REPORT ON CERTAIN BRANCHES OF ANALYSIS. 313 



as might be expected, perfectly agrees with the conclusions 

 which are derived from more direct, and, perhaps, more ge- 

 neral considerations. If n, or the number of roots a:,, x^, x^, 

 &c., be a prime number, then the dimensions of the final re- 

 ducing equation will be 1 . 2 ...(?«— 2) ; and if w be a compo- 

 site number = mp, then the dimensions of the final reducing 

 equation will be 



1.2...W ' 1 .2.. .« 



or 



{m-\)m.{\ .2...py" (^ - l)p . (1 . 2 . . . mY 

 according as we arrive at it, by grouping the terms of the ex- 

 pression 



into m periods of ^ terms, or into p periods of m terms. It thus 

 appears, that for an equation of 5 dimensions, the final reducing 

 equation is of 6 dimensions ; for an equation of 6 dimensions, 

 the final reducing equation is of 10 dimensions in one mode of 

 derivation and 15 in the other ; and the higher the dimensions 

 of the equation are, the greater will be the excess of the dimen- 

 sions of the final reducing equation. And in as much as there 

 exist no periodical or other relations amongst the roots of these 

 reducing equations, it is obvious that the application of this 

 process, and therefore also of any of those primary methods 

 which lead to the assumption of the form of the roots of the 

 reducing equation, must increase instead of diminishing the 

 difficulties of the solution which was required to be found. 



It was the imagined discovery of a cyclical period amongst 

 the roots of this reducing equation which induced Meyer Hirsch, 

 a mathematician of very considerable attainments, to believe 

 that he had discovered methods for the general solution of equa- 

 tions of the fifth and higher degrees. Amongst the different 

 methods which Lagrange has analysed in the Berlin Memoirs 

 is that which Tschirnhausen proposed in the Acta Eruditorvm 

 for 1683. It proposed to exterminate, by means of an auxiliary 

 equation, all the terms of the original equation except the first 

 and the last, and thus to reduce it to a binomial equation. 

 Thus, in order to exterminate the second term of a;^ + ax 

 + 6 = 0, we must employ the auxiliary equation y -\- A + x 

 = 0, and then eliminate x. To exterminate simultaneously 

 the second and third terms of the cubic equation x^ -^ a x^ 

 + b X + c = 0, we must employ the auxiliary equation y +A 

 + B ar + a;^ = 0, and then eliminate x ; and more generally, to 

 destroy all the intermediate terms of an equation of n dimen- 

 sions, 



X" + ff, X"-^ + «2 ^"'^ +...<?„ = 0, 



