Sir William Rowan Hamilton on Fluctuating Functions. 287 



Now it often happens that if the function y^ be obliged to satisfy conditions which 

 determine all its values by means of the arbitrary values which it may have for a 

 given finite range, from a :=a to a = b, the integral relative to a in the formula 

 (f) can be shown to vanish at the limit Ar = 0, for all real and positive values of 

 /3, except those which are roots of a certain equation 



Qp = ; (g) 



while the same integral is, on the contrary, infinite, for these particular values of 

 j8 ; and then the integration relatively to /3 will in general change itself into a 

 summation relatively to the real and positive roots p of the equation (g), which is 

 to be combined with an integration relatively to a between the given limits a and 

 b ; the resulting expression being of the form 



/x = 2,(<^a0.,„,X (h) 



For example, in the case where p is a cosine, and f a negative exponential, if 

 the conditions relative to the function y be supposed such as to conduct to expres- 

 sions of the forms 



in which h is any real or imaginary quantity, independent of a, and having its 

 real part positive ; it will follow that 



1 



S 



dae-''^' (cos /3a — v/ — I sin /3a)/. 



_Vr(/3v/-l+^) ^(/3v/-l-A;) 



(P'") 



0(^/-l+A;) cpip^-l-k) 



in which v^a* is = a or = — a, according as a is > or < 0, and the quantities 

 ^ and k are real, and k is positive. The integral in (p'"), and consequently 

 also that relative to a in (f), in which, now. 



p„ = cos a, F„ = e **^'•^ 



