Convergency of Fourier s Series. 131 



increases without limit, p diminishes without limit. For each of 

 the above m elements can be broken up into two parts of equal 

 range it, in one of which sin <$> is positive, in the other nega- 

 tive. The value of each element will therefore be of the form 



2 {pi—p 2 ) where p x is some value of % ( " * \ taken between the 



limits of the first portion of the element, and p 2 between those 



of the second. But as n increases, the change in %( 9 , i ) 



when (/> changes by 2tt diminishes ; and since % is everywhere 

 finite and continuous, p x and p 2 tend to the same value. Hence 

 by increasing n sufficiently, we can make p\—p 2 as small as 

 we please; and therefore in the limit when n = co it vanishes. 



In other words, since as n increases v | L1 , | tends to 



remain constant during the integration of any element while 

 sin <f) passes through all the values included between + 1, each 

 element tends to the value zero, the value it would really have 



if X ( i) i ) remained absolutely constant during the inte- 

 gration. 



15. This holds for all finite values of h however small. 

 When h is very small, p± and p 2 will have their greatest 

 values in the neighbourhood of (j) = (2n-\-l)-%h, in which case 

 (putting x = for convenience, the reasoning being applicable 

 for any value of a) 



2-7T 



where t is some value lying between and x -. and is in- 



J & 2w+l 



finitely small compared with. h. (pi—p 2 ) can therefore be 



made as small as we please for values of h as small as we 



please provided h is so chosen that jr 2 and — — 



are both infinitely small. But since F is everywhere conti- 

 nuous, and n is to be increased without limit, this condition 

 can always be satisfied. Hence the limit of p, and therefore 

 of C, is zero for values of h as small as we please ; and in 

 the same manner we may show that the limit of A is zero. 

 The value of S*, therefore depends only upon the value of 

 the infinitely thin strip B of breadth 2 A within which the 

 function integrated becomes infinite, and is independent of 

 the values of F(0-Kr) outside this strip. Consequently, we 



