NOTICED BY SIR WILLIAM HAMILTON. 319 



be found which will render lim^^o] ,1 [ a finite quantity. A corollary 

 to this definition immediately offers itself, viz. that if Vim ,,^„ {f{x)\ is 

 an infinitesimal of the w/'' order, the function '— ^ increases or decreases 



indefinitely, while x converges indefinitely towards zero, according as »/, 



is greater or less than ???. For 



,/W ^,f(x) 1 



x"-' X" ' X'"'-"'' 



an(j as ^Lizl converges towards a finite limit while x converges indefinitely 



f(x) 

 towards zero, it depends upon the sign of n/i — m whether '- — ^ increases 



or decreases indefinitely at the same time. But if no finite and positive 



value of m can be found which will render lim^^oj^^^l a finite quan- 



f{x\ 

 tity, there are two cases to be considered. 1st. \i —~- converges inde- 

 finitely towards zero along with x however great m may be taken, it 

 follows from the general definition of an infinitesimal of the ??«"" order 

 that lim r = o \fkx)\ is an infinitesimal of an infinitely high order. 2d. If 



J—^ increases indefinitely towards - for values of x which converge in- 



definitely towards zero, however small m may be taken, it follows from 

 the same general definition that lim^ = ol/(^)} is an infinitesimal of an 

 infinitely low order. 



2. Of infinitesimals in general I may enunciate the following 



theorem. 



THEOREM. 



If \\xa. ,^t,\f {x)\ is an infinitesimal of the »«"' order, and if 

 limx=o{0(^)S is an infinitesimal of the m^^ order, the equation /(a;) = 0(a-) 

 cannot exist for the general value of x. 



Dem. For if /(a-) can be equal to (p{x) for the general values of 



f{x\ (b {x\ 



X, dividing by x", we find that '^-^ can be equal to ^--^, which is 



