OP THE CENTRE OF IKF.aTIA, AND OF MOMENTUM. 



35 



chord of the generating circle, the di- 

 ameter being I ; and the arc being 

 perpendicular to that chord, its flux- 

 ion, by similar triangles, is to that of 

 the absciss as the diameter to •/}/ : 

 therefore the cycloid answers the con- 

 ditions in every part, and consequently in the whole curve. 

 Scholium. The demonstration implies that the origin 

 ■of the curve must coincide with the uppermost given point : 

 row only one cycloid can fulfil this condition and pass 

 through the other point, and it will often happen that the 

 curve must descend below the second point and rise again. 

 26'2. Theorem. The time of vibration 

 of a simple circular pendulum in a small arc 

 is ultimately the same as that of a cycloi- 

 dal pendulum of the same length ; but in 

 larger arcs the times are greater. 



For in small cycloidal arcs the radius of curvature is 

 nearly constant, but, at greater distances from the lowest 

 point, the circular arc falls without the cycloidal, and is 

 less inclined to the horizon. 



263. Theorem. If a body suspended by 

 a thread revolve freely round the vertical 

 line, the times of revolution will be the same 

 when the height of the point of suspension 

 above the plane of revolution is the same, 

 whatever be the length of the thread. 



For by the resolution offerees, the force urging the body 

 towards the vertical line is to that- of gravity as the distance 

 from that line to the vertical height ; the other part of the 

 force being counteracted by the effect of the thread ; and 

 when the forces are as the distances, the times are equal 



(242). 



264. Theorem. The time of a revolu- 

 tion of a body suspended by a thread is equal 

 to the time occupied by a cycloidal pendulum 

 of which the length is equal to the height of 

 the point of suspension above the plane of re- 

 <Brution,in vibrating once forwards and once 

 backwards to the point at which its motion 

 began ; and if the revolutions be small, and 

 the thread nearly vertical, they will be nearly 

 isochronous, whatever be their extent. 



For, supposing the distance equal to the height, the cen- 

 tral force will be equal to the force of gravity, and while the 



body describes a distance equal to the radius, another body 

 would fall through half that radius (240), and the whole tim* 

 of revolution Is therefore to this time as the circumference to 

 the radius, and is therefore equal to the time of four semi- 

 vibrations of a cycloidal pendulum of which the length is 

 equal to the given height. And since the time varies, in 

 the same revolving pendulum, only as the square root of the 

 cosine of the angle of inclination, it will be nearly constant 

 for all small revolutions. 



265. Theorem. The vibrations of a cy- 

 cloidal pendulum will be performed in the 

 same time, whether they be without resist- 

 ance, or retarded by a uniform force. 



Let the relative force of 

 gravity, at the distance AB ^ 

 in the curve from its low- 

 est point, be always repre- 

 sented by the ordinate 

 AC ; then CB will be a right line : now the resistance may 

 always be represented by the equal ordinates AD, BE, and 

 DC will express the remaining force, which becomes neu- 

 tral at F, and then negative : therefore the force is always 

 the same at equal distances on each side of F, as in the 

 simple pendulum on each side of B, and the vibration will 

 be perfectly similar to the vibration of th.e simple pendulum 

 in a smaller arc ; but it will extend only to G. In the 

 return of the body from G, the neutral point will be deter- 

 mined by the intersection of HI parallel to AB, and as much 

 below it as DE was above il : this vibration will terminate in 

 a point as much above H as G is below it : so that the ex- 

 tent of each vibration will be less than that of the preced- 

 ing one by twice the length of FE, until the whole force 

 is exhausted, the time remaining unaltered. 



SECTION VI. OF THE CENTRE OF INERTIA, 

 AND OF MOMENTUM. 



266. Definition. A moveable body, is 

 to be imagined as a point, composed of 

 single points or particles equally moveable, 

 which, as they differ in number, constitute 

 the proportionally different mass or bulk of 

 the body. 



267. Definition. A reciprocal action 

 between two bodies is an action whicli affects 



