respecting Eq nations of the Fifth Degree. 223 



Let x„ x^, ir^, x^ be any four arbitrary quantities, or roots of the general 

 biquadratic, 



x'' -f a, ar* -j- Oj ar* -}- «3 a; + 04 = ; 



and let ^{x„ x^, x^, x^), or, more concisely, (1, 2, 3, 4), denote any rational 

 function of them. By altering the arrangement of these four roots, we shall in 

 general obtain twenty-four different but syntypical functions ; of which each, 

 according to the analogy of the foregoing notation, may be denoted by any one 

 of the four following symbols : 



(a, ^, 7, g) = F (X„, X^, X^, X^) 

 ^ (1, 2, 3, 4)^ =: ^j (^i> ^2> ^3> "^v- 



In passing from any one to any other of these twenty-four syntypical functions 

 F,, . . . F24> hy a change of arrangement of the four rm)ts, some one of these roots, 

 such as the first in order, must be changed to some other, such as the second ; 

 and this second must at the same time be changed either to the first or to a diffe- 

 rent root, such as the third ; while, in the former case, the third and fourth roots 

 may either be interchanged among themselves or not ; and, in the latter case, 

 the third root may be changed either to the first or to the fourth. We have 

 therefore four and only four distinct sorts of changes of arrangement,, which 

 may be typified by the passages from the function (a, /3, 7, 8) to the four 

 following : 



I. . (i3, a, 7, 5) ; II. . (^ a, g, 7) ); III. . (/3, 7, a, I) ; IV. . (P, 7, I, a) ; 



and may be denoted by the four characteristics 



Vl> V2> V3> V4; 



or more fully by the following, 



Of h a^b a,h,c a,h,c . ^ 



Vi> V2, V3> V4; 



a, h 



V 1 implying, when prefixed to any function (a, ^, 7, I), that we are to inter- 



a,b 



change the a'* and ¥'' of the roots on which it depends ; v 2> that we are to 

 interchange among themselves, not only the a'* and /!»'*, but also the c'* and d"" ; 



a,b,c 



V3, tliat we are to change the a" to the ¥", the h"' to the (f, and the c'* to the 



VOL. XVIII, 2 H 



