132 Mr. Murphy on the 



and expanding each term of this series according to the descend- 

 ing powers of (3, the value of the root is 



2 3 4 



. a a a a 



«%, „ « ,3.4 a" 4.5 a 5 



a 5J> 



2'(i ll ~ " fi 1.2 ' fi" 





D.a a . a 



, 8.7.6 a 5 



&c. &c. 



All the terms here mutually strike out with the exception of the 

 first a; but since a and /3 are similarly involved in the given 

 series, it follows that if we expanded according to the descending- 

 powers of a, all the terms would strike out except /3; thus this 

 series analytically represents a or (Z indifferently ; but if we stop 



( a \" 

 Ti) ' 



(3\ n 

 -) ; and if a < fi, the former is very small, 



and the latter great; the series therefore, arithmetically, gives the 

 least root (abstracting from its sign) ; and generally, whatever is 

 the proposed equation, the series expressing the root is manifestly 

 a symmetrical function of all the roots, and therefore does not 

 analytically express one more than another, but comprehends 

 alike all the roots ; but as each term may be expanded in a con- 



