424 Mr DENISON, ON SOME RECENT 



the tooth on the stop keeps the arm there until the pendulum comes and carries it off; 

 and consequently the difference between the angle at which the arm begins to ascend, 

 and that to which it descends with the pendulum (which difference is the angle of impulse), 

 becomes larger than it ought to be, and the swing of the pendulum is increased. This 

 defect of most gravity escapements seems to have escaped the notice of clockmakers alto- 

 gether. 



Thirdly, the escapement should have no friction sufficient to require oil on any of 

 the parts affecting the pendulum. I do not mean that it is necessary that no oil should 

 be used, but that the friction should be so small that no difference can be observed in 

 the arc of the pendulum whether oil is applied or not. So far as any gravity escape- 

 ment fails in this respect, it descends to the condition of a common impulse escapement. 

 And on the other hand, any contrivance for diminishing the friction below the point at 

 which the pendulum is indifferent to the absence or presence of oil, is a merely super- 

 fluous refinement, and is generally made at the expense of more important objects, as 

 was the case in several of the escapements I mentioned just now. 



The fourth condition which a gravity escapement ought to satisfy, or at least not to 

 deviate far from, is the mathematical condition investigated in my former paper, which may 

 be shortly stated thus, with respect to what I there shewed to be the best form of gravity 

 escapement, viz. that in which the pendulum takes up one arm just as it leaves the other : — 

 If y is the arc of the pendulum from zero when this change of arms takes place, and a the 



extreme arc, then y ought to = — — = .7] a nearly ; for that is the proportion which makes 



the difference of the time from that of a free pendulum a maximum, and therefore makes the 

 variation of that difference evanescent for any small variation of a. And now that we know 

 how small a moving force is really required to keep up the vibration of the pendulum when it 

 is delivered from the incumbrance of friction, it will be found by calculation that the error will 



still be insignificant if -y deviates from the above proportion so far as to = - instead of . 



2 y/9. 



For if W is the clock-weight (or rather so much of it as arrives at the pendulum clear of 

 friction) and h its daily fall, and M and I the weight and length of the pendulum, then 

 the daily rate 



a_ 2 _ 

 Wh y* 2 da 



MU 



v^ 



y 



a 

 I 



In the Westminster clock Wh is certainly not above lOlbs. x 6ft., and Ml is 680lbs. x 13ft. 



Consequently, taking y = - or 1° in this last expression, and assuming a variation of arc 



2 



of 5', the daily rate due to the escapement would not be more than — sec. ; and that is much 



less than the daily rate due to the circular error corresponding to such a variation of arc, 



