636 



Mr. DENISON, ON CLOCK ESCAPEMENTS. 



length of the pendulum, since one of the arms is, in this form of escapement, always acting on the 

 pendulum. And expanding sin (^ + G) we may put 1 for cos 6, and 9 for sin 6 as before, 



d-e gif Pp cos S\ ^ Pp sin ^) 



which is of the form -— + - {mO + <p) = ; 



/I e- w' 



.-. if d) were = the time would = tt \/ , or for a second's pendulum ^ must = — = tt'" say, 



' gm I m ^ 



which is a very little less than tt^ The only part of the force which produces an effect involving 

 the arc is tt'^c/), and it is a constant force. Therefore we may apply to it Mr. Airy's expression for 

 the increase of time due to such a force acting from a down to - -y ; and we have 



and it is the same from - -y to the other extremity of the arc ; therefore for the whole oscillation 



20 ,- • 



vra 



This is, in fact, Mr. Airy's result for a recoil escapement ; and if the pallets of a recoil escape- 

 ment were made of any regular form, so that we could separate the force into one part varying as the 

 arc and the other part constant, it would be the same thing as a gravity escapement, only with 

 much greater friction, and the important difference, that the force depends upon the train, whereas 

 in a gravity escapement it is independent, and therefore uniform. Mr. Airy proceeds to remark, 

 that " the differential coefficient of this quantity with respect to a is 



2(p a' - 27' dA 



Hence it appears that the vibrations are quicker " than they would be without the maintaining force ; 



but if the arc be increased while the maintaining force remains the same, the vibrations are slower. 



If while the arc remains the same the force be increased, the vibrations are quicker." 



a d A 

 But something else appears also : viz. the important fact, that if 7 be made = —7^5 -j — = 0, 



provided the force remains the same, as it does in a gravity escapement. And luckily this is a per- 

 fectly practicable value for 7, though it is larger than a clockmaker would probable make it 

 without knowing anything of this result ; for if a = 120', 7 will = .7 x 120' = 84', and a - y, 

 or the space in which the unlocking has to take place = 36', which with p = 4 or 5 inches will do 

 very well for a clock which is liable to such small changes of arc as these are. Therefore a gravity 

 escapement may be made, in which the error will be nothing for a small alteration of the arc ; and 

 in such an escapement there is no such variation in either force or friction as can cause any material 

 change in the arc. 



In a recoil escapement we should have to differentiate A with respect both to a and <p ; 



2(h(a'-2y' da , ddTi 



