioner. Such an institution would undoubtedly prevent many gross impositions by unlettered millwrights and pump-makers, who now seldom appear under any name but that of engineer, although they are frequently ignorant even of the elements of mechanical science, and are totally unacquainted with the higher mathematics; without which it is absolutely impossible for them to contrive a machine well suited to the intended purpose, or to say with any tolerable precision what will be the performance of the engine they have erected. Yet there are questions susceptible of accurate solution, because they depend on the unalterable laws of matter and motion.

All who have a just view of the unspeakable advantages which this highly favoured land possesses in the superiority and activity of its manufacturers, and who know how much of this superiority should be ascribed to the great improvements which have been made in practical mechanics within these last thirty years, will join us in willing succours to some such institution as that now mentioned.

We were naturally led to these reflections when we turned our thoughts to machinery in general, and observed what is done in this country by the native energy of its inhabitants, unassisted by such scientific instructions as they might have expected from the pupils of a Newton, their countryman, under the patronage of the best of Sovereigns, eminently knowing in these things, and ever ready to encourage those sciences and arts which have so highly contributed to the national prosperity. What might not be reasonably expected from British activity, if those among ourselves who have knowledge and leisure had been at the same pains with the members of the foreign academies to cultivate the Newtonian philosophy, and particularly the more refined branches of mechanics, and to deduce from their speculations maxims of construction fitted to our situation as a great manufacturing nation? But such knowledge is not attainable by those who are acquainted only with the imperfect elements contained in the publications read by the bulk of our practitioners. Much to this purpose has been done on the continent by the most eminent mathematicians; but from want of individual energy, or perhaps of general security and protection, the patriotic labours of those gentlemen have not done the service to their country which might have been reasonably expected. Indeed, their dissertations have generally been so composed, that only the learned could see their value. They seem addressed only, or chiefly, to such; but it is to those authors that our countrymen generally have recourse for information concerning every thing in their profession that rises above mere elementary knowledge. The books in our language which profess to be systems of mechanics rarely go beyond this; they contain only the principles of equilibrium. These are absolutely necessary for the knowledge of machines; but they are very far indeed from giving what may be called a practical knowledge of working machinery. This is never in a state of equilibrium. The machine must move in order to work. There must be a superiority of impelling power, beyond what is merely sufficient for balancing the resistance or contrary action of the work to be performed. The reader may turn to the article Statics in the Encyclopaedia Britannica, and he will there find some farther observations on this head. And in the article Mechanics he will find a pretty ample detail of all the usual doctrines, and a description of a considerable variety of machines or engines, accompanied by such observations as are necessary for tracing the propagation or transmission of pressure from that part of the machine to which the natural power is applied to the working part of the machine. Along with these two articles, it will be proper to read with peculiar attention the article Rotation.

By far the greatest number of our most serviceable engines consist chiefly of parts which have a motion of rotation round fixed axes, and derive all their energy from levers virtually contained in them. And these acting parts are also material, requiring force to move them over and above what is necessary for producing the acting force at the working part of the machine. The modifications which this circumstance frequently makes of the whole motions of the machine, are indicated in the article Rotation in an elementary way; and the propositions there investigated will be found almost continually involved in the complete theory of the operation of a machine. Lastly, it will be proper to consider attentively the propositions contained in the article Strength of Materials, that we may combine them with those which relate wholly to the working of the machine; because it is from this combination only that we discover the strains which are excited at the various points of support, and of communication, and in every member of the machine. We suppose all these things already understood.

Our object at present is to point out the principles which enable us to ascertain what will be the precise motion of a machine of given construction, when actuated by a natural power of known intensity, applied to a given point of the machine, while it is employed to overcome a known resistance acting at another point. To abbreviate language, we shall call that the repelling point of the machine to which the pressure of the moving power is immediately applied; and we may call that the working point, where the resistance arising from the work to be performed immediately acts.

To consider this important subject, even in its chief varieties, requires much more room than can be allowed in an undertaking like ours, and therefore we must content ourselves with a very limited view; but at the same time, such a view as shall give sufficient indication of the principles which should direct the practical reader in every important case. We shall consider those machines... which perform their motions round fixed axes; these being by far the most numerous and important, because they involve in their construction and operations all the leading principles.

That we may proceed securely, it is necessary to have a precise and adequate notion of moving force, as applied to machinery, and of its measures. We think this peculiarly necessary. Different notions have been entertained on this subject by Mr Leibnitz, Des Cartes, and other eminent mechanicians of the last century; and their successors have not yet come to an agreement. Nay, some of the most eminent practitioners of the present times (for we must include Mr Smeaton in the number) have given measures of mechanical power in machinery which we think inaccurate, and tending to erroneous conclusions and maxims.

We take for the measure (as it is the effect) of exerted mechanical power the quantity of motion which it produces by its uniform exertion during some given time. We say uniform exertion, not because this uniformity is necessary, but only because, if any variation of the exertion has taken place, it must be known, in order to judge of the power. This would needlessly complicate the calculations; but in whatever way the exertion may have varied, the whole accumulated exertion is still accurately measured by the quantity of motion existing at the end of the exertion. The reader must perceive that this is the same thing that is expressed in the article Dynamics of this Supplement, no. 90, by the area of the figure whose abscissa or axis represents the time of exertion; and the ordinates are as the pressures in the different instants of that time, the whole being multiplied by the number of particles (that is, by the quantity of matter), because that figure represents the quantity of motion generated in one particle of matter only. All this is abundantly clear to persons conversant in these disquisitions; but we will to carry along with us the distinct conceptions of that useful class of readers whose profession engages them in the construction and employment of machines, and to whom such disquisitions are not so familiar. We must endeavour therefore to justify our choice of this measure by appealing to familiar facts.

If a man, by pressing uniformly on a mass of matter for five seconds, generates in it the velocity of eight feet per second, we obtain an exact notion of the proportion of this exertion to the mechanical exertion of gravity, when we say that the man's exerted force has been precisely one-twentieth part of the action of gravity on it; for we know that the weight of that body (or, more properly, its heaviness) would, in five seconds, have given it the velocity of 160 feet per second, by acting on it during its fall. But let us attend more closely to what we mean by saying that the exerted force is one-twentieth of the exertion of gravity. The only notion we have of the exertion of gravity is what we call the weight of the body—the pressure which we feel it make on our hand. To say that this is 20 pounds weight, does not explain it; because this is only the action of gravity on another piece of matter. Both pressures are the same. But if the body weighs 20 pounds, it will draw out the rod of a steelyard to the mark 20. The rod is so divided, that the 20th part of this pressure will draw it out to 1. Now the fact is, that if the man presses on the mass of 20 pounds weight with a spring steelyard during five seconds, and if during that time the rod of the steelyard was always at the mark 1, the body will have acquired the velocity of eight feet per second. This is an acknowledged fact. Therefore we were right in saying, that the man's exertion is one-twentieth of the exertion of gravity. And since we believe the weight of bodies to be proportional to their quantity of matter, all matter being equally heavy, we may say, that the man's exertion was equal to the action of gravity on a quantity of matter whose weight is one pound. We express it much more familiarly, by saying, that the man exerted on it the pressure of one pound of matter, or the force of one pound.

In this manner, the motion communicated to a mass of matter, by acting on it during some time, informs us with accuracy of the real mechanical force or pressure which has been exerted. This is judged to be double when twice the velocity has been generated in the same mass, or where the same velocity has been generated in twice the mass; because we know, that a double pressure would have done either the one or the other.

But farther: We know that this pressure is the exertion; we have no other notion of our own force; and our notion of gravity, of elasticity, or any other natural force, is the same. We also know that the continuance of this exertion fatigues and exhausts our strength as completely as the most violent motion. A dead pull, as it is called, of a horse, at a post fixed in the ground, is a usual trial of his strength. No man can hold out his arm horizontally for much more than a quarter of an hour; and the exertion of the last minutes gives the most distressing fatigue, and disables the shoulder from action for a considerable time after. This is therefore an expenditure of mechanical power, in the strict primitive sense of the word. Of this expenditure we have an exact and adequate effect and measure in the quantity of motion produced; that is, in the product of the quantity of matter by the velocity generated in it by this exertion. And it must be particularly noticed, that this measure is applicable even to cases where no motion is produced by the exertion; that is, if we know that the exertion which is just unable to start a block of stone lying on a smooth stone pavement, but would start it, if increased by the smallest addition; and if we know that this would generate in a second 32 feet of velocity in 100 pounds of matter—we are certain that it was a pressure equal to the weight of this 100 pounds. It is a good measure, though not immediate, and may be used without danger of mistake when we have no other.

The celebrated engineer Mr Smeaton, in his excellent dissertation on the power of water and wind to drive machinery, and also in two other dissertations, all three published in the Philosophical Transactions, and afterwards in a little volume, has employed another measure, both of the expenditure of mechanical power, and of the mechanical effect produced. He says, that the weight of a body, multiplied by the height through which it descends, while driving a machine, is the only proper measure of the power expended; and that the weight, multiplied by the height through which it is uniformly raised, is the only proper measure of the effect produced. And he produces a large train of accurate experiments. We have already noticed the complete expenditure of animal power by continued pressure, even when motion is not produced; the only difficulty is to connect this in a measurable way with the power which the same exertion has of generating motion in a body.

When a man supports a weight for a single instant, he certainly balances the pressure or action of gravity on that body; and he continues this action as long as he continues to support it; and we know that if this body were at the end of a horizontal arm turning round a vertical axis, the same effort which the man exerted in merely carrying the weight, if now exerted on the body, by pushing it horizontally round the axis, will generate in it the same velocity which gravity would generate by its falling freely. On this authority therefore we say, that the whole accumulated action of a man, when he has just carried a body whose weight is 30 pounds for one minute, is equal to the whole exertion of gravity on it during that minute; and if employed, not to counteract gravity, but to generate motion, would generate, during that minute, the same motion that gravity would, that is, $60 \times 32$ feet velocity per second, in a mass of 30 pounds. There would be 30 pounds of matter moving with the velocity of 1920 feet per second. We would express this production or effect by $30 \times 180$, or by 57600, as the measure of the man's exertion during the minute.

But, according to Mr Smeaton, there is no expenditure of power, nor any production of mechanical effect, in thus carrying 30 pounds for a minute; there is no product of a weight by a height through which it is equally raised; yet such exertion will completely exhaust a man's strength if the body be heavy enough. Here then is a case to which Mr Smeaton's measure does not readily apply; and this case is important, including all the actions of animals at a dead pull.

But let us consider more narrowly what a man really does when he performs what Mr Smeaton allows to be the production of a measurable mechanical effect. Suppose this weight of 30 pounds hanging by a cord which passes over a pulley, and that a man, taking this cord over his shoulder, turns his back to the pulley, and walks away from it. We know, that a man of ordinary force will walk along, raising this weight, at the rate of about 60 yards in a minute, or a yard every second, and that he can continue to do this for eight or ten hours from day to day; and that this is all that he can do without fatigue. Here are 30 pounds raised uniformly 180 feet in a minute; and Mr Smeaton would express this by $30 \times 180$, or 5400, and would call this the measure of the mechanical effect, and also of the expenditure of power. This is very different from our measure 57600.

But this is not an accurate and complete account of the man's action on the weight, and of the whole effect to be inproduced. To be convinced of this, suppose that a curate, man A has been thus employed, while another B, walking along side of him at the same rate, suddenly takes the rope out of his hand, frees him of the task, and continues to raise the weight without the smallest change on its velocity of ascent. What is the action of B, and whether is it the same with that of A or not? It is acknowledged by all, that the exertion of B against the load is precisely equal to 30 pounds. If he holds the rope by a spring fleelyard, it will stand constantly at the mark 30. B exerts the same action on the load as when when he simply supports it from falling back into the pit. It was moving with the velocity of three feet per second when he took hold of the rope, and it would continue to move with that velocity if anything could annihilate or counteract its gravity. If therefore there was no action when a person merely carried it, there is none at present when it is rising 180 feet in a minute. The man does indeed work more than on that occasion, but not against the load; his additional work is walking, the motion of his own body, as a thing previously necessary that he may continue to support the load, that he may continue his mechanical effort as it follows him. It appears to yield to him; but it is not to his efforts that it yields; its weight completely balances those efforts, and is balanced by them. It was to a greater effort of the man A that it yielded. It was then lying on the ground. He pulled at the cord, gradually perhaps increasing his pull till it was just equal to its weight. When this obtains, the load no longer presses on the ground, but is completely carried by the rope. But it does not move by this effort of 30 pounds; but let him exert a force of 31 pounds, and continue this for three seconds. He will put it in motion; will accelerate that motion; and at the end of three seconds the load is rising with the velocity of three feet per second. The man feels that this is as much speed as he can continue in his walk; he therefore slackens his pull, reducing his action to 30 pounds, and with this action he walks on. All this would be distinctly perceived by means of a fleelyard. The rod would be pulled out beyond 30, till the load acquired the uniform velocity intended, and after this it would be observed to shrink back to 30.

More is done therefore than appears by Mr Smeaton's measure. Indeed, all that appears in it is the exertion necessary for continuing a motion already produced, but which would be immediately extinguished by a contrary power, which must therefore be counteracted. This measure will not apply to numberless cases of the employment of machines, where there is no such opposing power, and where, notwithstanding, mechanical power must be expended, even according to Mr Smeaton's measure. Such are corn mills, boring mills, and many others.

