38 DEFINITE INTEGRALS. [INT. III. 



If L s is the greatest value of f(x) in the interval 8 S , l s the 

 least, so that the oscillation in that interval is D s = L 8 - 1 8 , the 



n n 



two sums $j = 2 S s L s and $ 2 = 2 S g s must approach limits as n 



increases indefinitely, for & is always greater than $ 2 , and as we 

 increase n, Si can never increase and $ 2 can never decrease. Now 



n 



the sum 2 $?/(&) always lies between $j and $ 2 , therefore if their 



n n 



difference 2 S s D approaches the limit zero the sum 2 & s f(Zs) 

 must approach a limit. 



Consider now two different modes of subdivision of the 

 interval ab, 



MI, # 2 a?n-i and a?/, a? 2 ' a/ n /-i, 



and the corresponding sums 



Let the points ^ a7 w _! and a?/ #V-i taken together 



form the system TI r p ^ and let 



Pi = n - 



In the interval a?^!, a7 there may or may not fall an r. In 

 general if as^. l = r h) x s will be r h+t (t = 1), so that 



$s = ph+i + ph+2 ...... + ph+t- 



Then 



V* = Ph+if"h+i + 0H4/"jM ..... + Ph+tf"h+t 



+ /OA +I (/ s /"ft+i) + P^+2 (/ -/"ft+a) ...... + />fc+ (/* f"h+t)> 



where the /&"'s are arbitrarily chosen values of/ in the intervals 

 p t , and 



where 



P = S 2 B [ P +1 (/. -/ V.) + /> 



5=1 



