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PREFACE.

In the following work, I have proposed to myself to apply the principles of mechanics to the discussion of the most important and obvious of those questions which present themselves in the practice of the engineer and the architect; and I have sought to include in that discussion all the circumstances on which the practical solution of such questions may be assumed to depend. It includes the substance of a course of lectures delivered to the students of King's College in the department of engineering and architecture, during the years 1840, 1841, 1842.*

In the first part I have treated of those portions of the science of STATICS, which have their application in the theory of machines and the theory of construction.

In the second, of the science of DYNAMICS, and, under this head, particularly of that union of a continued pressure with a continued motion which has received from English writers the various names of "dynamical effect," "efficiency," "work done," "labouring force," "work," &c.; and "moment d'activité," "quantité d'action," "puissance mécanique," "travail," from French writers.

Among the latter this variety of terms has at length given place to the most intelligible and the simplest of them,

The first 170 pages of the work were printed for the use of my pupils in the year 1840. Copies of them were about the same time in the possession of several of my friends in the Universities.

"travail." The English word "work" is the obvious translation of "travail," and the use of it appears to be recommended by the same considerations. The work of overcoming a pressure of one pound through a space of one foot has, in this country, been taken as the unit, in terms of which any other amount of work is estimated; and in France, the work of overcoming a pressure of one kilogramme through a space of one metre. M. Dupin has proposed the application of the term dyname to this unit.

I have gladly sheltered myself from the charge of having contributed to increase the vocabulary of scientific words, by assuming the obvious term " unit of work" to represent concisely and conveniently enough the idea which is attached to it.

The work of any pressure operating through any space is evidently measured in terms of such units, by multiplying the number of pounds in the pressure by the number of feet in the space, if the direction of the pressure be continually that in which the space is described. If not, it follows, by a simple geometrical deduction, that it is measured by the product of the number of pounds in the pressure, by the number of feet in the projection of the space described,* upon the direction of the pressure; that is, by the product of the pressure by its virtual velocity. Thus, then, we conclude at once, by the principle of virtual velocities, that if a machine work under a constant equilibrium of the pressures applied to it, or if it work uniformly, then is the aggregate work of those pressures which tend to accelerate its motion equal to the aggregate work of those which tend to retard it; and, by the principle of vis viva, that if the machine do not work under an equilibrium of the forces impressed upon it, then is the aggregate work of those which tend to accelerate the motion of the machine greater or less

*If the direction of the pressure remain always parallel to itself, the space described may be any finite space; if it do not, the space is understood to be so small, that the direction of the pressure may be supposed to remain parallel to itself whilst that space is described.

than the aggregate work of those which tend to retard its motion by one half the aggregate of the vires viva acquired or lost by the moving parts of the system, whilst the work is being done upon it. In no respect have the labours of the illustrious president of the Academy of Sciences more contributed to the development of the theory of machines than in the application which he has so successfully made to it of this principle of vis viva.* In the elementary discussion of this principle, which is given by M. Poncelet, in the introduction to his Mécanique Industrielle, he has revived the term vis inertia (vis inertia, vis insita, Newton), and, associating with it the definitive idea of a force of resistance opposed to the acceleration or the retardation of a body's motion, he has shown (Arts. 66. and 122.) the work expended in overcoming this resistance through any space, to be measured by one half the vis viva accumulated through the space; so that throwing into the consideration of the forces. under which a machine works, the vires inertia of its moving elements, and observing that one half of their aggregate vis viva is equal to the aggregate work of their vires inertiæ, it follows, by the principle of virtual velocities, that the difference between the aggregate work of those forces impressed upon a machine, which tend to accelerate its motion, and the aggregate work of those which tend to retard the motion, is equal to the aggregate work of the vires inertia of the moving parts of the machine: under which form the principle of vis viva resolves itself into the principle of virtual velocities. So many difficulties, however, oppose themselves to the introduction of the term vis inertia, associated with the definitive idea of a force, into the discussion of questions of mechanics, and especially of practical and elementary mechanics, that I have thought it desirable to avoid it. It is with this view that I have given a new interpretation to that function of the velocity of a moving body which is known as its vis viva. One half that function I have interpreted to represent the number of units of work accumulated

* See Poncelet, Mécanique Industrielle, troisième partie.

in the body so long as its motion is continued. This number of units of work it is capable of reproducing upon any resistance opposed to its motion. A very simple investigation (Art. 66.) establishes the truth of this interpretation, and gives to the principle of vis viva the following more simple enunciation:"The difference between the aggregate work done upon the machine, during any time, by those forces which tend to accelerate the motion, and the aggregate work, during the same time, of those which tend to retard the motion, is equal to the aggregate number of units of work accumulated in the moving parts of the machine during that time if the former aggregate exceed the latter, and lost from them during that time if the former aggregate fall short of the latter." Thus, then, if the aggregate work of the forces which tend to accelerate the motion of a machine exceeds that of the forces which tend to retard it, then is the surplus work (that done upon the driving points, above that expended upon the prejudicial resistances and upon the working points) continually accumulated in the moving elements of the machine, and their motion is thereby continually accelerated. And if the former aggregate be less than the latter, then is the deficiency supplied from the work already accumulated in the moving elements, so that their motion is in this case continually retarded.

The moving power divides itself whilst it operates in a machine, first, into that which overcomes the prejudicial resistances of the machine, or those which are opposed by friction and other causes, uselessly absorbing the work in its transmission. Secondly, into that which accelerates the motion of the various moving parts of the machine, and which accumulates in them so long as the work done by the moving power upon it exceeds that expended upon the various resistances opposed to the motion of the machine. Thirdly, into that which overcomes the useful resistances, or those which are opposed to the motion of the machine at the working point, or points, by the useful work which is done by it.

Between these three elements there obtains in every machine a mathematical relation, which I have called its MODULUS. The general form of this modulus I have discussed in a memoir on the "Theory of Machines" published in the Philosophical Transactions for the year 1841. The determination of the particular moduli of those elements of machinery which are most commonly in use, is the subject of the third part of the following work. From a combination of the moduli of any such elements there results at once the modulus of the machine compounded of them.

When a machine has acquired a state of uniform motion, work ceases to accumulate in its moving elements, and its modulus assumes the form of a direct relation between the work done by the motive power upon its driving point and that yielded at its working points. I have determined by a general method* the modulus in this case, from that statical relation between the driving and working pressures upon the machine which obtains in the state bordering upon its motion, and which may be deduced from the known conditions of equilibrium and the established laws of friction. In making this deduction I have, in every case, availed myself of the following principle, first published in my paper on the theory of the arch, read before the Cambridge Philosophical Society in Dec. 1833, and printed in their Transactions of the following year:-"In the state bordering upon motion of one body upon the surface of another, the resultant pressure upon their common surface of contact is inclined to the normal, at an angle whose tangent is equal to the coefficient of friction."

This angle I have called the limiting angle of resistance. Its values calculated, in respect to a great variety of surfaces of contact, are given in a table at the conclusion of the second part, from the admirable experiments of M. Morin,t into the mechanical details of which precautions have been introduced hitherto unknown to experiments of this class,

Art. 152. See Phil. Trans., 1841, p. 290.

+ Nouvelles Expériences sur le Frottement, Paris, 1833.

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