312 Sir William Rowan Hamilton on Fluctuating Functions. 



and a rigorous expression for it may be obtained by the process of the fourth 

 article, namely 



4" 0^' (a„^ „ — a„) cl ; 



in which, as before, a„, a„^„ are suitably chosen roots of the equation (g) ; c is 

 a finite constant; 6 is included between the limits ±1 ; and I is the difference 

 between two values of the function ^^, corresponding to two values of the varia- 

 ble 7 of which the difference is less than ^~'b, b being another finite constant. 

 The integral (a^^) therefore diminishes indefinitely when ^ increases indefinitely ; 

 and thus, or simply by the theorem (z) combined with the expression (e"), we 

 have, rigorously, at the limit, without supposing here that n^ vanishes. 



i 



rfaN,^F„ = 0. (w) 



The same conclusion is easily obtained, by reasonings almost the same, for the 

 case where 7 continually decreases from 7„ to 74, in such a manner as to give, 

 within this extent of variation, a finite and determined and negative value to the 

 differential coefficient (c^'''). And with respect to the case where the function 7 

 is for a moment stationary in value, so that its differential coefficient vanishes 

 between the limits of integration, it is sufficient to observe that although ^ in 

 (e") becomes then infinite, yet f in (a^'^) remains finite, and the integral of the 

 finite product das^^F^, taken between infinitely near limits, is zero. Thus, 

 generally, the theorem (w), which is an extension of the theorem (z), holds good 

 between any finite limits a and b, if the function f be finite between those limits, 

 and if, between the same limits of integration, the function 7 never remain un- 

 changed throughout the whole extent of any finite change of a, 



[26.] It may be noticed here, that if j3 be only very large, instead of being 

 infinite, an approximate expression for the integral (a^^) may be obtained, on the 

 same principles, by attending only to values of a which differ very little from 

 those which render the coefficient (c^^) infinite. For example, if we wish to find 

 an approximate expression for a large root of the equation (p ''^^^ ), or to express 

 approximately the function 



