138 Mr. Murphy on the 



In comparing thus the magnitude of the roots, we go on the 

 hypothesis of their reality. When the roots are imaginary how- 

 ever, a similar order may be supposed to subsist ; thus - a + \/ - — b 



4 



■ a— V -r — 



is a less numerical root than -a—\/-—b (abstracting from the 

 sign) when — > b, it holds therefore the same rank when — < b. 



Since the whole theory of imaginary quantities results from 

 an extension of the properties of real quantities, we have this 

 advantage resulting from the theorem above proved, that it esta- 

 blishes an order when quantities are some real and some ima- 

 ginary — analogous to the relation of greater and less amongst real 

 quantities. 



To find the m th least root of a proposed equation <f> (x) = o. 



From the coefficient of - in 1. %r-l subtract the coefficient of - 



x x'"- x 



<b (x) 

 in 1. -~r > tne remainder will be the »»"' least root. This theorem 



x'" 



is an immediate consequence of what has been already proved in 

 this Section. 



To find the sum of any function /{a^ J'(a s ) &c. of the m least 

 roots. 



1 (b (x) 



From m/(0) subtract the coefficient of - in /' (x) . 1. ±—^- , 



the remainder will be the required quantity. 



The proof of this theorem is similar to that given for a function 

 of one root, it will be unnecessary therefore to add it. 



To find the value of any function of the m th least root. 



