Friction affects the Motions of Time-Keepers. 133 



nates of an arc described from M, with the radius MD, would 

 clearly represent the velocities at different points of that por- 

 tion of the oscillation. Supposing C to be the point at which 

 the maintaining pressure ceases to act, CE would be the velo- 

 city which the balance has when left to the agency of the fric- 

 tion and of the balance spring. The remainder of the oscilla- 

 tion has f for its vertex, so that, since the motion is to cease at 

 A, the distance fE must be equal to fA. The point E, then, 

 is determined by the intersection of the arc described from M, 

 with the radius MD, and that described fromj^ with the radius 

 fA. The perpendicular drawn from E thus indicates the ter- 

 mination of the arc of impulsion. 



The extent of the arc of impulsion can be determined from 

 the principle of virtual velocities ; and must be the fourth pro- 

 portional to the maintaining pressure, the friction, and the en- 

 tire extent (twice AA 7 ) of the oscillation.* The time of the 

 description of BC will be represented by the angle DME, 

 while that of the description of C A will be measured by E/A. 



It thus appears that the entire time of return from A, to A 

 is represented by the sum of the three angles A y fD, DME, 

 and E fA. But, had the maintaining pressure not acted, that 

 time would have been represented by a half revolution, or by 

 the sum of the three angles AjfD, D/E, and E/A ; whence 

 the acceleration caused by the maintaining power is propor- 



* This leads to the following 

 beautiful proposition, which I do 

 not recollect of ever having seen 

 before : — 



Porism. 



If from any eccentric point M an 

 arc be described cutting two con- 

 centric circles, twice the rectangle i£ 

 under the excentricity M/, and the 

 distance between the chords T)d and 

 F,e of the intercepted arcs, is equi- 

 valent to the difference between 

 the squares of the radii of the con- 

 centric circles, and that indepen- 

 dently of the radius of the secant 

 circle. 



