Sir William Rowan Hamilton on Fluctuating Functions. 311 



for thus, by inverting the order of the two remaining integrations, that is by 

 writing 



^ da^ de... = ^ d6\ da.., {f") 



we find first 



lim P^ sin ((4^ + 2) (g-^) sine) _ ' 



for every value of 6 between and tt, and of x between a and b ; and finally. 



[25.] The results of the foregoing articles may be extended by introducing, 

 under the functional signs n, p, a product such as §r^, instead of j3«, 7 being an 

 arbitrary function of a. ; and by considering the integral 





in which f is any function which remains finite between the limits of integration. 

 Since 7 is a function of a, it may be denoted by 7^, and a will be reciprocally a 

 function of 7, which may be denoted thus : 



While a increases from a to b, we shall suppose, at first, that the function 7^ in- 

 creases constantly and continuously from 7„ to 74, in such a manner as to give 

 always, within this extent of variation, a finite and determined and positive value 

 to the differential coefficient of the function 0, namely, 



We shall also express, for abridgment, the product of this coefficient and of the 

 function f by another function of 7, as follows, 



0'.Fa = ^ (d«) 



Then the integral (a"^) becomes 



