112 Professor Airy on the Disturbances of Pendulums, 



Tliis ^ves f, cos'"^' n< + 6 = — ^ ^_^ ■ - when m is odd. 



and = »'-^-2....2 2 ^^^^ ^ j^ ^^^^ 

 ?n + 1 . »M - 1 2 M 



Thus the decrease of the are 



m.m-2....l /,^„».-.„,»(^odd). 



m + l . m— 1 ... .2 



^^, ^ wt->»-2....2 2 A: . n'"-' . a"' (wi even). 

 7nH-l.m-1....3 



4 

 When m = 2 this = ~ ka^. 



The proportionate increase of the time of vibration 



Ic 



= - -n^'^.a"'' .ft cos^ni + ft.sin nt + b, 



IT 



which between the same limits = o, whether m be whole or 

 fractional. 



Ex. 4. The resistance is expressed by any function of the 

 velocity. Here f= — <P {v), and the increase of the arc 



= — - ft (b(v) . cos nt + b 



to be taken from 



nt + b = ~ - to nt + b = -. 

 Since v = na cos nt + b, -j^ = - n'a .sin nt + b, 



and the increase of the arc 



_ ^ r w. ^ ^ cos w< + 6 - _L r ^^(") 

 - n'a J' "P ^"^ sin »^«Ti " «'« •^' ^nW-v' ' 



