268 Sir William Rowan Hamilton on Fluctuating Functions. 



N„ <: 0, 



we shall have 



\ ' ^'rfaN„(F„ — F^) < 0; 

 \ rfaN„(F^^ — F„) <0; 



(k) 



s. 



the integral (i) will therefore be not < *„ j f\ and not > *„,( f'\ if we put, for 

 abridgment, 



and consequently this integral (i) may be represented by *„ , f', in which 



f' < v\ f' D> f", 

 because, with the suppositions already made, s„_i > 0. We may even write 



f' > f\ f' < f\ 



unless it happen that the function f„ has a constant value through the whole 

 extent of the integration ; or else that it is equal to one of its extreme values, 

 f' or f'\ throughout a finite part of that extent, while, for the remaining part of 

 the same extent, that is, for all other values of a between the same limits, the 

 factor N„ vanishes. In all these cases, f' may be considered as a value of the 

 function f„, corresponding to a value a'„i of the variable a which is included 

 between the limits of integration ; so that we may express the integral (i) as 

 follows : 



in which 



In like manner, the expression (m), with the inequalities (n), may be proved to 

 hold good, if N„ be never > 0, and sometimes < 0, within the extent of the 

 integration, the integral «„_j being in this case < ; we have, therefore, rigo- 

 rously. 



r«""4-i 



\ rfa N. F, = *„,„ F,; -f5„,,F^ +... + *„,tF,i . 



(0) 