How then comes it that Mr Smeaton's valuable experiments concur so exactly in shewing that the same quantity of water descending from the same height, always produces the same effect (as he measured it), whatever be the velocity? In the first place, all his experiments are cases where the power expended and the work performed are of the same kind: A heavy body descends, and by its preponderance raises another heavy body. But even this would not ensure the precise agreement observed in his experiments, if Mr Smeaton were not careful to exclude from his calculations all that motion where there is any acceleration, and all the expenditure of water during the acceleration, and to admit only those motions that are sensibly uniform. In moderate velocities, the additional pressure required for the first acceleration is but an insignificant part of the whole; and to take these accelerated motions into the account, would have embarrassed the calculations, and perhaps confused many of the readers. We see, in the instance now given, that the addition of one pound continued for three seconds only, was all that was necessary.

Mr Smeaton's measurement is therefore abundantly exact for practice; and being accommodated to the circumstances most likely to engage the attention, is very proper for the instruction of the numerous practitioners in all manufacturing countries who are employed for ordinary erections; but it is improperly proposed as an article essential to a full theory of mechanics, and therefore it was proper to notice it in this place. Besides, there frequently occur most important cases, in which the motion of a machine is, of necessity, deftly alternately accelerated, and retarded. We should not derive all the advantages in our power from the first mover, if we did not attend particularly, and chiefly, to the accelerating forces. And in every case, the improvement, or the proper employment of the machine, is not attained, if we are not able to discriminate between the two parts of the mechanical exertion; one of them, by which the motion is produced and accelerated to a certain degree; and the other, by which that motion is continued. We must be able to appreciate what part of the effect belongs to each.—But it is now time to proceed to the important question,

What will be the precise motion of a machine of given construction, actuated by a power of known intensity and manner of acting, and opposed by a known resistance?

In the solution of this question, much depends on things to the nature of both power and resistance. In the statical consideration of machines, no attention is paid to any differences. The intensity of the pressures is all work that it is necessary to regard, in order to state the proportion of pressure which will be exerted in the various parts of the machine. The pressures at the impelled and working points, combined with the proportions of the machine, necessarily determine all the rest. Pressure being the sole cause of all mechanical action among bodies, any pressure may be substituted for another that is equal to it; and the pressure which is most familiar, or of easiest consideration, may be used as the representative of all others. This has occasioned the mechanical writers to make use of the pressure of gravity as the standard of comparison, and to represent all powers and resistances by weights. However proper this may be in their hands, it has hurt the progress of the science. It has rendered the usual elementary treatises of mechanics very imperfect, by limiting the experiments and illustrations to such as can be so represented with facility. This has limited them to the state of equilibrium (in which condition a working machine is never found), because illustrations by experiment out of this state are neither obvious nor easy. It has also prevented the students of mechanics from accomplishing themselves with the mathematical knowledge required for a successful prosecution of the study. The most elementary geometry is sufficient for a thorough understanding of equilibrium, or the doctrines of statics; but true mechanics, the knowledge of machines as instruments by which work is performed, requires more refined mathematics, and is inaccessible without it.

Had not Newton or others improved mathematics by the invention of the infinitesimal analysis and calculus, we must have been contented with the discoveries (really great) of Galileo and Huyghens. But New- There is also an important distinction in the manner in which this external motion is kept up. In a weight employed as the moving power, the actuating pressure seems to reside in the matter itself; and all that is necessary for continuing this pressure is merely to continue the connection of it with the machine. But in the action of animals, it may be very different: A man pushing at a capstan bar, must first of all walk as fast as the bar moves round, and this requires the expenditure of his muscular force. But this alone will not render his action an effective power: He must also press forward the capstan bar with as much force as he has remaining over and above what he expends in walking at that rate. The proportion of these two expenditures may be very different in different circumstances; and in the judicious selection of such circumstances as make the first of these as inconsiderable as possible, lies much of the skill and sagacity of the engineer. In the common operation of thrashing corn, much more than half of the man's power is expended in giving the necessary motion to his own body, and only the remainder is employed in urging forward the flail with a momentum sufficient for shaking off the ripe grains from the stalk. We had sufficient proof of this, by taking off the swifte of the flail, and putting the same weight of lead on the end of the flail, and then causing the bind to perform the usual motions of thrashing, with all the rapidity that he could continue during the ordinary hours of work. We never could find a man who could make three motions in the same time that he could make two in the usual manner, so as to continue this for half an hour. Hence we must conclude, that half (some will say two-thirds) of a thrasher's power is expended in merely moving his own body. Such modes of animal action will therefore be avoided by a judicious engineer: but to be avoided, their inconvenience must be understood. More of this will occur hereafter.—In other cases, we are almost (never wholly) free from this unprofitable expenditure of power. Thus, in the steam engine, the operation requires that the external air follow the piston down the cylinder, in order to continue its pressure. But the force necessary for feeding in this rare fluid into the cylinder with the necessary velocity, is such an insignificant part of the whole force which is at our command, that it would be ridiculous affectation in any engineer to take it into account; and this is one great ground of preference to this natural power. The same thing may be said of the action of a strong and light spring, which is therefore another very eligible first mover for machinery. The ancient artillerymen had discovered this, and employed it in their warlike engines.

We must also attend to the nature of the resistance which the work to be performed opposes to the motion of our machine. Sometimes the work opposes, not a simple obstruction, but a real resistance or reaction, which, if applied alone to the machine, would cause it to move the contrary way. This always obtains in cases where a heavy body is to be raised, where a spring is to be compressed, and in some other cases. Very often, however, there is no such contrary action. A flour mill, a saw mill, a boring mill, and many such engines, exhibit no reaction of this kind. But although such machines, when at rest or not impelled by the first mover, sustain no pressure in the opposite direction, yet... they will not acquire any motion whatever, unless they be impelled by a power of a certain determinate intensity. Thus in a saw mill, a certain force must be impressed on the teeth of the saw, that the cohesion of the fibres of the timber may be overcome. This requires that a certain force, determined by the proportions of the machine, be impressed on the impelled point. If this, and no more, be applied there, a force will be excited at the teeth of the saw, which will balance the cohesion of the wood, but will not overcome it. The machine will continue at rest, and no work will be performed. Any addition of force at the impelled point, will occasion an addition to the force excited in the teeth of the saw. The cohesion will be overcome, the machine will move, and work will be performed. It is only this addition to the impelling power that gives motion to the machine; the rest being expended merely in balancing the cohesion of the woody fibres. While therefore the machine is in motion, performing work, we must consider it as actuated by a force impressed on the impelled point by the natural power, and by another acting at the working point, furnished by or derived from the resistance of the work.

Again: It not unfrequently happens, that there is not even any such resistance or obstruction excited at the working point of the machine; the whole resistance (if we can with propriety give it that name) arises from the necessity of giving motion to a quantity of inert and inactive matter. This happens in urging round a heavy fly, as in the coining press, in the punching engine, in drawing a body along a horizontal plane without friction, and a few similar cases. Here the smallest force whatever, applied at the impelled point, will begin motion in the machine; and the whole force so applied is consumed in this service. Such cases are rare, as the ultimate performance of a machine; but occasionally, and for a farther purpose, they frequently occur; and it is necessary to consider them, because there are many of the most important applications of machinery where a very considerable part of the force is expended in this part of the general task.

Such are the chief circumstances of distinction among the mechanical powers of nature which must be attended to, in order to know the motion and performance of a machine. These never occur in the statical consideration of the machine, but here they are of chief importance.

But farther: The action of the moving power is transferred to the working point through the parts of a machine, which are material, inert, and heavy. Or, to describe it more accurately, before the necessary force can be excited at the working point of the machine, the various connecting forces must be exerted in the different parts of the machine; and in order that the working point may follow out the impression already made, all the connecting parts or limbs of the machine must be moved, in different directions, and with different velocities. Force is necessary for thus changing the state of all this matter, and frequently a very considerable force. Time must also elapse before all this can be accomplished. This often consumes, and really wastes, a great part of the impelling power. Thus, in a crane worked by men walking in a wheel, it acquires motion by slow degrees; because, in order to give sufficient room for the action of the number of men or cattle that are necessary, a very capacious wheel must be employed, containing a great quantity of inert matter. All of this must be put in motion by a very moderate preponderance of the men. It accelerates slowly, and the load is raised. When it has attained the required height, all this matter, now in considerable motion, must be flopped. This cannot be done in an instant with a jolt, which would be very inconvenient, and even hurtful; it is therefore brought to rest gradually. This also consumes time; nay, the wheel must get a motion in the contrary direction, that the load may be lowered into the cart or lighter. This can only be accomplished by degrees. Then the tackle must be lowered down again for another load, which also must be done gradually. All this wastes a great deal both of time and of force, and renders a walking wheel a very improper form for the first mover of a crane, or any machine whose use requires such frequent changes of motion. The same thing obtains, although in a lower degree, in the steam engine, where the great beam and pump rods, sometimes weighing very many tons, must be made to acquire a very brisk motion in opposite directions twice in every working stroke. It obtains, in a greater or a less degree, in all engines which have a reciprocating motion in any of their parts. Pump mills are of necessity subjected to this inconvenience. In the famous engine at Marly, about 4% of the whole moving power of some of the water wheels is employed in giving a reciprocating motion to a set of rods and chains, which extend from the wheels to a cistern about three-fourths of a mile distant, where they work a set of pumps. This engine is, by such injudicious construction, a monument of magnificence, and the struggle of ignorance with the unchangeable laws of Nature. In machines, all the parts of which continue the direction of their motions unchanged, the inertia of a great mass of matter does no harm; but, on the contrary, contributes to the steadiness of the motion, in spite of small inequalities of power or resistance, or unavoidable irregularities of force in the interior parts. But in all reciprocations, it is highly prejudicial to the performance; and therefore constructions which admit such reciprocation without necessity, are avoided by all intelligent engineers. The mere copying artist, indeed, who derives all his knowledge from the common treatises of mechanics, will never suspect such imperfections, because they do not occur in the statical consideration of machines.

Lastly, no machine can move without a mutual rubbing of its parts, at all points of communication; such friction, as the teeth of wheelwork, the wipers and lifts, and the guides of its different axes. In many machines, the ultimate task performed by the working point, is either friction, or very much resembles it. This is the case in polishing mills, grinding mills, nay in boring mills, saw mills, and others. A knowledge of friction, in all its varieties, seems therefore absolutely necessary, even for a moderate acquaintance with the principles of machinery. This is a very arbitrary subject; and although a good deal of attention has been paid to it by some ingenious men, we do not think that a great deal has been added to our knowledge of it; nor do the experiments which have been made seem to us well calculated to lead us to a distinct knowledge of its nature and modifications. It has been considered chiefly with a view to diminish it as much as possible in the communicating parts. parts of machinery, and to obtain some general rules for ascertaining the quantity of what unavoidably remains. Mr Amontons, of the Royal Academy of Sciences at Paris, gave us, about the beginning of this century, the chief information that we have on the subject. He discovered, that the obstruction which it gave to motion was very nearly proportional to the force by which the rubbing surfaces are pressed together. Thus he found, that a smooth oaken board, laid on another smooth board of the same wood, requires a force nearly equal to one third of what presses the surfaces together. Different substances required different proportions.

He also found, that neither the extent of the rubbing surfaces, nor the velocity of the motion, made any considerable variation on the obstruction to motion. These were curious and unexpected results. Subsequent observations have made several corrections necessary in all these propositions. This subject will be more particularly considered in another place; but since the deviations from Mr Amontons's rule are not very considerable, at least in the cases which occur in this general consideration of machines, we shall make use of it in the meantime. It gives us a very easy method of estimating the effect of friction on machines. It is a certain proportion of the mutual pressure of the rubbing surfaces, and therefore must vary in the same proportion with this pressure. Now, we learn from the principles of statics, that whatever pressures are exerted on the impelled and working point of the machine, all the pressures on its different parts have the same constant proportion to these, and vary as these vary: Therefore the whole friction of the machine varies in the same proportion. But farther, since it is found that the friction does not sensibly change with the velocity, the force which is just sufficient to overcome the friction, and put the loaded machine in motion, must be very nearly the same with the force expended in overcoming the friction while the machine is moving with any velocity whatever, and performing work. Therefore if we deduct from the force which just puts the loaded machine in motion that part of it which balances the reaction of the impelled point occasioned by the resistance of the work, or which balances the resistance of the work, the remainder is the part of the impelling power which is employed in overcoming the friction. If indeed the actual resisting pressure of the work varies with the velocity of the working point, all the pressures, and all the frictions in the different communicating parts of the machine, vary in the same proportion. But the law of this variation of working resistance being known, the friction is again ascertained.

We can now state the dynamical equilibrium of forces in the working machine in two ways. We may either consider the efficient impelling power as diminished by all that portion which is expended in overcoming the friction, and which only prepares the machine for performing work, or we may consider the impelling power as entire, and the work as increased by the friction of the machine; that is, we may suppose the machine without friction, and that it is loaded with a quantity of additional resistance acting at the working point. Either of these methods will give the same result, and each has its advantages. We took the last method in the flight view which we took of this subject in the Encyclopædia, art. Rotation, no. 64, and shall therefore use it here.

Supposing now this previous knowledge of all these variable circumstances which affect the motion of machines of the rotative kind, so that, for any momentary position of it while performing work, we know what are the precise pressures acting at the impelled and working points, and the construction of the machine, on which depend the friction, and the momentum of its inertia (expressed in the article Rotation by $\frac{1}{2}mr^2$); we are now in a condition to determine its motion, or at least its momentary acceleration, competent to that position. Therefore,

Let there be a rotative machine, so constructed, that while it is performing work, the velocity of its impelled point is to that of its working point as $m$ to $n$. It is easy to demonstrate, from the common principles of the mechanics, that if a simple wheel and axle be substituted for it, having the radius of the wheel to that of the axle in the same proportion of $m$ to $n$, and having the same momentum of friction and inertia, and actuated by the same pressures at the impelled and working points, then the velocities of these points will be precisely the same as in the given machine.

