DYNAMICS. 



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 O, fig. 8, to the 

 bottom H of the curve OPS, would have 

 power to reascend (excluding friction) to 

 S, which is level with O ; and it would 

 rise from P to S in the same space of time 

 .it occupied in falling from O 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. 



It was for a long time supposed, that a 

 body would move more rapidly in a direct 

 line, not being vertical, than by any other 

 course ; but it has been ascertained, that 

 any curve, not exceeding 60 degrees, is 

 a quicker descent than its chord ; and that 

 a curve of 90 degrees is a quicker de- 

 scent than any tangent laying between 

 the same parallels. Thus, in fig. 9, the 

 curve A B C is of quicker descent than 

 the chord A C, and the curve M A H C 

 gives a quicker descent then the tangent 

 F G, or H I, or K L, laying between the 

 same parallels M K, and C D. 



With regard to the motions of projec- 

 tiles, \ve refer the reader to that article. 



Of central forces we have already given 

 an article, but shall observe here, in ad- 

 dition, that all bodies, when put in mo- 

 tion, would preserve their respective ve- 

 locities, and their original directions, 

 were they not acted upon by other forces. 

 When a force acts equally for a limited 

 distance, and then is superseded by the 

 actions of another force, the body will de- 

 scribe a polygon in its track ; but if the 

 original force be gradually weakened, 

 the body will then describe a curve, 

 bending towards the centre of attraction, 

 resulting from the operation of a deflect- 

 ing force, which, by its pressure, causes 

 the body to bend from its original direc- 

 tion. When a body is constantly attract- 

 ed towards a centre, it is under the influ- 

 ence of a centripetal force ; and when it 

 is disposed to fly from that centre, it is 

 under the bias of a centrifugal force. 

 These two latter constitute what are 

 termed central forces. The projectile 

 force is the original direction of an im- 

 pelled body, forming a tangent with the 

 curve occasioned by the deflecting force. 

 The track of a body under the influence 

 of a centripetal force, is called its trajec- 

 tory, or orbit. The radius vector is a line 

 drawn from the centre to which the force 

 is referred, or wherein it is supposed to 

 act, to any point in the trajectory where 



VOL. IV. 



the body is found. A body moving regu- 

 larly on a trajectory, or orbit, which re- 

 turns into itself, is, on its return to the 

 incipient point whence the motion began, 

 said to have made a period ; and the time 

 occupied is called its periodical time. It 

 must be understood, that a body can nei- 

 ther set itself in motion, nor avert its own 

 course ; such effects must be the result 

 of forces exteriorly applied ; also we must 

 state, that the motion of each body is na- 

 turally in a right line, but by the impulse 

 of some one or more powers its course 

 will deviate into a curve. Thus, a peb- 

 ble in a sling, or a glass full of water 

 placed within a hoop that is turned swift- 

 ly round, will follow the course of the 

 sling or hoop, respectively; but when li- 

 beral ed, or improperly checked, they 

 will fly off in a right line, which they 

 must preserve, if not opposed by the air, 

 &c. 



We invariably find, that, when a boat 

 is pushed off from the shore, a certain 

 bias towards the place quitted is felt by 

 every person on board. If any thing 

 should be overset at that instant, unless 

 pressed towards any other point, it will 

 fall towards that shore. On arriving at 

 the opposite bank, if the boat is allowed 

 to run against it, a disposition to fall to- 

 wards that bank will be manifested by 

 every person, and by every matter at li- 

 berty, within the boat. Hend we find, 

 that all bodies at rest are disposed to re- 

 main so ; and that, when bodies are set in 

 motion, they would continue to move, 

 were they not obstructed by either a me- 

 chanical, or an invisible, agent. All bo- 

 dies moving in orbits have a disposition 

 to fly out of them ; and those which de- 

 scribe orbits of the smallest diameter have 

 rotatory motions quicker than those 

 which take a greater range. If one body 

 moves round another, both will describe 

 curves round their common centre of 

 gravity. The centrifugal force of a re- 

 volving body is in direct proportion to 

 the quantity of matter multiplied into the 

 velocity. The centripetal forces in cir- 

 cles are as the squares of the velocities 

 directly, and of the radii inversely .- 

 therefore, when the centripetal force, 

 and the distance from the centre, are gi- 

 ven, the velocity is given. The mutual 

 attraction of bodies does not affect their 

 centre of gravity ; and if, while two bo- 

 dies act on each other, they be projected 

 in opposite and parallel directions, with 

 velocities in proportion to their respec- 

 tive distances from the centre of gravity, 

 they will describe similar figures around 

 that centre. It is on these principles 



Mm 



