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



degree, if n > 3, is, alas ! all new and not true. My friend Pro- 

 fessor Harley has shown to me that the irrationals i H 1 j. ... i H n j- 

 are not symmetrical functions of then + 1 variables x ,x„x i2 ,. ..,x n , 

 although the values of H x are invariable by their cyclical permu- 

 tation. I hope the scientific reader will pardon the nonsense of 

 the latter part of my preceding communication, and I wish that 

 he may live to confess, like myself, that his mathematical powers 

 are the worse for wear. 



P.S. — Let me venture one word more. All that is required 

 for the solution of S = of the (n + l)th degree, is the trans- 

 formation of H,, H 2 , . . . , H n in the expression for x in my pre- 

 ceding paper, into symmetrical functions of the n -j- 1 variables 

 2#. By writing H? instead of J" in 2^ first above written, we 

 can obtain, by solution of a given quadratic, |Hj| instead of 

 -fJ.J- and we can write this 



showing that Hj is transformed into an irrational symmetric in 

 the n variables 2<# — oc Q . But the Tin values of Hj are obtain- 

 able by the permutations of any n of the n + \ variables. We 

 have got -jo^-i j- by permuting all but x , which is undisturbed in 

 the above process. We can repeat the process, permuting the 

 n } 2<# — x l} %i being now undisturbed. The result will be 



an irrational symmetric in the n roots ^x—x v In the same 

 way we can transform H l into the irrational i 2 H 1 j-, symmetric 

 in the n roots 2,2? — a? 2 . And thus we transform Hj into LHj j- 

 symmetric in the n roots £a? — a? r , whatever r may be, obtaining 

 the n-\-\ results 



H^loHJ, H 1 ={ 1 HJ, ^ 1 ={ a H 1 },...H 1 ={„H 1 }. 



From these it follows that 



H»+' = {o H 1 } {l H 1 } {2 H 1 }{ 3 H,}... { „H 1 } = [H 1 ] ) 

 which is symmetrical in all the n-\-\ roots Xx ; and 



is the required transformation of the rational asymmetric H, 

 into an irrational symmetric in all the roots, and therefore a 

 function of the coefficients of © = 0. 



If the reader will assist me so to handle the ambiguities of the 

 implicated surds that we may be quite sure of all this (and if we 

 can do it with Hj we can do it with all the n IPs in the expres- 



