Mr. DENISON, ON CLOCK ESCAPEMENTS. 637 



This is always negative ; for it cannot be + unless 



J— IS < ;; '- , 



da a" — y^ 



a 

 which is impossible, since (p always increases faster than a. Therefore the recoil escapement always 

 gains as the arc increases, as is well known ; and the cause of its infeiiority to either of the others 

 is evident. 



But still we want to ascertain what the error of a gravity escapement with y of the proper value 

 will amount to, for some definite value of da, which the clock is not likely to exceed. Therefore we 

 must find the value of <p. 



Now the work done by the clock-weight is raising the weight P through 



p\cos {S — y) — cos (^ +y)\ '= ~P sin S sin y, 86400 times a day. 



Then assuming W and p the same as before (though this clock evidently does not require the same 

 maintaining power as the dead escapement with its large amount of friction, I believe not half as 

 much), 



2 Pp sin d sin -y = =— = ; 



' 86400 86400 



2 2Pp am S 4x9 .01 



TT 3Ilir 7rl4 X 39 X 864OO7 864OO7 



.-. 86400 A = - ^-7-^ \\/a^ - y'\ = - 20 sec, if — be made = 2, and a = a" 

 a"7 7 



= when y = a. 

 as a clockmaker would probably make it, in ignorance of the fact that — should = -^2 ; 



a' 



-^-2 

 . .012rfa a' - 27- 7- da 



.: 86400^ A = 5- ,^-= - = -012 \ — 



7' 



= .577 sec, for the last mentioned value of - , if da = 5'. 



7 

 This then is the daily error of a gravity escapement made, as we may say, at random, for an increase 

 of the arc of 5', remembering that we have taken (p twice as large as it need be. 



But if - is made of the proper magnitude, so as to make — - = 0, we must differentiate again, 

 a da 



and put n' = 27^, in order to find the actual error for a small increase of « : tlien we have 



d^'A .012 2 a 



86400 — -— = ; 



da' a' 7 y/„^ - y* 



, , ., .012 Qada da 1 „ .... 



or tlie daily rate = . — = — of a second, it da = H . 



7 



Vol. VIII. Paut V. 4N 



