Sir William Rowan Hamilton on Fluctuating Functions. 



305 



should either receive no change, or simply change its sign, when x is increased 

 by p, according as j3 tends to co by coinciding with large and odd or with large 

 and even numbers. 



[23.] In all the examples hitherto given to illustrate the genei'al formulas of 

 this paper, it has been supposed for the sake of simplicity, that the function p is 

 a cosine ; and this supposition has been sufficient to deduce, as we have seen, a 

 great variety of known results. But it is evident that this function p may receive 

 many other forms, consistently with the suppositions made in deducing those 

 general formulas ; and many new results may thus be obtained by the method of 

 the foregoing articles. 



For instance, it is permitted to suppose 



p„=l, ifa^<l; (k''") 



p, = 0; {V") 



* n l_0 — ■ 



and then the equations (t^^) of the last article, with all that were deduced from 

 them, will still hold good. We shall now have ^^ 



and the definite integral denoted by zr, and defined by the equation (r'), may 

 now be computed as follows. Because the function n„ changes sign with «, we 

 have 



T3- = 2C rfaN„a-'; (o''^^) 



but 



and 

 Hence 



\ rfa N„ a ' = 6 log 3 — 4 log 4, 



(P^") 



(q''") 



(r-O 



the logarithms being Napierian ; and generally, if m be any positive integer num- 

 ber, or zero, 



VOL. XIX. 2 R 



