Mr. T. P. Kirkman on the Resolution of Algebraic Equations. 263 



whatever be the integers i and k, is invariable by the substitu- 

 tions of the group P. 

 Now the function 



rlJ?=(j?+ 2 J?) f =K^ 



is invariable by the substitutions of the group T 2 ; and as 



p=r 3 r 2 =r 2 + 1 r 2 + 2 r 2 =G+G 1 -hG 2 , 



the function 



RlijJ=(K* + 1 Kj+ t K*) i, =:Ei 



is invariable by the six substitutions of the group T 3 T q , what- 

 ever be the integers v, f, a, i. Again, writing G = r 3 r 2 , 



p=r 4 r 3 r 2 =G+G 1 +G 2 +G 3; 



whence 



rtr;ilJj=(L|+ 1 i4+ 1 i4+a^)' 1 =M; 



is a function invariable by the twenty-four substitutions of 

 P = r 4 r 3 r 2 . Precisely in the same way, writing r 4 r 3 r 2 = G, 

 we prove that N£, the Xth power of the sum of five values of M£, 

 is invariable by the 115 substitutions of P = r 5 r 4 r 3 r 2 , and 

 finally we see that 



is invariable by the lift substitutions of 



X ^ 1 n I n _ i 1 n —o . . . 1 3 1 2 . 



Hence the sum of /3th powers 2^ is a rational and symme- 

 trical function of the n variables x v x 2 , x s , . . . , x n ; the sum 

 of 7th powers Z y is the issuing root {Z } of a given equa- 

 tion of the 71th degree ; the sum of 8th powers Y s is the issuing 

 root { Y s ] of a given equation of the degree n — 1 , . . . the sum 

 of vth powers M v is the issuing root {M„} of a given quintic ; 

 the sum of fth powers L^ is the root {L^} of a given quartic 

 obtained by using v—^, v = 2, v = 3, j/ = 4; the sum of ath. 

 powers K a is the issuing root {K a } of a given cubic, obtained by 

 using £= 1, f = 2, £=3 ; the sum of ith powers J t is the issuing 

 root {J { } of a given quadratic, got by using a = l and « = 2; 

 and from ^ J^ _, using n values of i, we obtain, by solving a given 

 equation of the wth degree, the required expression 



of Hj, H 2 , . . . , H n , as irrational functions symmetrical in the 

 n variables x v x 2 , x 3 , x 4) . . . , x n . 



This restatement and proof of the theorem of my former paper 

 is, I believe, all true, and in part, I hope, new. The application 

 that I mide of it to the solution of equations of the (n + l)th 



