1 10 Professor Airy on the Disturbances of Pendulums 



If the circumstances were such that it was necessary to in- 

 tegrate through two vibrations, we should have 



proportionate increase of time of vibration = ftf .sin nt + b. 



These formulae are convenient when the disturbing forces 

 (;au be expressed in terms of t. If however, they are expressed 

 in terms of x (as is the case jjarticularly in clock escapements), 



da 



da da dt dt f 



since T- - -77 . -T- = - 



2/. ' 



dx dfdx na cos nt + b «'<J 



db 



. db dt f sinn<+6 f 



and - - - ' 



dx na cos nt+b wa"' cos nt+b ntf ' Ja^-x^' 



we have 



increase of arc of semi-vibration = -j- /I f. 



n a"^ "^ 



1 f fv 

 proportionate increase of time of vibration = — 3—, / : ' . 



Example 1. Instead of vibrating in a cycloid, the pendulum 

 vibrates in a circle. Here the force 



• X /X .t' \ , SX 2" 



= -gr.sm^=-g(^-^) nearly = -Y+ 6^ ^%- 

 and the proportionate increase of the time of vibration 



= 6V^-^'^'"'-"«+^- 



