FOURIER'S THEOREM 103 



This is of course merely a restatement of the theorem of 

 34, with the necessary changes of notation. It will be noticed 

 that A represents the mean value of the function. 



We have already been led to formulae of the type (1) as 

 expressing the motion at any assigned point of a freely vibrat- 

 ing string, the period r being equal to 2l/c. Another important 

 acoustical application is to the analysis of a periodic current of 

 air, as in the siren, or the reed-stops of an organ ( 90). Again, 

 in the case of electromagnetically driven tuning forks, a periodic 

 current can powerfully excite, not only a fork in unison with 

 itself, but also others whose natural frequencies are respectively, 

 twice, three times, ... as great. This is due to the fact that the 

 disturbing force is of the type (1), the selective resonance taking 

 place according to the principles of 9. 



Various mechanical contrivances for resolving a given 

 periodic curve into its simple-harmonic constituents, and 

 conversely for compounding a number of independent sine- 

 and cosine-curves whose periods are as 1, , J, ..., have been 

 devised by Lord Kelvin and others. From the standpoint of 

 the present subject the most remarkable of these is perhaps 

 the machine constructed by Prof. A. A. Michelson, in which 

 provision is made for as many as 80 constituents*. 



It is hardly necessary to say explicitly that the resolution 

 of a periodic function of t in the form (1) can only be effected 

 in one way, the values of the coefficients as given by (2) and (3) 

 being determinate. In particular, a series of the above type 

 cannot vanish for all values of t unless its coefficients severally 

 vanish. Thus in a freely vibrating string, if the motion at any 

 given point x be prevented, as by touching with a camel-hair 

 pencil, the coefficients of cos(s7rct/l) and sin(s7rct/l) in the 

 general formula (7) of 25 must be zero, i.e. we must have 



. ' . STTX -. . STTX _ 

 A 8 smj- = Q, B 8 sm-j- = (4) 



for all values of s. Unless x be commensurable with I this 

 requires that A a = Q, B a = Q, and the whole string will be 



* Phil. Mag. (5), vol. XLV. (1898) ; this paper contains a number of most 

 interesting examples of results obtained. The construction is also explained in 

 his book On Light Waves and their Uses, Chicago, 1903. 



