DAMPING OF THE OSCILLATIONS. 21 



If in expanding the value of T' we only consider the first term 

 that is, if we suppose a = we find the time of oscillation obtained 

 before ; this is what is called the time of an infinitely small oscillation. 



When the angle a is very small, we may restrict the series to its 

 first two terms, and, neglecting the fourth power of the amplitude, 

 may write 



(10) 



-V 



i6/ 



In all cases the time of oscillation increases with the amplitude, but 

 at first very slowly. This motion is that of the pendulum in vacuo. 



678. DAMPING OF THE OSCILLATIONS. In practice the oscil- 

 lations become more or less rapidly extinct, or they deaden them- 

 selves, owing to retarding actions, such as friction, the resistance of 

 the medium, etc., and in the case of magnets in motion, owing to the 

 effects of induction developed in adjacent masses of metal. All the 

 forces of this kind are functions of the velocity. The simplest 

 hypothesis is to consider them as proportional to a whole power m of 

 the velocity. If C x is the moment of these forces for an angular 

 velocity equal to unity, the equation of the oscillatory motion (i) in 

 the case of a directive couple proportional to the angle of deflection 

 becomes 



'dx\ 



*) 



or, putting 



2_ C _ C 1 



If the principal couple was proportional to the sine of the de- 

 viation, the last term should be replaced by n 2 sin x ; but we shall 

 more particularly consider equation (n), and therefore, in dealing 

 with pendulum motions, the case of very small oscillations. 



679. RESISTANCE PROPORTIONAL TO THE SQUARE OF THE 

 VELOCITY. Poisson has examined the hypothesis m = 2* He 

 observes, first of all, that if the movable body is displaced through 

 an angle a, from its position of rest, and then left to itself without 

 initial velocity, the deflection at any given time is necessarily a 



* POISSON. Mecanique. First edition, Vol. I., p. 405. 1811. 



