Sir William Rowan Hamilton on Fluctuating Functions. 269 

 But also, we have, by (h) 



— Sn,o + Sn,y-\- •■•-{■ Sn,k; (p) 



the integral in (o) may therefore be thus expressed, without any loss of rigour : 



k 



in which 



n'hi + i 



\ rfaN<.F„ = S„.„ A„,„ + ...4-*n.iA„,*, (q) 



»JCL. 



so that A„,i is a finite difference of the function f„, corresponding to the finite 

 diflference a'„i — a„ of the variable a, which latter difference is less than a„+i — 

 a„, and therefore less than the finite constant b of the last article. The theorem 

 (q) conducts immediately to the following, 



\^_, c?aN^„F„ = /3 '(s„,„8„,„ + ... + ;?^a8„,*), (s) 



in which 



8„,i = F^-,„.^_. — F^-,„„; (t) 



so that, if /3 be large, ?„_; is small, being the difference of the function f„ corres- 

 ponding to a difference of the variable a, which latter difference is less than 

 /3~'b. Let±8„be the greatest of the/c-l-l differences 2„,oj-'^n,*> or let it 

 be equal to one of those differences and not exceeded by any other, abstraction 

 being made of sign ; then, since the k-\-l factors 5„,o> •'■ \k are alternately posi- 

 tive and negative, or negative and positive, the numerical value of the integral 

 (s) cannot exceed that of the expression 



But, by the definition (1) of 5„_i, and by the Umits ±c of value of the finite func- 

 tion N„, we have 



±«n,i > (a«,i + l — «n,Oc; (v) 



therefore 



± (*»,o — «n,, + •■• + (— 1)* *n,*) > («« + , — a„) c ; (w) 



and the following rigorous expression for the integral (s) results : 



