and on the Theory of Escapements. 115 



If then an impulse be given when the pendulum is at its lowest 

 point, c is 0, and the time of vibration is unaffected. 



Ex. 8. A force / which is equal at equal distances from the 

 lowest point on both sides accelerates the pendulum. By the 

 general formulae it will be found that the action on both sides 

 of the lowest point tends to increase the arc : but that the action 

 before reaching the lowest point tends to diminish the time of 

 oscillation, and that after it to increase the time, and that on the 

 whole the time of oscillation is not altered. 



Ex. 9. A force M which is equal at equal distances retards 

 the pendulum as it ascends from the distance c to its highest 

 l)oint, and accelerates it as it descends to tire same jilace. Upon 



taking —^f^M from x = c till x again = c, we find that the length 



of the arc is not altered. In rising the time of vibration is in- 

 creased by 



l_ r Mx (x = c\ 



vc'd^'J '' ^dr-x- U = a^ ■ 



To find the effect produced as the pendulum descends we must 



remark that si o: — x^ was introduced as equal to a cos nt -j- h, 

 which then becomes negative; and the I'adical must therefore 

 be taken with a negative sign. We must, therefore, take 



^ r Mj? tx = ay _^_ r Mx ix = c]^ 



~ ^'^ ''' ~ JlF^^x' \x = c\' ^^ ~ Wa'^ ^ >/^^^^ \x = a\' 



The whole decrement in the time is, therefore, 



2 p Mx ,x = c\ 



A force of this kind then does not alter the arc of vibration, but 

 tends during the whole of its action to diminish the time. 



p 2 



