272 Sir William Rowan Hamilton on Fluctuating Functions. 



we shall also suppose that No = 0, and that n„ a"' tends to a finite limit as a. tends 

 to 0, whether this be by decreasing or by increasing ; although the limit thus 

 obtained, for the case of infinitely small and positive values of «, may possibly 

 differ from that which corresponds to the case of infinitely small and negative 

 values of that variable, on account of the discontinuity which the function n„ may 

 have. We are then to investigate, with the help of these suppositions, the value 

 of the double limit : 



lim . lim . (•' + ' . ^_, ^ , ,. 



6 = /3 = 00 \ f" ^pu-x^ (« - ^) /a ; (g ) 



this notation being designed to suggest, that we are first to assume a small but 

 not evanescent value of e, and a large but not infinite value of /3, and to effect 

 the integration, or conceive it effected, with these assumptions ; then, retaining 

 the same value of e, make /3 larger and larger without limit ; and then at last 

 suppose 6 to tend to 0, unless the result corresponding to an infinite value of j8 

 shall be found to be independent of e. Or, introducing two new quantities y 

 and »7, determined by the definitions 



yzz^{a~x), »7 = /3e, (h') 



and eliminating a and ^ by means of these, we are led to seek the value of the 

 double limit following : 



lim . lira . c " , _, . 



in which rj tends to oo, before e tends to 0. It is natural to conclude that since 

 the sought limit (g') can be expressed under the form (1'), it must be equivalent 

 to the product 



/,X^ dyTfyy-'; ^ (k') 



and in fact it will be found that this equivalence holds good ; but before finally 

 adopting this conclusion, it is proper to consider in detail some difficulties which 

 may present themselves. 



[6.] Decomposing the function yV+t^-'s i^^to two parts, of which one is inde- 

 dent of y, and is =^x» while the other part varies with y, although slowly, and 



