318 Mk MOORE, OX A DIFFICULTY IX ANALYSIS 



of this principle itself we commit the error of passing directly from 

 finite to infinite states of functions and variables, instead of estimating 

 and comparing their relations in the different stages of their convergency. 

 I have attempted in what follows to give a rigorous demonstration of 

 the principle in question, at the same time fixing the precise limits of 

 its application, and enumerating the different classes of functions to which 

 it necessarily does not apply. 



1. To effect this object I shall begin by explaining what is inider- 

 stood by infinitesimals of different orders. If a function of a; converges 

 indefinitely towards zero along with x, in such a manner that, for a 

 very small value of a; J'{-r) shall be less than any given magnitude, 

 the function J'{x) at the limit of those values of x which converge in- 

 definitely towards zero is called an infinitely small quantity or infinitesimal. 

 But as for similar decreasing values of x the ratio of convergency may 

 be much higher in one function than in another, we are led naturally 

 to consider indefinitely decreasing quantities of different degrees or orders 

 of convergency. And having fixed upon some one function whose 

 ratio of decrease we assume as the unit of convergency, we call a 

 second function which for similar decreasing values of x decreases in 

 m times as fast a ratio as the first, an indefinitely decreasing quantity 

 of the ?«* order. We extend this definition to the infinitesimals which 

 are the limits of these quantities, and call the infinitesimal which is 

 the limit of the former of the quantities, an infinitesimal of the first, 

 and the infinitesimal which is the limit of the latter, an infinitesimal 

 of the »«'" order. Choosing x itself for the function whose ratio of 

 decrease is taken as the unit of convergency, we see clearly that when 

 X is less than unity Ax" is an indefinitely decreasing quantity of the 

 m"' order, where m may be integer or fractional. From this we infer 

 that Ax" may represent an indefinitely decreasing quantity of anj^ order, 

 and that the limit of Ax" for values of x which converge indefinitely 

 towards zero may represent an infinitesimal of any order. This we 

 shall designate by the notation lim, = o (^^«'"). A very wide generaliza- 

 tion, which only suggests itself from the study of the different analytical 

 functions, is given to this definition by defining lim ^=„ {_/(.r)', to be 

 an infinitesimal of the ?«"" order, if any finite and positive value of m can 