Let $p$ represent the intensity (which may be measured by pounds weight) of the pressure exerted in the moment at the impelled point; and $r$ express the pressure exerted at the working point by the resistance opposed by the work that is then performing. This may arise from the weight of a body to be raised, from the cohesion of timber to be sawed, &c. Any of these resistances may also be measured by pounds weight; because we know, that a certain number of pounds hung on the saw of a saw mill, will just overcome this cohesion, or overcome it with any degree of superiority. Therefore the impelling power $p$, and the resistance $r$, however differing in kind, may be compared as mere pressures.

Let $x$ represent the quantity of inert matter which must be urged by the impelling power $p$, with the same velocity as the impelled point, in order that this pressure $p$ may really continue to be exerted on that point. Thus, if the impelling power is a quantity of water in the bucket of an overshot wheel, acting by its weight, this weight cannot impel the wheel except by impelling the water. In this way, $x$ may be considered as representing the inertia of the impelling power, while $p$ represents its pressure on the machine. In like manner, let $y$ represent the quantity of external inert matter which is really moved with the velocity of the working point in the execution of the task performed by the machine.

Whatever be the momentum of the inertia of the machine, we can always ascertain what quantity of matter, attached to the impelled point, or the working point of the wheel and axle, will require the same force to give the wheel the same angular motion; that is, which shall have the same momentum of inertia. Let the quantity $a$, attached to the working point, give this momentum of inertia $a'$.

Lastly, supposing that the wheel and axle have no friction, let $f$ be such a resistance, that if applied to the working point, it shall give the same obstruction as the friction of the machine, or require the same force at the impelled point to overcome it. These things being thus established, the angular velocity of the wheel and axle, that is, the number of turns, or the portion of a turn, which it will make in a given time, will be proportional to the fraction

\[ \frac{p \cdot m - r + f \cdot n}{x \cdot m^2 + a + y \cdot n^2} \]

(1.) — See Rotation, no. 64, &c.

Since the whole turns together, the velocities of the different points are as their distances from the axis, and may be expressed by multiplying the common angular velocity by these distances. Therefore the above formula, multiplied by \(m\) or \(n\), will give the velocity of the impelled or of the working point. Therefore,

Velocity of impelled point = \[ \frac{p \cdot m^2 - r + f \cdot n}{x \cdot m^2 + a + y \cdot n^2} \] (II.)

Velocity of working point = \[ \frac{p \cdot m \cdot n - r + f \cdot n^2}{x \cdot m^2 + a + y \cdot n^2} \] (III.)

In order to obtain a clear conception of these velocities, we must compare them with motions with which we are well acquainted. The proposition being universally true, we may take a case where gravity is the sole power and resistance; where, for example, \(p\) and \(r\) are the weights of the water in the bucket of a wheel, and in the tub that is raised by it. In this case, \(p = x\), and \(r = y\). We may also, for greater simplicity, suppose the machine without inertia and friction. The velocity of \(p\) is now \[ \frac{p \cdot m^2 - r \cdot m}{p \cdot m^2 + r \cdot n^2} \]

Let \(g\) be the velocity which gravity generates in a second. Then it will generate the velocity \(g \cdot i\) in the moment \(i\). Let \(v\) be the velocity generated during this moment in \(p\), connected as it is with the wheel and axle, and with \(r\). This connection produces a change of condition \(= g \cdot i - v\). For, had it fallen freely, it would have acquired the velocity \(g \cdot i\); whereas it only acquires the velocity \(v\). In like manner, had \(r\) fallen freely, it would have acquired the velocity \(g \cdot i\). But, instead of this, it is raised with the velocity \[ \frac{n}{m} \cdot v \] The change on it is therefore \(= g \cdot i + \frac{n}{m} \cdot v\). These changes of mechanical condition arise from their connection with the corporeal machine. Their pressures on it bring into action its connecting forces, and each of the two external forces is in immediate equilibrium with the force exerted by the other. The force excited at the impelled point, by \(r\) acting at the working point, may be called the momentum or energy of \(r\). These energies are precisely competent to the production of the changes which they really produce, and must therefore be conceived as having the same proportions. They are therefore equal and opposite, by the general laws observed in all actions of tangible matter; that is, they are such as balance each other. Thus, and only thus, the remaining motions are what we observe them to be.

That is, \(p \cdot g \cdot i - v \cdot m = r \cdot g \cdot i + \frac{n}{m} \cdot v\)

Or \(p \cdot m \cdot g \cdot i - p \cdot m \cdot v = r \cdot m \cdot g \cdot i + r \cdot n^2 \cdot v\)

Or \(p \cdot m^2 \cdot g \cdot i - p \cdot m^2 \cdot v = r \cdot m \cdot g \cdot i + r \cdot n^2 \cdot v\)

Or \(p \cdot m^2 \cdot r \cdot m \cdot n \cdot g \cdot i = p \cdot m^2 + r \cdot n^2 \cdot v\)

That is, the denominator of the fraction expressing the velocity of the impelled point, is to the numerator as the velocity which a heavy body would acquire in the moment \(i\), by falling freely, is to the velocity which the impelled point acquires in that moment. The same thing is true of the velocity of the working point.

This reasoning suffers no change from the more complicated nature of the general proposition. Here the impelling power is still \(p\), but the matter to be accelerated by it at the working point is \(a + y\), while its reaction, diminishing the impelling power, is only \(r\). We have only to consider, in this case, the velocity with which \(a + y\) would fall freely when impelled, not by \(a + y\), but only by \(r\). The result would be the same; \(g\) would still be to \(v\) as the denominator of the same fraction to its numerator.

Thus have we discovered the momentary acceleration of our machine. It is evident, that if the pressures \(p\) and \(r\), and the friction and inertia of the machine, and the external matter, continue the same, the acceleration will continue the same; the motion of rotation will be uniformly accelerated, and \(p \cdot m^2 + a + y \cdot n^2\) will be to \(p \cdot m^2 - r + f \cdot n\) as the space \(r\), through which a heavy body would fall in any given time \(t\), is to the space through which the impelled point will really have moved in the same time. In like manner, the space through which the working point moves in the same time is \[ \frac{p \cdot m \cdot n - r + f \cdot n^2}{p \cdot m^2 + a + y \cdot n^2} \]

Thus are the motions of the working machine determined. We may illustrate it by a very simple example. Suppose a weight \(p\) of five pounds, descending from a pulley, and dragging up another weight \(r\) of three pounds on the other side. \(m\) and \(n\) are equal, and each may be called 1. The formula becomes \[ \frac{p - r}{p + r} \] or \[ \frac{5 - 3}{5 + 3} = \frac{1}{8} \] Therefore, in a second, the weight \(p\) will descend \(\frac{1}{8}\)th of 16 feet, or 2 feet; and will acquire the velocity of 8 feet per second.

Having obtained a knowledge of the velocity of every point of the machine, we can easily ascertain its performance. This depends on a combination of the quantity of resistance that is overcome at the working point, and the velocity with which it is overcome. Thus, in raising water, it depends on the quantity (proportional to the weight) of water in the bucket or pump, and the velocity with which it is lifted up. This will be had by multiplying the third formula by \(r\), or by \(r \cdot g \cdot i\), or by \(r \cdot s\). Therefore we obtain this expression,

Work done = \[ \frac{p \cdot m \cdot r \cdot n - r + f \cdot r \cdot n^2}{p \cdot m^2 + a + y \cdot n^2} \cdot g \cdot i \] (IV.)

Such is the general expression of the momentary performance of the machine, including every circumstance which can affect it. But a variation of those circumstances produces great changes in the results. These must be distinctly noticed.

Cor. 1. If \(p \cdot m \cdot n\) be equal to \(r + f \cdot r \cdot n^2\), there will be no work done, because the numerator of the fraction is annihilated. There is then no unbalanced force, and the the natural power is only able to balance the pressure propagated from the working point to the impelled point.

2. In like manner, if \( n = 0 \), no work is done although the machine turns round. The working point has no motion. For the same reason, if \( m \) be infinitely great, although there is a great prevalence of impelling momentum, there will not be any sensible performance during a finite time. For the velocity which \( p \) can impel is a finite quantity, and the impelled point cannot move faster than \( x \) would be moved by it if detached from the machine. Now when the infinitely remote impelled point is moved through any finite space, the motion of the working point must be infinitely less, or nothing, and no work will be done.

Remark. We see that there are two values of \( n \), viz., \( v \) and \( m \times \frac{p}{r} \), which give no performance. But in all other proportions of \( m \) and \( n \) some work is done. Therefore, as we gradually vary the proportion of \( m \) to \( n \), we obtain a series of values expressing the performance, which must gradually increase from nothing, and then decrease to nothing. There must therefore be some proportion of \( m \) to \( n \), depending on the proportion of \( p \) to \( r + f \), and of \( x \) to \( a + y \), which will give the greatest possible value of the performance. And, on the other hand, if the proportion of \( m \) to \( n \) be already determined by the construction of the machine already erected, there must be some proportion of \( p \) to \( r + f \), and of \( x \) to \( a + y \), by which the greatest performance of the machine may be ensured. It is evident, that the determination of these two proportions is of the utmost importance to the improvement of machines. The well-informed reader will pardon us for endeavouring to make this appear more forcibly to those who are less instructed, by means of some very simple examples of the first principle.

Suppose that we have a stream of water affording three tons per minute, and that we want to drain a pit which receives one ton per minute, and that this is to be done by a wheel and axle? We wish to know the best proportion of their diameters \( m \) and \( n \). Let \( m \) be taken \( = 6 \); and suppose,

1. That \( n = 5 \).

Then \( \frac{p \cdot m \cdot n - r \cdot n^2}{p \cdot m^2 + r \cdot n^2} = \frac{3.6 \cdot 1.5 - 1.25}{3.36 + 1.36} = 0.4887 \)

2. Let \( n = 6 \). The formula is \( = 0.5 \).

3. Let \( n = 7 \). The formula is \( = 0.49045 \). Hence we find, that the performance is greater when \( n \) is 6, than when it is either 5 or 7.

As an example of the second principle, suppose the machine a simple pulley, and let \( p \) be 10.

1. Let \( r \) be \( = 3 \). The formula is \( = \frac{10 \times 3 - 9}{10 + 3} = \frac{21}{13} = 1.6154 \).

2. Let \( r \) be \( = 4 \). The formula is \( = \frac{10 \times 4 - 16}{10 + 4} = \frac{24}{14} = 1.7143 \).

3. Let \( r \) be \( = 5 \). The formula is \( = \frac{10 \times 5 - 25}{10 + 5} = \frac{25}{15} = 1.6666 \). Here it appears, that more work is done when \( r \) is 4 than when it is 5 or 3.

It must therefore be allowed to be one of the most important problems in practical mechanics to determine that construction by which a given power shall overcome a given resistance with the greatest advantage, and the proportion of work which should be given to a machine already constructed so as to gain a similar end.

I. The general determination of the first question has proportion but little difficulty. We must consider \( n \) as the variable magnitude in the formula \( \frac{p \cdot m \cdot n - r + f \cdot r \cdot n^2}{p \cdot m^2 + r \cdot n^2} \), which expresses the work done; and find its value when the formula is a maximum. Taking this method, we shall find that the formula IV. is a maximum when \( n \) is

\[ \frac{\sqrt{x^2 (r + f)^2 + p \cdot x \cdot (a + y)}}{p \cdot (a + y)} \]

This expression of the performance, in its best state, appears pretty complex; but it becomes much more simple in all the particular applications of it, as the circumstances of the case occur in practice.

We have obtained a value of \( n \) expressed in parts of \( m \). If we substitute this for \( n \) in the third formula, we obtain the greatest velocity with which the resistance \( r \), connected with the inertia \( y \), can be overcome by the power \( p \), connected with the inertia \( x \), by the intervention of a machine, whose momentum of inertia and friction are \( a \cdot n^2 \) and \( f \cdot n \). This is \( = \frac{r + f}{2 \cdot a + y} \times \left( \sqrt{\frac{p \cdot a + y}{r + f \cdot n}} + 1 \right) \cdot i \). This expresses the velocity of the working point in feet per second, and therefore the actual performance of the machine.

But the proper proportion of \( m \) to \( n \), ascertained by this process, varies exceedingly, according to the nature both of the impelling power, and of the work to be performed by the machine.

1. It frequently happens that the work exerts no contrary strain on the machine, and consists merely in impelling a body which resists only by its inertia. This is the case in urging round a millstone or a heavy fly; in urging a body along a horizontal plane, &c. In this case \( r \) does not enter into the formula, which now becomes \( m \times \frac{\sqrt{x^2 (a + y)} + p \cdot x \cdot (a + y)}{p \cdot (a + y)} \). If the friction be insignificant we may take \( n = m \times \frac{\sqrt{x \cdot (a + y)}}{p \cdot (a + y)} \).

The velocity of the working point is nearly \( = \frac{p}{2 \cdot \sqrt{x \cdot a + y}} \). In this case, it will be found that the velocity acquired at the end of a given time will be nearly in the proportion of the power applied to the machine.

2. On the other hand, and more frequently, the inertia of the external matter which must be moved in performing the work need not be regarded. Thus, in the grinding of grain, sawing of timber, boring of cylinders, &c., the quantity of motion communicated to the flour, to the saw dust, &c. is too insignificant to be taken into the account. In this case, \( y \) vanishes from the formula, which becomes extremely simple when the friction and inertia of the machine are inconsiderable. We shall shall not be far from the truth if we make \( m \) to \( n \) as

\[ \frac{2r}{p} \text{ or } n = m \times \frac{p}{2r + f}. \]

In this case, the velocity of the working point is

\[ \frac{f^2}{4(r+f)+a} \times \frac{p}{4(r+f)}. \]

But it is rare that machines of this kind have a small inertia. They are generally very ponderous and powerful; and the force which is necessary for generating even a very moderate motion in the unloaded machine (that is, unloaded with any work), bears a great proportion to the force necessary for overcoming the resistance opposed by the work. The formula must therefore be used in all the terms, because \( a \) is joined with \( y \). It would have been simpler in this particular, had \( a \) been joined with \( x \) in the expression of the angular velocity.

