FOURIER'S THEOREM 95 



litions laid down the values of the coefficients A 8 and B 8 

 must ultimately diminish indefinitely as s increases, owing to 

 the more and more rapid fluctuation in sign of cos so; and 

 sin sx, and the consequent more complete cancelling of the 

 various elements in the definite integrals of 32 (5) and 

 33 (3). 



More definite results have been formulated by Stokes. 

 The following statement must be understood to refer to the 

 function as continued in the manner above explained; and 

 care is necessary, in particular cases, to see whether discon- 

 tinuities of f(x) or its derivatives are introduced at the 

 terminal points of the various segments: 



If f(x) have (in a period) a finite number of isolated 

 discontinuities, the coefficients converge ultimately towards 

 zero like the members of the sequence 



1, 2> 3> 4> ! 



This is exemplified by 32 (16) and Fig. 36. 



If f(x) is everywhere continuous, whilst its first derivative 

 f'(x) has a finite number of isolated discontinuities, the con- 

 vergence is ultimately that of the sequence 



ill 1 



' 2 2 ' 3 2 ' 4 2 ' * 



This is illustrated by 32 (12) and Fig. 35. 



If f(x\ f(x) are continuous, whilst /"(#) is discontinuous 

 at isolated points, the sequence of comparison is 



l > %3> ^3> #> 



as in the case of 32 (8). And, generally, if f(x) and its 

 derivatives up to the order n 1 inclusive are continuous, whilst 

 the nth derivative has (in a period) a finite number of isolated 

 discontinuities, the convergency is ultimately as 



1 1 1 



The nature of the proof, which is simple, may be briefly 



