346 THIKli KtroRT lout/. 



admitting the correctness of that form of it which Poisson ha^ 



assigned*. 



* If we transform e-*' by replacing a by ^, we shall get the expression 



c •"'*, which may be easily shown as above, and also by other means, to be 

 equal to zero when x' is equal to zero, and equal to 1 when x' is equal to in- 

 finity. 



Professor Hamilton of Dublin, in a paper in the Irish Transactions for 1830, 



has quoted the expression e -^ as possessing some very peculiar properties, 

 which are inconsistent with the universality of a very commonly received 

 principle of analysis. It is commonly assumed that " if a real function of a 

 positive variable x approaches to zero with the variable, and vanishes along 

 with it, then that function can be developed in a real series of the form 



Aa« +Ba;^ + Cx'' + &c. (I.) 



where «, /3, y, &c., are constant and positive. A, B, C, &c., constant, and all 

 those coefficients diflferent from zero : but if we put the equation under the 

 form 



^-« e"'^^ = A + B a;^-" + C xV-^^ &c., 



supposing » the least of the several indices », /3, y, &c., then if x = 0, we 



_— ] 



shall find x""* e •''^ = or A equal to zero ; for if we replace — by y, we 



shall get 



1 i. 



4 — a 6 — a, 



" " 1.2 1.2.3 



all whose terms are positive, and which, when a!= ory = oo , will necessarily 



become equal to infinity : it follows, therefore, that the function e * is not ca- 

 pable of developement in a series of the assumed form (1.). The same ex- 

 pression, as has been remarked by Professor Hamilton, has been noticed by 

 Cauchy as an example of the vanishing of a function and of all its differential 

 coefficients, for a particular value of the variable, without the function va- 

 nishing for other values of the variable, thus forming an exception to another 

 principle generally received in analysis. In his Lemons sur le Calcul Infinite- 

 simal, Cauchy has produced this last anomaly as a sufficient reason for not 

 founding the principles of the differential calculus upon the developement of 

 functions, as effected by or exhibited in, the series of Taylor. 



It is possible that more enlarged views of the analytical relations of zero 

 and infinity, and of the interpretation of the circumstances of their occurrence, 

 as well as of the principles and applications of Taylor's series, may enable 

 us to explain these and other anomalies, and to show that they arise natu- 

 rally and necessarily out of the very framework of analysis ; but it must be 

 confessed that there are many other difficulties, which are yet unexplained, 

 which are connected with the developement of e*' when x is negative or ima- 