3. In some cases we need not attend to the inertia of the power, as in the steam engine. In this case, if taken strictly, \( n \) appears to have no value, because \( x \) is a factor of every term of the numerator. But the formula gives this general indication, that the more insignificant the inertia of the moving power is supposed, the larger should \( m \) be in proportion to \( n \); provided always, that the impelling power is not, by its nature, greatly diminished, by giving too great a velocity to the impelled point. This circumstance will be particularly considered afterwards.

4. If the inertia of the power and the resistance be proportional to their pressures, as when the impelling power is water lying in the buckets of an overshot wheel, and the work is the raising of water, minerals, or other heavy body, acting only by its weight; then \( p \) and \( r \) may be substituted for \( x \) and \( y \), and the formula expressing the value of \( n \), when the performance is a maximum, becomes

\[ n = m \frac{\sqrt{p^2 + f^2 + pr + ra + r}}{p \times a + r}. \]

If, in this case, the inertia and friction of the machine may be disregarded, as may often be done in pulleys, we have

\[ n = m \frac{\sqrt{p + r}}{r - 1}. \]

If we make \( m \) the unit of the radii, and \( r \) the unit of force, we have

\[ n = \sqrt{p + r} - 1, \text{ in parts of } m = 1. \]

Or, making \( p = r \), we have \( n = \sqrt{\frac{1}{r} + 1} - 1 \).

These very simple expressions are of considerable use, even in cases where the inertia of the machine is very considerable, provided that it have no reciprocating motions. A simple wheel and axle, or a train of good wheelwork, have very moderate friction. The general results, therefore, which even very unlettered readers can deduce from these simple formulas, will give notions that are useful in the cases which they cannot so thoroughly comprehend. Some service of this kind may be derived from the following little table of the belt proportions of \( m \) to \( n \), corresponding to the proportions of the power furnished to the engineer, and the resistance which must be overcome by it. The quantity \( r \) is always \( = 10 \), and \( m = 1 \).

| \( p \) | \( n \) | |-------|-------| | 1 | 0.9488 | | 2 | 0.9554 | | 3 | 0.9602 | | 4 | 0.9632 | | 5 | 0.9646 | | 6 | 0.9649 | | 7 | 0.9638 | | 8 | 0.9616 | | 9 | 0.9584 | | 10 | 0.9442 |

This must suffice for a very general view of the first problem.

II. The next question is not less momentous, namely, to determine for a machine of a given construction that portion of proportion of the resistance at the working point to the impelling power which will ensure the greatest performance of the machine; that is, the proportion of \( m \) to \( n \) being given, to find the best proportion of \( p \) to \( r \).

This is a much more complicated problem than the other; for here we have to attend to the variations both of the pressures \( p \) and \( r \), and also of the external matters \( x \) and \( y \), which are generally connected with them. It will not be sufficient therefore to treat the question by the usual fluxionary process for determining the maximum, in which \( r \) is considered as the only varying quantity. We must, in this cursory discussion, be satisfied with a comprehension of the circumstances which most generally prevail in practice.

It must either happen, that when \( r \) changes, there is no change (that is, of moment) in the mass of external matter which must be moved in performing the work, or that there is also a change in this circumstance. If no change happens, the denominator of the fourth formula, expressing the performance, remains the same; and then the formula attains a maximum when the numerator \( p \times m \times n - r + f \times r \times n^2 \) is a maximum. Also, we may include \( f \) without complicating the process, by the consideration, that \( f \) is always in nearly the same ratio to \( r \); and therefore \( r + f \) may be considered as a certain multiple of \( r \), such as \( b \times r \). We may therefore omit \( f \) in the fluxionary equations for obtaining the maximum, and then, in computing the performance, divide the whole by \( b \). Thus if the whole friction be \( \frac{1}{20} \)th of the resisting pressure \( r \), we have \( r + f = \frac{21}{20} \) of \( r \), and \( b = \frac{21}{20} \). Having ascertained the best value for \( r \), we put this in its place in the fourth formula, and take \( \frac{20}{21} \) of this for the performance. This will never differ much from the truth.

This process gives us \( p \times m \times n = 2 \times n^2 \times r \), and \( r = \frac{p \times m}{2 \times n} \); and if we further simplify the process, by making \( p = 1 \), and \( m = 1 \), we have \( r = \frac{1}{2 \times n} \); a most simple expression, directing us to make the resistance one half of what would balance the impelling power by the intervention of the machine.

This will evidently apply to many very important cases, cases, namely, to all those in which the matter put in motion by the working point is but trifling.

But it also happens in many important cases, that the change is at least equally considerable in the inertia of the work. In this case it is very difficult to obtain a general solution. But we can hardly imagine such a change, without supposing that the inertia of the work varies in the same proportion as the pressure excited by it at the working point of the machine; for since r continues the same in kind, it can rarely change but by a proportional change of the matter with which it is connected. Yet some very important cases occur where this does not happen. Such is a machine which forces water along a long main pipe. The resistance to motion and the quantity of water do not follow nearly the same ratio. But in the cases in which this ratio is observed, we may represent y by any multiple b r of r, which the case in hand gives us; b being a number, integer, or fractional. In the farther treatment of this case, we think it more convenient to free r from all other combinations; and instead of supposing the force f (which we made equivalent with the friction of the machine) to be applied at the working point, we may apply it at the impelled point, making the effective power q = p - f. For the same reasons, instead of making the momentum of the machine's inertia = am², we may make it am³, and make a + n = s. Now, supposing q, or p - f, = 1, and also m = 1, our formula expressing the performance becomes

\[ r = \frac{\sqrt{a^2 + b^2}}{b} \]

This is a maximum when

\[ r = \frac{\sqrt{a^2 + b^2}}{b} \]

Cor. 1. If the inertia of the work is always equal to its pressure, as when the work consists wholly in raising a weight, such as drawing water, &c., then b = 1, and the formula for the maximum performance becomes

\[ r = \frac{\sqrt{a^2 + 1}}{n} \]

2. If the inertia of the impelling power is also the same with its pressure, and if we may neglect the inertia and friction of the machine, the formula becomes

\[ r = \frac{\sqrt{a^2 + 1}}{n} \]

Example. Let the machine be a common pulley, so that the radii m and n are equal, and therefore n = 1. Then, \( r = \frac{\sqrt{1 + 1}}{1} = \sqrt{2} - 1 = 0.4142 \)

&c. more than 3ths of what would balance it.

Here follows a series of the best values of r, corresponding to different values of n. m and p are each = 1. The numbers in the last column have the same proportion to 1 which r has to the resistance which will balance p.

| n | r | p | |---|-----|-----| | 1 | 1.8885 | 0.4724 | | 2 | 1.3928 | 0.4639 | | 3 | 0.8086 | 0.4493 | | 4 | 0.4142 | 0.4142 | | 5 | 0.1830 | 0.3660 | | 6 | 0.1111 | 0.3333 | | 7 | 0.0772 | 0.3030 |

From what has now been established, we see with sufficient evidence the importance of the higher mathematics to the science of mechanics. If the velocities of the impelled and working points of an engine are not properly adjusted to the pressures, the inertia, and the friction of the machine, we do not derive all the advantages which we might from our situation. Hence also we learn the fallacy of the maxim which has been received as well founded, that the augmentation of intensity of any force, by applying it to the long arm of General a lever, is always fully compensated by a loss of time; but erroneous maxim, as it is usually expressed, "what we gain by a machine in force we lose in time." If the proportion of m to n is well chosen, we shall find that the work done, when it refills by its inertia only, increases nearly in the proportion of the power employed; whereas when the inertia of the work is but a small part of the resistance, it increases nearly in the duplicate ratio of the power employed.

It was remarked, in the setting out in the present problem, that the formula do not immediately express the velocity of any point of the machine, but its momentary acceleration. But this is enough for our purpose; because, when the momentary acceleration is a maximum, the velocity acquired, and the space described, in any given time, is also a maximum. We also showed how the real velocities, and the spaces described, may be ascertained in known measures. We may say in general, that if g represent the pressure of gravity on any mass of matter w, then \( \frac{g}{w} \) is to \( \frac{p}{m} \) as \( \frac{r}{n} \) to \( \frac{a}{m^2 + a + r} \).

16 feet to the space described in a second by the working point in a second, or as 32 feet per second is to the velocity acquired in that time.

A remark now remains to be made, which is of the cause why greatest consequence, and gives an unexpected turn to machines the whole of the preceding doctrines. It appears, from all that has been said, that the motion of a machine must be uniformly accelerated, and that any point will describe spaces proportional to the squares of the times; for while the pressures, friction, and momentum of inertia remain the same, the momentary acceleration must also be invariable. But this seems contrary to all experience. Such machines as are properly constructed, and work without jolts, are observed to quicken their pace for a few seconds after starting; but all of them, in a very moderate time, acquire a motion that is sensibly uniform. Is our theory erroneous, or what are the circumstances which remain to be considered, in order to make it agree with observation? The science of machines is imperfect, till we have explained the causes of this deviation from the theory of uniform acceleration.

These causes are various.

1. In some cases, every increase of velocity of the machine produces an increase of friction in all its communicating parts. By these means, the accelerating force, which is \( p = r + f \), or \( p = f - m - r \), is diminished, and consequently the acceleration is diminished. But it seldom happens that friction takes away or employs the whole accelerating force. We are not yet well instructed in the nature of friction. Most of the kinds of friction which obtain in the communicating parts of machines, are such as do not sensibly increase by an increase of velocity; some of them really diminish. Yet even the most accurately constructed machines, unloaded with work, attain a motion that is sensibly uniform. If we take off the pallets from a pendulum...

Gulum clock, and allow it to run down again, it accelerates for a while, but in a very moderate time it acquires an uniform motion. So does a common kitchen jack. These two machines seem to bid the fairest of any for an uniformly accelerated motion; for their impelling power acts with the utmost uniformity. There is something yet unexplained in the nature of friction, which takes away some of this acceleration.

But the chief cause of its cessation in these two instances, and others of every rapid motion, is the resistance of the air. This arises from the motion which is communicated to the air displaced by the swift moving parts of the machine. At first it is very small; but it increases nearly in the duplicate ratio of the velocity (see Resistance of Fluids, Encycl.) Thus \( r \) increases continually; and, in a certain state of motion, \( r + f_n \) becomes equal to \( p_m \). Whenever this happens, the accelerating power is at an end. The acceleration also ceases; and the machine is in a state of dynamical equilibrium; not at rest, but moving uniformly, and performing work.

Still, however, this is not one of the general causes of the uniform motion attained by working engines. Rarely is the motion of their parts so rapid, as to occasion any great resistance from the air. But in the most frequent employments of machines, every increase of velocity is accompanied by an increase of resistance from the work performed. This occurs at once to the imagination; and few persons think of inquiring farther for a reason. But there is perhaps no part of mechanics that is more imperfectly understood, even in our present improved state of mechanical science. In many kinds of work, it is very difficult to state what increase of labour is required in order to perform the work with twice or thrice the speed. In grinding corn, for instance, we are almost entirely ignorant of this matter. It is very certain, that twice the force is not necessary for making the mill grind twice as fast, nor even for making it grind twice as much grain equally well. It is not easy to bring this operation under mathematical treatment; but we have considered it with some attention, and we imagine that a very great improvement may still be made in the construction of grist mills, founded on the law of variation of the resistance to the operation of grinding, and a scientific adjustment of \( m \) to \( n \), in consequence of our knowledge of this law. We may make a similar observation on many other kinds of work performed by machines. In none of those works where the inertia of the work is inconsiderable, are we well acquainted with the real mechanical process in performing it. This is the case in sawing mills, boring mills, rolling mills, fluting mills, and many others, where the work consists in overcoming the strong cohesion of a small quantity of matter. In sawing timber (which is the most easily understood of all these operations), if the saw move with a double velocity, it is very difficult to say how much the actual resisting pressure on the teeth of the saw is increased. Twice the number of fibres are necessarily torn asunder during the same time, because the same number are torn by one deflection of the saw, and it makes that stroke in half the time. But it is very uncertain whether the resistance is double on this account; because if each fibre be supposed to have the same tenacity in both cases, it resists with this tenacity only for half the time. The parts of bodies resist a similar change of condition in different manners; and there is another difference in their resistance of different changes—the resistance of red hot iron under the roller may vary at a very different rate from that of its resistance to the cutting tool. The resistance of the spindles of a cotton mill, arising partly from friction, partly from the inertia of the heaped bobbins, and partly from the resistance of the air, is still more complicated, and it may be difficult to learn its law. The only case in which we can judge with some precision is, when the inertia of matter, or a constant pressure like that of gravity, constitutes the chief resistance. Thus in a mill employed to raise water by a chain of buckets, the resistance proceeds from the inertia only of the water. The buckets are moving with a certain velocity, and the lowest of them takes hold of a quantity of water lying at rest in the pit, and drags it into motion with its acquired velocity. The force required for generating this motion on the quiescent water must be double or triple, when the velocity that must be given to it is so. This absorbs the surplus of the impelling power, by which that power exceeds what is necessary for balancing the weight of the water contained in all the ascending buckets. This is a certain determinate quantity which does not change; for in the same instant that a new bucket of water is forced into motion below, and its weight added to that of the ascending buckets, an equal bucket is emptied of its water at top. The ascending buckets require only to be balanced, and they then continue to ascend, with any velocity already acquired. While the machine moves slow, the motion impressed on the new bucket of water is not sufficient to absorb all the surplus of impelling power. The quantity not absorbed accelerates the machine, and the next bucket must produce more motion in the water which it takes up. This consumes more of the surplus. This goes on till no surplus of power is left, and the machine accelerates no more. The complete performance of the machine now is, that "a certain quantity of water, formerly at rest, is now moving with a certain velocity." Our engineers consider it differently; "as a certain weight of water lifted up." But while the machine is thus moving uniformly, it is really not doing so much as before; that is, it is not exerting such great pressures as before the motion was rendered uniform; for at that time there was a pressure at the working point equal to the weight of all the water in the ascending buckets; and also an surplus of pressure, by which the whole was accelerated. In the state of uniform motion, the pressure is no more than just balances the weight of the ascending chain. We shall learn by and bye how the pressures have been diminishing, although the mill has been accelerating; a thing that seems a paradox.

