100 KINEMATICS. [199. 



As applied to the theory of wave motion this means of course that any 

 'wave motion, however complex, can be regarded as nude up of a series 

 of superposed simple harmonic vibrations of periods T t T/2, 7/3, , 

 or since T= K/V, of wave lengths A, A/a, A/3, .... 



199. A full discussion of Fourier's theorem cannot be given in this 

 place. We wish, however, to show its practical application in an 



example. 

 The equation (ai) can be written in the form 



/(/) = ! cosej sin. /+ a* cos e 8 sin. a /+ <* 3 cos e 3 sin- 3/+ 



or putting a irfT'ss #, <*! sin j 

 2 cos e a = /?a, , 



a + -^ cos x -f At cos *x + ^ 8 cos 3* + 



-f ^sin^e -f^ a sin 2x 4-^? 3 sin3Jc 4- . (aa) 



This is known as Fourier's series. According to the nature of the func- 

 tion to be expanded, it is often sufficient to use the sine series or the 

 cosine series alone. As the method of determining the coefficients is 

 always the same, it will be sufficient to consider the simple sine series : 



200. The problem before us can now be stated as follows : Given any 



single-valued function of je, either by its analytical expression or by the 



trace of the curve representing it, to determine the coefficients B it 



(33) so as to make the right-hand member of this equation represenf 



the values of the given function between certain finite limits of x. 



. We shall assume these limits to be x = o and x = * : and we shall 



*The student is referred to THOMSON and TAIT, Natural philosophy* I. I, 1879, 

 pp. 55-60; also to B. RIBMANN, Partielk Di/trtHtialgieickHHgv* % herausgegeben votf 

 K. Hattendorff, 3d ed., Braunschweig, Vieweg, 1882, pp. 44-95, and to G. M. 

 MINCHIN, L-ttiplanar tintma(tcs, Oxford, Clarendon Press, 1882, p. 13 sq. 



