DYNAMICS. 



first part of the motion would be more 

 rapid than the latter ; because the com- 

 mencement of the concave space being 

 more horizontal than the latter, or upper 

 part, would permit the force to act more 

 powerfully on the body D. On the other 

 hand, if the line of ascent were convex, 

 as at A E B, the first part of the ascent, 

 being steepest, would be slowest; and 

 the latter, in consequence of its regular 

 approach to the horizontal, would be 

 proportionately rapid. Thus we see, 

 that C is an uniform force on the plane 

 A B ; an accelerating force on the con- 

 vex ascent A E B ; and a retarding force 

 on the concave ascent A D B : we there- 

 fore deduce, that accelerating and retard- 

 ing motions may be described by arcs of 

 which the axes are easily ascertained. 



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 ready 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 32 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 96$ for its increase, and the 

 fourth second will give 128|. 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 

 allowea 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. If a 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 descending, and 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- 

 certained, 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 planes 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 be equal ; 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 AB. 



When a body descends along any num- 

 ber of contiguous planes, it wUl 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 suffered to fall 

 down a curved surface which is perfect- 

 ly smooth, the velocity acquired will be 

 equal to that which would result from 



