Sir William Rowan Hamilton on Fluctuating Functions. 303 



Consequently, if the equations (t^^) be satisfied, the multiples (by whole num- 

 bers) of p will all be roots of the equation (a^^) ; and reciprocally that equation 

 will have no other real roots, if we suppose that the function p.., which vanishes 



when a is any odd multiple of ^, preserves one constant sign between any one 



P 

 such multiple and the next following, or simply between a = and « = ^- We 



may then, under these conditions, write 



Vi = ip, (x''') 



i being any integer number, positive or negative, and vi denoting generally, as 

 in (b''^), any root of the equation (a''^). And we shall have 



^"</aN. + ,pa-' = (-l)*^, if) 



k being any integer number, and w still retaining the same meaning as in the 

 former articles. Also, for any integer value of k, 



P^ = (-1)*P.. (z"') 



These things being laid down, let us resume the integral (e''^, and let us sup- 

 pose that the law by which j3 increases to co is that of coinciding successively with 

 the several uneven integer numbers 1, 3, 5, &c., as was supposed in deducing the 

 formula (c). Then §v in (e ^^) will be an odd or even multiple ofj), according 

 as V is the one or the other, so that we shall have by (x''^), (y^^, the following 

 determined expression for the sought limit (f '^^) : 



^, = (-l)V; (a-0 



but also, by {x"'), (z^O. 



P.,. = (-1)'P,; (b"") 



therefore 



^.pr' = ^Po-', (c'-^O 



the value of this expression being thus the same for all the roots of (a^^). At 

 the same time, in (i''^), 



the equation (t) becomes therefore now 



