46, 47] NORMAL FUNCTIONS. 173 



series. We may now make use of it to obtain the constants in the 

 series above. Putting t = and writing B s cos a s = A g , we have 



122) 



The problem is, / being an arbitrary function of x to find the 

 coefficients in the development in the trigonometric series. To find 

 the coefficients A r multiply the equation 122) by the r ih normal 

 function and integrate from to ?, giving 



i i 



/ S = cc f~ 



, N . ricx -. >n A I . snx . ritx -, 



f(x) sm -y- dx = ^ A s I sin sin ~- dx, 



o o 



and by the property just found the integral on the right vanishes in 

 every term except that in which r = s. But 



i 



I 



o 



Therefore we have the value of the coefficient 



i 



124) A r = jff(%) sin ^dx. 



o 



We are thus led to a particular case of the remarkable trigono- 

 metric series associated with the name of Fourier. Such series were 

 first considered by Daniel Bernoulli in connection with this very 

 problem of a vibrating string. This determination of the coefficients 

 was given by Euler in 1777. The importance of the series in 

 analysis was first brought out by Fourier who insisted that such a 

 series was capable of representing an arbitrary function, as had been 

 maintained by Bernoulli, but doubted by Euler and Lagrange. 



47. Forced Vibrations of General System. Let us now 



briefly consider the question of forced vibrations of the general 

 system of 45. 



Suppose that there is impressed upon each coordinate a harmo- 

 nically varying force, 



F r = E r cospt, 



the period and phase being the same for all, the amplitude E r being 

 taken at pleasure. The equations are most easily dealt with if, instead 

 of proceeding as we did in treating equations 41) and 42) we make 

 use of the principle that, in an equation involving complex quantities, 

 the real and the imaginary parts must be equated separately. Let 



