Eoss — Verh-Fimctions. 



63 



Taking the series three at a time, 



The only term which remains after reduction is the foui-th one ; 

 so that 



= nt^H_.^ -n{n- 2%tQt.^ + — t^^ 



li 



= - V-z- 



From the general symmetry of the expressions, we may infer that 

 the sums of higher products are equal to the remaining coefficients 

 of in order. The disappearance of the various tenns is due to the 

 relations which exist between the coefficient of [}l/nP and the sums 

 denoted by >S'„,. 



In the product of all the quantities Xi, X2, . . . only the first 

 term remains after reduction, and this gives 



X1X2X2 . . . = abc . . . ; 



and the product of all the values of together = + 



Hence the value of as calculated by descending operative 



division is theoretically the complete solution of the equation 



when its lowest power of ^ is not less than 



As is an infinite series, the argument of Abel, Sir "William 



Hamilton, and others is not concerned with it, except as showing 

 that it cannot he summed in finite terms} Except when ^=1, the 

 solution is a transcendental one. But it is perhaps entitled to be 

 called the general solution, and would appear to be the only general 

 solution possible. "We should be scarcely justified in calling the series 

 the expansion of [i/'u]"^ — it is rather [«/^»]~^ itself. 



- 23. Notes on the Solution of Numerical Equations hy Operative 

 Division. — The expression for [i/^n]"^ Diay be of some theoretical 

 interest because it appears to be the complete invert of a linear 

 algebraic operation of any degree ; and it will probably be of service 

 in the Theory of Equations, and in other branches of analysis. 

 Further discussion of it would be out of place in a paper which aims 



^ See Sir William Hamilton's article on Abel's argument, Transactions of the 

 Royal Irish Academy, vol. xviii., 1839. 



PROC. K.I. A., VOL. XXV., SEC, A.] E 



