and on the Theory of Escapements. 109 



that we please : in some cases it will be convenient to take the 

 integral from one extremity of the vibration to the other: in 

 others it will be preferable to take it from the time at which 

 the pendulum passes its lowest position to the time at which 

 it again arrives there. In some cases it will be necessary to 

 integrate for two vibrations. 



To find the alteration in the lengf^h of time occupied by 

 a vibration, produced in one oscillation, let B be the value of h 

 at the first limit, and B' that at the second : and let T and T' 

 be the times. Then the first time at which the pendulum passes 

 its lowest point is found by making nT + B = O: the second time 

 is found by making nT' + B' = w. Hence 



n{T'-T) + B' - B^^, and T'-T=- -:?^—^. 

 But T' - T - time occupied by one vibration : and 



between the proper limits. Consequently the time of oscillation 

 is increased from 



to - + A ~ sin nt i-b: 



n n ^ ' ria 



it is therefore increased by the proportional part 



ftf. sin nt + b. 



Recapitulating, then, we have 



increase of arc of .semi-vibration = - /(./. cos nt + b, 



n-' " 



proportionate increase of time of vibration = ftf. sin nt-\-h. 



ttTICI 



