FOURIER'S THEOREM 105 



may be written 



provided n' 2 = n 2 -JA; 2 ...................... (8) 



Hence, by the formula referred to, 



cos n'tdt 



~~ **' 



e~ **' cos n't je* kt f() sin n'tdt. . . .(9) 



If we have x 0, x for t oo , the limits of integration 

 are oo and t. For instance, the value of x when t = 

 becomes 



x = -l,r e* kt f(i)sinn'tdt. (10) 



Owing to the presence of the exponential factor it is only for 

 a certain range of negative values of t that the function under 

 the integral sign has as a rule an appreciable value. In other 

 words, the effects of the action of the force prior to a certain 

 antecedent epoch have practically disappeared. 



