of Algebraic Equatioyis of the Fifth Degree. 559 



seen independently of the form of that equation, or of the na- 

 ture of its roots*, may be deduced anew, in conjunction with 

 some very remarkable properties of the roots in question, from 

 the theorems given in the last section. 



Designating by P one of the quantities 2hi P<2> ^^ perceive 

 from the equation (f.), or rather from 



(in which /3^ may be supposed to be equal to any one of the 

 four indices, /3, y, 8, e; and y , s to be each of them different 

 from a), that the expression for P, considered as a function of 

 a?„, Xji, . . «■,, will be such as to assume all its values whilst one of 

 the roots 57^, remains fixed; and Xf^^ a-^, x-^, x^ undergo among 

 themselves all the different changes of arrangement to which 

 they can be subjected. It appears therefore already that the 

 equation on which P depends cannot rise above the [\.2.SA)\h 

 degree. 



Again, it is evident from the next theorem (g.) that of the 

 four pairs of equations to which all those which include/^, or 



Vi + yii— (' + ''') y«' ^^'^ reducible, 



A = 0, 1 AC^j-) = 0, 1 /, (y 5) = 0, 1 /, CiOCxD = 0, J 



the first will furnish the same expression for P as the fourth ; 



and the second as the third. 



And since^^ will not, whilst u remains fixed, admit of more 



4x3. 



than :; t: different expressions, the number of different values 



1x2' 



4i 4x3 

 which may be assigned to P cannot exceed — x . 



^ 1 X <6 



P thei'efore will depend on an equation of the 12th degree, 

 or rather on an equation of the form 



(Pi2 + Bi P'l + B^Pio + .. + Bi2)io = 0; 

 in which B„ Bg, . . Bjg are symmetric relatively to x^, x^, . . x^ 

 and may consequently, as is well known, be expressed as ra- 

 tional functions of A J, A2, . . Ag, the coefficients of the original 

 equation. 



27. To obtain the roots of this equjitioo in terms of a^j, x^, 

 . . Xf^, let us suppose that ' '^ •• -' 



« = 1; 



* From considering that the equations 



A\=!P, A'3=0, A'4-iAV.= 

 o 



are of the first, third, and fouyth degrees relatively to pi, p^, p^. 



