Friction affects the Motions of' Time-Keepers. 137 



If A denote the inclination at the beginning, A n that at the 

 end of n oscillations, and <p the angle GSF, we have 

 A — - A = 2n<p 



n r 



and thus the value of the angle <p can be obtained by direct ex- 

 periment. 



From this proposition, it is clear that the friction does acce- 

 lerate the oscillations of a pendulum ; but then, in every well 

 made instrument, the value of the angle GSF is so exceedingly 

 small, that its secant will not differ, by any appreciable quan- 

 tity, from the radius, and thus the usual deductions are left 

 intact. 



Having now ascertained the extents of the oscillations, it re- 

 mains to compute the time during which any given number of 

 them will be performed. For this purpose I must have recourse 

 to the integral calculus. The algebraic method would indeed 

 have easily conducted us to the conclusions already arrived at ; 

 but from a partiality to the perspicuity of actual representation, 

 as well as from a wish to render this important subject as gene- 

 rally known as possible, the geometric form has been preferred. 



Let I be the length of the pendulum, a the angle which at 

 any time it makes with the vertical line, A the value of a at the 

 commencement of the oscillation, and <p the angle of friction, we 

 have 



^~ = — g { sin a — cos a tan 



0 t 



= — g sec (p {sin a cos <p — cos a sin <p} 

 — — gsecip. sin (a — cp) 



But v =l~ whence 



d t 



v } v = — g I sec <p . sin (a — <p) % a 



Or integrating 



v 2 = l2^}±y = __2glsecq>. ver(a — <p) + C 



Now the velocity is zero when a=r. A whence 



!£«Ef { ver(A-*)_ver( a _,) } 



2 V gsecf \\ 2 / V 2/J 



