respecting Equations of the Fifth Degree. 235 



Reciprocally, if these four last conditions can be satisfied, by any suitable 

 arrangement of the four roots, and by any suitable choice of those coefficients or 

 functions which have hitherto been left undetermined, we shall have the four ex- 

 pressions just now mentioned, for the four roots of the general biquadratic, as 

 the four values of an irrational and irreducible function V", of the third order. 

 Now, these four conditions are satisfied when we suppose 



^a — "^V "^13 — ^'ii ^y — *''3> ■^S — ^' 



4 1 



II _ — a, ," _ 1 7" _ 1 



Oo.o-^-; 01.0-4^,; ^'-'"iv' 



and finally 



/; 



h..= 



16 64 



vww 



2«, 2 ' 



but not by any suppositions essentially distinct from these. It is therefore 

 possible to express the four roots of the general biquadratic equation, as the four 

 values of an irrational and irreducible expression of the third order b'", namely 

 as the following : 



^ _ z,'" _ -fli . ar , ai" , 16V&/'e4 



X - h" =z ZflL 4. J!l1 _ <1 _ 16&r6."e4 

 ' "■' 4 4 6," 4V' ar<' ' 



*3 — "1,0 T- -TTT/ "T" 



4 4 6," 462" ai"'a, 



in „ III > 



^ ,"' _ -a, a,'" a/' , \&b^'b^'e^ 



** — ''1,1 T- 



46," 462" a{"a^" ' 



and there exists no system of expressions, essentially distinct from these, which 

 can express the same four roots, without the introduction of some radical, such 

 as o/^, of an order higher than the third. We must, however, remember that 

 these expressions involve several arbitrary symmetric functions of x^, x^, x , i\, 

 or arbitrary rational functions of a^, a^, a^, a^, which enter into the composition 

 of the radicals o/, o/', a/" a/", though only in the way of multiplying a function 

 by an exact square or cube before the square-root or cube-root is extracted : 



