124 Professor Airy on the Disturbances of Pendulums, 



for clocks, with this exception, that the force varies as the 

 distance from the middle position. Putting ex for this force, 

 and observing that it retards the balance from the distance c to 

 a, and then accelerates it from a to o, we find for the pro- 

 portionate increase in the time 



— ^-5 I - a- - + — • sin ^^— ) : 



■Kii-a^ V 2 2 a 2 / 



which, if c be not very large, is nearly equal to 



e ec^ 



2n i'Tvna 



The first of these terms is large, but invariable : the second, which 

 is variable, is not so small as at first sight it appears. For it will 



be found that the arc of semi-vibration is increased by ^ .. : and 



•' 2n"a 



this must be equal to A, the quantity by which the arc of semi- 

 vibration is diminished by friction, &c. The product ec"- is, there- 



ec' 

 fore, of th^ same order as A, and, therefore, the term r^ is of 



the order cA, which is not to be neglected. The only variable 

 quantity in this term is a : if from an alteration in the quantity 

 of friction, &c., the length of the arc be increased, the time of 

 vibration will be increased, but c will be a multiplier of the ex- 

 pression for the increase. This, therefore, so far as the theory 

 is concerned, is a pretty good construction. Graham's cylinder 

 escapement, and Mudge's lever escapement (now extensively used 

 under the name of the detached lever), possess almost exactly 

 the same properties as the dead-beat escapement of clocks, and 

 are, therefore, very good. The duplex escapement is an instance 

 of a class differing from all those which we have mentioned in 

 one important respect — the maintaining power acts on the balance 

 only once in two vibrations. For the rest, its action (like that 



