Sm William Rowan Hamilton on Fluctuating Functions. 271 

 and we may easily deduce from it the following : 



jSzToo J/aN^(a-x,F<. = 0; (V) 



the function f, being here also finite, within the extent of the integration, and :v 

 being independent of a. and j3. For the reasonings of the last article may easily 

 be adapted to this case ; or we may see, from the definitions in article [3.], that 

 if the function n„ have the properties there supposed, then N„_a; will also have 

 those properties. In fact, if n„ be always comprised between given finite limits, 

 then N„_x will be so too ; and we shall have, by (f ), 



^ rfaN„_^=\ c;aN<. = M„_^— M_,; (c') 



in which M_a; is finite, because the suppositions of the third article oblige m„ to 

 be always comprised between the limits a ± be ; so that the equation 





c?aN„_^ = a — M_^, (d') 



which is of the form (g), has infinitely many real roots, of the form 



a = a;-\-a„ (e') 



and therefore of the kind assumed in the two last articles. Let us now examine 

 what happens, when, in the first member of the formula (b'), we substitute, 

 instead of the finite factor f„, an expression such as (a — ^)~ Va? which becomes 

 infinite between the limits of integration, the value of x being supposed to be 

 comprised between those limits, and the function y^ being finite between them. 

 That is, let us inquire whether the integral 



i' 



(in which ^ > a, < b), tends to any and to what finite and determined limit, as j8 

 tends to become infinite. 



In this inquiry, the theorem (b') shows that we need only attend to those 

 values of a. which are extremely near to x, and are for example comprised be- 

 tween the limits orqie, the quantity e being small. To simplify the question, we 

 shall suppose that for such values of «, the function/^ varies gradually in value ; 



