266 Sir William Rowan Hamilton on Fluctuating Functions. 



which thus changes sign indefinitely often, within a finite range of a variable on 

 which it depends, may be called a fluctuating function. We shall consider now 

 a class of cases, in which such a function may present itself. 



[3.] Let N„ be a real function of a, continuous or discontinuous in value, 

 but always comprised between some finite limits, so as never to be numerically 

 greater than ± c, in which c is a finite constant ; let 



M„= ^ rfaN^; (f) 



and let the equation 



M« = a, (g) 



in which a is some finite constant, have infinitely many real roots, extending 

 from — CO to -j- oc, and such that the interval a„^, — a„, between any one root 

 a„ and the next succeeding a„4.,, is never greater than some finite constant, b. 

 Then, 



and consequently the function n must change sign at least once between the 

 limits a-=. a^ and a = a„^j ; and therefore at least m times between the limits 

 az=an and az=.a,^j^my this latter limit being supposed, according to the analogy 

 of this notation, to be the m"' root of the equation (g), after the root a„. Hence 

 the function n^„, formed from n„ by multiplying a by /3, changes sign at least m 

 times between the limits a = \, a =. n, if * 



\ > P~^a„, /i < ^~' a„^„ ; 

 the interval /x — \ between these limits being less than |3~' (m -\- 2) b, if 



\ > ^~'a„_„ /x < p~'a„^™^,; 

 so that, under these conditions, (j3 being >0,) we have 



m > — 2 + |3b~'(/x — A). 

 However small, therefore, the interval /x — A may be, provided that it be greater 



* These notations >• and -< are designed to signify the contradictories of > and < ; so that 

 " a > V is equivalent to " a not > b," and " a < b" is equivalent to " a not < b." 