In this instance, then, we see clearly, why a machine must attain a uniform motion. A pumping machine gives us the same opportunity, but in a manner so different as to require explanation. The piston may be supposed at the very surface of the pit water, and the impelling power may be less than will support a column in the pipe as high as can be raised by the pressure of the atmosphere. Suppose the impelling power to be the water lying in the buckets of an overshot wheel. Let this water be laid into the buckets by a very small stream. It will fill the buckets very slowly; and as this gives them a preponderance, the mill loses its balance, the wheel begins to move, and the piton to rise, and the water to follow it. The water may be delivered on the wheel drop by drop; the piton will rise by insensible degrees, always standing still again as soon as the atmospheric pressure on it just balances the water on the wheel. The water in the rising pipe is always a balance to the pressure of the atmosphere on the cistern; therefore the pressure of the atmosphere on the piton (which is the \( r \) in our formula) is equal to the weight of this water. Our pump-makers therefore (calling themselves engineers) say, that the weight of water in the pipe balances the water on the wheel. It does not balance it, nor is it raised by the wheel, but by the atmosphere; but it serves us at present for a measure of the power of the wheel. At last, all the buckets of the wheel are full, and the water is (for example) 25 feet high in the pipe. Now let the stream of water run its full quantity. It will only run over from bucket to bucket, and run off at the bottom of the wheel; but the mill will not move, and no work will be performed. (N.B. We are here excluding all impulse or stroke on the buckets, and supposing the water to act only by its weight.) But now let all be emptied again, and let the water be delivered on the wheel in its full quantity at the first. The wheel will immediately acquire a preponderance, which will greatly exceed the first small pressure of the atmosphere on the piton. It will therefore accelerate the piton, overcoming the pressure of the air with great velocity. The piton rises fast; the water follows it, by the pressure of the atmosphere; and when it attains the former utmost height, it attains it with a considerable velocity. If allowed to run off there, it will continue to run off with that velocity; because there is the same quantity of water pressing round the wheel as before, and therefore enough to balance the pressure of the atmosphere on the piton. The pressure of the same atmosphere on the water in the cistern, raised the water in the pipe with this velocity; therefore it will continue to do so, and the mill will deliver water by the pump with this velocity, although there is no more pressure acting on it than before, when the water ran to waste, doing no work whatever.

This mode of action is extremely different from the former example. The mill is not acting against the inertia of the water to be moved, but against the pressure of the atmosphere on the piton. The pressure of the same atmosphere on the cistern is employed against the inertia of the water in the pipe; and the use of the mill is to give occasion, by raising the piton, to the exertion of this atmospheric pressure, which is the real raiser of the water. The maxim of construction, and the proper adjustment of \( m \) to \( n \) in this case, are different from the former; and we should run the risk of making an imperfect engine were we to confound them.

We must mention another case of a pumping mill, seemingly the same with this, but essentially different. Suppose the pipe of this pump to reach 30 feet below the surface of the pit water, and that the piton is at the very bottom of it. Suppose also, that the wheel buckets, when filled with water, only enable it to sup-

Suppl. Vol. II. Part I.

The draught may be increased till he is reduced to a trot, to a walk, nay, till he is unable to draw it. Now, just inverting this process, we see, that there is a certain strain which will sufficiently tire the horse without stirring from the spot, but which he could continue to exert for hours. This is greater than the load that he can just crawl along with, employing his strength as much as would be prudent to continue from day to day. And, in like manner, every lesser draught has a corresponding rate, at which the horse, employing his whole working strength, can continue to draw during the working hours of a day. At setting out, he pulls harder, and accelerates it.* Following his pull, he walks faster, and therefore pulls less (because we are still supposing him to employ his whole working strength). At last he attains that speed which occupies his whole strength in merely continuing the pull. Other animals act in a similar manner; and it becomes a general rule, that the pressure actually exerted on the impelled point of a machine diminishes as its velocity increases.

From the concurrence of so many facts, we perceive we must that we must be careful to distinguish between the quantity of power expended, and the quantity that is used between the power fully employed, which must be measured solely by the pressure exerted on the machine. When a weight of five pounds is employed to drag up a weight of three pounds by means of a thread over a pulley, it descends, with a motion uniformly accelerated, four feet in the first second. Mr Smeaton would call this an expenditure of a mechanical power 20. The weight three pounds is raised four feet. Mr Smeaton would call this a mechanical effect 12. Therefore the effect produced is not adequate to the power expended. But the fact is, that the pressure, strain, or mechanical power really exerted in this experiment, is neither five nor three pounds; the five pound weight would have fallen 16 feet, but it falls only 4. A force has therefore acted on it sufficient to make it describe 12 feet in a second, with a uniformly accelerated motion; for it has communicated so much of its weight. The thread was strained with a force equal to 3½ pounds, or ¾ of 5 pounds. In like manner, the three pound weight would have fallen 16 feet; but it was raised 4 feet. Here was a change precisely equal to the other. A force of 3½ pounds, acting on a mass whose matter is only 3, will, in a second, cause it to describe 12 feet with a uniformly accelerated motion. Now, \(5 \times 12\), and \(3 \times 20\), give the same product 60. And thus we see, that the quantity of motion extinguished or produced, and not the product of the weight and height, is the true unequivocal measure of mechanical power really expended, or the mechanical effect really produced; and that these two are always equal and opposite. At the same time, Mr Smeaton's theorem merits the attention of engineers; because it generally measures the opportunities that we have for procuring the exertion of power. In some sense Mr Smeaton may say, that the quantity of water multiplied by the height from which it descends in working our machines, is the measure of the power expended; because we must raise this quantity to the dam again, in order to have the same use of it. It is expended, but not employed; for the water, at leaving the wheel, is still able to do something.

It requires but little consideration to be sensible, that The preceding account of the cessation of accelerated motion in our principal machines, must introduce different maxims of construction from those which were expressly adapted to this acceleration; or rather, which proceeded on the erroneous supposition of the constancy of the impelling power and the resistance. The examination of this point has brought into view the fundamental principle of working machines, namely, the perfect equilibrium which takes place between the impelling power and the simultaneous resistance. It may be expressed thus:

The force required for preserving a machine in uniform motion, with any velocity whatever, is that which is necessary for balancing the resistance then actually exerted on the working point of the machine. We saw this distinctly in the influence of the two weights acting against each other by the intervention of a thread over a fixed pulley. It is equally true of every case of acting machinery; for if the force at the impelled point be greater than what balances the resistance acting at the same point, it must accelerate that point, and therefore accelerate the whole machine; and if the impelling force be less than this, the machine must immediately retard its motion. When the machine has once acquired this degree of motion, every part of it will continue in its present state of motion, if only the two external forces are in equilibrium, but not otherwise. But when the pressure of the external power on the impelled point balances the resistance opposed by that point, it is, in fact, maintaining the equilibrium with the external power acting at the working point; for this is the only way that external forces can be set in opposition to each other by the intervention of a body. The external forces are not in immediate equilibrium with each other, but each is in equilibrium with the force exerted by the point on which it acts. This force exerted by the point is a modification of the connecting forces of the body, all of which are brought into action by means of the actions of the external forces, and each is accompanied by a force precisely equal and opposite to it.

Now, the principles of statics teach us the proportions of the external pressures which are thus set in equilibrium by the intervention of a body; and therefore teach us what proportion of power and resistance will keep a machine of a given construction in a state of uniform motion.

This proposition appears paradoxical, and contrary to common observation; for we find, that, in order to make a mill go faster, we must either diminish the resistance, or we must employ more men, or more water, or water moving with greater velocity, &c. But this arises from some of the causes already mentioned. Either the resistance of the work is greater when the machine is made to move faster, or the impulsion of the power is diminished, or both these changes obtain. Friction and resistance of air also come in for their share, &c. The actual pressure of a given quantity of the external power is diminished, and therefore more of it must be employed. When a weight is uniformly raised by a machine, the pressure exerted on it by the working point is precisely equal to its weight, whatever be the velocity with which it rises. But, even in this simplest case, more natural power must be expended in order to raise it faster; because either more natural power must be employed to accelerate the external matter which is to press forward the impelled point, or the relative motion of the pressing matter will be diminished.

It is well known, that, in the employment of the mechanic powers, whether in their state of greatest simplicity, or any how combined in a complicated machine, if the machine be put in motion, the velocities of the extreme points (which we have called the impelled and working points) are inversely proportional to the forces which are in equilibrium when applied to these points in the direction of their motion. This is an inductive proposition, and has been used as the foundation of systems of mechanics. It is unnecessary to take up time in proving what is so familiarly known; consequently, the products of the pressures at these points by the velocities of the motions are equal; that is, the product of the pressure actually exerted at the impelled point of a machine working uniformly, multiplied by the velocity of that point, is equal to the product of the resistance actually exerted at the working point, multiplied by the velocity of that point, that is, by the velocity with which the resistance is overcome,

\[ p \cdot m = r \cdot n. \]

Now, the product of the resistance, by the velocity with which it is overcome, is evidently the measure of the performance of the machine, or the work done. The product of the actual pressure on the impelled point, by the velocity of that point, may be called the momentum of impulse.

Hence we deduce this proposition:

In all working machines which have acquired a uniform motion, the performance of the machine is equal to the momentum of impulse (a).

---

(a) The truth of this proposition has been long perceived in every particular instance that happened to engage equal attention; but we do not recollect any mechanician before Mr Euler considering it as a general truth, expressing in a few words a mechanical law. This celebrated mathematician undertook, about the year 1735 or 1736, a general and systematic view of machines, in order to found a complete theory immediately conducive to the improvement of practical mechanics. In 1743 he published the first propositions of this useful theory in the 10th volume of the Comment. Petropolitani, containing the excellent dynamical theorems of which we have given the substance. In the 3d volume of the Comment. Novi Petropol. he prosecuted the subject a little farther; and in the 8th volume, he entered on what we are now engaged in, and formally announces this fundamental proposition, calling these two products the momentum of impulse, and the momentum of effect. It is much to be regretted, that this consummate mathematician did not continue these useful labours; his ardent mind being carried away by more abstruse speculations in all the most refined departments of mathematics and philosophy. No man in Europe could have prosecuted the subject with more judgment and success.—See also Mem. Acad. Berlin, 1747 and 1752. This is a proposition of the utmost importance in the science of machines, and leads to the fundamental maxim of their construction. Since the performance of a machine is equal to the momentum of impulse, it increases and diminishes along with it, and is a maximum when the momentum of impulse is a maximum; therefore, the fundamental maxim in the construction of a machine is to fashion it in such a manner, that the momentum of impulse shall be a maximum, or that the product of the pressure actually exerted on the impelled point of the machine by the velocity with which it moves may be as great as possible. Then are we certain that the product of the resistance, by the velocity of the working point, is as great as possible, provided that we take care that none of the impulse be needlessly wasted by the way by injudicious communications of motion, by friction, by unbalanced loads, and by reciprocal motions, which irrecoverably waste the impelling power. This maxim holds good, whether the resistance remains constantly the same, or varies by any law whatever.

But much remains to be done for the improvement of mechanical science before we can avail ourselves of this maxim, and apply it with success. The chief thing, and to this we should give the most unremitting attention, is, to learn the changes which obtain in the actual pressure exerted by those natural powers which we can command; the changes of actual pressure produced by a change of the velocity of the impelled point of the machine. These depend on the specific natures of those powers, and are different in almost every different case. Nothing will more contribute to the improvement of practical mechanics than a series of experiments, well contrived, and accurately made, for discovering those laws of variation, in the cases of those powers which are most frequently employed. Such experiments, however, would be costly, beyond the abilities of an individual; therefore, it were greatly to be wished that public aid were given to some persons of skill in the science to institute a regular train of experiments of this kind. An experimental machine might be constructed, to be wrought either by men or by cattle. This should be loaded with some kind of work which can be very accurately measured, and the load varied at pleasure. When loaded to a certain degree, the men or cattle should be made to work at the rate which they can continue from day to day. The number of turns made in an hour, multiplied by the load, will give the performance corresponding to the velocities; and thus will be discovered the most advantageous rate of motion. The same machine should also be fitted for grinding, for sawing, boring, &c., and similar experiments will discover the relation between the velocities with which these operations are performed, and the resistances which they exert. The laws of friction may be investigated by the same machine. It should also be fitted with a walking wheel, and the trial should be made of the slope and the velocity of walking which gives the greatest momentum of impulse. It is not unreasonable to expect great advantages from such a train of experiments.

Till this be done, we must content ourselves with establishing the above, in the most general terms, applicable to any case in which the law of the variation of force may hereafter be discovered.

