Friction affects the Motions of Time-Keepers. 1 31 



Supposing A to be the point at which the motion commen- 

 ced, the body would descend with increasing velocity until it 

 reached the point F ; there its motion would begin to be re- 

 tarded, and it would finally be arrested at a distance FA, equal 

 to FA. As soon as the returning oscillation is commenced, the 

 friction changes its direction, the vertex of the motion is trans- 

 ferred toy, and the distance t /A 2 measured equal to fA, defines 

 the limit of the oscillation. Denoting by A the extreme dis- 

 tance at the beginning, and by A n that at the end of n oscilla- 

 tions, also putting <p for the arc VF, we have 



A — A n = 2n<p, 

 as is evident from the above construction : In words, The ex- 

 tents of the oscillations decrease in arithmetical progression. 



As the motion of the cycloidal pendulum represents iso- 

 chronous motions in general, it follows that a constant friction, 

 while it diminishes the extents of the oscillations in arithmetical 

 progression, leaves unaltered the time in which each oscillation 

 is performed ; and there would result this remarkable conclu- 

 sion, that variations in the viscidity of the oil should not in- 

 fluence the rate of a chronometer. Such, in fact, would be the 

 case were it not that the maintaining power supplies, at some 

 part of each oscillation, the motion which has been consumed 

 by the friction during the rest of it. 



Before proceeding to consider the effect of the stroke of the 

 escapement, I may recal to the recollection of those who have 

 not lately been examining this subject, the well-known law of 

 isochronous oscillations. 



Conceiving the path of the oscillating body to be expanded 

 into a straight line, let AVA, — 

 be the entire extent of the mo- 



tion, and suppose a to be an / \ \ 

 intermediate position. Having / \ \ 



described a semicircle on A A,, / \ \ 



erected the perpendicular aB, A a v a 

 and joined VB, «B will represent the velocity acquired at a, 

 while the angle AVB will be proportional to the time of de- 

 scription of the path Aa, and the supplement BVA to the time 

 of describing aA r 



i % 



