102 DYNAMICAL THEORY OF SOUND 



sin (STTX/I), or cos (sirx/l), respectively. More complicated forms 

 will be met with when we come to the theory of transverse 

 vibrations of bars, and to that of membranes ; and even in the 

 cases just mentioned the simplicity of type would at once 

 disappear if the uniformity of line-density, or of cross-section, 

 respectively, were departed from. In some problems, indeed, 

 of considerable interest, e.g. that of the vibrations of a rectangular 

 plate, the precise form of the functions has still to be discovered. 

 But in any case the functions theoretically exist ; and on the 

 principle that any free motion whatever of the system consists 

 of some combination or other of the various normal modes, it 

 must be possible to express any arbitrary initial state, and 

 therefore any arbitrary function of position in the system, by a 

 series of normal functions. Such preeminence as attaches to 

 Fourier's theorem is, from the present point of view, due merely 

 to the fact that in it we have the simplest exemplification of 

 this principle in the case of a continuous system, and the one 

 where the physical induction has been most fully corroborated 

 by independent mathematical proof. It may also be added that 

 it is only in the case of strings that the calculation of the 

 effect of particular initial conditions has any great interest. 



There is however another point of view from which the 

 resolution of a function into a series of sines or cosines of the 

 variable is of peculiar importance, viz. when we are dealing with 

 functions of the time. The dynamical reason for this has 

 already been dwelt upon ( 19). 



When a function f(t) is known to be periodic, of period T, 

 its resolution by Fourier's theorem is 



/v,x r 



j (t) = A + A l cos -- f- A z cos -- h A a cos -- [- . . . 



-f B l sin - - + J5 2 sin h B 3 sin ^-^ + . . . , (1) 



T T T 



where A =*( T f(t)dt, (2) 



whilst for s > 0, 



/A p nH fl4 T) / f(t\ oin /O\ 



^t^ oos ac, xj, - i y ^c^ sin . v* 5 / 



