respecting Equations of the Fifth Degree. 221 



to that . 



(a, A 7) • (<h y, /3) = (A y, «) (A «, 7) = (7» «, ^) (7. A «) = ^, 



this product e being some symmetric function; at the same time, the sum 

 (o, j3, 7) 4" («j 7> /3) is a three-valued function 1/^, which may be put under 

 the form 



a, b, and c being symmetric, and b and c being obliged not both to vanish. 

 Attending therefore to that cubic equation of which x^, x^ and x are the roots, 

 we have 



y/ = a^« + 6«'a;„ + c^^'V, 



a'*', b^, and c® denoting here some symmetric functions, and c, c'^' being obliged 

 not both to vanish ; and consequently, by eliminating x^, we obtain an equation 

 of the form 



in which the coefficients of ^„ and _y„2 cannot both vanish, and in which therefore 

 the coefficient of x^ cannot vanish, because the three-valued function ^^ must not 

 be a root of any equation with symmetric coefficients, below the third degree ; we 

 have therefore an expression of the form 



2 



fa' 



in which, p, q, r are symmetric, and q and r do not both vanish. But 



i'a = («> ^' 7) + («' 7> /3) = («. A 7) + {a,^,y) ' 



and the cube of (a, /3, 7) is a two-valued function ; therefore 



^a=i'' + ?'(«.A7)+^'(«,A7)^ 



the functions p', q', r' being either symmetric or two-valued, and consequently 

 undergoing no change, when we pass successively from the first to the second, or 

 from the second to the third, of the three functions (a, j8, 7), (p, 7, a), (7, a, /3), 

 by changing at each passage, x^ to Xn, x^ to x , and x to x^ ; and we have seen 



