638 Mr. DENISON, ON CLOCK ESCAPEMENTS. 



And we see that the clock will gain if a be either increased or diminished from y \/2. Therefore if 

 the pendulum be adjusted when the clock is clean to vibrate 2' or s' more than y^/s, the arc may 

 diminish (and it will never spontaneously increase) as much as 5 or 6' with even a much less error 

 than that above deduced, which it is to be remembered was already too large in consequence of our 

 assuming the same maintaining force as in the dead escapement. 



I have omitted in this calculation the effect of the impact of the pendulum against the arms, and 

 the small friction at unlocking, as I found in the calculations which I made retaining them, that 

 they only introduced a very small term of a lower order than (p. 



There is another form of the gravity escapement, in which, instead of one arm being taken 

 up just as the other is left, the pendulum is free for some space in the middle of its arc. This 

 is evidently inferior to tiie other, for if the arc varies, the proportion between the time during 

 which the pendulum has only its own moment of inertia, and that during which it has that of 

 one of the arms also, will vary. And it will be seen that the inferiority is still greater from 

 another cause. 



For if we put -y for the angle at which each arm is met by the pendulum, and f3 for that to 

 which it descends ; then for the two portions of the arc in which the pendulum is acted on by the 

 arms, we may integrate the same expression as before, only from y to a, and down again to y3 ; 



d A _ _^ f g' - 2 /3^ g- - 2 7- 1 



ft' s/a' - y' 



One value of /3 and 7 that will make this = is evidently /3 = =t 7 = — ^; but if ^ = 7 there 



nes the former kind of clock. In order to find 



g' - 2 /3- = 2 0!- - a\ 



= 0,i{^=xy = v/g-' - /3- x/a" - y'. 

 da 2 



Since the value of — -= for /3 and 7 is useless, let us take the highest value for ^ that will leave 

 V2 

 a — /3 of a sufficient size to secure the unlocking always taking place ; which can hardly be less 

 than 30' with arms of moderate length : then /3 will be 90' and 7 = 78'. And this leaves only 12' 

 for the maintaining power to act tlirough. An escapement of this sort is therefore barely practi- 

 cable ; and in it the weight of the arms, and consequently the errors of the clock, must be much 

 larger than in one where the action takes place through 84' on each side of zero. This kind of 

 escapement, however, would do for such a clock in which the force acts on the bob of the pendulum 

 for a short distance at each extremity of the arc — the worst possible place, unless the arc through 

 which it acts satisfies the above condition. However, the object of this paper was to shew that, 

 mathematically speaking, gravity escapements may be made very superior to the dead escapement 

 with its large amount of friction and variation of arc, and to remove the cloud which has hitherto 

 lain over them in consequence of it being supposed that whatever mechanical improvements might 

 be made in them, they must remain liable to an insuperable mathematical objection. 



E. B. DENISON. 



