314 THIRD REPORT — 1833. 



we must employ the auxiliary equation 



7/ + A + AjX + Ac^x^ + . . . ;r«-' = 0, 



whose dimensions are less by 1 than those of the given equation. 



Such a process is apparently very simple and uniform and 

 equally applicable to all equations ; and so it appeared to its 

 author. But it will be found that the equations upon which 

 the determination of A, Aj, Ag, depend, in an equation of the 

 fourth degree, will rise to the sixth degree, which are subse- 

 quently reducible to others of the third degree ; and that for 

 an equation of the fifth degree, it will be impossible to reduce 

 them below the sixth degree. Such was the decision of La- 

 grange, who has subjected this process to a most laborious 

 analysis, and who has actually calculated one of the coefficients 

 of the final reducing equation, and shown the mode in which 

 the others may be determined *. 



Meyer Hirsch, however, though fully adopting the conclu- 

 sions of Lagrange to this extent, attempted to proceed further ; 

 and, deceived by the form which he gave to his types of combina- 

 tion, imagined that he had discovered cyclical periods amongst 

 the roots of this final equation, by which it might be resolved 

 into two equations of the third degree. If such a distribution 

 of the roots was practicable in tlie case of the final equation cor- 

 responding to equations of the fifth degree, it would be practi- 

 cable in that corresponding to equations of higher degrees. 

 But some consequences of this discovery, and particularly the 

 multiplicity of solutions which it gave, would have startled an 

 analyst whose prudence was not laid asleep by the excitement 

 consequent upon the expected attainment of a memorable ad- 

 vancement in analysis, which had eluded the grasp even of 

 Lagrange. Its author, however, was too profound an analyst 

 to continue long ignorant at once of the consequences of his 

 error and of the source from which it sprung. In the Preface 

 to his Integraltafeln, an excellent work, which was published 

 in 1810, within two years of the announcement of his discovery, 

 he acknowledges with great modesty and propriety, that he 

 had not succeeded in effecting general solutions of equations 

 in the sense in which the problem was understood by Euler, 

 Lagrange, and the greatest analysts. 



The well known Hoene de Wronski, in a short pamphlet pub- 

 lished in 181 1, announced a method for the general resolution of 

 equations. He assumes hypothetical expressions for the roots of 

 the given equation in terms of the n roots of 1, and of the {m — V) 



* In the Berlin Memoirs for 1771. p- 170 : it forms a work of prodigious 

 labour, such as few persons would venture to undertake or to repeat. 



