Note to a Paper on the Error of a received Principle of Analysis. 

 By WILLIAM R. HAMILTON, &c. 



Read April 18, 1831. 



The Royal Irish Academy having done me the honour to pubUsh, 

 in the First Part of the Sixteenth Volume of their Transactions, a 

 short Paper, in which I brought forward a certain exponential func- 

 tion as an example of the Error of a received Principle respecting 

 Developments, I am desirous to mention that I have since seen (with- 

 in the last few days) an earlier Memoir by a profound French Mathe- 

 matician, in which the same function is employed to prove the fal- 

 lacy of another usual principle. In the French Memoir, (tom. v. of 

 the Royal Academy of Sciences, at page 13, of the History of the 



Academy,) the exponential \e **/ is given by M. Cauchy, as an 

 example of the vanishing of a function and of all its differential 

 coefficients, for a particular value of the variable {x), without the 

 llinction vanishing for other values of the variable. In my Paper 

 the same exponential is given as an example of a function, which 

 vanishes with its variable, and yet cannot be represented by any 

 development according to powers of that variable, with constant 

 positive exponents, integer or fractional. There is therefore a diffe- 

 rence between the purposes for which this function has been em- 

 VOL. XVI. 2 ^ 