There is a certain velocity of the impelled point of a machine which puts an end to the action of the moving power. Thus, if the floats of an undershot wheel be moving with the velocity of the stream, no impulse is made on them. If the arm of a gin or capstan be moving with that velocity with which a horse or a man can just move, so as to continue at that speed from day to day, employing all his working strength, but not fatiguing himself in this state of motion, the animal can exert no pressure on the machine. This may be called the extinguishing velocity, and we may express it by the symbol \( e \). Let \( f \) be that degree of force or pressure which the animal can exert at a dead pull or thrust, as it is called. We do not mean the utmost strain of which the animal is capable, but that which it can continue unremittingly during the working hours of a day, fully employing, but not fatiguing itself. And let \( p \) be the pressure which it actually exerts on the impelled point of a machine, moving with the velocity \( m \).

Let \( e - m \) be called the relative velocity, and let it be expressed by \( v \). And let it be supposed, that it has been discovered, by any means whatever, that the actual pressure varies in the proportion of \( v^q \), or \( e - m = v^q \). This supposition gives us \( e^q : v^q = f : p \), and \( p = f \times \frac{v^q}{e^q} \). For the machine must be at rest, in order that the agent may be able to exert the force \( f \) on its impelled point. But when the machine is at rest, what we have named the relative velocity is \( e \), the whole of the extinguishing velocity.

The momentum of impulse is \( p \times m \), that is \( \frac{v^q}{e^q} \times f \times m \), or \( f \times \frac{v^q}{e^q} \times e - v \) (because \( m = e - v \)). Therefore \( f \times \frac{v^q}{e^q} \times e - v \) must be made a maximum. But \( f \) and \( e \) are two quantities which suffer no change. Therefore the momentum of impulse will be a maximum when \( v^q \times e - v \) is a maximum. Now \( v^q \times e - v = v^q e - v^q v = v^q e - v^{q+1} \). The fluxion of this is \( q \times v^{q-1} \times e - q + 1 \times v^q \). This being supposed \( = 0 \), we have the equation

\[ q \times v^{q-1} = q + 1 \times v^q \]

And \( q \times v = q + 1 \times v^q \)

Therefore \( v = \frac{q}{q + 1} \)

And \( m \), which is \( e - v \), becomes \( \frac{e}{q + 1} \). Therefore we must order matters so, that the velocity of the impelled point of the machine may be \( \frac{e}{q + 1} \). Now \( p \) is \( f \times \frac{v^q}{e^q} \), and therefore \( f \times \frac{q^q}{q + 1} \). And \( p \times m = f \times \frac{q^q}{q + 1} \times \frac{e}{q + 1} = f \times \frac{q^q e}{q + 1} \), the momentum of impulse, and therefore = the momentum of effect, or the performance of the machine, when in its best state.

Thus may the maxim of construction be said to be brought to a state of great simplicity, and of most easy recollection. A particular case of this maxim has been long known, having been pointed out by Mr Parent. Since the action of bodies depends on their relative velocity,

Locality, the impulse of fluids must be as the square of the relative velocity. From which Mr Parent deduced, that the most advantageous velocity of the floats of an undershot wheel is one-third of that of the stream. This maxim is evidently included in our general proposition; for in this case, the index \( q \) of that function of the relative velocity \( v \), which is proportional to the impulse, is \( \frac{2}{3} \). Therefore we have the maximum when \( v = \frac{2}{3} e \), and \( m = \frac{1}{3} e \), the extinguishing velocity, is evidently the velocity of the stream. Our proposition also gives us the precise value of the performance. The impulse of the stream on the float at rest being supposed \( f \); its impulse on the float moving with the velocity \( \frac{1}{3} e \) must be \( \frac{4}{9} f \). This is the measure of the actual pressure \( p \). This being multiplied by \( m \), or by \( \frac{1}{3} e \), gives \( \frac{4}{27} f \). Now \( f \) is considered as equal to the weight of a column of water, having the surface of the floatboard for its base, and the depth of the sluice under the surface of the reservoir (or, more accurately, the fall required for generating the velocity of the stream) for its height. Hence it has been concluded, that the utmost performance of an undershot wheel is to raise \( \frac{4}{27} \) of the water which impels it, to the height from which it falls. But this is not found very agreeable to observation. Friction, and many imperfections of execution in the delivery of the water, the direction of its impulse, &c. may be expected to make a deflection from this theoretical performance. But the actual performance, even of mills of acknowledged imperfection, considerably exceeds this, and sometimes is found nearly double of this quantity. The truth is, that the particular fact from which Mr Parent first deduced this maxim (namely, the performance of what is called Parent's or Dr Barker's mill), is, perhaps of all that could have been selected, the least calculated for being the foundation of a general rule, being of a nature to abstruse, that the first mathematicians of Europe are to this day doubtful whether they have a just conception of its principles. Mr Smeaton's experiments show very distinctly, that the maximum of performance of an undershot wheel corresponds to a velocity considerably greater than one-third of the stream, and approaches nearly to one-half; and he affirms some reasons for this which seem well founded. But, independent of this, the performance of Mr Smeaton's model was much greater than what corresponds with the velocity by the above mentioned estimation of \( f \). The theory of the impulsion of fluids is extremely imperfect; and Daniel Bernoulli shews, from very unquestionable principles, that the impulse of a narrow vein of fluid on an extended surface is double of what was generally supposed; and his conclusions are abundantly confirmed by the experiments adduced by him.

It is by no means pretended, that the maxim of construction is reduced to the great simplicity enounced in the proposition now under consideration. We only suppose, that a case had been observed where the pressure exerted by some natural agent did follow the proportions of \( v^q \). This being admitted, the proposition is strictly true. But we do not know any such case; yet is the proposition of considerable use: for we can affirm, on the authority of our own observations, that the action both of men and of draught horses does not deviate very far from the proportions of \( v^q \). The observations were made on men and horses tracking a lighter along a canal, and working several days together, without having any knowledge of the purpose of the observations. The force exerted was first measured by the curvature and weight of the track rope, and afterwards by a spring fleckyard. This was multiplied by the number of yards per hour, and the product considered as the momentum. We found the action of men to be very nearly as \( e - m^2 \). The action of horses, loaded so as not to be able to trot, was nearly as \( e - m^6 \).

The practitioner can easily avail himself of the maxim, although the function \( q \) should never be reduced to any algebraic form. He has only to institute a train of experiments on the natural agent, and select that velocity which gives the highest product when multiplied by its corresponding pressure.

When this selection has been made, we have two ways of giving our working machines the maximum of their effect, having once ascertained the pressure \( f \) which our natural power exerts on the impelled point of the machine when it is not allowed to move.

1. When the resistance arising from the work, and from friction, is a given quantity; as when water is to be raised to a certain height by a piston of given dimensions.

Since the friction in all the communicating parts of the machine vary in the same proportion with the pressure, and since these vary in the same proportion with the resistance, the sum of the resistance and friction may be represented by \( b r \), \( b \) being an abstract number. Let \( n \) be the undetermined velocity of the working point; or let \( m : n \) be the proportion of velocities at the impelled and working points. Then, because the pressures at these points balance each other, in the case of uniform motion, they are inversely as the velocities at those points. Therefore we must make \( b r : p = m : n \),

\[ \frac{q^2}{q + 1} \times b r = g^2 f, \]

and \( n = \frac{b r}{q + 1} \times \frac{g^2 f}{q + 1} \), or \( m : n = \frac{q^2 f}{q + 1} \times b r : g^2 f \).

2. On the other hand, when \( m : n \) is already given, by the construction of the machine, but \( b r \) is susceptible of variation, we must load the machine with more and more work, till we have reduced the velocity of its impelled point to \( \frac{e}{q + 1} \).

In either case, the performance is expressed by what expresses \( p m \), that is, by \( f e \times \frac{q^2}{q + 1} \). But the useful performance, which is really the work done, will be had by dividing the value now obtained by the number \( b \), which expresses the sum of the resistance overcome by the working point and the friction of the machine.

What has been now delivered contains, we imagine, the chief principles of the theory of machines, and points out the way in which we must proceed in applying them to every case. The reader, we hope, sees clearly the imperfection of a consideration. tion of machines which proceeds no farther than the statement of the proportions of the simultaneous pressures which are excited in all the parts of the machine by the application of the external forces, which we are accustomed to call the power and the weight. Unless we take also into consideration, the immediate effect of mechanical force applied to body, and combine this with all the pressures which statical principles have enabled us to ascertain, and by this combination be able to say what portion of unbalanced force there is acting at one and all of the pressing points of the machine, and what will be the motion of every part of it in consequence of this overplus, we have acquired no knowledge that can be of service to us. We have been contemplating, not a working machine, but a fort of balance. But, by reasoning about these unbalanced forces in the same simple manner as about the fall of heavy bodies, we were able to discover the momentary accelerations of every part, and the sensible motion which it would acquire in any assigned time, if all the circumstances remain the same. We found that the results, although deduced from unquestionable principles, were quite unlike the observed motions of most working machines. Proceeding still on the same principles, we considered this deviation as the indication, and the precise measure, of something which we had not yet attended to, but which the deviation brought into view, and enabled us to ascertain with accuracy. These are the changes which happen in the exertions of our actuating powers by the velocity with which we find it convenient to make them act. Thus we learn more of the nature of those powers; and we found it necessary to distinguish carefully between the apparent magnitude of our actuating power and its real exertion in doing our work. This consideration led us to a fundamental proposition concerning all working machines when they have attained an uniform motion; namely, that the power and resistance then really exerted on the machine precisely balance each other, and that the machine is precisely in the condition of a steelyard loaded with its balanced weights, and moved round its axis by some external force distinct from the power and the weight. We found that this force is the previous overplus of impelling power, before the machine had acquired the uniform motion; and on this occasion we learned to estimate the effect produced, by the momentum (depending on the form of the machine) of the quantity of motion produced in the whole assemblage of power, resistance, and machine.

The theory of machines seemed to be now brought back to that simplicity of equilibrium which we had said was so imperfect a foundation for a theory; but in the availing ourselves of the maxim founded on this general proposition, we saw that the equilibrium is of a very different kind from a quiescent equilibrium. It necessarily involves in it the knowledge of the momentary accelerations and their momenta; without which we should not perceive that one state of motion is more advantageous than another, because all give us the same proportion of forces in equilibrium.

But this is not the only use of the previous knowledge of the momentary accelerations of machines; there are many cases where the machine works in this very state. Many machines accelerate throughout while performing their work; and their efficacy depends entirely on the final acceleration. Of this kind is the coining press, the great forge or tilt mill, and some other capital engines. The steam engine, and the common pump, are necessarily of this class, although their efficacy is not eliminated by their final acceleration. A great number of engines have reciprocating motions in different subordinate parts. The theory of all such engines requires for its perfection an accurate knowledge of the momentary accelerations; and we must use the formulæ contained in the first part of this article.

Still, however, the application of this knowledge has many difficulties, which make a good theory of such machines a much more intricate and complicated matter than we have yet led the reader to suppose. In most of these engines, the whole motion may be divided into two parts. One may be called the working stroke, and the other in which the working points are brought back to a situation which fits them for acting again, may be called the returning stroke. This return must be effected either by means of some immediate application of the actuating power, or by some other force, which is counteracted during the working stroke, and must be considered as making part of the resistance. In the steam engine, it is generally done by a counterpoise on the outer end of the great working beam. This must be accounted a part of the resistance, for it must be raised again; and the proportions of the machine for attaining the maximum must be computed accordingly. The quantity of this counterpoise must be adjusted by other considerations. It must be such, that the descent of the pump rods in the pit may just employ the whole time that is necessary for filling the cylinder with steam. If they descend more briskly (which an unskilful engineer likes to see), this must be done by means of a greater counterpoise, and this employs more power to raise it again. Desaguliers describes a very excellent machine for raising water in a bucket by a man's stepping into an opposite bucket, and defending by his preponderancy. When he comes to the bottom, he steps out, goes up a flight, and finds the bucket returned ready to receive him again. This machine is extremely simple, and perhaps the best that can be contrived; and yet it is one of the most likely to be a very bad one. The bucket into which the man steps must be brought up to its place again by a preponderancy in the machine when unloaded. It may be returned sooner or later. It should arrive precisely at the same time with the man. If sooner, it is of no use, and wastes power in raising a counterpoise which is needlessly heavy; if later, time is lost. Therefore, the perfection of this very simple machine requires the judicious combination of two maximums, each of which varies in a ratio compounded of two other ratios. Suppose the man to employ a minute to go up stairs 50 feet, which is very nearly what he can do from day to day as his only work, and suppose him to weigh 150 pounds, and that he acts by means of a simple pulley—the maximum for a lever of equal arms would require him to raise about 60 pounds of water. But when all the other circumstances are calculated, it will be found that he must raise 138 pounds (neglecting the inertia of the machine). He should raise 542 pounds to feet in a minute; and this is nearly the most exact valuation of a man's work.

There is the same necessity of attending to a variety of of circumstances in all machines which reciprocate in the whole or any considerable part of their motion. The force employed for bringing the machine into another working position must be regulated by the time necessary for obtaining a new supply of power; and then the proportion of m to n must be so adjusted, that the work performed, divided by the whole time of the working and returning strokes, may give the greatest quotient. It is still a difficult thing, therefore, to construct a machine in the most perfect manner, or even to say what will be the performance of a machine already constructed; yet we see that every circumstance is susceptible of accurate computation.

With respect to machines which acquire a sort of uniform motion in general, although subject to partial reciprocations, as in a pumping, stamping, forging engine, it is also difficult to assign the rate even of this general uniform motion. We may, however, say, that it will not be greater than if it were uniform throughout. Were it entirely free from friction, it would be exactly the same as if uniform; because the accelerations during the advantageous situations of the impelling power would compensate the retardations. But friction diminishes the accelerations, without diminishing the retardations.

