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XXIII. On the General Solution of Algebraic Equations, 

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



ALL algebraic equations can be solved as clearly as the cubic 

 and the quartic, that of the (w-{-l)th degree after that of 

 the nth, by the use of the following theorem, and of familiar 

 algebraic operations. 



Let any of the various ways of writing the group fl of Tin 

 substitutions made with the n elements 1,2,3, ... , (n — l),n } as 

 a product of n — 1 groups, be this, 



H = G 2 . G 3 . G 4 . . . G n -- 2 . Gr w _i. G„, 



where G,. is a group of the rth order, in which the n—r final 

 elements are undisturbed. 



Let Hj be any given asymmetric function of the n variables 

 #j, ,r 2 , x 3 , . . . , x n which has Tin values by their permutations. 



Let G n . IT be the sum of the n values of H\ made by the 

 substitutions of G n , and let 



be the ath power of that sum. It is here convenient to write 

 the exponent a in an algebraical sense over G n ; and it can ob- 

 scure nothing, since G\, considered as the group product G n .G n , 

 is simply G n . 

 Also let 



?l _l. Ky n . ti =br n _i. Ji 



be the bth power of the sum of n — 1 values of V a which are 

 made on the group G ?J _j ; and so on. 



And let r, H be the group of the m cyclical permutations of the 

 first m elements, the remaining n — m being undisturbed. 



We can construct without any difficulty, except tedious alge- 

 braic routine, the function following, for any given system of 

 positive exponents that we desire, 



y —v 7^ — v f 3 v v v^ y^ a p ci m ci l ci c n^ n a it* 



which is the sum of n given values of the /3th power of 



Z y = I n _! I $, 



which is the sum of n — 1 given values of the 7th power of 



Ys = r ?l _ 2 X e , &c, 

 till we come to read the function 



* Communicated by the Author. 



