[ 264 ] 



XXXIV. Note on the Resolution of Algebraic Equations. 

 By Thomas P. Kirkman, M.A., F.R.S* 



r |^HE function 2^ of my communication in the preceding 

 -■- Number is needlessly complex. If H x be any asymmetric 

 function of the 7i-\- 1 variables x , x v . . . , x n) which is invariable 

 by the cyclical permutation of all those variables, and which con- 

 sequently receives all its values, suppose U.n values, by the per- 

 mutations of x v x 2 , . . . , x n ._ l , x n , and if T r denote the group of r 

 cyclical permutations of the first r of 1, 2, 3, . . . , n } the remain- 

 ing n— r being undisturbed, and if 



r w Hi = Hi + H2 + H3+ . . . +Hrc=Ji 



be the n values of Hi made by the substitutions of T m then I say 

 that 



V P 7^ P T^ "TV ■pA.piapvp^Ta 



^/3 — -L n^y — L n l n-1 1 n-2 ••• 1 5 i 4 i 3 1 2 J i 



is a rational and symmetrical function of oo v x 2 , . . . , x m whatever 

 be the positive integers (3,<y,h,. . . ,v, %, a, i. Here, as before, 

 the exponent over T r is intended to be written over the sum of r 

 values made by its substitutions. This theorem is sufficiently 

 proved in my paper of last month, and can be made further evi- 

 dent thus. 



If P = G + G 1 + G a +...+G w 



be any group consisting of a group G of any order followed by 

 m — 1 derivates of G, and if 6 be any substitution of P, the ope- 

 ration 6P can change nothing in the right member except the 

 order of the terms G, G 15 . . . , G, n . If (P) be any rational function 

 invariable by the substitutions of the group P, we can write it 



(P) = (G) + (G 1 ) + (G 2 )+...+(GJ, 



where (G. r ) is the value of (G) which is formed on G r . It 

 follows that if (P). denote the sum of the 2th powers of the ra-f- 1 

 values (G r ), we have 



(P) ! .=(G) i +(G 1 ) i + (G 2 ) , + ...+(G») i , 



a function invariable by the substitutions of the group P. For 

 as the operation 0(P) can do nothing more than change the 

 order of the m + 1 functions (G r ), and as the substitution which 

 changes (G„) into (G m ) will change 



(G W )(G W ) into (GJ(GJ, (G n )(G„)(G B ) into (G m )(G w )(GJ, &c, 

 the operation #(P). can make no algebraic change in the right 

 member of (P). ; that is, 



{(G) , + (G 1 ) i + (G 2 ) i + ... + (Gj'}*=(P)f, 



* Communicated by the Author. 