We may conclude this article with some observations tending to the general improvement of machines.

Nothing contributes more to the perfection of a machine, especially such as is massive and ponderous, than great uniformity of motion. Every irregularity of motion wastes some of the impelling power; and it is only the greatest of the varying velocities which is equal to that which the machine would acquire if moving uniformly throughout; for while the motion accelerates, the impelling force is greater than what balances the resistance then actually opposed to it, and the velocity is less than what the machine would acquire if moving uniformly; and when the machine attains its greatest velocity, it attains it because the power is then not acting against the whole resistance. In both of these situations, therefore, the performance of the machine is less than if the power and resistance were exactly balanced; in which case it would move uniformly.

Every attention should therefore be given to this, and we should endeavour to remove all cause of irregularity. The communications of motion should be so contrived, that if the impelled point be moving uniformly, by the uniform pressure of the power, the working point shall also be moving uniformly. Then we may generally be certain, that the massy parts of the machine will be moving uniformly. When this is not done through the whole machine, there are continual returns of strains and jolts; the inertia of the different parts acting in opposite directions. Although the whole momenta may always balance each other, yet the general motion is hobbling, and the points of support are strained. A great engine so constructed, commonly causes the building to tremble; but when uniform motion pervades the whole machine, the inertia of each part tends to preserve this uniformity, and all goes smoothly. It is also deserving of remark, that when the communications are so contrived that the uniform motion of one part produces uniform motion on the next, the pressures at the communicating points remain constant or invariable. Now the accomplishing of this is always within the reach of mechanics.

One of the most useful communications in machinery belt forms is by means of toothed wheels acting on each other. It is of importance to have the teeth so formed, that the pressure by which one of them A urges the other B round its axis shall be constantly the same. It can easily be demonstrated, that when this is the case, the uniform angular motion of the one will produce a uniform angular motion of the other; or, if the motions are thus uniform, the pressures are invariable. This is accomplished on this principle, that the mutual actions of solid bodies on each other in the way of pressure are perpendicular to the touching surfaces. Therefore let the tooth a press on the tooth b in the point C; and draw the line FCDE perpendicular to the touching surfaces in the point C. Draw AF, BE perpendicular to FE, and let FE cut the line AB in D. It is plain, from the common principles of mechanics, that if the line FE, drawn in the manner now described, always pass through the same point D, whatever may be the situation of the acting teeth, the mutual action of the wheels will always be the same. It will be the same as if the arm AD acted on the arm BD. In the treatises on the construction of mills, and other works of this kind, are many instructions for the formation of the teeth of wheels; and almost every noted millwright has his own methods. Most of them are egregiously faulty in respect of mechanical principle. Indeed they are little else than instructions how to make the teeth clear each other without sticking. Mr de la Hire first pointed out the above mentioned principle, and justly condemned the common practice of making the small wheel or pinion in the form of a lantern (whence it also took its name), consisting of two round discs, having a number of cylindrical spaces (fig. 2.). The slightest inspection of this construction shows, that, in the different situations of the working teeth, the line FCE continually changes its intersection with AB. If the wheel B be very small in comparison of the other, and if the teeth of A take deep hold of the cylindrical pins of B, the line of action EF is sometimes so disadvantageously placed, that the pressure of the one wheel has scarcely any tendency at all to turn the other. Mr de la Hire, or Dr Hooke, was, we think, the first who investigated the form of communication which procured this constant action between the mended by wheels; and in a very ingenious dissertation, published among the Memoirs of the Academy of Sciences at Paris 1668, the former of these gentlemen shows, that this will be ensured by forming the teeth into epicycloids. Mr Camus of the same Academy has published an elaborate dissertation on the same subject, in which he prosecutes the principle of Mr de la Hire, and applies it to all the variety of cases which can occur in practice. There is no doubt as to the goodness of the principle; and it has another excellent property, "that the mutual action of the teeth is absolutely without any friction." The one tooth only applies itself to the other, and rolls on it, but does not slide or rub in the smallest degree. This makes them last long, or rather does not allow them to wear in the least. But the construction is subject to a limitation which must not be neglected. The teeth must be so made, that the curved part of the tooth b is acted on by a flat part of the tooth a till it comes to the line AB in the course of its action; after which... which the curved part of \(a\) acts on a flat part of \(b\); or the whole action of \(a\) on \(b\) is either completed, or only begins at the line \(AB\), joining the centres of the wheels.

Another form of the teeth secures the perfect uni- formity of action without this limitation, which requires very nice execution. Let the teeth of each wheel be formed by evolving its circumference; that is, let the acting face \(GCH\) of the tooth \(a\) have the form of the curve traced by the extremity of the thread \(FC\), un- lapped from the circumference. In like manner, let the acting face of the tooth \(b\) be formed by unlapping a thread from its circumference. It is evident, that the line \(FCE\), which is drawn perpendicularly to the touch- ing surfaces in the point \(C\), is just the direction or posi- tion of the evolving threads by which the two acting faces are formed. This line must therefore be the com- mon tangent to the two circles or circumferences of the wheels, and will therefore always cut the line \(AB\) in the same point \(D\). This form allows the teeth to act on each other through the whole extent of the line \(FCE\), and therefore will admit of several teeth to be acting at the same time (twice the number that can be admitted in Mr de la Hire's method). This, by divi- ding the pressure among several teeth, diminishes its quantity on any one of them, and therefore diminishes the dents or impressions which they unavoidably make on each other. It is not altogether free from sliding and friction, but the whole of it can hardly be said to be sensible. The whole slide of a tooth three inches long, belonging to a wheel of ten feet diameter, acting on a tooth of a wheel of two feet diameter, does not amount to \(\frac{1}{25}\)th of an inch, a quantity altogether insig- nificant.

In the formation of the teeth of wheels, a small de- viation from these perfect forms is not perhaps of very great importance, except in cases where a very large wheel drives a very small one (a thing which a good engineer will always avoid). As the construction, how- ever, is exceedingly easy, it would be unpardonable to omit it. Well formed teeth, and a great number of them acting at once, make the communication of mo- tion extremely smooth and uniform. The machine works without noise, and the teeth last a very long time without sensibly changing their shape. But there are cases, such as the pallets of clocks and watches, where the utmost accuracy of form is of the greatest importance for the perfection of the work.

When heavy tampers are to be raised, in order to drop on the matters to be pounded, the wipers by which they are lifted should be made of such a form, that the tapper may be raised by a uniform pressure, or with a motion almost perfectly uniform. If this is not attended to, and the wiper is only a pin sticking out from the axis, the tapper is forced into motion at once. This occasions violent jolts to the machine, and great strains on its moving parts and their points of support; whereas when they are gradually lifted, the inequality of de- fultory motion is never felt at the impelled point of the machine. We have seen pilons moved by means of a double rack on the pilon rod. A half wheel takes hold of one rack, and raises it to the required height. The moment the half wheel has quitted that side of the rack, it lays hold of the other side, and forces the pilon down again. This is proposed as a great improvement; cor-

recting the unequal motion of the pilon moved in the common way by a crank. But it is far inferior to the crank motion. It occasions such abrupt changes of motion, that the machine is shaken by jolts. Indeed if the movement were accurately executed, the machine would be shaken to pieces, if the parts did not give way by bending and yielding. Accordingly, we have always observed that this motion soon failed, and was changed for one that was more smooth. A judicious engineer will avoid all such sudden changes of motion, especially in any ponderous part of a machine.

When several tampers, pilons, or other reciprocal movers, are to be raised and depressed, common sense teaches us to distribute their times of action in a uni- form manner, so that the machine may always be equa- lly loaded with work. When this is done, and the ob- servations in the preceding paragraph attended to, the machine may be made to move almost as smoothly as if there were no reciprocations in it. Nothing shows the ingenuity of the author more than the artful yet simple and effectual contrivances for obviating those difficul- ties that unavoidably arise from the very nature of the work that must be performed by the machine, and of the power employed. The inventive genius and sound judgment of Watt and Boulton are as perceptible to a skilled observer in these subordinate parts of some of their great engines, as in the original discovery on which their patent is founded. In some of those engines the mass of dead matter which must be put into motion, and this motion destroyed and again restored in every stroke, is enormous, amounting to above an hundred tons. The ingenious authors have even contrived to draw some advantages from it, by allowing a great want of equilibrium in certain positions; and this has been condemned as a blunder by engineers who did not see the use made of it.

There is also great room for ingenuity and good choice in the management of the moving power, when it is such as cannot immediately produce the kind of motion required for effecting the purpose. We men- tioned the conversion of the continued rotation of an be com- axis into the reciprocating motion of a pilon, and the improvement which was thought to have been made on the common and obvious contrivance of a crank, by substituting a double rack on the pilon-rod, and the inconvenience arising from the jolts occasioned by this change. We have seen a great forge, where the engi- neer, in order to avoid the same inconvenience arising from the abrupt motion given to the great fledge ham- mer of seven hundred weight, resisting with a five-fold momentum, formed the wipers into spirals, which com- municated motion to the hammer almost without any jolt whatever; but the result was, that the hammer rose no higher than it had been raised in contact with the wiper, and then fell on the iron bloom with very little effect. The cause of its inefficiency was not guel- fed at; but it was removed, and wipers of the common form were put in place of the spirals. In this opera- tion, the rapid motion of the hammer is absolutely ne- cessary. It is not enough to lift it up; it must be so- fed up, so as to fly higher than the wiper lifts it, and to strike with great force the strong oaken spring which is placed in its way. It compresses this spring, and is reflected by it with a considerable velocity, so as to hit the iron as if it had fallen from a great height. Had it been allowed to fly to that height, it would have fallen upon the iron with somewhat more force (because no caken spring is perfectly elastic); but this would have required more than twice the time.

In employing a power which of necessity reciprocates, to drive machinery which requires a continuous motion (as in applying the steam engine to a cotton or a grist mill), there also occur great difficulties. The necessity of reciprocation in the first mover wastes much power; because the instrument which communicates such an enormous force must be extremely strong, and be well supported. The impelling power is wasted in imparting, and afterwards destroying, a vast quantity of motion in the working beam. The skilful engineer will attend to this, and do his utmost to procure the necessary strength of this first mover, without making it a vast load of inert matter. He will also remark, that all the strains on it, and on its supports, are changing their directions in every stroke. This requires particular attention to the manner of supporting it. If we observe the steam engines which have been long erected, we see that they have uniformly shaken the building to pieces. This has been owing to the ignorance or inattention of the engineer in this particular. They are much more judiciously erected now, experience having taught the most ignorant that no building can withstand their defultory and opposite jolts, and that the great movements must be supported by a framework independent of the building of masonry which contains it (a).

The engineer will also remark, that when a single stroke steam engine is made to turn a mill, all the communications of motion change the direction of their pressure twice every stroke. During the working stroke of the beam, one side of the teeth of the intervening wheels is pressing the machinery forward; but during the returning stroke, the machinery, already in motion, is dragging the beam, and the wheels are acting with the other side of the teeth. This occasions a rattling at every change, and makes it proper to fashion both sides of the teeth with the same care.

It will frequently conduce to the good performance of an engine, to make the action of the resisting work unequal, accommodated to the inequalities of the impelling power. This will produce a more uniform motion in machines in which the momentum of inertia is inconsiderable. There are some beautiful specimens of this kind of adjustment, in the mechanism of animal bodies.

It is very customary to add what is called a Fly to machines. This is a heavy disk or hoop, or other mass of matter, balanced on its axis, and so connected with the machinery as to turn briskly round with it. This may be done with the view of rendering the motion of the whole more regular, notwithstanding unavoidable inequalities of the accelerating forces, or of the resistances occasioned by the work. It becomes a Regulator.

Suppl. Vol. II. Part I.

(a) The gudgeons of a water-wheel should never rest on the wall of the building. It shakes it; and if set up soon after the building has been erected, it prevents the mortar from taking firm bond; perhaps by shattering the calcareous crystals as they form. When the engineer is obliged to rest the gudgeons in this way, they should be supported by a block of oak laid a little hollow. This softens all tremors, like the springs of a wheel carriage. This practice would be very serviceable in many other parts of the construction. weight of the fly, the irregularity of the motion may be rendered as small as we please. It is much better to enlarge the diameter. This preserves the friction more moderate, and the pivot wears less. For these reasons, a fly is in general a considerable improvement in machinery, by equalising many exertions that are naturally very irregular. Thus, a man working at a common windlass, exerts a very irregular pressure on the winch. In one of his positions in each turn he can exert a force of near 70 pounds without fatigue, but in another he cannot exert above 25; nor must he be loaded with much above this in general. But if a large fly be connected properly with the windlass, he will act with equal ease and speed against 30 pounds.

This regulating power of the fly is without bounds, and may be used to render uniform a motion produced by the most defective and irregular power. It is thus that the most regular motion is given to mills that are driven by a single stroke steam engine, where for two or even three seconds there is no force pressing round the mill. The communication is made through a massive fly of very great diameter, whirling with great rapidity. As soon as the impulse ceases, the fly, continuing its motion, urges round the whole machinery with almost unabated speed. At this instant all the teeth, and all the joints, between the fly and the first mover, are heard to catch in the opposite direction.

If any permanent change should happen in the impelling power, or in the resistance, the fly makes no obstacle to its producing its full effect on the machine; and it will be observed to accelerate or retard uniformly, till a new general speed is acquired exactly corresponding with this new power and resistance.

