PREFACE TO THE SECOND EDITION.

I HAVE added in this Edition articles:-first, "On the Dynamical Stability of Floating Bodies;" secondly, "On the Rolling of a Cylinder;" thirdly, "On the descent of a body upon an inclined plane, when subjected to variations of temperature, which would otherwise rest upon it;" fourthly, "On the state bordering upon motion of a body moveable about a cylindrical axis of finite dimensions, when acted upon by any number of pressures.'

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The conditions of the dynamical stability of floating bodies include those of the rolling and pitching motion of ships. The discussion of the rolling motion of a cylinder includes that of the rocking motion to which a locomotive engine is subject, when its driving wheels are falsely balanced, and that of the slip of the wheel due to the same cause. The descent of a body upon an inclined plane when subjected to variations in temperature, which otherwise would rest upon it, appears to explain satisfactorily the descent of glaciers.

The numerous corrections made in the text, I owe chiefly to my old pupils at King's College, to whom the lectures of which it contains the substance, were addressed. For

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several important ones I am, however, indebted to Mr. Robinson, Master of the School for Shipwrights' Apprentices, in Her Majesty's Dockyard, Portsea; to whom I have also to express my warm acknowledgments for the care with which he has corrected the proof sheets whilst going through

the press.

May, 1855

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,' quantité d'action," "puissance mécanique, "travail," from French writers.

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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 vivæ 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 inertiæ, 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 inertiæ, 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.