288 Sir William Rowan Hamilton on Fluctuating Functions. 



will therefore, under these conditions, tend to vanish with k, unless ^ be a root p 



of the equation 



</>(pv/-l)=0, (O 



which here corresponds to (g) ; but the same integral will on the contrary tend 

 to become infinite, as k tends to 0, if /3 be a root of the equation (q'")- Making 

 therefore |3 = p -J~ ^^' ^"<^ supposing k\ to be small, while p is a real and posi- 

 tive root of (q'"), the integral (p'") becomes 



k-' 



1+V 

 in which A^ and b^ are real, namely. 



,(A,-v/-lBj. (r'") 



' ^'{p^-l)^<t>'(-pv'-iy 



(n 



(f) being the differential coefficient of the function 0. Multiplying the expres- 

 sion (r'") by 7r~' d^ (cos ^x -^ \/ — 1 sin ^x), which may be changed to 

 Tr~^ kd\ {cos px -\- \/ — 1 sin pa:) ; integrating relatively to X between indefi- 

 nitely great limits, negative and and positive ; taking the real part of the result, 

 and summing it relatively to p ; there results, 



/x=2p(ApCospar-HBpSinp^); (t'") 



a development which has been deduced nearly as above, by Poisson and Liou- 

 viLLE, from the suppositions (n'"), (o'"), and from the theorem of Fourier 

 presented under a form equivalent to the following ; 



/x = ^^™Q • '^"^ J ^^ S "^^ «'* "^"^cos i^a - ^x)f^ ; (u'") 



and in which it is to be remembered that if be a root of the equation (q'")) the 

 corresponding terms in the development ofy^; must in general be modified by 

 the circumstance, that in calculating these terms, the integration relatively to A 

 extends only from to oo. 



For example, when the function y is obliged to satisfy the conditions 



