Ferguson's Lectures on Select Subjects in Mechanics, Hydrostatics, Hydraulics, Pneumatics, Optics, Geography, Astronomy, and Dialing

ON THE CONSTRUCTION AND EFFECTS OF MA-
CHINES,'

By Mr. John Leslie, Professor of Mathematics in the University of Edinburgh.

1. Ir is a principle in statics, that, if a body act upon another by the intervention of machinery, an equilibrium will obtain when their potential velocities are reciprocally as their masses. the power exerted be augmented beyond what is barely sufficient to maintain the balance, a motion will immediately commence, and if it be still increased, the velocity will continually

This excellent paper, which Professor Leslie was so kind as to communicate to the editor, was written at London so early as February 1790. The same subject was afterwards (in 1801) treated at great length by the late Dr. Robison, in the art. Machinery, Sup. Eucycl. Britan.; and that we do not conceive that we are derogating in the least from the talents of that learned and good man, when we say, the present paper is written with greater perspicuity, and gives a more elementary and connected view of this interesting subject. At some future period Mr. Leslie intends to resume the investigation of this subject.-ED.

increase. But this velocity will increase in a smaller ratio than the power; and there will, therefore, be a certain point of augmentation, at which the force employed will produce the greatest proportional effect. Such is the grand object that we ought to have always in view in the construction of machines.

2. Forces have been divided into two kinds those whose action is supposed to be instantaneous, and those whose action is continued and incessant. The former have been termed impulsive, the latter accelerating or retarding. Though accelerated or retarded motions perpetually occur to our observation, the ancients seem to have admitted no other force but that of impulsion. It is difficult, indeed, to conceive, that a body can act at a distance; and the idea that motion is always communicated by contact, is one of our earliest and strongest prejudices. Sir Isaac Newton himself was in this instance carried away by the current of opinion. His theory of æther was an attempt to explain gravitation by impulsive forces.-But there are many facts and experiments which satisfactorily prove, that between the particles of matter there subsists a repulsion, increasing as the distance diminishes, and that no absolute contact can ever take place. A body does not acquire its celerity in an instant. Nothing material can exist but what is finite; and the beautiful law of continuation, by which changes are produced by imperceptible shades, can never be violated. But an amazing force may be exerted, and an effect may be produced, in a time so small as to elude the acuteness of our senses. Hence the origin of our idea that motion is derived from impulse. If, however, we consider the subject with more

attention, we shall find that it is really as difficult to conceive action in contiguity as at a distance. In neither case can we deduce the consequences a priori. The connexion which subsists between cause and effect is not necessary and absolute; it is founded upon the invariable experience of our senses. We may, therefore, conclude, that there is only one kind of force, and that is the accelerating or the retarding. Hence it will always be possible to determine the proportional intensity of any given force, compared with that of gravity, and to assign a weight, which, by its pressure alone, would in a given time produce the same effect.

3. If the gravity of an elementary point at the earth's surface be denoted by 1, the whole attractive force will be as the number of points, or as M, the mass of the body. Let F express the intensity of another force urging the same body; then MxF will denote the quantity of force exerted, or ø; but =F; wherefore, M

the intensity of a force is directly as its quantity, and inversely as the mass which is urged.

4. Let the velocity of a body, S the space described, and T the time of description; the velocity that is acquired may be conceived to be composed of all the successive augmentations which are produced by the continued exertion of the force, and which are proportional to the intensity of its action. But the force may for 2 moment be conceived to be uniform; whence the increment of velocity is compounded of the force, and of the increment of the time, or VFT. Suppose the force to be constant, then VFT; and when the time is

given, the velocity must be as the accelerating force. Let V-CFT, and Vand T denote feet and seconds of time. When F-1, and T-1, we shall have V-C the velocity acquired by descent at the surface of the earth at the end of the first second. Put d=16.1 feet, then

C=2d; whence V=2dFİ' and İ=

2dF

5. We may conceive that the velocity is uniform for an indefinitely small portion of time. Whence S=VT and Ï=& ; hence also

V=; but, by the last article, V=2dFİ, con

sequently VV= 2dFS; from which equation the velocity may be determined, when the relation is given between the accelerating force and the space described. If F be constant, then by integration, VV=2dFS and V4dFS; wherefore V-2/dFS. At the same time, because V=2dFT, $ = 2dFTT; whence, S=2d>FT3,

6. We may divide machines into two general kinds; into those where the action is interrupted and renewed at short intervals, and into those where the action is continued for a certain period. In the former, the effect of friction not having time to accumulate, may generally be disregarded, and the motion may be considered as uniformly accelerated. With regard to the lat ter, if a machine be constructed so that the resistance is great, it increases rapidly with the celerity; it soon counterbalances the accelerating force, and produces a motion which is equal and constant.

7. Let us abstract the momenta and friction

of the parts of communication, and consider the effects of a machine which is uniformly impelled. Suppose the motion of a power, equal to the gravity of the mass p, be connected to that of a weight w, so that the potential velocity of the former be constantly to that of the latter as v: 1, and let v be termed the advantage. It is manifest, from the principles of statics, that a part of the power is able to maintain the

the power and the weight.

equilibrium, and that the remaining part þonly is employed in producing the motion. But the action of this force p- is divided between and the weight. Put y the part which urges the power, and z= the part which is exerted against the weight. But (3) the intensity of the force y, which impels the power, is denoted by; and therefore the velocity acquired in a given time is (4) also. But the influence of the force z upon the weight, will, in consequence of the mechanism, be equal to the direct action of a force uz; whence the velocity acquired by the weight in the same. time will be 22. Wherefore, by hypothesis, y ,༧ ::v: 1; consequently and y=