Mr. DENISON, ON CLOCK ESCAPEMENTS. 



635 



dA='^ — '(dF ■ 



3F(y'- 7) 



da. 



or the "daily rate" = - .8= (^ - — ] . 



We see, therefore, that the real merit of this escapement arises from the two causes of error tending 

 to counteract each other ; for, though no exact relation can be determined between the chano-es of 

 the arc and of the force, since they depend on the changes in the friction of different parts of the 

 clock, yet it is easy to see that a will diminish when F does, under the influence of increasing 



friction as the clock gets dirty. It appears that — is not generally so much as and 



a 3 F ^ 



therefore the clock gains as the arc diminishes. Moreover, the circular error, which is never com- 

 pletely corrected by the pendulum-spring, I understand, tends to make the clock gain as the arc 



diminishes ; since d A for the circular error = , as may be seen from any book on pendulums. 



I have in one instance seen the contrary effect take place, where a church-clock, soon after it was 

 put up, spontaneously increased its arc by more than a degree, from the pallets polishing themselves 

 more perfectly than had been done by the maker, and at the same time it gained considerably, as 

 we see it ought to have done. The tendency to gain as the arc diminishes has led to the practice 

 of making turret-clocks, which are liable to great changes both in the force and the arc, with a 

 slight recoil in the place of the dead part of the pallets, as the effect of the recoil is to diminish 

 the time as the arc increases. 



The principle of nearly all the gravity or remontoir escapements is this : There are two small 

 arms three or four inches long on each side of the pendulum suspended separately on an axis coinci- 

 dent with that of the pendulum and moving in the same plane with it : these arms carry a small 

 weight at their lower ends, and also a detent to stop a tooth of the escape-wheel, and a pallet of some 

 kind by means of which the arms are alternately raised by the wheel at every beat. The pendulum 

 in ascending, at an angle y from the vertical, impinges on one of the arms, unlocks the wheel and 

 carries the arm with it as far as it swings ; the arm then descends with the pendulum, not only to y, 

 but farther to an angle /3, less than y. The maintaining force of the arms therefore acts on the 

 pendulum through y — fi, and the work which the clock has to do is raising the arms from /3 to -y. 

 This is the way in which these escapements have been usually made, I suppose with the view of keep- 

 ing the pendulum free during as much of its arc as possible ; but we shall see that it is much better 

 to make /3 = - 7, or one arm to be taken up by the pendulum just when the other is left ; and as 

 it is also more simple, I shall consider that case first. 



In order to find the errors of such an escapement, let p be the length of the arms supposed to 

 be without weight ; Pg the weight they carry at their lower end ; S the angle which the arm in con- 

 tact makes with the pendulum when it is vertical. Then the equation of motion will be 



df- I 



[e^^il^o)] 



1 + 



pj/ 



MP 



= 0. 



We may, in considering the error in the going of the clock, neglect the denominator I + 



Pf 



Ml"' 



not 



only because it very nearly =■ I, but because it only causes a permanent change in tiie effective 



• It muBt be remembered that experiments of altering the 

 dock'Weight, to find what cfi'cct in produred on the arc, do not 

 reprenciit what taken place in the elock naturally ; for when the 



clock is left to itself the arc varies probably more from the varyinjj 

 friction and slate of the oil on the pullets than frutii the chunffc of 

 force in the train. 



