300 Sir William Rowan Hamilton on Fluctuating Functions. 



fx = •a--' Po V ^ \ da. s„_^,^/„ ; (s) 



which includes the transformations (c) and (r), and in which the notation V„ is 

 designed to indicate that after performing the operation V/3 we are to make the 

 variable /3 infinite, according to some given law of increase, connected with the 

 form of the operation denoted by v • 



[21.] In order to deduce the theorems (c), (r), (s), we have hitherto sup- 

 posed (as was stated in the twelfth article), that the equation n„ = has no real 

 root different from between the limits :+:(& — a), in which a and h are the 

 limits of the integration relative to a, between which latter limits it is also sup- 

 posed that the variable x is comprised. If these conditions be not satisfied, the 

 factor N„"l'j, in the formula (m"), may become infinite within the proposed extent 

 of integration, for values of a and x which are not equal to each other ; and it 

 will then be necessary to change the first member of each of the equations (m"), 

 (c), (r), (s), to a function different fromy^, but to be determined by similar 

 principles. To simplify the question, let it be supposed that the function n„ re- 

 ceives no sudden change of value, and that the equation , 



N„ = 0, (a'-O 



which coincides with (w"), has all its real roots unequal. These roots must here 

 coincide with the quantities a„^j, of the fourth and other articles, for which the 

 function n„ changes sign ; but as the double index is now unnecessary, while the 

 notation a„ has been appropriated to the roots of the equation (g), we shall denote 

 the roots of the equation (a''^), in their order, by the symbols 



and choosing v^ for that root of (a^^) which has already been supposed to vanish, 

 we shall have 



v, = (\ (c'O 



while the other roots will be > or < 0, according as their indices are positive or 

 negative. If the differential coefficient p„ be also supposed to remain always finite, 

 and to receive no sudden change of value in the immediate neighbourhood of any 

 root V of (a''^), we shall have, for values of a in that neighbourhood, the limiting 

 equation : 



