156 OF DEFLECTIVE FORCES. 



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. 



Scholium. The near approach of these revolutions 

 to isochronism has sometimes been applied to the measure- 

 ment of time, but more frequently, and more successfully, 

 to the regulation of the motions of machines. Thus, in 

 Mr. Watt's steam engines, two balls are fixed at the 

 ends of rods in continual revolution, and as soon as the 

 motion becomes a little too rapid, the balls rise consider- 

 ably, and turn a cock, which regulates the quantity of 

 of steam admitted. 



293. Theorem. " 265/' The vibrations 

 of a cycloidal pendulum will be performed in 

 the same time, whether they be without re- 

 sistance, or retarded by a uniform force. 



Let the relative force of 

 gravity, at the distance AB in 



XE — H_ — the curve from its lowest point, 



be always represented 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 neutral 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 the simple pendulum in a smaller arc, 

 but it will extend only to G, where the ordinate HI is 

 equal to DC, and FHzzFD. In the return of the body 

 from G, the neutral point will be determined by the inter- 



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K 



