304 Sir William Rowan Hamilton on Fluctuating FuJictions. 



^ifr + ip = •=^~' Po V ^ ) da S„_^,^/„, (u) 



/3 tending to infinity by passing through the successive positive odd numbers, and 

 i receiving all integer values which allow x-\-ip to be comprised between the 

 limits a and b. If any integer value of i render x -\-ip equal to either of these 

 limits, the corresponding term of the sum in the first member of (u) is to be \fa, 

 or ^/"j ; and if the function y receive any sudden change of value between the 

 same limits of integration, corresponding to a value of the variable which is of the 

 form X -\- ip, the term introduced thereby will be of the form ^/^ -j- ^J'^\ 

 For example, when 



P„ = cos a, sr =. TT, p — -n, i^'^") 



we obtain the following known formula, instead of (r"), 



^i/.+i. = T-> 2(„) .:( da COS (2na - <2nx)f^ ; {{"") 



which may be transformed in various ways, by changing the limits of integration, 

 and in which halves of functions are to be introduced in extreme cases, as above. 

 On the other hand, if the law of increase of j8 be, as in (r), that of coinciding 

 successively with larger and larger even numbers, then 



and the equation (t) becomes 



2i(-l)'/x+.v = ^''Po V,J c?«s<._,,^/„. (v) 



For example, in the case {e^^'), we obtain this extension of the formula (b''), 



2i(-iy/x + ,v = 7r-'2w_:^'rfacos(2^m.^T:::i:)/„. (h''^^) 



We may verify the equations ({^") (h^") by remarking that both members of 

 the former equation remain unchanged, and that both members of the latter are 

 changed in sign, when x is increased by tt. A similar verification of the equa- 

 tions (u) and (v) requires that in general the expression 



