respecting Equations of the Fifth Degree. 217 



since it is unnecessary, on account of that equation, to retain in evidence the 

 cube or any higher powers of :r , we may write simply 



Oy b, c being here symmetric functions of the three quantities ;?•,, x^, Xj : so that, 

 in this case, the six syntypical functions, or values of the function f, reduce 

 themselves to the three following 



a-\-bXi-\-cXi% a -^ bx^ -\- cx^% a -\- bXj-\- cx^^ . 



Nor can these three reduce themselves to any smaller number, without their all 

 becoming equal and symmetric, by the vanishing of 6 and c. 

 In the second case, the form of f^ being such that 



it must also be such that 



(A 7i «)i = (r. «» ^)i ; 



for the same reason we must have 



(^, a, 7). =z (a, y, /3). = (7, /3, a)., 



so that the function changes when any two of the three indices are interchanged, 

 but returns to its former value when any two are interchanged again ; from 

 which it results that the two following combinations 



(«. A 7)i + (A «. y)i and 



(x —xJlx ~x )(x.~x ) 



remain unchanged, after all interchanges of the indices, and are therefore sym- 

 metric functions, such as 2 a and 2 b, of the three quantities ar^ x^, x^ : so that 

 we may write - 



F. (x^, x^ x^) = (a, ft 7). = a + 6 {x~ xp) {x^- x^) (x^- x^) ; 



and consequently the six syntypical functions, or values of the function r, reduce 

 themselves in this case to the two following, 



a ± 6(a:,-jrj) (or.-^g) (jTs-jrj), 



2g2 



