32 MEASUREMENT OF OSCILLATIONS. 



the greatest deflection beyond the position of equilibrium takes 

 place at the time 



t = I i (o ~ a ( 6 ~y) c +y 



2 27 ' a>-a(e + y) ' -7' 



which corresponds to the condition -j- = 0. The system returns to 



its position of rest after an infinite time. 



In order that the value of t l be real, the two terms of the 

 fraction, the logarithm of which is taken, must be positive ; that is to 

 say, that 



w>a(e-f 7). 



687. Let us assume, finally, that for / = 0, x = 0, and -- = u> ; 

 we find then, for the constants, 



A = ^o A '=--^ 

 27' 27 



and the equation of the motion becomes 



C7) -5*- 



The greatest deflection is reached at the time / p given by the 

 equation 



27 -7 



The value of the elongation is not capable of a simple expression; 

 but the time / x being independent of the initial velocity, equation 

 (27) shows that the elongation is proportional to this velocity. 



688. The case in which e = n, and therefore 7 = 0, is particularly 

 interesting. It will be sufficient in the formulas (25), (26), and (29), 

 to develop in a series the exponentials &*, stopping at the two first 

 terms, and to suppose 7 = 0. 



