On the Error of a received Principle of Analysis, respecting Functions 

 which vanish with their Variables. By WILLIAM R. HAMIL- 

 TON, Royal Astronomer of Ireland, SfC. 



Read January 25, 1830. 



It appears to be a received principle of analysis, that if a real 

 function of a positive variable (x) approaches to zero with the varia- 

 ble, and vanishes along with it, then that function can be developed 

 in a real series of the form 



Ax ■{■ Bx^ + Cx^ A. kc . 



the exponents a, /3, y, . . being constant and positive, and the coeffi- 

 cients A, B, C, . . being constant, and all these constant exponents 

 and coefficients being finite and different from zero. This principle 

 has been made the foundation of important theories, and has not 

 ever, so far as I know, been questioned; but I believe that the fol- 

 lowing example of exception, which it would be easy to put in a 

 more general form, will sufficiently prove it to be erroneous ; since 

 if the principle be true, it is by its nature universal. 



The real function e , in which e is the base of the neperian 



logarithms, approaches to zero along with x and vanishes along with 

 it. Yet if we could develope this function in a series of the kind 

 described, we should have 



— 2 



"e~ " = A + Bx^~'+Cx ^■^•+ &c. 



VOL. XVI. 



