XIII. On the Explanation of a Difficultij in Analysis noticed hy Sir William 

 Hamilton. By Arthur Augustus Moore, Esq. of Trinity College. 



[Read Maij 1, 1837.] 



In the Memoirs of the Royal Irish Society, Sir William Hamilton 

 has made an important observation upon a general principle of Analysis, 

 which has been used by La Grange as the basis of his Calculus of 

 Functions. Sir W. H. remarks that there is a case in which this prin- 

 ciple (which had till then been considered axiomatic and universally 

 true) does not hold good. The case which Sir W. H. cites is the 

 function ^-^s which M. Cauchy had already in his Calcul Differentiel 

 shown to be an exception to another generally received principle of 

 analysis*. M. Cauchy seems to be of opinion that the existence of this 

 anomaly is a sufficient reason for rejecting the mode of exposition of 

 the Differential Calculus of which La Grange is the author, and which 

 is certainly based upon the assumption of both these principles, the 

 latter however of which is comprised in the former as a particular case. 

 But the function e~^' is, only one of a general class of functions which 

 with another constitute the only known exceptions to La Grange's principle. 

 The latter class has no apparent analogy with the former, but on ex- 

 amination we shall find that both these apparent anomalies are immediate 

 consequences of the fundamental conditions of analytical developement, 

 and that the only reason why they were not at once recognized, a priori, 

 as exceptions to the general principle was that in the demonstration 



* M. Cauchy remarks that this function and all it differential coefficients vanish for the 

 particular value of the variable j; = 0, although the function itself does not vanish for any 

 other value of the variable, thus constituting an exception to a generally received analytical 

 principle. 



