of Algebraic Equations. 173 



ever r may be, it receives all its values by the permutations of 

 the n variables x v x 2 , # 3 , . . ., x n . 



"Wherefore, by what precedes, the n values of H^ 



can be transformed into the issuing roots 



{H,K {H 2 }, {H 3 } ) ...,{H„}, 



which are symmetric irrational functions of x {i x 2 , x 3 , . . . <27«, and 

 also of necessity symmetrical irrational functions of the n + 1 va- 

 riables # , a? la ..., #». These functions {H} are consequently 

 transformable into given irrational functions of the coefficients 

 of © = 0, and the algebraic solution of ® = is infallibly com- 

 pleted by the substitution of these issuing roots so transformed, 

 for the n asymmetric functions H ls H 2 , . . ., H n in our value of 

 x Q . All this can be verified for the simple cases of n + 1 = 3, 

 w + l=4, and ra + l = 5. 



It may be supposed that these processes will be encumbered 

 with hopeless ambiguity by the multiplicity of values of the 

 issuing roots 



{Z r }, {Y,}, {Xr},...,{Jr}, 



which appear in the coefficients of the equations of the nth and 

 lower degrees which have to be solved. 



But the truth is that there is no ambiguity at all, if we take 

 care that the issuing roots 



{Z,}, {Z 2 },...{Z„_ l } 



are all corresponding roots (that is, if we read them all with the 

 same radicals affected by the same signs and imaginaries), and 

 if we take care that the issuing roots 



{Y,}, {Y 2 },...{Y„_ 2 }, 



as well as 



{J,}, {3,}, . . . {J,,}, 



are in each set of solutions a set of corresponding roots. For 

 the rest, it matters not which of the n issuing roots we take for 

 {Z y }, nor, after that, which of n — 1 issuing roots we take for 

 {Y5} &c. By whatever path we choose to march, we shall assu- 

 redly arrive at the symmetrical expression of {J ,} the sum of n 

 values of H l ], formed either on G n , or on one of its U(n — 1) 

 — 1 derivatives; and a? is correct if expressed in n values of 

 the Hj above found, which are formed on any one of those deri- 

 vatives. In other words, we may write for Pj, P 2 , . . . , ~P n in our 

 expression of x Q the n results of operation on them by any sub- 

 stitution, not in G rt , made with the n elements 1, 2, 3, . . . , n. 



