136 



where C, E, G are rational and integral functions of the coefficients 

 of the given equation, being in fact seminvariants, and F is a mere 

 numerical multiple of the square root of the discriminant. 



The roots of the given quintic equation are each of them rational 

 functions of the roots of the auxiliary equation, so that the theory of 

 the solution of an equation of the fifth order appears to he now 

 carried to its extreme limit. We have in fact 



where (K^XV l) 4 > &c. are the values, corresponding to the roots 

 a?!, &c. of the given equation, of a given quartic function. And com- 

 bining these equations respectively with the quintic equations satisfied 

 by the roots x v &c. respectively, it follows that, conversely, the roots 

 ae lt x 2> &c. are rational functions of the combinations 0^ + 2 4 + 3 </> 5 , 

 0i^2 ~t" 0304 ~^~ 0506 & C> respectively, of the roots of the auxiliary 

 equation. 



It is proper to notice that, combining together in every possible 

 manner the 6 roots of the auxiliary equation, there are in all 1 5 com- 

 binations of the form ^a+^s^+^e- But tne combinations oc- 

 curring in the above-mentioned equations are a completely determi- 

 nate set of five combinations : the equation of the order 1 5, whereon 

 depend the combinations ^0., + 3 4 + 5 06> * s not rationally decom- 

 posable into three quintic equations, but only into a quintic equation 

 having for its roots the above-mentioned five combinations, and into 

 an equation of the tenth order, having for its roots the other ten 

 combinations, and being an irreducible equation. Suppose that the 

 auxiliary equation and its roots are known ; the direct method of 

 ascertaining what combinations of roots correspond to the roots of 

 the quintic equation would be to find the rational quintic factor of 

 the equation of the fifth order, and observe what combinations of the 

 roots of the auxiliary equation are also roots of this quintic factor. 

 The direct calculation of the auxiliary equation by the method of 

 symmetric functions would, "I imagine, be very laborious. But the 



