and on the Theory of Escapements. 113 



the integral being taken from v = o to v — O again. But it must 

 be observed that from v = o to v = na, the radical must be taken 



with a negative sign, because sin nt-^b is then negative. The 

 increase of the arc is, therefore, the sum of 



1_ r v(p(v) {v = \ ,1 ^. V(j> (v) (v = na\ 



that is, the decrease of the arc is 



2 ' /• V(f) (v) (V — ) 



n'ajv ^n'a^-v" U = nJ ' 

 The proportional increase of the time of vibration is 





^f,^iv).sin nt + b =--^^f,^i,) . - = __,y:^(^). ___^(,). 



{V = O) 

 _ I is in all cases = o. A re- 

 sistance, therefore, vt'hich is constant, or which depends on the 

 velocity, does not alter the time of vibration. 



Ex. 5. The resistance is that produced by a current of air 

 moving in the plane of vibration with a velocity V greater than 

 the greatest velocity of the pendulum ; and varies as the square 

 of their relative velocity. In this case, when the pendulum 

 moves in the direction of the current 



and when it moves in the opposite direction, 



0(1;) = k{r+ vy. 



By the formula above we find that when the pendulum moves 

 in the direction of the current, the arc is increased by 



, /2r' Fair 4a\ 

 Vol. III. Part I. P 



