FOUKIER'S THEOREM 



91 



Fig. 36. 



The preceding illustrations, with the diagrams, afford at 

 all events a presumption in favour of the theorem in question, 

 but shew at the same time that it is subject to some restric- 

 tions. The theorem admits of independent mathematical proof 

 under certain conditions as to the nature of the "arbitrary" 

 function f(x). We shall, however, not enter upon this, but 

 shall content ourselves with the following formal statement : 



If we form the sum of the first m terms of the series (2), 

 and write 



fm (#) = AI sin x + A z sin 2x + ... + A m sin mx, (18) 



where 



A 8 = I f(x)sinsxdx, ............ (19) 



it may be shewn that, for any assigned value of x in the range 

 from to TT, the sum f m (x) will tend with increasing m to the 

 limit f(x), provided the function f(x) is continuous throughout 

 the above range, has only a finite number of maxima and 

 minima, and vanishes for x = and x = IT. 



It will be noticed that the conditions here postulated are 