Many machines include in their construction movements which are equivalent with this intentional regulator. A flour mill, for example, cannot be better regulated than by its millstone; but in the Albion mills, a heavy fly was added with great propriety; for if the mills had been regulated by their millstones only, then at every change of stroke in the steam engine, the whole train of communications between the beam, which is the first mover, and the regulating millstone, which is the very last mover, would take in the opposite direction. Although each drop in the teeth and joints be but a trifle, the whole, added together, would make a considerable jolt. This is avoided by a regulator immediately adjoining to the beam. This continually presses the working machinery in one direction. So judiciously were the movements of that noble machine contrived, and so nicely were they executed, that not the least noise was heard, nor the slightest tremor felt in the building.

Mr Valon's beautiful pile engine employed at Westminster Bridge is another remarkable instance of the fly-regulating power of a fly*. When the ram is dropped, and its follower disengaged immediately after it, the horses would instantly tumble down, because the load, against which they had been straining hard, is at once taken off; but the gin is connected with a very large fly, which checks any remarkable acceleration, allowing the horses to lean on it during the descent of the load; after which their draught recommences immediately. The spindles, cards, and bobbins, of a cotton mill, are also a sort of flies. Indeed all bulky machines of the rotative kind tend to preserve their motion with some degree of steadiness, and their great momentum of inertia is as useful in this respect as it is prejudicial to the acceleration or any reciprocation when wanted.

There is another kind of regulating fly, consisting of a ball compassed with a ring round till the resistance of the air prevents any great acceleration. This is a very bad one for a working machine, for it produces its effect by really wasting a part of the moving power. Frequently it employs a very great and unknown part of it, and robs the proprietor of much work. It should never be introduced into any machine employed in manufactures.

Some rare cases occur where a very different regulator is required; where a certain determined velocity is found necessary. In this case the machine is furnished with its extreme mover, with a conical pendulum consisting of two heavy balls hanging by rods, which move in very nice and steady joints at the top of a vertical axis. It is well known, that when this axis turns round, with an angular velocity suited to the length of those pendulums, the time of a revolution is determined. Thus, if the length of each pendulum be 39½ inches, the axis will make a revolution in two seconds very nearly. If we attempt to force it more swiftly round, the balls will recede a little from the axis, but it employs as long time for a revolution as before; and we cannot make it turn swifter, unless the impelling power be increased beyond all probability; in which case the pendulum will fly out from the centre till the rods are horizontal, after which every increase of power will accelerate the machine very sensibly. Watt and Boulton have applied this contrivance with great ingenuity to their steam engines, when they are employed for driving machinery for manufactures which have a very changeable resistance, and where a certain speed cannot be much departed from without great inconvenience. They have connected this recess of the balls from the axis (which gives immediate indication of an increase of power or a diminution of resistance) with the cock which admits the steam to the working cylinder. The balls flying out, cause the cock to close a little, and diminish the supply of steam. The impelling power diminishes the next moment; and the balls again approach the axis, and the rotation goes on as before, although there may have occurred a very great excess or deficiency of power. The same contrivance may be employed to raise or lower the feeding sluice of a water mill employed to drive machinery.

A fly is sometimes employed for a very different purpose from that of a regulator of motion—it is employed as a collector of power. Suppose all resistance removed from the working point of a machine furnished with a very large or heavy fly immediately connected with the working point. When a small force is applied to the impelled point of this machine, motion will begin in the machine, and the fly begin to turn. Continue to press uniformly, and the machine will accelerate. This may be continued till the fly has acquired a very rapid motion. If at this moment a resisting body be applied to the working point, it will be acted on with very great force; for the fly has now accumulated in its circumference a very great momentum. If a body were exposed immediately to the action of this circumference, it would be violently struck. Much more will it be so, if the body be exposed to the action of the working... working point, which perhaps makes one turn while the fly makes a hundred. It will exert a hundred times more force there (very nearly) than at its own circumference. All the motion which has been accumulated on the fly during the whole progress of its acceleration is exerted in an instant at the working point, multiplied by the momentum depending on the proportion of the parts of the machine. It is thus that the coining press performs its office; nay, it is thus that the blacksmith forges a bar of iron. Swinging the great fledge hammer round his head, and urging it with force the whole way, this accumulated motion is at once extinguished by impact on the iron. It is thus we drive a nail; and it is thus that by accumulating a very moderate force exerted during four or five turns of a fly, the whole of it is exerted on a punch set on a thick plate of iron, such as is employed for the boilers of steam engines. The plate is pierced as if it were a bit of cheese. This accumulating power of a fly has occasioned many who think themselves engineers to imagine, that a fly really adds power or mechanical force to an engine; and, not understanding on what its efficacy depends, they often place the fly in a situation where it only added a useless burden to the machine. It should always be made to move with rapidity. If intended for a mere regulator, it should be near the first mover. If it is intended to accumulate force in the working point, it should not be far separated from it. In a certain sense, a fly may be said to add power to a machine, because by accumulating into the exertion of one moment the exertions of many, we can sometimes overcome an obstacle that we never could have balanced by the same machine unaided by the fly.

It is this accumulation of force which gives such an appearance of power to some of our first movers. When a man is unfortunately caught by the teeth of a paltry country mill, he is crushed almost to mummy. The power of the stream is conceived to be prodigious; and yet we are certain, upon examination, that it amounts to the pressure of no more than fifty or sixty pounds. But it has been acting for some time; and there is a millstone of ten weight whirling twice round in a second. This is the force that crushed the unfortunate man; and it required it all to do it, for the mill stopped. We saw a mill in the neighbourhood of Elbingroda in Hanover, where there was a contrivance which disengaged the millstone when any thing got entangled in the teeth of the wheels. It was tried in our sight with a head of cabbage. It crushed it indeed, but not violently, and would by no means have broken a man's arm.

It is hardly necessary to recommend simplicity in the construction of machines. This seems now sufficiently recommended. Multiplicity of motions and communications increases frictions; increases the unavoidable losses by bending and yielding in every part; exposes to all the imperfections of workmanship; and has a great chance of being indistinctly conceived, and therefore constructed without science. We think the following construction of a capstan or crab a very good example of the advantages of simplicity. It is the invention of an untaught but very ingenious country tradesman.

EAB is the barrel of the capstan, standing vertically in a proper frame, as usual, and urged round by bars such as EF. The upper part A of the barrel is 17 inches in diameter, and the lower B is 16. C is a strong pulley 16 inches in diameter, having a hook D, simple and which holds hold of a hawser attached to the load. The rope ACB is wound round the barrel A, passes over the pulley C, and is then wound round the barrel B in the opposite direction. No farther description is necessary, we think, to show that, by heaving by the bar F, so as to wind more of the rope upon A, and unwind it from B, the pulley C must be brought nearer to the capstan by about three inches for each turn of the capstan; and that this simple capstan is equivalent to an ordinary capstan of the same length of bar EF, and diameter of barrel B, combined with a 16 fold tackle of pulleys; or, in short, that it is 16 times more powerful than the common capstan; free from the great loss by friction and bending of ropes, which would absorb a third of the power of a 16 fold tackle; and that whereas all other engines become weaker as they multiply the power to a greater degree (unless they are proportionally more bulky), this engine becomes really stronger in itself. Suppose we wanted to have it twice as powerful as at present; nothing is necessary but to cover the part B of the barrel with laths a quarter of an inch thick. In short, the nearer the two barrels are to equality, the more powerful does it become. We give it to the public as an excellent capstan, and as suggesting thoughts which an intelligent engineer may employ with great effect. By this contrivance, and using an iron wire instead of a catgut, we converted a common eight day clock into one which goes for two months.

We intended to conclude this article with some observations on the chief classes of powers which are employed to drive machinery; such as water, wind, atmospheric pressure, gunpowder, and the force of men and other animals, giving some notion of their absolute magnitudes, and the effect which may be expected from them. We should then have mentioned what has been discovered as to their variation by a variation of velocity. And we intended to conclude with an account of what knowledge has been acquired concerning friction, and the loss of power in machinery arising from this cause, and from the slippage of ropes, and some other causes: But we have not yet been able to bring these matters into a connected form, which would suggest the methods and means of farther information thereon. We must endeavour to find another opportunity of communicating to the public what we may yet learn on those subjects.

We have now established the principles on which machines must be constructed, in order that they may produce the greatest effect; but it would be improper to dismiss the subject without stating to our readers Mr Bramah's new method of producing and applying a more considerable degree of power to all kinds of machinery requiring motion and force, than by any means at present practised for that purpose. This method, for which on the 31st of March 1796 he obtained a patent, consists in the application of water or other dense fluids to various engines, so as, in some instances, to cause them to act with immense force; in others, to communicate the motion and powers of one part of a machine to some other part of the same machine; and, lastly, to communicate the motion and force of one machine to another, where their local situations preclude the application of all other methods of connection.

The first and most material part of this invention will be clearly understood by an inspection of fig. 4, where "A is a cylinder of iron, or other materials, sufficiently strong, and bored perfectly smooth and cylindrical; into which is fitted the piston B, which must be made perfectly water-tight, by leather or other materials, as used in pump making. The bottom of the cylinder must also be made sufficiently strong with the other part of the surface, to be capable of resisting the greatest force or strain that may at any time be required. In the bottom of the cylinder is inserted the end of the tube C; the aperture of which communicates with the inside of the cylinder, under the piston B, where it is fitted with the small valve D, the same as the suction pipe of a common pump. The other end of the tube C communicates with the small forcing pump or injector E, by means of which water or other dense fluids can be forced or injected into the cylinder A, under the piston B. Now, suppose the diameter of the cylinder A to be 12 inches, and the diameter of the piston of the small pump or injector E only one quarter of an inch, the proportion between the two surfaces or ends of the said pistons will be as 1 to 2304; and supposing the intermediate space between them to be filled with water or other dense fluid capable of sufficient resistance, the force of one piston will act on the other just in the above proportion, viz. as 1 is to 2304. Suppose the small piston in the injector to be forced down when in the act of pumping or injecting water into the cylinder A, with the power of 20 cwt., which could easily be done by the lever H; the piston B would then be moved up with a force equal to 20 cwt., multiplied by 2304. Thus is constructed a hydro-mechanical engine, whereby a weight amounting to 2304 tons can be raised by a simple lever, through equal space, in much less time than could be done by any apparatus constructed on the known principles of mechanics; and it may be proper to observe, that the effect of all other mechanical combinations is counteracted by an accumulated complication of parts, which renders them incapable of being usefully extended beyond a certain degree; but in machines acted upon or constructed on this principle, every difficulty of this kind is obviated, and their power subject to no finite restraint. To prove this, it will be only necessary to remark, that the force of any machine acting upon this principle can be increased ad infinitum, either by extending the proportion between the diameter of the injector and the cylinder A, or by applying greater power to the lever H.

Fig. 5. represents the section of an engine, by which very wonderful effects may be produced instantaneously by means of compressed air. AA is a cylinder, with the piston B fitting air-tight, in the same manner as described in fig. 4. C is a globular vessel made of copper, iron, or other strong materials, capable of resisting immense force, similar to those of air guns. D is a strong tube of small bore, in which is the flop-cock E. One of the ends of this tube communicates with the cylinder under the piston B, and the other with the globe C. Now, suppose the cylinder A to be the same diameter as that in fig. 4, and the tube D equal to one quarter of an inch diameter, which is the same as the injector fig. 4.; then, suppose that air is injected into the globe C (by the common method), till it presses against the cock E with a force equal to 20 cwt., which can easily be done; the consequence will be, that when the cock E is opened, the piston B will be moved in the cylinder AA with a power or force equal to 2304 tons; and it is obvious, as in the case fig. 4, that any other unlimited degree of force may be acquired by machines or engines thus constructed.

"Fig. 6. is a section, merely to show how the power and motion of one machine may, by means of fluids, be transferred or communicated to another, let their distance and local situation be what they may. A and B are two small cylinders, smooth and cylindrical; in the inside of each of which is a piston, made water and air tight, as in figs 4. and 5. CC is a tube conveyed underground, or otherwise, from the bottom of one cylinder to the other, to form a communication between them, notwithstanding their distance be ever so great; this tube being filled with water or other fluid, until it touch the bottom of each piston; then, by depressing the piston A, the piston B will be raised. The same effect will be produced vice versa: thus bells may be rung, wheels turned, or other machinery put invisibly in motion, by a power being applied to either.

"Fig. 7. is a section, shewing another instance of communicating the action and force of one machine to another; and how water may be raised out of wells of any depth, and at any distance from the place where the operating power is applied. A is a cylinder of any required dimensions, in which is the working piston B, as in the foregoing examples; into the bottom of this cylinder is inserted the tube C, which may be of less bore than the cylinder A. This tube is continued, in any required direction, down to the pump cylinder D, supposed to be fixed in the deep well EE, and forms a junction therewith above the piston F; which piston has a rod G, working through the stuffing box, as is usual in a common pump. To this rod G is connected, over a pulley or otherwise, a weight H, sufficient to overcome the weight of the water in the tube C, and to raise the piston F when the piston B is lifted: thus, suppose the piston B is drawn up by its rod, there will be a vacuum made in the pump cylinder D, below the piston F; this vacuum will be filled with water through the suction pipe, by the pressure of the atmosphere, as in all pumps fixed in air. The return of the piston B, by being pressed downwards in the cylinder A, will make a stroke of the piston in the pump cylinder D, which may be repeated in the usual way by the motion of the piston B, and the action of the water in the tube C. The rod G of the piston F, and the weight H, are not necessary in wells of a depth where the atmosphere will overcome the water in the suction of the pump cylinder D, and that in the tube C. The small tube and cock in the cistern I, are for the purpose of charging the tube C."

That these contrivances are ingenious, and may occasionally prove useful, we are not inclined to contradict; but we must confess, that the advantages of them appear