Sir William Rowan Hamilton on Fluctuating Functions. 289 



the suppositions (n'") (o'") are satisfied ; the functions and ^ being here such 

 that 



^ (A) = C rfa («*('-") — e*(»-'))/„ ; 



therefore the equation (q'") becomes in this case 



cos pi = 0, (w'") 



and the expressions (s'") for the coefficients of the development (t'") reduce 

 themselves to the following : 



2 c' 

 ^(, = Y^ da cos /Ja/„ ; B„ rz ; (x'") 



so that the method conducts to the following expression for the function y^ which 

 satisfies the conditions (v'"), 



/. = ^2,.-cose^^;i::^(.«cos e^il^/.; if) 



in which y^ is arbitrary from a = to a = /, except that fi must vanish. The 

 same method has been applied, by the authors already cited, to other and more 

 difficult questions ; but it will harmonize better with the principles of the present 

 paper to treat the subject in another way, to which we shall now proceed. 



[15.] Instead of introducing, as in (e) and (f), a factor which has unity for 

 its limit, we may often remove the apparent indeterminateness of the formula (d) 

 in another way, by the principles of fluctuating functions. For if we integrate 

 first relatively to a between indefinitely great limits, negative and positive, then, 

 under the conditions which conduct to developments of the form (ii), we shall 

 find that the resulting function of j3 is usually a fluctuating one, of which the 

 integral vanishes, except in the immediate neighbourhood of certain particular 

 values determined by an equation such as (g) ; and then, by integrating only in 

 such immediate neighbourhood, and afterwards summing the results, the develop- 

 ment (h) is obtained. For example, when p is a cosine, and when the conditions 

 (v'") are satisfied by the function yj it is not difficult to prove that 



VOL. XIX. 2 p 



