114 Professor Airy on the Disturbances of Pendulums, 

 and when it returns, the arc is diminished by 



\ n- n J / 



The diminution in two vibrations is, therefore, A- . . The 



n 



time of vibration is unaltered. 



Ex. 6. The resistance is that produced by a current of air 

 whose velocity is not equal to the greatest velocity of the pen- 

 dulum. Here, when the pendulum moves in the direction of 

 the current (p{v) = — k [F- v)- when i) is < V, and </> (f) = k {v— Vy 

 when V is >J'. By the formula above, the increase of the arc is 



n'al ^ 3^ ^ \ na 4/j 



The time is not altered. The motion in the opposite direction 

 is the same as in the last Example. 



Ex. 7. The force F acts through a very small space s at the 

 distance c from the lowest point. For the increase of the arc we 

 must take 



Tliis is plainly = —r- . 

 *^ ' n'a 



The proportionate increase of the time of vibration 



_ 1 /* Fx (X = c ^ 



~ TTTi'a- -^ • ^a--.^ \x = c+s]' 



if the general value of the integral be <p (x), the value bet^veen 

 these limits will be (p{c + s) - <p (c) = (p' (c) . s nearly 



_ Fs c 



