Sir William Rowan Hamilton on Fluctuating Functions. 275 



assumed in the fifth article, 



e ™0 \ =00 ' i_fi ^yy~' (/-+--' y — /x) = 0, (s') 



if f be any finite and determined quantity, however large, we are conducted to 

 examine whether this double limit vanishes when the integration is made to 

 extend from y=^ioy=.'q. It is permitted to suppose that f^ continually 

 increases, or continually decreases, from a ■=. x to az=L x -{- e ; let us therefore 



consider the integral 



SI 

 C?aN„F„G<., (f) 



in which the function f„ decreases, while g„ increases, but both are positive and 

 finite, within the extent of the integration. 



By reasonings similar to those of the fourth article, we find under these con- 

 ditions, 



and therefore 



\ p^n + m • 



+ (^«n+, - ^-n^-a) «<•« + .+ (^»« + 3 - ^°. + .) «"« + 4 + ^'^- - 



This inequality will still subsist, if we increase the second member by changing, 

 in the positive products on the second and third lines, the factors g to their 

 greatest value g„ ; and, after adding the results, suppress the three negative 



terms which remain in the three lines of the expression, and change the functions 

 F, in the first and third lines, to their greatest value F„ . Hence, 



±\ rfaN„F„G„<3bcF g ; (w') 



this integral will therefore ultimately vanish, if the product of the greatest values 

 of the functions f and g tend to the limit 0. Thus, if .we make 



2n 2 



