172 Mr. T. P. Kirkman on the General Solution 



H 1 + H 2 + H 3 +...+H M =={J 1 } 

 H?+Hl+H§+...+H* = {J 2 }, 



H?+H? + H? + ... + H:={J n }; 



and the n asymmetric values of H t made on G n are thus finally 

 transformed, by the solution of a given equation of the nth de- 

 gree, into the irrational shapes 



{H 1 UH 2 },{H 3 },...{H,,}, 



which are all symmetric functions of x v x 2 , . . ., x n . 



From this position to the solution of the general equation 

 © = of the (/z-|-l)th degree, there is but one, little, easy step. 



If a be any imaginary root of y n+l = \ ) and if x , x v x qj . . . Xn 

 be the roots of = 0, we have 



tf = 2<£ + (« + « 2 + a 3 + ... + a l+w )(,r 1 + tf 2 + tf 3 + ...+^ 



or, which is the same thing, 



1 "I K + l -. n+\ -j n+1 



u n + 1 w + 1 l n+l 2 ' n+1 



where 



T 1 = x + ax Y + a?x 2 + A 3 + . . . + u n x n , 



P 2 = cc + ctx 2 + a 2 # 3 + u 3 x 4 + . . . + oC l x v 

 P g = x + a# 3 + a 2 # 4 + a 3 <2? 5 + . . . + ct n x 2 , 



V n = x Q + ax n + a 2 x l + a 3 ^ 2 + . . . + u n X n -\> 



P 2 , P 3 , P 4 , . . ., V n being here formed with P, on Q n =:T n by the 

 cyclical permutations of x l} x v . . ., x n , while x and «, a 2 . . . are 

 undisturbed. 



This expression of # becomes, after raising P ]; P 2 , . . ., P n 

 each to the (n + l)th power by ordinary involution, 



2ff+^(S + H^i + ^(S + H a )*Ti 



° 71 + 1 72+1 V " 72+1 



where S is a symmetric function of x , x v x q , . . ., x n , and there- 

 fore a rational function of the coefficients of © = 0, and where 

 H l is a function of the same 72 + 1 variables which has Tin values 

 by their permutations. And since H, is invariable by the cyclical 

 permutations of x 0) x ]} x v . . ., x n) because P n+1 = (a r P) w+1 , what- 



