FOURIER'S THEOREM " *101 



where <f> (x) vanishes except between the limits a e and a + e, 

 say. The formula (7) of 36 then gives 



2 [ l . , N . STTX , 2 f a+t , , . . S7TX , , . 



C g = j <f> O) sm r <te = 7 # W sm ~T~ * ( 10 ) 



& ./ ^ *J a-t 



If be small, the series thus obtained converges at first very 

 slowly, and a great many terms might have to be taken to 

 secure a reasonable approximation. In the terms of lower 

 order we have 



2 . STra f a+e . . , 2u . sira 

 0,-jgUi-y-J <0r)<fo=-^sm-y-, ...(11) 



/a+c 

 <f>(x)dx, (12) 

 a 



i.e. /Lt represents the total impulse. The corresponding term in 

 the value (3) of y is 



2/4 1 .. . STra . STTX . sirct /1Q , 



.- e-*"* 11 . sm -j- . sm =- . sm = . . . .(13) 

 PTTC s III 



But however small e may be, so long as it is not evanescent, 

 the value of G 8 given by (10) will ultimately tend to zero with 

 increasing 5, owing to the more and more complete mutual 

 destruction of positive and negative elements under the integral 

 sign. This shews the effect of diffusing the impulse over a 

 small but finite portion of the string. 



The case of an instantaneous local impulse is obtained by 

 putting r=0(cf. 26). 



39. General Theory of Normal Functions. Har- 

 monic Analysis. 



The space which has been devoted to Fourier's theorem is 

 no more than is warranted by its importance, especially in rela- 

 tion to the theory of strings, but it is well to remember that 

 from the standpoint of the theory of vibrations the theorem is only 

 one out of an infinite number which can be based on the same 

 kind of physical considerations. Every vibratory system has its 

 own series of "normal functions," as they are called, which 

 express the configuration of the system in the various normal 

 modes. In the case of a uniform string, or of the doubly-open 

 organ pipe, these functions happen to have he simple form 



