Friction affects the Motions of Time-Keepers. 155 



escapement accelerates; but when it is removed nearer A, the 

 escapement retards the rate of the chronometer. 



It is almost unnecessary to remark, that the foregoing pro- 

 positions enable us either to compute the influence of such an 

 escapement on the rate of a clock pendulum, or to place the arc 

 of impulse so that that influence may be zero. It would indeed 

 be worth while to try to what degree of precision such a process 

 would enable us to attain ; seeing that we would thereby be 

 saved the tedious observation of the coincidences of one pen- 

 dulum with another, and that we would reap the additional ad- 

 vantage of retaining the arc of oscillation nearly uniform for a 

 great length of time. 



Another very important question here presents itself : To de- 

 termine the position of the arc of impulse, in order that a slight 

 change in the viscidity of the oil may not influence the rate of 

 the watch. The solution of this question requires merely an 

 ordinary operation in maxima, but unfortunately that operation 

 leads to an enormously complex equation, which I have not yet 

 been able to reduce to manageable dimensions. I shall give the 

 investigation as it at present stands, if no better can be made 

 out, in the succeeding Number of this Journal. 



The investigation into the motions of the common or circular 

 pendulum, present difficulties of an order higher than those we 

 have yet encountered. The friction on the knife edge is pro- 

 portional to the pressure of that edge against the plate on which 

 it rests. Now, the pressure diminishes as the cosine, of the in- 

 clination, but increases with the centrifugal force, so that one 

 part of the inquiry must refer to the retardation caused by the 

 statical part of the friction ; another to the effect of a friction 

 proportional to the square of the velocity. The latter part of 

 the inquiry may evidently be engrossed into the computation of 

 the effects of the air's resistance ; and I shall, therefore, confine 

 my attention to the first part of it. 



Let SV represent the vertical position of a pendulum, pro- 

 duce it till SG represent gravity, and erect the perpendicular 

 GF to represent the value of the friction when the pendulum 

 presses the knife edge against the plate with its whole weight. 

 Suppose the pendulum to be turned aside into the position SA ; 



