of Algebraic Equations of the Fifth Degree. 547 



«> /3, y, 8, s, we substitute 1, 2, 3, 4, 5, in all the diflFerent 

 arrangements which they can assume. But these 120 systems 

 will be found to furnish only twelve different sets of values for 

 Pii Ihi Ps- ^"^' fi''*' object will be to express /)j, p^^ p^ as 

 functions of j;,, x^^ . . x^ without t and u. 



3. By combining any three of the five e(juations of the 

 system (a.), we see that we may eliminate t and u ; and that 

 therefore, if we replace x^ + P\^a "^ • • > ^it + Pv ^n + • • j 

 .. x^ + p^x^ -^ .. , by y^t y^j, . . y,, we may arrive at a final 

 equation of the first degree with respect to y^, y^, . . y^, 



where jot^, [in, . . /otj are functions of i which have no common 

 factor different from 1, and are such that two of them must 

 admit of being equated to zero. 



5 . 4< . 3 



4. Il is clear that /*^, jw.^, . . jw.^ will be susceptible of ' ' — 



1 • Z • 3 



or ten differently derived sets of values. The ten equations 

 which may thus arise I shall for the moment represent by 



'V«j/« + 'V/jy/j + . . + ^V.2/s=o. 



AH these will belong to the same system. 



5. Further, if we consider that y^ + 2//3 + ^^ + yg + y,, which 

 must be equated to zero, will give 



•^-^ + V + • • + -^^.^ + K"" + '^A^ + • • + -^^i^)/-! 



+ K + ^,3 + • ._+ ^i)Pi + 5P3 = 0; 

 and that consequently, if denoting 



oC + ^l + " +^r by S„ 

 we eliminate p^ from the equation 



^r^ + Pi ^r + Pa^r + ^3 = ^r* 



there will result 



we shall perceive that any two of those ten equations will, if 

 incapable of being made coincident by any transformation and 

 reduction, be sufficient for enabling us to express p^ and p^ as 

 rational functions of the roots of the original equation. 



6> It is not difficult actually to obtain such a pair of equa« 

 2 02 



