respecting Equations of the Fifth Degree. 181 



and the functions yj", f^',, which enter into the equations of definition of the 

 radicals a,'", o^'", namely into the equations 



- rt '"2 4-11 „ llli fll 



»l — J I > "2 — 72 > 



may in like manner be expressed so as to involve no radicals in denominators, 

 namely thus : 



a,"- = .3 + <' + («.-<)(^y, 



«2 



«\2 



1112. 



It would be easy to give other instances of the same sort of transformation, 

 but it seems unnecessary to do so. 



[4.] It is important in the next place to observe, that any term of the fore- 

 going general development of the general irrational function b , may be isolated 



from the rest, and expressed separately, as follows. Let b (^) (m) denote a 



new irrational function, which is formed from b by changing every radical 



(m) 



such as a. to a corresponding product such as /) ' a. , in which p („) is, as 

 before, a root of unity ; so that 



^ (m) W 



(m) (m) /^^^ (m) 



(m) (m) 



and let any isolated term of the corresponding development of 6 or 6 be 

 denoted by the symbol 



(m) • _ (m-1) W/3; W/3(„) 



^^(•"), . . . /3« ~ /^C""), . . . /3W ■ ^l • ■ • "Z™) " ' 



