98 DYNAMICAL THEORY OF SOUND 



the squares of the errors involved in the substitution of (2) for 

 (3) as small as possible. Thus if, for shortness, we replace I by 

 TT, we have to choose the coefficients so as to make the integral 



/(#) ( A l sin x + AZ sin 2# 4- . . . + A m sin mx)} z dx (4) 

 a minimum. If we differentiate with respect to A 8 we get . 

 )(A 1 smx+A z sin2x + ...+A m smmx)}smsxdx=(), (5) 



or, by 32 (4), 



2 /**' 

 A 8 =- \ f(x)smsxdx ................ (6) 



TTjQ 



Hence this method of least squares, applied to the expression 

 (2) consisting of a finite number of terms, gives precisely the 

 values of the coefficients which were obtained by Fourier's 

 process*. Each coefficient is determined by itself, and the 

 effect of adding more terms to (2) is to improve the approxima- 

 tion, without affecting the values of the coefficients already 

 found. If we revert to general units, the formula (6) is 

 replaced by 



2 l / s S7TX 

 S= ~ 



In the case of the plucked string, the form to which we 

 endeavour to approximate is 



y = Pxla [0<x<a], y = # (I -x)l(l- a) [a<x<l\ (8) 



The result is obtained at once from 32 (11) if we write jrx/l 

 for x, and therefore irajl for a, I for TT, and introduce the factor fi. 

 Thus 



, 2/9Z 2 . STTO, 



as stated in 26 (1). The nature of the approximation is 

 illustrated in Fig. 35. 



37. Application to Violin String. 



To apply the method to the problem of the violin string 

 ( 27), we take as origin of t the instant when the point Q in 



* This theorem is due to A. Toepler (1876). 



