and on the Theory of Escapements. 119 



at all in maintaining the arc of vibration, but always diminishes 

 the time. The latter, by Example 8, if it be equal on both sides 

 of the lowest point has no effect on the time of vibration, but 

 increases the arc of vibration if there be no resistance, or maintains 

 it if there be resistance. The former of these, then, is of no use 

 whatever, and is prejudicial as affecting the rate of the clock : 

 the latter does not affect the rate, and fulfils the office of an 

 escapement by maintaining the motion of the pendulum. Thus 

 we see that the principle of this escapement is radically bad. 

 The force during the greater part of its action is disturbing the 

 rate of the clock without maintaining the motion of the pen- 

 dulum. 



If the pallets have such a form that the force is constant 

 and = F,. we find by Example 9, that the time of vibration is 

 diminished by 



The differential coefficient of this quantity with respect to a is 



— 2F d'-2c^ 



Hence it appears that the vibrations are quicker than they would 

 be without the maintaining force: but that if from a diminution 

 of friction, &c. the arc be increased while the maintaining force 

 remains the same, the vibrations are slower. If while the arc 

 remains the same the maintaining force be increased, the vibrations 

 are quicker. 



If the resistance varied as the square of the velocity, the 

 diminution of the arc from that cause (see Example 3.) would be 



