I 74 On the General Solution of Algebraic Equations. 



We have, in our expression of the values of IT, taken Q n iden- 

 tical with r„, which is a course by no means necessary. It sim- 

 plifies our view of the group ft of II?i substitutions to suppose 

 G m everywhere identical with T m ; and if ft be constructed even 

 in this unskilful way, the theorem will support us and bring us 

 safely to our goal. 



No selection of a form for ft can save us from the necessity of 

 constructing the symmetric function 5^ for every combination of 

 the positive exponents a, b, c, . . . , /, m,p that satisfies 



a < n + 1 , b < n, c < n— 1, . . . I < 5, m < 4, p < 3 ; 



but the Greek exponents need not always all be positive, as for cer- 

 tain systems 5^ is symmetrical when one or more or all of them 

 are zero, which reduces the Y carrying the zero to a factor unity. 

 The best choice of form for H will introduce in 2/3 the greatest 

 number of zeros among the Greek exponents, and thus cause an 

 enormous reduction of the algebraic labour. This labour will be 

 abridged by the computer in proportion to his command of the 

 theory of groups. 



I have thus the pleasure of accomplishing a reasonable number 

 of demonstrated impossibilities, mingled with regret at the ruin 

 of one of the most charming of ancient mysteries. It is a comfort 

 to know that no true Briton for the next twenty years will so far 

 sin against all that is dignified and venerable in mathematics as to 

 suffer himself cither to understand or to believe one word of this. 

 To the scientific reader who thinks of taking some day a second 

 peep at this paper, I would say — don't ; you will only step out 

 of that enchanted fog of analytical sublimity into a boundless 

 and commonplace bog of algebraic inutility. There is nothing 

 here which adds to my right or to my wish to detain the philo- 

 sopher. I am quietly satisfied, but far from being elated, by 

 these results. In the opinion of myself and of my respected 

 friend Posterity (the only section of the scientific public in these 

 practical islands whose mathematical reading will ever qualify 

 them to form anything like a judgment on the precise points of 

 comparison), this finishing contribution to the rich theory of 

 equations is a trifle compared with my extensive additions to the 

 skeleton theory of groups, which are again a trifle compared with 

 my creation of the entire theory of the polyedra. How saga- 

 ciously amused our profundities will find themselves, if ever they 

 hear of these burrowings of a country mouse ! 



Croft, near Warrington. 



