276 Sir William Rowan Hamilton on Fluctuating Functions. 



the upper sign being taken wheny^ increases from az=a:toa=:3;-\-e; and if 

 we suppose that f and rj are of the forms a„ and On+m ; we see that the integral 

 (t') is numerically less than 3 be a„~' (/"«+. — f^), and therefore that it vanishes 

 at the limit 6 = 0. It is easy to see that the same conclusion holds good, when 

 we suppose that rj does not coincide with any quantity of the form a„^„„ and 

 when the limits of the integration are changed to — tj and — f . We have 

 therefore, rigorously, 



lim . lim .(*»», _,. .. 

 6 = ^=00 3_/^N*3/ '(/x+«,-»— /x) = 0, (x') 



nowithstanding the great and ultimately infinite extent over which the integration 

 is conducted. The variable part of the functiony may therefore be suppressed 

 in the double limit (i'), without any loss of accuracy ; and that limit is found to 

 be exactly equal to the expression (k') ; that is, by the last article, to the deter- 

 mined product -sr/j;. Such, therefore, is the value of the limit (g'), from which 

 (i) was derived by the transformation (h') ; and such finally is the limit of the 

 integral (f), proposed for investigation in the fifth article. We have, then, 

 proved that under the conditions of that article, 



B zToo " W« N^ (a-x) (« - ^r'/a = ■=[/■- ; (y') 



and consequently that the arbitrary but finite and gradually varying functiony"j.> 

 between the limits x ^a, x=: b, may be transformed as follows : 



f. = ^~' ^ rf«N.(„_^) (a — .r)-'/„ ; (z') 



which is a result of the kind denoted by (d) in the second article, and includes 

 the theorem (a) of Fourier. For all the suppositions made in the foregoing arti- 

 cles, respecting the form of the function n, are satisfied by assuming this function 

 to be the sine of the variable on which it depends ; and then the constant sy, 

 determined by the formula (r'), becomes coincident with tt, that is, with the 

 ratio of the circumference to the diameter of a circle, or with the least positive 

 root of the equation 



