CENTRAL FORCES. 



font on their posts ; neither are they to 

 sing, smoke, or suffer any noise to be made 

 near them. They are not to sit down, 

 lay their arms out of their hands, or sleep ; 

 but keep moving about their posts during 

 the two hours they stand, if the weather 

 will allow of it. No centry to move more 

 than 50 paces to the right, and as many to 

 the left of his post; and let the weather be 

 ever so bad, he must not get under any 

 other cover but that of the centry box. 

 No one to be allowed to go from his post 

 without leave from his commanding offi- 

 cer ; and to prevent desertion or maraud- 

 ing, the Gentries and videttes must be 

 charged to let no soldier pass. 



CENTRAL forces, the powers which 

 cause a moving body to tend towards, or 

 recede from, the center of motion. 



If a body A (plate III. Miscel. fig. 10.) 

 be suspended at the end of a string A C, 

 moveabie about a point C, as a centre, 

 and in that position it receive an impulse 

 in an horizontal direction, it will be there- 

 by compelled to describe a circle about 

 the central point. While the circular mo- 

 tion continues, the body will certainly en- 

 deavour to recede from the center, which 

 is called its centrifugal force, and arises 

 from the horizontal impetus. With this 

 force it acts upon the h'xed center-pin, 

 and that, by its immobility, re-acts with 

 an equal force on the body, by means 

 of the string, and solicits it towards the 

 center of motion; whence it is called 

 the centripetal force ; and when we speak 

 of either or both indefinitely, they are 

 called the central forces of the revolving 

 body. 



The doctrine of central forces makes 

 a considerable branch of the Newtonian 

 philosophy, and has been greatly cultiva- 

 ted by mathematicians, on account of its 

 extensive use in the theory of gravity, 

 and other physical and mathematical 

 sciences. 



In this doctrine it is supposed, that 

 matter is equally indifferent to motion or 

 rest ; or that a body at rest never moves 

 itself; and that a body in motion never of 

 itself changes either the velocity or the 

 direction of its motion ; but that every 

 motion would continue uniformly, and its 

 direction rectilinear, unless some exter- 

 nal force or resistance should affect it, or 

 act upon it. Hence, when a bodyat rest 

 always tends to move, or when the veloci- 

 ty; of any rectilinear motion is continually 

 accelerated or retarded, or when the di- 

 rection of a motion is continually changed, 

 and a curve line is thereby described, it 

 Is supposed that these circumstances pro- 



\OT, m. 



ceed from the influence of some power 

 that acts incessantly ; which power may 

 be measured, in the first case, by the 

 pressure of the quiescent body against 

 the obstacle which prevents it from mov- 

 ing, or by the velocity gained or lost 

 in the second case, or by the flexure of 

 the curve described in the third case ; 

 due regard being had to the time in 

 which these effects are produced, and 

 other circumstances, according to the 

 principles of mechanics. Now the power 

 or force of gravity produces effects of 

 each these kinds, which fall under our 

 constant observation near the surface of 

 the earth : for the same power which 

 renders bodies heavy, while they are at 

 rest, accelerates their motion when they 

 descend perpendicularly ; and bends the 

 track of the motion into a curve line, 

 when they are projected in a direction 

 oblique to that of their gravity. But we 

 can judge of the forces or powers that 

 act on the celestial bodies by effects of 

 the last kind only. And hence it is, that 

 the doctrine of central forces is of so 

 much use in the theory of the planetary 

 motions. 



Sir Isaac Newton has tr ited of central 

 forces in his Principia, and has demon- 

 strated this fundamental theorem, viz. that 

 the areas which .evolving bodies describe, 

 by radii drawn to an immoveable centre, 

 lie in the same immoveable planes, and 

 are proportional to the times in which 

 they are described. 



The theory of this species of motion is 

 comprised in the following propositions. 

 1 When two or more bodies revolve at 

 equal distances from the center of the 

 circle they describe, but with unequal 

 velocities, the central forces necessary 

 to retain them will be to each other as the 

 squares of their velocities. That is, if one 

 revolves twice as fast as the other, it will 

 require four times the retaining force the 

 other does ; if with three times the ve- 

 locity, it will require nine times the force 

 to retain it in its orb, 8cc. 



2. When two or more bodies move with 

 equal velocities, but at unequal distances 

 from the center they revolve about, their 

 central forces must be inversely as their 

 distances. That is, by how many times 

 greater the distance a body revolves at 

 is from the center, so many times less 

 force will retain it. 



3. When two or more bodies perform 

 their revolutions in equal times, but at 

 different distances from the center they 

 revolve about, the forces requisite to re- 

 tain them in their orbs will le to eacfo other 



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