92 DYNAMICAL THEOKY OF SOUND 



fulfilled as a matter of course by any function which it is 

 natural to assume as representing the initial form, or the 

 initial velocity, of a tense string. We also see that the 

 difficulty met with in the case of (16) can be accounted for 

 by the fact that the function does not vanish for x = 0. 

 An extension of the statement to meet such cases will be 

 given presently ( 34). 



33. The Cosine- Series. 



The theory of the longitudinal vibrations of rods, or of 

 columns of air, leads, in addition, to a similar theorem relating 

 to the expansion of an arbitrary function in a series of cosines. 

 The formal statement is now as follows : 



If we write 



fm (#) = A + A l cos x + AZ cos 2% + . . . + A m cos mx, (1) 



where 



whilst for s > 



2 

 = -l f(x)co$sxdx, (3) 



it may be shewn that as m increases the sum f m (x) will tend to 

 the limit /(#), provided f(x) is continuous throughout the 

 range from to TT, and has at most a finite number of 

 maxima and minima. There is now no restriction as to the 

 values of /(O) and /(TT). 



If the determination of the effect of special initial conditions 

 in a longitudinally vibrating bar which is free at both ends 

 were as interesting a problem as it is in the case of strings 

 we should have recourse to the cosine-series. 



34. Complete Form of Fourier's Theorem. Discon- 

 tinuities. 



The question arises as to what is represented by the 

 sine-series or the cosine-series, supposed continued to infinity, 

 when x lies outside the limits and TT. The answer is supplied 

 by the consideration that both series are periodic functions 

 of x } the period being 2?r, whilst the former is an odd, the 



