1 8 MEASUREMENT OF OSCILLATIONS. 



equilibrium. The motion is periodic, and the system makes a series 

 of oscillations which are perfectly identical. 



In all cases, whether the successive oscillations are or are not 

 identical, the position of the system when it makes the greatest angle 

 with the position of equilibrium is called the elongation; the ampli- 

 tude of oscillation is the sum of two successive angles of deflection 

 that is to say, the angle of two extreme positions ; the duration of an 

 oscillation is the time between two consecutive elongations. 



The only cases useful for consideration from the experimental 

 point of view, are those in which the directing couple is proportional 

 to the angle of deflection, or to the sine of the angle. In that case 

 we must replace the moment / (x) by C x or C sin x, the factor C 

 representing the couple which corresponds to an angle equal to unity 

 in the first case, or to a rotation of 90 in the second. 



Let us first suppose that the retarding actions are null, and put 



according to the case considered, equation (i) becomes, 



or 

 (3) 



The formulas (2) and (3) are the same for very small arcs. 



676. ISOCHRONOUS OSCILLATIONS. The general integral of the 

 equation (2) is 



x = A sin nt + A' cos nt ; 



from which is deduced 



o> = = n ( A cos nt A' sin /) . 



dt 



If we count the time / from the moment at which the movable 

 body passes through its position of equilibrium, and if we call o> the 

 initial angular velocity, we shall have for the constants A and A', 



A' = 0, w = nA ; 



