94 DYNAMICAL THEORY OF SOUND 



whilst for s > 



If* 1 1 f" \ 



A s = - I {/(#) +/( a))} cos sxdx = I f(x) cos s#cfo, 



TTJo " TTJ - n I ^ 



i t w i r* 1 



B 8 =-\ \f(x) f(- x)} sin s#cfo? = - I /(#) sin s#d#, 



TTJo * TTJ _ ff 



tends with increasing m to the limit /(a?), provided f(x) is 

 continuous from X TT to # = TT and has at most a finite 

 number of maxima and minima, and provided also that 

 /( TT) =/(TT). For values of x outside this range the limit 

 represents, under these conditions, a periodic function of 

 period 2?r. This is the complete form of Fourier's Theorem, 

 and includes the others as special cases. 



We should be led directly, on physical grounds, to this form 

 of the theorem if we were to investigate the "longitudinal" 

 vibrations of the column of air in a reentrant circular tube. 



We have so far supposed the function /(#) to be continuous, 

 as well as finite, even when continued beyond the original 

 range as a periodic function. But the theorems hold, with 

 a modification to be stated immediately, even if f(x) have 

 a finite number of isolated discontinuities. In such a case 

 the series f m (x) still converges, with increasing m, to the value 

 of /(#), except at the points of discontinuity. But if a be 

 a point where f(x) abruptly changes its value, the sum f m (a) 

 tends to the limit 



where f(a 0) and/(a + 0) represent the values of f(x) at 

 infinitesimal distances to the left and right, respectively, of the 

 point a. For example, in the case of the sine-series 32 (3), 

 if f(x) does not vanish when x = or when X = TT, there is 

 discontinuity at these points in the periodic function, and 

 the series / m (0), for example, has the limit 0, which is 

 the arithmetic mean of the values of the continued function 

 on the two sides of the point #=0. This is illustrated in 

 Fig. 36. 



35. Law of Convergence of Coefficients. 

 It remains to say something as to the law of decrease of 

 the successive terms. It is evident at once that under the 



