Short Arc of Vibration Jbr Pendulums. 139 



Conceive now, that, by an augmentation of the weight of the 

 pendulum, and a consequent increase of friction on the knife 

 edge, the maintaining power becomes sufficient for a sweep of 

 only one degree on each side of the vertical Hne ; and, having 

 again adjusted the clock to go in true time, let the same dimi- 

 nution take place in the maintaining power, the arc of vibration 

 will then be contracted to 19-20ths of a degree on each side, 

 while the daily rate will experience an acceleration of (y'130, or 

 rather less than the 400th part of the former. 



When we consider, then, the mere action of gravity, the su- 

 periority of the heavy pendulum with the small arc of vibration, 

 over the light pendulum with the long sweep, is obvious ; the 

 variation in the state of the oil, and the other inequalities of the 

 escapement exerting less influence on the former in a ratio du- 

 plicate of that of the arcs of vibration. 



That part of the error of a clock's going which arises from 

 variations in the buoyancy of the air, attaches alike to all pen- 

 dulums of the same material; but that part which arises from 

 the variable resistance of the atmosphere is much less felt on the 

 heavy pendulum. Returning to our former case, the velocity 

 of the heavy pendulum will be twenty times less than that of the 

 light one, so that the resistance of the air on a given extent of 

 surface will be 400 times less ; the quantity of surface, however, 

 is 7368 times greater, while the distance through which the re- 

 sistance acts is 20 times less, so that, in all, the influence which 

 that resistance exerts in counteracting the maintaining power is 

 lessened 1085 times ; and hence that irregularity in the going of 

 the clock, which arises from the variable resistance of the air, 

 will be less 400 times 1085, or upwards of 400,000 times. 



On account of the increased weight, the friction on the knife- 

 edge is increased twenty times, while the distance through which 

 it acts is diminished as often, so that, in the case of the heavy 

 pendulum, the friction on the knife-edge interferes with the 

 maintaining power just as much as in the case of the light one; 

 variations, then, in this friction, will only produce the 400th 

 part of the disturbance on the clock's rate when the heavy pen- 

 dulum is used ; the edge, however, will require to be somewhat 

 strengthened, so that this circumstance will interfere a little with 

 these proportions. 



