XXI. Researches in the theory of machines - Moseley, Henry (Philosophical transactions of the Royal Society of London, 1841)
The work of a mechanical agent may be defined as the union of a continual pressure with a continual motion. The work of overcoming a pressure of one pound through a space of one foot, is in this country taken as the unit in terms of which any other amount of work is estimated. 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 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, however, of this principle, which is given by M. Poncelet in the Introduction to his Mécanique Industrielle, he has revived the term vis inertiæ (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 inertiæ 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 inertiæ 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 an opposing force, into the discussion of questions of mechanics, and especially of practical and elementary mechanics, that it has appeared to the author of this paper desirable to avoid it. It is with this view, that in the researches which form the subject of the paper now submitted to the Society, a new interpretation is given to that function of the velocity of a moving body which is known as its vis viva; one-half that function being interpreted to represent the number of units of work accumulated in the body so long as its motion is continued, and which number of units of work it is capable of reproducing upon any resistance which may be opposed to its motion, and bring it to rest.