108 Professor Airy on the Disturbances of Pendulums, 



on tlie supposition that a and b are constant. As a and b are 

 variable, this process is erroneous. But as their variation depends 

 on /, the error depends on /\ or on quantities of that order. 

 Our approximation then will include all terms depending on 

 the first power of/, and no more, an approximation sufficiently 

 exact for all the cases to which we shall have to apply it. 



Since the expression for the distance of the pendulum from 

 its lowest position, and the expression for the velocity of the 

 l>endulum, are the same as those in an undisturbed cycloidal 

 pendulum where « is the extent of the vibration, on each side, 



and - the time which elapsed from the instant at which the 

 n 



pendulum passed its lowest position to the instant from which t 



is measured, it is plain that if the disturbing force ceased, the 



l)endulum would move in the same manner: that is, the extent 



of its vibration would be a, and it would move as if it had 



passed its lowest point at the time - - , a and b having the 



values which they had at the instant when the force ceased to 

 act. And generally, considering a and b as functions oft, the time 

 of arriving at the lowest point will be determined by making 



sin n< + i = 0, 

 and the time of reaching the highest point by making 



cos nt + b = o. 



In order to find the alteration in the length of the arc of 

 vibration which takes place in one oscillation, we must integrate 



— cos nt + b through the limits of t corresponding to one os- 

 cillation ; that is, from a value of <, which gives nt-^b—a, to the 

 value of t, which gives nt + b = a -\- ir. Here « may be any thing 



