CHAPTER III 



FOURIER'S THEOREM 



32. The Sine-Series. 



The study of the transverse vibrations of strings has 

 already suggested a remarkable theorem of pure mathematics, 

 to which some further attention must now be given. The 

 theory of the normal modes has led us ( 25) to the conclusion 

 that the free motion of a string of length I, started in any 

 arbitrary manner, can be expressed by a series of the form 



STTCt n . S7TCt\ . STTX 



8 cosj- + 5 f sm Jsin-j-, ...... (1) 



where s = 1, 2, 3, ..., provided the constants A 8) B 8 be properly 

 determined. In particular if the string be supposed to start 

 from rest at the instant t = in the arbitrary form y = f(x), it 

 should be possible to determine the coefficients A s so that 



(2) 



for values of x ranging from x = to x = I. This is a particular 

 case of "Fourier's Theorem*." Since I is at our disposal we 

 may conveniently replace it (for general purposes) by TT, and 

 the statement then is that an arbitrary function f(x) can be 

 expressed, for values of x ranging from to TT, in the form 



f(x) = A l sin x + A 2 sin 2# -f . . . + A 8 sin sx + ....... (3) 



* J. B. J. Fourier (17681830). The history of the theorem is closely 

 interwoven with that of the theory of strings, and of the theory of heat- 

 conduction. Fourier's own researches are expounded in his Theorie de la 

 Chateur, Paris, 1822. An outline of the history is given in Prof. Carslaw's 

 book cited on p. 96. The subject is treated most fully by H. Burkbardt in his 

 report entitled Entwickelungen nach oscillirenden Funktionen..., Leipzig, 1908. 



