There is a kind of fluctuating or alter nate force to be found in the action of forcing pumps. In these the compres sion of the' fluid, and of the intermediate air, demands a greater force, in propor tion as the piston descends ; and, vice versa, as it ascends ; the gradual increase of compression, furnished by the return of the fluid into the lower part' of the cylinder, causes a gradual diminution of resistance to the upward motion, and consequently. acts as an accelerating force.
When bodies at rest fall from heights, the times employed are, respectively, as the square roots of the cubes of the heights from -which those bodies fall. We may, perhaps, form a reedy estimate of this circumstance, when we recollect that gravity giVes to all falling bodies an accelerating force. In the latitude of London, a heavy body falls nearly sixteen one-twelfth feet in the first second ; which velocity is not only doubled in the next second, so as to amount to 321 feet, but quadrupled by means of the addi tional force gained by the continued ac tion of gravity ; the third second of time will give 961 for its increase, and the fourth second will give 1281, In this we suppose the bodies not to be impell ed downwards by any force, (exclusive of gravity) but to be in a state of rest, and allowed to descend simply by their own weight ; for instance, by cutting the line that suspends a weight As bodies gain velocity in falling, so they lose velocity when projected up wards. Ha body be impelled upwards, with the same force it had acquired in falling, its velocity • upwards would gra dually decrease in the exact ratio that it increased in with such a power it would reach to that height whence it 'had fallen, but no further. Hence, by ascertaining either the time of ascent, or of descent, the height will be discovered. It is worthy of notice, that the acquired velocities are as fol low: 2, 4, 6, 8, 10, &c. upon each pre ceding second respectively ; the spaces for each time being 1, 3, 5, 7, &c. re spectively, and their constant differences 2. These laws of acceleration were as. certansed, by Galileo, to prevail equally in the motion of bodies along inclined planes, which may, indeed, be fully proved by observing the progress of a ball as it descends a hill : but this will not hold good unless the body be at per fect liberty ; for in cases of interruption, each gradation must be considered as the incipient motion ; were it other wise, no carriage could descend a long declivity without the certainty of being dashed to pieces, nor ascend a hill at the same pace throughout.
The force which accelerates or re tards the motion of a body upon an in clined plane is, to the force of gravity, as the height of the plane to its length ; or as the sine of the plane's elevation to the radius ; and if the diameter of a circle be perpendicular to the horizon, and chords be drawn from either extremity, the time of descent down all the chords will beequal ; and each will be equal to the time of free descent through the ver tical diameter. In the circle, fig. 7, the line A B is a vertical diameter,- the chords A C, A D, A E, and E B, are all of different lengths, and of different in clinations from the horizon. It is evident that, in every instance,. the shortest . chords have the least deviation from the horizontal ; consequently they must re tard the progress of a body passing down them much more than those which approach to the vertical ; the latter, •ob viously, give a more free passage to the' body ; and as they become more verti cal, approach nearer to the free descent at.A. B. , When a body descends along any num ber of contiguous planes, it will ultimate ly acquire the same velocity as would have been acquired by falling perpen dicularly through the height of the whole number of planes. The times of. descents along similar arcs, similarly si tuated, are as the square roots of those. arcs ; or as the square roots of the radii of their respective circles. And if a body, being at rest, is to fall: down a curved surface whiell is perfectly ly smooth, the velocity acquired will equal to that which would result from falling the same perpendicular height. It will require an incipient force to urge the body back to the height whence it fell, equal to that which it had acquired on reaching the bottom of the curve. There fore a body falling from 0, fig. 8, to the bottom P of the curve 0 P S, would have power to re-ascend(excluding friction) to S, which is level with 0 ; and it would rise from P to S in the same space of time it occupied in falling from 0 to P; in either case the time occupied would be the same as from X to P. When treating of the pendulum this will be more ob viously demonstrated.