respecting Equations of the Fifth Degree. 173 



can be expressed as a finite irrational function of the two arbitrary coefficients 

 a„ a^, namely, as the following function, which is of the first order : 



x = b' —f «, a^, Oj) = -g-^ + «i'> 

 the radical a,' being such that 



< =/i («n ««) = X "~ *2 ; 



insomuch that, with this form of the irrational function 5', the equation 



h'^ _|_ a^y + a, = 



is satisfied, independently of the quantities a, and a^, which remain altogether 

 arbitrary. 



Again, it is well known that for w = 3, that is, in the case of the general 

 cubic equation 



:r' -J- a,.r* -\- a^x -\- a^-=.% 



a root jc may be expressed as an irrational function of the three arbitrary co- 

 efficients, a,, Oj, Oj, namely as the following function, which is of the second 

 order : 



or = 6" =/' «', <, a„ a„ a^ 



the radical of highest order, o/', being defined by the equation 



= c, + a,', 

 and the subordinate radical a,' being defined by this other equation 



while c, and c.^ denote for abridgment the two following rational functions : 



c. = - 7¥ (2 V - 9o. o, + 2703), 

 Ca = i (a,* — 302) ; 



so that, with this form of the irrational function 6", the equation 



