Sib William Rowan Hamilton on Fluctuating Functions. 283 



rc?ap„=0, (w") 



should have no real root a different from 0, between the limits qr (& — a). But 

 it is permitted to suppose, consistently with this restriction, that a is < 0, and 

 that 5 is > 0, while both are finite and determined ; and then the formula (m"), 

 or (c) which is a consequence of it, may be transformed so as to receive new 

 limits of integration, which shall approach as nearly as may be desired to negative 

 and positive infinity. In fact, by changing a to \a, j; to Xx, and^^; to y^;, the 

 formula (c) becomes 



/, = \^-' p„ (J^-,^ <^«/a + 2(„ri J;^-!^ da Q,._,,,„/„) ; (x") 



in which \~'a will be large and negative, while X~^b will be large and positive, 

 if \ be small and positive, because we have supposed that a is negative, and b 

 positive ; and the new variable x is only obliged to be > \~*a, and < X''^, if 

 the new function y*t be finite and vary gradually between these new and enlarged 

 limits. At the same time, the definition (o") shows that PaQx„_x,,„ will tend 

 indefinitely to become equal to 2P2„^(„_,) 5 in such a manner that 



lim . PflQxa— Xj.n ■■ /„"^ 



\ = 2"7 ; ~ ' ^^ ^ 



at least if the function p be finite and vary gradually. Admitting then that we 

 may adopt the following ultimate transformation of a sum into an integral, at least 



under the sign \ rfo, 



*^ CO 



X^i^'o ■ ^ ^ (^ ^» + ^^'•" ^-M»-.)) = j^ d^ P.(a-.). (Z") 



we shall have, as the limit of (x"), this formula : 



fx — ^~^\ G?« W^P;9(a_x)/a; (d) 



which holds good for all real values of the variable x^ at least under the conditions 

 lately supposed, and may be regarded as an extension of the theorem (b), from 

 finite to infinite limits. For example, by making p a cosine, the theorem (d) 



2o2 



