22 MEASUREMENT OF OSCILLATIONS. 



function of the initial deflection a. If we only consider small values 

 of a, the arc may be developed in a very converging series, which 

 contains no term independent of a ; for as the initial velocity is 

 null, x ought to vanish at the same time as a. Calling x v x 2 . . . . 

 functions of /, we may put 



x = ax^ + c 



Replacing this value in equation (u), and making the coefficients 

 of the various powers of a equal to zero, we obtain a series of 

 differential equations which will serve to determine the functions 

 x v x, 2 . . . Neglecting powers of a higher than the second, we have 

 the two equations 



^+2* 



dt 

 The integral of the first is 



If we suppose that for /=0, we have x= a and r ^t it follows 

 that x l = i and - = 0, and therefore A = - i and /? = ; which 



gives 



x 1 = - cos nt. 



Substituting this value of x l in the equation in x 2 , and observing 

 that 2 sin 2 /= i -cos 2#/, this becomes 



70 



-7-2 + CW 2 (i - COS 2Ht} + n 2 X 2 = 



dt* 

 the general integral of this latter equation is 



x 2 A' cos (nt + ft'} - e - - cos znt. 

 o 



4 

 The initial conditions give A' = and /5' = 0; therefore 



\J 



A 



x = - e H -- cos nt cos 2nt. 



