Sir William Rowan Hamilton on Fluctuating Functions. 273 



vanishes with that variable ; it is clear that the formula (i') will be decomposed 

 into two corresponding parts, of which the first conducts immediately to the 

 expression (k') ; and we are now to inquire whether the integral in this expres- 

 sion has a finite and determinate value. Admitting the suppositions made in 

 the last article, the integral 



^ ^^N,^ ' 



•^-i 



will have a finite and determinate value, if f be finite and determinate ; we are 

 therefore conducted to inquire whether the integrals 



are also finite and determinate. The reasonings which we shall employ for the 

 second of these integrals, will also apply to the first ; and, to generalize a little 

 the question to which we are thus conducted, we shall consider the integral 



0«N„F„J (!') 



F„ being here supposed to denote any function of a which remains always positive 

 and finite, but decreases continually and gradually in value, and tends indefinitely 

 towards 0, while a increases indefinitely from some given finite value which is 

 not greater than a. Applying to this integral (1') the principles of the fourth 

 article, and observing that we have now Fa„i<f<.„j «'7.,i being > a,„ and a„ being 

 assumed <; a ; and also that 



we find 



± 5J'"rfaN„ FX^bc (F„^ - F„„^_) ; (n') 



and consequently 



p^n + tn 

 -3a„ «^«N„F„<^bc(F<,„-F„^^,^). (O') 



This latter integral is therefore finite and numerically less than g- be f„ , however 

 great the upper limit a„^„maybe; it tends also to a determined value as m 



VOL. XIX. 2 N 



