96 DYNAMICAL THEORY OF SOUND 



indicated for the case of the sine-series. We have, by a partial 

 integration, 



2 f 77 

 A s I f(x) sin sx dx 



1 F2 "1 2 C n 



= -- -/O)coss# + I f(x)w$sxdx, ...(4) 



where the integrated term is to be calculated separately for 

 each of the segments lying between the points of discontinuity 

 of /(#), if any, which occur in the range extending from x = to 

 # = TT inclusively. For example, if as in 32 (14) the only 

 discontinuity is at x 0, its value is 2/(0)/S7r. In any case 

 there is, for all values of s, an upper limit to the coefficient of 

 l/s in the first part of (4); we denote this limit by M. The 

 definite integral in the second term tends ultimately to 

 zero, as s increases, owing to the fluctuations in sign of cos sx. 

 Hence A 8 is ultimately comparable with M/s. If there is no 

 discontinuity of /(#), even at the points x = 0, x TT, the first 

 term in the above value of A 8 vanishes, and continuing the 

 integration we find 



A s = -- \- f O)sin sx\ - -f- fjT<*) sin ** dx. . . .(5) 

 r \_Tr J s "^J Q 



In the first part, regard must be had to the discontinuities of 

 /'(#), if any. Denoting by M the upper limit of the coefficient 

 of l/s 2 , we see that A s is ultimately comparable with M/s*, the 

 second term in (5) vanishing in comparison, by the principle 

 of fluctuation. The further course of the argument is now 

 sufficiently apparent. 



36. Physical Approximation. Case of Plucked String. 



It has been thought worth while to state Fourier's theorem 

 with some care, although we do not enter into the details of the 

 mathematical proof, which is necessarily somewhat intricate, 

 owing to the various restrictions which are involved*. 



From a physical point of view the matter may be dealt with, 

 and perhaps adequately, in a much simpler manner. To explain 

 this, it is best to take a definite problem, for instance that of 



* The most recent English treatise on the subject is that of Prof. 

 H. S. Carslaw, Fourier's Series and Integrals, London, 1906. 



