466 



THE STEAM-ENGINE. 



vibrations be attentively considered, it will be perceived that the effect on the 

 ball will be a circular motion, precisely similar to the circular motion of the 

 balls of the governor already described. 



Now the time of vibration of the pendulum S P between east and west will 

 not in any way be affected by the second vibration, which it is supposed to 

 receive between north and south, and therefore the time the pendulum takes 

 in moving from P to P' and back again from P' to P will be the same whether 

 it shall have simultaneously or not the other vibration between north and south. 

 Hence it follows that the time of revolution of the circular pendulum will be 

 equal to the time of similar vibrations of the same pendulum, if, instead of 

 having a circular motion, it were allowed to vibrate in the manner of a common 

 pendulum. 



If this point be understood, and if it also be remembered that the time of 

 vibration of a common pendulum is necessarily the same whether the arch of 

 vibration be small or great, it will be easily perceived that the revolving 

 pendulum or governor will have nearly the same time of revolution whether it 

 revolve in a large circle or a small one : in other words, whether the balls 

 revolve at a greater or a less distance from the central spindle or axis. This, 

 however, is to be understood only approximately. When the angle of diverg- 

 ence of the balls is as considerable as it usually is in governors, the time of 

 revolution at different distances from the axis will therefore be subject to some 

 variation, but to a very small one. 



The centrifugal force (which is the name given in mechanics to that influence 

 which makes a body revolving in a circle fly from the centre) depends con- 

 jointly on the velocity of revolution, and on the distance of the revolving body 

 from the centre of the circle. If the velocity of revolution be the same, then 

 the centrifugal force will increase in the same proportion as the distance of 

 the revolving body from the centre. If, on the other hand, the distance of the 

 revolving body from the centre remain the same, the centrifugal force will in- 

 crease in the same proportion as the square of the time of vibration diminishes, 

 or, in other words, it will increase in the same proportion as the square of the 

 number of revolutions per minute. It follows from this, therefore, that the 

 greater is the divergence of the balls of the governor, and the more rapidly 

 they revolve, the greater will be their centrifugal force. Now this centrifugal 

 force, if it were not counterbalanced, would give the balls a constant tendency 

 to recede from the centre ; but from the construction of the apparatus, the 

 further they are removed from the centre the greater will be the effect of their 

 gravitation in resisting the centrifugal force. 



It is evident that the ball at P will have a greater tendency to fall by gravita- 

 tion toward O than it would have at p, because the acclivity of the arch descend- 

 ing toward O at P is greater than its acclivity at p. The gravitation, there^ 

 fore, or tendency of the ball to fall toward the central axis being greater at P 

 than at p it will be able to resist a greater centrifugal force. This increased 

 centrifugal force, which the ball would have revolving at the distance P above 

 what it would have at the distance p, is produced partly by the greater distance 

 of the ball from the central axis, and partly by the greater velocity of its 

 motion. But it will be evident that the time of its revolution may neverthe- 

 less be the same, or nearly the same, at both distances. If it should appear 

 that the actual velocity of its motion of revolution at P be greater than its 

 velocity at p, in the same proportion as the circles in which they revolve, then 

 it is evident that the time of revolution would be as much increased by the 

 greater space which P will have to travel over, as it will have to be diminish- 

 ed by the greater speed with which that space is traversed. The time of 

 revolution, therefore, may be the same, or nearly the same, in both cases. 



