634 Mn. DENISON, ON CLOCK ESCAPEMENTS. 



Then if a be the extreme value of 6, y the angle at which the impulse begins, and y on the 

 other side of zero at which it ends, Mr. Airy shews that A, the increase of the time of an oscil- 

 lation due to the escapement, 



TTO 



= ~-^ (7 + 7) (7 - 7) nearly, 

 -i 7r ct 



if y and 7 are so small, that -^ may be neglected. 



a 



Mr. Airy remarks : " This is a quantity extremely minute ; for 7 and 7' are generally small, 

 and 7' - 7 may be made almost as small as we please. It cannot, however, be made absolutely ; 

 for the wheel must be so adapted to the pallets, that when it is disengaged from one it may strike 

 the other not on the acting surface, but a little above it ; therefore 7' must be greater than 7 ; but 

 the difference may be made so small that the effect on the clock's rate shall be almost impercep- 

 tible. This escapement therefore approaches nearly to absolute perfection ; and in this respect theory 

 and practice are in exact agreement." 



Since A is only the increase in the time of one vibration, and there are 86,400 vibrations in a 

 day, (assuming the clock to have a second's pendulum,) and a second a day is a large error, it is 

 worth while to see what A really is. If fVg be the clock-weight, and h its fall in a day ; then, since 

 p (^ + y') tan /3 is the thickness of the pallets, or the drop of a tooth in one beat, 



Pp Wh 



_(7 + 7') tan /3, or ,^ (7 + 7') = W^ei^o' 



and this quantity (which we may call F), will be the same for all clocks of the same Hind, whatever 

 /3 or 7 + 7' may be ; and 



Now a weight of 2lbs. falling 9 inches a day will keep a well-made clock of this kind vibrating 

 2" on each side of zero, ^s 39 inches, and M is usually about Ulbs. Therefore (allowing nothing 

 for the friction of the train), 



2x9 .033 



^ = 



14. >i 39-< 86400 86400 



.005 'y — 'v . 



and 86400 A = smce a = 2° = .035. 



.001 a 



I understand from clockmakers that 7' - 7 can hardly be made less than 20', and is seldom so 



little; .-. "^ ~'^ =-, and 86400 A = .8 of a second, nearly. This is the amount of A in a day 



a 6 



But it is not the error of the clock, being only the difference between the rate of a free pendulum 

 and one disturbed by this escapement. The error, or, as it is called, the " rate,^'' of the clock, 

 with the sign changed from what it would naturally have, is the variation of A, which depends 

 on the variation of a and of F, according to the friction of the train and the pallets. 



Differentiating A with regard both to a and F, 



